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PART 3 CHAPTER FINANCIAL ASSETS 6 Interest Rates 7 Bonds and Their Valuation 8 Risk and Rates of Return 9 Stocks and Their Valuation ª NICHOLAS KAMM/AFP/GETTY IMAGES CHAPTER 6 Interest Rates Low Interest Rates Encourage Investment and Stimulate Consumer Spending The U.S. economy performed well from the early 1990s through 2007. Economic growth was positive, unemployment was fairly low, and inflation remained under control. One reason for the economy’s good performance was the low level of interest rates over most of that period, with the rate on 10-year Treasury bonds generally at or below 5%, a level last seen in the 1960s, and rates on most other bonds correspondingly low. These low interest rates reduced the cost of capital for businesses, which encouraged corporate investment. They also stimulated consumer spending and helped produce a massive growth in the housing market. The drop in interest rates was due to a number of factors—low inflation, foreign investors’ purchases of U.S. securities (which drove their rates down), and effective management of the economy by the Federal Reserve and other government policy makers. However, some shocks hit the system in 2007, including $100 per barrel oil and massive write-offs by banks and other institutions that resulted from the subprime mortgage 162 debacle. Higher oil prices and a weakening dollar could lead to higher inflation, which, in turn, would push interest rates up. Likewise, the growing federal budget deficit, combined with the weakening dollar, could cause foreigners to sell U.S. bonds, which would put more upward pressure on rates. At the same time, though, the economy seems to be weakening, which has led the Federal Reserve to lower its key short-term rate in hopes of staving off a general recession. So some forces are trying to drive rates higher, but other forces are operating to keep rates low. Because corporations and individuals are greatly affected by interest rates, this chapter takes a closer look at the major factors that determine those rates. As we will see, there is no single interest rate—various factors determine the rate that each borrower pays—and in some cases, rates on different types of debt move in different directions. For example, in the aftermath of the recent subprime mortgage crisis, investors rushed to put their money in liquid securities with little or no default risk. This “flight Chapter 6 Interest Rates to quality” led to a decline in the rate the government had to pay when it borrowed. At the same time, investors demanded much higher rates from corporate borrowers— particularly those thought to be especially risky. The subprime mortgage crisis demonstrates how major shocks to the economy can have profound effects on 163 interest rates in a wide number of markets, all of which are interconnected. Looking ahead, it will be interesting to see if interest rates can continue to remain low and if not, whether the economy can continue to perform as well as it has in the past. PUTTING THINGS IN PERSPECTIVE Companies raise capital in two main forms: debt and equity. In a free economy, capital, like other items, is allocated through a market system, where funds are transferred and prices are established. The interest rate is the price that lenders receive and borrowers pay for debt capital. Similarly, equity investors expect to receive dividends and capital gains, the sum of which represents the cost of equity. We will take up the cost of equity in a later chapter, but our focus in this chapter is on the cost of debt. We begin by examining the factors that affect the supply of and demand for capital, which, in turn, affects the cost of money. We will see that there is no single interest rate—interest rates on different types of debt vary depending on the borrower’s risk, the use of the funds borrowed, the type of collateral used to back the loan, and the length of time the money is needed. In this chapter, we concentrate mainly on how these various factors affect the cost of debt for individuals; but in later chapters, we delve into cost of debt for a business and its role in investment decisions. As you will see in Chapters 7 and 9, the cost of debt is a key determinant of bond and stock prices; it is also an important component of the cost of corporate capital, which we take up in Chapter 10. When you finish this chapter, you should be able to: List the various factors that influence the cost of money. Discuss how market interest rates are affected by borrowers’ need for capital, expected inflation, different securities’ risks, and securities’ liquidity. Explain what the yield curve is, what determines its shape, and how you can use the yield curve to help forecast future interest rates. l Production Opportunities The investment opportunities in productive (cash-generating) assets. l l 6-1 THE COST OF MONEY The four most fundamental factors affecting the cost of money are (1) production opportunities, (2) time preferences for consumption, (3) risk, and (4) inflation. To see how these factors operate, visualize an isolated island community where people live on fish. They have a stock of fishing gear that permits them to survive reasonably well, but they would like to have more fish. Now suppose one of the island’s inhabitants, Mr. Crusoe, had a bright idea for a new type of fishnet that would enable him to double his daily catch. However, it would take him a year to perfect the design, build the net, and learn to use it efficiently. Mr. Crusoe would probably starve before he could put his new net into operation. Therefore, he might suggest to Ms. Robinson, Mr. Friday, and several others that if they would give him one fish each day for a year, he would return two fish a day the next year. If someone accepted the offer, the fish that Ms. Robinson and the others gave to Mr. Crusoe would constitute savings, these savings would be invested in the Time Preferences for Consumption The preferences of consumers for current consumption as opposed to saving for future consumption. Risk In a financial market context, the chance that an investment will provide a low or negative return. Inflation The amount by which prices increase over time. Part 3 Financial Assets fishnet, and the extra fish the net produced would constitute a return on the investment. Obviously, the more productive Mr. Crusoe thought the new fishnet would be, the more he could afford to offer potential investors for their savings. In this example, we assume that Mr. Crusoe thought he would be able to pay (and thus he offered) a 100% rate of return—he offered to give back two fish for every one he received. He might have tried to attract savings for less—for example, he might have offered only 1.5 fish per day next year for every one he received this year, which would represent a 50% rate of return to Ms. Robinson and the other potential savers. How attractive Mr. Crusoe’s offer appeared to a potential saver would depend in large part on the saver’s time preference for consumption. For example, Ms. Robinson might be thinking of retirement, and she might be willing to trade fish today for fish in the future on a one-for-one basis. On the other hand, Mr. Friday might have a wife and several young children and need his current fish; so he might be unwilling to “lend” a fish today for anything less than three fish next year. Mr. Friday would be said to have a high time preference for current consumption; Ms. Robinson, a low time preference. Note also that if the entire population was living right at the subsistence level, time preferences for current consumption would necessarily be high, aggregate savings would be low, interest rates would be high, and capital formation would be difficult. The risk inherent in the fishnet project (and thus in Mr. Crusoe’s ability to repay the loan) also affects the return that investors require: The higher the perceived risk, the higher the required rate of return. Also, in a more complex society, there are many businesses like Mr. Crusoe’s, many goods other than fish, and many savers like Ms. Robinson and Mr. Friday. Therefore, people use money as a medium of exchange rather than barter with fish. When money is used, its value in the future, which is affected by inflation, comes into play: The higher the expected rate of inflation, the larger the required dollar return. We discuss this point in detail later in the chapter. Thus, we see that the interest rate paid to savers depends (1) on the rate of return that producers expect to earn on invested capital, (2) on savers’ time preferences for current versus future consumption, (3) on the riskiness of the loan, and (4) on the expected future rate of inflation. Producers’ expected returns on their business investments set an upper limit to how much they can pay for savings, while consumers’ time preferences for consumption establish how much consumption they are willing to defer and, hence, how much they will save at different interest rates.1 Higher risk and higher inflation also lead to higher interest rates. SE 164 LF TEST What is the price paid to borrow debt capital called? What are the two items whose sum is the cost of equity? What four fundamental factors affect the cost of money? 1 The term producers is too narrow. A better word might be borrowers, which would include corporations, home purchasers, people borrowing to go to college, and even people borrowing to buy autos or to pay for vacations. Also, the wealth of a society and its demographics influence its people’s ability to save and thus their time preferences for current versus future consumption. Chapter 6 Interest Rates 6-2 INTEREST RATE LEVELS Borrowers bid for the available supply of debt capital using interest rates: The firms with the most profitable investment opportunities are willing and able to pay the most for capital, so they tend to attract it away from inefficient firms and firms whose products are not in demand. Of course, the economy is not completely free in the sense of being influenced only by market forces. For example, the federal government has agencies that help designated individuals or groups obtain credit on favorable terms. Among those eligible for this kind of assistance are small businesses, certain minorities, and firms willing to build plants in areas with high unemployment. Still, most capital in the United States is allocated through the price system, where the interest rate is the price. Figure 6-1 shows how supply and demand interact to determine interest rates in two capital markets. Markets L and H represent two of the many capital markets in existence. The supply curve in each market is upward-sloping, which indicates that investors are willing to supply more capital the higher the interest rate they receive on their capital. Likewise, the downward-sloping demand curve indicates that borrowers will borrow more if interest rates are lower. The interest rate in each market is the point where the supply and demand curves intersect. The going interest rate, designated as r, is initially 5% for the low-risk securities in Market L. Borrowers whose credit is strong enough to participate in this market can obtain funds at a cost of 5%, and investors who want to put their money to work without much risk can obtain a 5% return. Riskier borrowers must obtain higher-cost funds in Market H, where investors who are more willing to take risks expect to earn a 7% return but also realize that they might receive much less. In this scenario, investors are willing to accept the higher risk in Market H in exchange for a risk premium of 7% – 5% ¼ 2%. Now let’s assume that because of changing market forces, investors perceive that Market H has become relatively more risky. This changing perception will induce many investors to shift toward safer investments—along the lines of the recent “flight to quality” discussed in the opening vignette to this chapter. As investors move their money from Market H to Market L, this supply of funds is increased in Market L from S1 to S2; and the increased availability of capital will push down interest rates in this market from 5% to 4%. At the same time, as investors move their money out of Market H, there will be a decreased supply in that market; and tighter credit in that market will force interest rates up from 7% to 8%. In this new environment, money is transferred from Market H to Market L and the risk premium rises from 2% to 8% – 4% ¼ 4%. Interest Rates as a Function of Supply and Demand for Funds FIGURE 6-1 Market L: Low-Risk Securities Market H: High-Risk Securities Interest Rate, r (%) Interest Rate, r (%) S2 S1 S 2 S1 8 rH = 7 rL = 5 4 D D 0 Dollars 0 Dollars 165 166 Part 3 Financial Assets There are many capital markets in the United States, and Figure 6-1 highlights the fact that they are interconnected. U.S. firms also invest and raise capital throughout the world, and foreigners both borrow and lend in the United States. There are markets for home loans; farm loans; business loans; federal, state, and local government loans; and consumer loans. Within each category, there are regional markets as well as different types of submarkets. For example, in real estate, there are separate markets for first and second mortgages and for loans on single-family homes, apartments, office buildings, shopping centers, and vacant land. And, of course, there are separate markets for prime and subprime mortgage loans. Within the business sector, there are dozens of types of debt securities and there are several different markets for common stocks. There is a price for each type of capital, and these prices change over time as supply and demand conditions change. Figure 6-2 shows how long- and shortterm interest rates to business borrowers have varied since the early 1970s. Notice that short-term interest rates are especially volatile, rising rapidly during booms and falling equally rapidly during recessions. (The shaded areas of the chart indicate recessions.) When the economy is expanding, firms need capital; and this demand pushes rates up. Also, inflationary pressures are strongest during business booms, also exerting upward pressure on rates. Conditions are reversed during recessions: Slack business reduces the demand for credit, inflation falls, and the Federal Reserve increases the supply of funds to help stimulate the economy. The result is a decline in interest rates. Long- and Short-Term Interest Rates, 1971–2007 FIGURE 6-2 Interest Rate (%) 18 16 14 Long-Term Rates 12 10 8 6 4 Short-Term Rates 2 0 1971 1975 1979 1983 1987 1991 1995 1999 2003 2007 Years Notes: a. The shaded areas designate business recessions. b. Short-term rates are measured by 3- to 6-month loans to very large, strong corporations; and long-term rates are measured by AAA corporate bonds. Source: St. Louis Federal Reserve web site, FRED database, http://research.stlouisfed.org/fred2. Chapter 6 Interest Rates These tendencies do not hold exactly, as demonstrated by the period after 1984. Oil prices fell dramatically in 1985 and 1986, reducing inflationary pressures on other prices and easing fears of serious long-term inflation. Earlier these fears had pushed interest rates to record levels. The economy from 1984 to 1987 was strong, but the declining fears of inflation more than offset the normal tendency for interest rates to rise during good economic times; the net result was lower interest rates.2 The relationship between inflation and long-term interest rates is highlighted in Figure 6-3, which plots inflation over time along with long-term interest rates. In the early 1960s, inflation averaged 1% per year and interest rates on highquality long-term bonds averaged 4%. Then the Vietnam War heated up, leading to an increase in inflation; and interest rates began an upward climb. When the war ended in the early 1970s, inflation dipped a bit; but then the 1973 Arab oil embargo led to rising oil prices, much higher inflation rates, and sharply higher interest rates. Inflation peaked at about 13% in 1980. But interest rates continued to increase into 1981 and 1982, and they remained quite high until 1985 because people feared another increase in inflation. Thus, the “inflationary psychology” created during the 1970s persisted until the mid-1980s. People gradually realized that the Federal Reserve was serious about keeping inflation down, that global competition was keeping U.S. auto producers and other corporations from raising prices as they FIGURE 6-3 Relationship between Annual Inflation Rates and Long-Term Interest Rates, 1972–2007 Interest Rate (%) 16 14 12 Long-Term Interest Rates 10 8 Inflation 6 4 2 0 1972 1977 1982 1987 1992 1997 2002 2007 Years Notes: a. Interest rates are rates on AAA long-term corporate bonds. b. Inflation is measured as the annual rate of change in the consumer price index (CPI). Source: St. Louis Federal Reserve web site, FRED database, http://research.stlouisfed.org/fred2. 2 Short-term rates are responsive to current economic conditions, whereas long-term rates primarily reflect longrun expectations for inflation. As a result, short-term rates are sometimes above and sometimes below long-term rates. The relationship between long-term and short-term rates is called the term structure of interest rates, and it is discussed later in this chapter. 167 Part 3 Financial Assets had in the past, and that constraints on corporate price increases were diminishing labor unions’ ability to push through cost-increasing wage hikes. As these realizations set in, interest rates declined. The current interest rate minus the current inflation rate (which is also the gap between the inflation bars and the interest rate curve in Figure 6-3) is defined as the “current real rate of interest.” It is called a “real rate” because it shows how much investors really earned after the effects of inflation were removed. The real rate was extremely high during the mid-1980s, but it has generally been in the range of 3% to 4% since 1987. In recent years, inflation has been about 2% a year. However, long-term interest rates have been volatile because investors are not sure if inflation is truly under control or is about to jump back to the higher levels of the 1980s. In the years ahead, we can be sure of two things: (1) Interest rates will vary, and (2) they will increase if inflation appears to be headed higher or decrease if inflation is expected to decline. We don’t know where interest rates will go, but we do know they will vary. SE 168 LF TEST What role do interest rates play in allocating capital to different potential borrowers? What happens to market-clearing, or equilibrium, interest rates in a capital market when the supply of funds declines? What happens when expected inflation increases or decreases? How does the price of capital tend to change during a boom? during a recession? How does risk affect interest rates? If inflation during the last 12 months was 2% and the interest rate during that period was 5%, what was the real rate of interest? If inflation is expected to average 4% during the next year and the real rate is 3%, what should the current rate of interest be? (3%; 7%) 6-3 THE DETERMINANTS OF MARKET INTEREST RATES In general, the quoted (or nominal) interest rate on a debt security, r, is composed of a real risk-free rate, r*, plus several premiums that reflect inflation, the security’s risk, its liquidity (or marketability), and the years to its maturity. This relationship can be expressed as follows: 6-1 Quoted interest rate ¼ r ¼ r þ IP þ DRP þ LP þ MRP Here r ¼ the quoted, or nominal, rate of interest on a given security.3 r* ¼ the real risk-free rate of interest. r* is pronounced “r-star,” and it is the rate that would exist on a riskless security in a world where no inflation was expected. 3 The term nominal as it is used here means the stated rate as opposed to the real rate, where the real rate is adjusted to remove inflation’s effects. If you had bought a 10-year Treasury bond in January 2008, the quoted, or nominal, rate would have been about 3.7%; but if inflation averages 2.5% over the next 10 years, the real rate would turn out to be about 3.7% − 2.5% ¼ 1.2%. Also note that in later chapters, when we discuss both debt and equity, we use the subscripts d and s to designate returns on debt and stock, that is, rd and rs. Chapter 6 Interest Rates 169 rRF ¼ r* þ IP. It is the quoted rate on a risk-free security such as a U.S. Treasury bill, which is very liquid and is free of most types of risk. Note that the premium for expected inflation, IP, is included in rRF. IP ¼ inflation premium. IP is equal to the average expected rate of inflation over the life of the security. The expected future inflation rate is not necessarily equal to the current inflation rate, so IP is not necessarily equal to current inflation as shown in Figure 6-3. DRP ¼ default risk premium. This premium reflects the possibility that the issuer will not pay the promised interest or principal at the stated time. DRP is zero for U.S. Treasury securities, but it rises as the riskiness of the issuer increases. LP ¼ liquidity (or marketability) premium. This is a premium charged by lenders to reflect the fact that some securities cannot be converted to cash on short notice at a “reasonable” price. LP is very low for Treasury securities and for securities issued by large, strong firms; but it is relatively high on securities issued by small, privately held firms. MRP ¼ maturity risk premium. As we will explain later, longer-term bonds, even Treasury bonds, are exposed to a significant risk of price declines due to increases in inflation and interest rates; and a maturity risk premium is charged by lenders to reflect this risk. Because rRF ¼ r* + IP, we can rewrite Equation 6-1 as follows: Nominal; or quoted; rate ¼ r ¼ rRF þ DRP þ LP þ MRP We discuss the components whose sum makes up the quoted, or nominal, rate on a given security in the following sections. 6-3a The Real Risk-Free Rate of Interest, r* The real risk-free rate of interest, r*, is the interest rate that would exist on a riskless security if no inflation were expected. It may be thought of as the rate of interest on short-term U.S. Treasury securities in an inflation-free world. The real risk-free rate is not static—it changes over time depending on economic conditions, especially on (1) the rate of return that corporations and other borrowers expect to earn on productive assets and (2) people’s time preferences for current versus future consumption. Borrowers’ expected returns on real assets set an upper limit on how much borrowers can afford to pay for funds, while savers’ time preferences for consumption establish how much consumption savers will defer—hence, the amount of money they will lend at different interest rates. It is difficult to measure the real risk-free rate precisely, but most experts think that r* has fluctuated in the range of 1% to 5% in recent years.4 The best estimate of r* is the rate of return on indexed Treasury bonds, which are discussed later in the chapter. 4 The real rate of interest as discussed here is different from the current real rate as discussed in connection with Figure 6-3. The current real rate is the current interest rate minus the current (or latest past) inflation rate, while the real rate (without the word current) is the current interest rate minus the expected future inflation rate over the life of the security. For example, suppose the current quoted rate for a one-year Treasury bill is 2.7%, inflation during the latest year was 1.2%, and inflation expected for the coming year is 2.2%. The current real rate would be 2.7% – 1.2% = 1.5%, but the expected real rate would be 2.7% – 2.2% = 0.5%. The rate on a 10-year bond would be related to the average expected inflation rate over the next 10 years, and so on. In the press, the term real rate generally means the current real rate; but in economics and finance (hence, in this book unless otherwise noted), the real rate means the one based on expected inflation rates. Real Risk-Free Rate of Interest, r* The rate of interest that would exist on default-free U.S. Treasury securities if no inflation were expected. 170 Part 3 Financial Assets 6-3b The Nominal, or Quoted, Risk-Free Rate of Interest, rRF ¼ r* þ IP Nominal (Quoted) RiskFree Rate, rRF The rate of interest on a security that is free of all risk; rRF is proxied by the T-bill rate or the T-bond rate. rRF includes an inflation premium. The nominal, or quoted, risk-free rate, rRF, is the real risk-free rate plus a premium for expected inflation: rRF ¼ r* þ IP. To be strictly correct, the risk-free rate should be the interest rate on a totally risk-free security—one that has no default risk, no maturity risk, no liquidity risk, no risk of loss if inflation increases, and no risk of any other type. There is no such security; hence, there is no observable truly risk-free rate. However, one security is free of most risks—a Treasury Inflation Protected Security (TIPS), whose value increases with inflation. TIPS are free of default, maturity, and liquidity risks and of risk due to changes in the general level of interest rates. However, they are not free of changes in the real rate.5 If the term risk-free rate is used without the modifiers real or nominal, people generally mean the quoted (or nominal) rate; and we follow that convention in this book. Therefore, when we use the term risk-free rate, rRF, we mean the nominal risk-free rate, which includes an inflation premium equal to the average expected inflation rate over the remaining life of the security. In general, we use the T-bill rate to approximate the short-term risk-free rate and the T-bond rate to approximate the long-term risk-free rate. So whenever you see the term risk-free rate, assume that we are referring to the quoted U.S. T-bill rate or to the quoted T-bond rate. 6-3c Inflation Premium (IP) Inflation Premium (IP) A premium equal to expected inflation that investors add to the real risk-free rate of return. Inflation has a major impact on interest rates because it erodes the real value of what you receive from the investment. To illustrate, suppose you saved $1,000 and invested it in a Treasury bill that pays a 3% interest rate and matures in one year. At the end of the year, you will receive $1,030—your original $1,000 plus $30 of interest. Now suppose the inflation rate during the year turned out to be 3.5%, and it affected all goods equally. If heating oil had cost $1 per gallon at the beginning of the year, it would cost $1.035 at the end of the year. Therefore, your $1,000 would have bought $1,000/$1 = 1,000 gallons at the beginning of the year, but only $1,030/$1.035 = 995 gallons at the end. In real terms, you would be worse off —you would receive $30 of interest, but it would not be sufficient to offset inflation. You would thus be better off buying 1,000 gallons of heating oil (or some other storable asset such as land, timber, apartment buildings, wheat, or gold) than buying the Treasury bill. Investors are well aware of all this; so when they lend money, they build an inflation premium (IP) equal to the average expected inflation rate over the life of the security into the rate they charge. As discussed previously, the actual interest rate on a short-term default-free U.S. Treasury bill, rT-bill, would be the real riskfree rate, r*, plus the inflation premium (IP): rTbill ¼ rRF ¼ r þ IP Therefore, if the real risk-free rate was r* = 1.7% and if inflation was expected to be 1.5% (and hence IP ¼ 1.5%) during the next year, the quoted rate of interest on one-year T-bills would be 1.7% þ 1.5% ¼ 3.2%. It is important to note that the inflation rate built into interest rates is the inflation rate expected in the future, not the rate experienced in the past. Thus, the 5 Indexed Treasury securities are the closest thing we have to a riskless security, but even they are not totally riskless because r* can change and cause a decline in the prices of these securities. For example, between its issue date in February 1998 and December 2004, the TIPS that matures on February 15, 2028 first declined from 100 to 89, or by almost 10%, but it then rose; and in February 2008, the bond sold for 130. The cause of the initial price decline was an increase in the real rate on long-term securities from 3.625% to 4.4%, and the cause of the subsequent price increase was a decline in real rates to 2.039%. Chapter 6 Interest Rates AN ALMOST RISKLESS TREASURY BOND Investors who purchase bonds must constantly worry about inflation. If inflation turns out to be greater than expected, bonds will provide a lower-than-expected real return. To protect themselves against expected increases in inflation, investors build an inflation risk premium into their required rate of return. This raises borrowers’ costs. To provide investors with an inflation-protected bond and to reduce the cost of debt to the government, the U.S. Treasury issues Treasury Inflation Protected Securities (TIPS), which are bonds that are indexed to inflation. For example, in 2004, the Treasury issued 10-year TIPS with a 2% coupon. These bonds pay an interest rate of 2% plus an additional amount that is just sufficient to offset inflation. At the end of each 6-month period, the principal (originally set at par or $1,000) is adjusted by the inflation rate. To understand how TIPS work, consider that during the first 6-month interest period, inflation (as measured by the CPI) was 2.02%. The inflation-adjusted principal was then calculated as $1,000(1 + Inflation) = $1,000 × 1.0202 = $1,020.20. So on July 15, 2004, each bond paid interest of 0.02/2 × $1,020.20 = $10.202. Note that the interest rate is divided by 2 because interest on Treasury (and most other) bonds is paid twice a year. This same adjustment process will continue each year until the bonds mature on January 15, 2014, at which time they will pay the adjusted maturity value. Thus, the cash income provided by the bonds rises by exactly enough to cover inflation, producing a real inflation-adjusted rate of 2% for those who hold the bond from the beginning to the r* (%) 3.00 end. Further, since the principal also rises by the inflation rate, it too is protected from inflation. Both the annual interest received and the increase in principal are taxed each year as interest income even though cash from the appreciation will not be received until the bond matures. Therefore, these bonds are not good for accounts subject to current income taxes; but they are excellent for individual retirement accounts (IRAs) and 401 (k) plans, which are not taxed until funds are withdrawn. The Treasury regularly conducts auctions to issue indexed bonds. The 2% rate was based on the relative supply and demand for the issue, and it will remain fixed over the life of the bond. However, after the bonds are issued, they continue to trade in the open market; and their price will vary as investors’ perceptions of the real rate of interest changes. Indeed, as we can see in the following graph, the real rate of interest on this bond has varied quite a bit since it was issued; and as the real rate changes, so does the price of the bond. Real rates fell in 2005, causing the bond’s price to rise; rates then rose to a peak in 2007, at which point the bond sold below its par value. They fell again in late 2007 and 2008 as investors sought safety in Treasury securities. Thus, despite their protection against inflation, indexed bonds are not completely riskless. The real rate can change; and if r* rises, the prices of indexed bonds will decline. This confirms again that there is no such thing as a free lunch or a riskless security. 10-Yr. 2% Treasury Inflation-Indexed Note, Due 1/15/2014 2.50 2.00 1.50 1.00 0.50 0.00 1-12-04 1-12-05 1-12-06 1-12-07 Source: St. Louis Federal Reserve web site, FRED database, http://research.stlouisfed.org/fred2. latest reported figures might show an annual inflation rate of 3% over the past 12 months, but that is for the past year. If people, on average, expect a 4% inflation rate in the future, 4% would be built into the current interest rate. Note also that the inflation rate reflected in the quoted interest rate on any security is the average inflation rate expected over the security’s life. Thus, the inflation rate built into a 1-year 1-12-08 171 172 Part 3 Financial Assets Students should go to www. bloomberg.com/markets/ rates to find current interest rates in the United States as well as in Australia, Brazil, Germany, Japan, and Great Britain. bond is the expected inflation rate for the next year, but the inflation rate built into a 30-year bond is the average inflation rate expected over the next 30 years.6 Expectations for future inflation are closely, but not perfectly, correlated with rates experienced in the recent past. Therefore, if the inflation rate reported for last month increased, people would tend to raise their expectations for future inflation; and this change in expectations would cause an increase in current rates. Also, consumer prices change with a lag following changes at the producer level. Thus, if the price of oil increases this month, gasoline prices are likely to increase in the coming months. This lagged situation between final product and producer goods prices exists throughout the economy. Note that Germany, Japan, and Switzerland have, over the past several years, had lower inflation rates than the United States; hence, their interest rates have generally been lower than those of the United States. Italy and most South American countries have experienced higher inflation, so their rates have been higher than those of the United States. 6-3d Default Risk Premium (DRP) The risk that a borrower will default, which means the borrower will not make scheduled interest or principal payments, also affects the market interest rate on a bond: The greater the bond’s risk of default, the higher the market rate. Treasury securities have no default risk; hence, they carry the lowest interest rates on taxable securities in the United States. For corporate bonds, the higher the bond’s rating, the lower its default risk and, consequently, the lower its interest rate.7 Here are some representative interest rates on long-term bonds in January 2008: U.S. Treasury AAA corporate AA corporate A corporate BBB corporate Default Risk Premium (DRP) The difference between the interest rate on a U.S. Treasury bond and a corporate bond of equal maturity and marketability. Rate DRP 4.28% 4.83 4.93 5.18 6.03 — 0.55 0.65 0.90 1.75 The difference between the quoted interest rate on a T-bond and that on a corporate bond with similar maturity, liquidity, and other features is the default risk premium (DRP). Therefore, if the bonds previously listed have the same maturity, liquidity, and so forth, the default risk premium will be DRP = 4.83% – 4.28% = 0.55% for AAAs, 4.93% – 4.28% = 0.65% for AAs, 5.18% – 4.28% = 0.90% for A corporate bonds, and so forth. If we had gone down into “junk bond” territory, we would have seen DRPs of as much as 8%. Default risk premiums vary somewhat over time, but the January 2008 figures are representative of levels in recent years. 6-3e Liquidity Premium (LP) A “liquid” asset can be converted to cash quickly at a “fair market value.” Real assets are generally less liquid than financial assets, but different financial assets 6 To be theoretically precise, we should use a geometric average. Also, since millions of investors are active in the market, it is impossible to determine exactly the consensus-expected inflation rate. Survey data are available, however, that give us a reasonably good idea of what investors expect over the next few years. For example, in 1980, the University of Michigan’s Survey Research Center reported that people expected inflation during the next year to be 11.9% and that the average rate of inflation expected over the next 5 to 10 years was 10.5%. Those expectations led to record-high interest rates. However, the economy cooled thereafter; and as Figure 6-3 showed, actual inflation dropped sharply. This led to a gradual reduction in the expected future inflation rate; and as inflationary expectations dropped, so did quoted market interest rates. 7 Bond ratings and bonds’ riskiness in general are discussed in detail in Chapter 7. For now, merely note that bonds rated AAA are judged to have less default risk than bonds rated AA, while AA bonds are less risky than A bonds, and so forth. Ratings are designated AAA or Aaa, AA or Aa, and so forth, depending on the rating agency. In this book, the designations are used interchangeably. Chapter 6 Interest Rates A 20% LIQUIDITY PREMIUM Since the yield curve is normally upward-sloping, short-term debt is normally less expensive than long-term debt. However, it’s dangerous to finance long-term assets with shortterm debt. To get around this problem, investment bankers created a new instrument, auction rate securities (ARS), which are long-term bonds with this wrinkle: Weekly (or monthly for some) auctions are held. The borrower buys back at par the bonds of holders who want to get out and simultaneously sells those reclaimed bonds to new lenders. Potential new lenders indicate the lowest interest rate they will accept, and the actual rate paid on the entire issue is the lowest rate that causes the auction to clear. Most of the bonds were insured by AAA insurance companies, which gave them a AAA rating. To illustrate, the total issue might be for $100 million and the initial rate might be 3%. One week later holders of $5 million of bonds might turn in their bonds, which would then be offered in an auction to potential buyers. To get the bonds resold, an annual rate of 3.1% might be required. Then for the next week, all $100 million of the bonds would earn 3.1%. There was a cap on the interest rate tied to an index of rates on regular long-term bonds. ON A 173 HIGH-GRADE BOND Investors liked the ARS because they paid a somewhat higher rate than money market funds and they were equally safe and almost as liquid. They were underwritten by major financial institutions such as Goldman Sachs, Merrill Lynch, and Citigroup, which would buy the excess if more bonds were turned in than were bid for at rates below the cap. The institutions would hold repurchased bonds in inventory and then sell them to their customers. Everything worked fine until the credit market meltdown of 2008. The banks who back-stopped the auction had lost billions in the subprime mortgage debacle, and they didn’t have the capital to step in. After a couple of failed auctions, many ARS holders became concerned about liquidity and tried to turn in their bonds. That rush to the exits caused the whole market to freeze up. Highly liquid securities suddenly became totally illiquid. Penalty rates for frozen securities kicked in, some as high as 20%. That’s much higher than “normal” liquidity premiums, but it does demonstrate that liquidity is valuable and that high liquidity premiums are built into illiquid securities’ rates. Source: Stan Rosenberg and Romy Varghese, “Auction-Rate Bonds May Come to Rescue,” The Wall Street Journal, February 15, 2008, p. C2. vary in their liquidity. Because liquidity is important, investors include a liquidity premium (LP) in the rates charged on different debt securities. Although it is difficult to measure liquidity premiums accurately, a differential of at least two and probably four or five percentage points exists between the least liquid and the most liquid financial assets of similar default risk and maturity. 6-3f Interest Rate Risk and the Maturity Risk Premium (MRP) U.S. Treasury securities are free of default risk in the sense that one can be virtually certain that the federal government will pay interest on its bonds and pay them off when they mature. Therefore, the default risk premium on Treasury securities is essentially zero. Further, active markets exist for Treasury securities, so their liquidity premiums are close to zero. Thus, as a first approximation, the rate of interest on a Treasury security should be the risk-free rate, rRF, which is the real risk-free rate plus an inflation premium, rRF ¼ r* + IP. However, the prices of long-term bonds decline whenever interest rates rise; and because interest rates can and do occasionally rise, all long-term bonds, even Treasury bonds, have an element of risk called interest rate risk. As a general rule, the bonds of any organization, from the U.S. government to Delta Airlines, have more interest rate risk the longer the maturity of the bond.8 Therefore, a maturity risk premium (MRP), 8 For example, if someone had bought a 20-year Treasury bond for $1,000 in October 1998, when the long-term interest rate was 5.3%, and sold it in May 2002, when long-term T-bond rates were about 5.8%, the value of the bond would have declined to about $942. That would represent a loss of 5.8%; and it demonstrates that longterm bonds, even U.S. Treasury bonds, are not riskless. However, had the investor purchased short-term T-bills in 1998 and subsequently reinvested the principal each time the bills matured, he or she would still have had the original $1,000. This point is discussed in detail in Chapter 7. Liquidity Premium (LP) A premium added to the equilibrium interest rate on a security if that security cannot be converted to cash on short notice and at close to its “fair market value.” Interest Rate Risk The risk of capital losses to which investors are exposed because of changing interest rates. Maturity Risk Premium (MRP) A premium that reflects interest rate risk. 174 Part 3 Financial Assets which is higher the greater the years to maturity, is included in the required interest rate. The effect of maturity risk premiums is to raise interest rates on long-term bonds relative to those on short-term bonds. This premium, like the others, is difficult to measure; but (1) it varies somewhat over time, rising when interest rates are more volatile and uncertain, then falling when interest rates are more stable and (2) in recent years, the maturity risk premium on 20-year T-bonds has generally been in the range of one to two percentage points.9 We should also note that although long-term bonds are heavily exposed to interest rate risk, short-term bills are heavily exposed to reinvestment rate risk. When short-term bills mature and the principal must be reinvested, or “rolled over,” a decline in interest rates would necessitate reinvestment at a lower rate, which would result in a decline in interest income. To illustrate, suppose you had $100,000 invested in T-bills and you lived on the income. In 1981, short-term Treasury rates were about 15%, so your income would have been about $15,000. However, your income would have declined to about $9,000 by 1983 and to just $2,700 by January 2008. Had you invested your money in long-term T-bonds, your income (but not the value of the principal) would have been stable.10 Thus, although “investing short” preserves one’s principal, the interest income provided by short-term T-bills is less stable than that on long-term bonds. SE Reinvestment Rate Risk The risk that a decline in interest rates will lead to lower income when bonds mature and funds are reinvested. LF TEST Write an equation for the nominal interest rate on any security. Distinguish between the real risk-free rate of interest, r*, and the nominal, or quoted, risk-free rate of interest, rRF. How do investors deal with inflation when they determine interest rates in the financial markets? Does the interest rate on a T-bond include a default risk premium? Explain. Distinguish between liquid and illiquid assets and list some assets that are liquid and some that are illiquid. Briefly explain the following statement: Although long-term bonds are heavily exposed to interest rate risk, short-term T-bills are heavily exposed to reinvestment rate risk. The maturity risk premium reflects the net effects of those two opposing forces. Assume that the real risk-free rate is r* = 2% and the average expected inflation rate is 3% for each future year. The DRP and LP for Bond X are each 1%, and the applicable MRP is 2%. What is Bond X’s interest rate? Is Bond X (1) a Treasury bond or a corporate bond and (2) more likely to have a 3month or a 20-year maturity? (9%, corporate, 20-year) 9 The MRP for long-term bonds has averaged 1.4% over the last 82 years. See Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Chicago: Morningstar Inc., 2008). 10 Most long-term bonds also have some reinvestment rate risk. If a person is saving and investing for some future purpose (say, to buy a house or to retire), to actually earn the quoted rate on a long-term bond, each interest payment must be reinvested at the quoted rate. However, if interest rates fall, the interest payments would be reinvested at a lower rate; so the realized return would be less than the quoted rate. Note, though, that reinvestment rate risk is lower on long-term bonds than on short-term bonds because only the interest payments (rather than interest plus principal) on a long-term bond are exposed to reinvestment rate risk. Non-callable zero coupon bonds, which are discussed in Chapter 7, are completely free of reinvestment rate risk during their lifetime. Chapter 6 Interest Rates 175 6-4 THE TERM STRUCTURE OF INTEREST RATES The term structure of interest rates describes the relationship between long- and short-term rates. The term structure is important to corporate treasurers deciding whether to borrow by issuing long- or short-term debt and to investors who are deciding whether to buy long- or short-term bonds. Therefore, both borrowers and lenders should understand (1) how long- and short-term rates relate to each other and (2) what causes shifts in their relative levels. Interest rates for bonds with different maturities can be found in a variety of publications, including The Wall Street Journal and the Federal Reserve Bulletin, and on a number of web sites, including Bloomberg, Yahoo!, CNN Financial, and the Federal Reserve Board. Using interest rate data from these sources, we can determine the term structure at any given point in time. For example, the tabular section below Figure 6-4 presents interest rates for different maturities on three different dates. The set of data for a given date, when plotted on a graph such as Figure 6-4, is called the yield curve for that date. U.S. Treasury Bond Interest Rates on Different Dates FIGURE 6-4 Interest Rate (%) 16 14 Yield Curve for March 1980 12 10 Yield Curve for February 2000 8 6 4 Yield Curve for January 2008 2 0 0 10 Short Term Intermediate Term 20 30 Years to Maturity Long Term INTEREST RATE Term to Maturity 1 year 5 years 10 years 30 years March 1980 14.0% 13.5 12.8 12.3 February 2000 6.2% 6.7 6.7 6.3 January 2008 2.7% 3.0 3.7 4.3 Term Structure of Interest Rates The relationship between bond yields and maturities. Yield Curve A graph showing the relationship between bond yields and maturities. 176 Part 3 Financial Assets As the figure shows, the yield curve changes in position and in slope over time. In March 1980, all rates were quite high because high inflation was expected. However, the rate of inflation was expected to decline; so short-term rates were higher than long-term rates, and the yield curve was thus downward-sloping. By February 2000, inflation had indeed declined; thus, all rates were lower, and the yield curve had become humped—medium-term rates were higher than either short- or long-term rates. By January 2008, all rates had fallen below the 2000 levels; and because short-term rates had dropped below long-term rates, the yield curve was upward-sloping. Figure 6-4 shows yield curves for U.S. Treasury securities; but we could have constructed curves for bonds issued by GE, IBM, Delta Air Lines, or any other company that borrows money over a range of maturities. Had we constructed such corporate yield curves and plotted them on Figure 6-4, they would have been above those for Treasury securities because corporate yields include default risk premiums and somewhat higher liquidity premiums. Even so, the corporate yield curves would have had the same general shape as the Treasury curves. Also, the riskier the corporation, the higher its yield curve; so Delta, which has been flirting with bankruptcy, would have a higher yield curve than GE or IBM. Historically, long-term rates are generally above short-term rates because of the maturity risk premium; so all yield curves usually slope upward. For this reason, people often call an upward-sloping yield curve a “normal” yield curve and a yield curve that slopes downward an inverted or “abnormal” curve. Thus, in Figure 6-4, the yield curve for March 1980 was inverted, while the one for January 2008 was normal. However, the February 2000 curve was humped, which means that interest rates on medium-term maturities were higher than rates on both short- and long-term maturities. We will explain in detail why an upward slope is the normal situation. Briefly, however, the reason is that short-term securities have less interest rate risk than longer-term securities; hence, they have smaller MRPs. So short-term rates are normally lower than long-term rates. Humped Yield Curve A yield curve where interest rates on medium-term maturities are higher than rates on both short-and long-term maturities. SE “Normal” Yield Curve An upward-sloping yield curve. Inverted (“Abnormal”) Yield Curve A downward-sloping yield curve. LF TEST What is a yield curve, and what information would you need to draw this curve? Distinguish among the shapes of a “normal” yield curve, an “abnormal” curve, and a “humped” curve. If the interest rates on 1-, 5-, 10-, and 30-year bonds are 4%, 5%, 6%, and 7%, respectively, how would you describe the yield curve? If the rates were reversed, how would you describe it? 6-5 WHAT DETERMINES THE SHAPE OF THE YIELD CURVE? Because maturity risk premiums are positive, if other things were held constant, long-term bonds would always have higher interest rates than short-term bonds. However, market interest rates also depend on expected inflation, default risk, and liquidity, each of which can vary with maturity. Expected inflation has an especially important effect on the yield curve’s shape, especially the curve for U.S. Treasury securities. Treasuries have essentially no default or liquidity risk, so the yield on a Treasury bond that matures in t years can be expressed as follows: T-bond yield ¼ rt þ IPt þ MRPt While the real risk-free rate, r*, varies somewhat over time because of changes in the economy and demographics, these changes are random rather than Chapter 6 Interest Rates predictable. Therefore, the best forecast for the future value of r* is its current value. However, the inflation premium, IP, varies significantly over time and in a somewhat predictable manner. Recall that the inflation premium is the average level of expected inflation over the life of the bond. Thus, if the market expects inflation to increase in the future (say, from 3% to 4% to 5% over the next 3 years), the inflation premium will be higher on a 3-year bond than on a 1-year bond. On the other hand, if the market expects inflation to decline in the future, long-term bonds will have a smaller inflation premium than will short-term bonds. Finally, since investors consider long-term bonds to be riskier than short-term bonds because of interest rate risk, the maturity risk premium always increases with maturity. Panel a of Figure 6-5 shows the yield curve when inflation is expected to increase. Here long-term bonds have higher yields for two reasons: (1) Inflation is expected to be higher in the future, and (2) there is a positive maturity risk premium. Panel b shows the yield curve when inflation is expected to decline. Such a downward-sloping yield curve often foreshadows an economic downturn because Illustrative Treasury Yield Curves FIGURE 6-5 a. When Inflation Is Expected to Increase b. When Inflation Is Expected to Decrease Interest Rate (%) Interest Rate (%) 8 Maturity Risk Premium 7 6 8 7 Inflation Premium 5 5 4 4 3 3 2 Inflation Premium 2 Real RiskFree Rate 1 0 Maturity Risk Premium 6 10 0 20 30 Years to Maturity Real RiskFree Rate 1 WITH INFLATION EXPECTED TO INCREASE 10 20 30 Years to Maturity WITH INFLATION EXPECTED TO DECREASE Maturity r* IP MRP Yield Maturity r* IP MRP Yield 1 year 5 years 10 years 20 years 30 years 2.50% 2.50 2.50 2.50 2.50 3.00% 3.40 4.00 4.50 4.67 0.00% 0.18 0.28 0.42 0.53 5.50% 6.08 6.78 7.42 7.70 1 year 5 years 10 years 20 years 30 years 2.50% 2.50 2.50 2.50 2.50 5.00% 4.60 4.00 3.50 3.33 0.00% 0.18 0.28 0.42 0.53 7.50% 7.28 6.78 6.42 6.36 177 178 Part 3 Financial Assets THE LINKS BETWEEN EXPECTED INFLATION AND INTEREST RATES: A CLOSER LOOK Throughout the text, we use the following equation to describe the link between expected inflation and the nominal risk-free rate of interest, rRF: rRF ¼ r þ IP Recall that r* is the real risk-free interest rate and IP is the corresponding inflation premium. This equation suggests that there is a simple link between expected inflation and nominal interest rates. It turns out, however, that this link is a bit more complex. To fully understand this relationship, first recognize that individuals get utility through the consumption of real goods and services such as bread, water, haircuts, pizza, and textbooks. When we save money, we are giving up the opportunity to consume these goods today in return for being able to consume more of them in the future. Our gain from waiting is measured by the real rate of interest, r*. To illustrate this point, consider the following example. Assume that a loaf of bread costs $1 today. Also assume that the real rate of interest is 3% and that inflation is expected to be 5% over the next year. The 3% real rate indicates that the average consumer is willing to trade 100 loaves of bread today for 103 loaves next year. If a “bread bank” were available, consumers who wanted to defer consumption until next year could deposit 100 loaves today and withdraw 103 loaves next year. In practice, most of us do not directly trade real goods such as bread—instead, we purchase these goods with money because in a well-functioning economy, it is more efficient to exchange money than goods. However, when we lend money over time, we worry that borrowers might pay us back with dollars that aren’t worth as much due to inflation. To compensate for this risk, lenders build in a premium for expected inflation. With these concerns in mind, let’s compare the dollar cost of 100 loaves of bread today to the cost of 103 loaves next year. Given the current price, 100 loaves of bread today would cost $100. Since expected inflation is 5%, this means that a loaf of bread is expected to cost $1.05 next year. Consequently, 103 loaves of bread are expected to cost $108.15 next year (103 × $1.05). So if consumers were to deposit $100 in a bank today, they would need to earn 8.15% to realize a real return of 3%. Putting this all together, we see that the 1-year nominal interest rate can be calculated as follows: rRF ¼ ð1 þ r Þð1 þ IÞ  1 ¼ ð1:03Þð1:05Þ  1 ¼ 0:0815 ¼ 8:15% Note that this expression can be rewritten as follows: rRF ¼ r þ I þ ðr  IÞ That equation is identical to our original expression for the nominal risk-free rate except that it includes a “cross-term,” r* × I. When real interest rates and expected inflation are relatively low, the cross-term turns out to be quite small and thus is often ignored. Because it is normally insignificant we disregard the cross-term in the text unless stated otherwise. One last point—you should recognize that while it may be reasonable to ignore the cross-term when interest rates are low (as they are in the United States today), it is a mistake to do so when investing in a market where interest rates and inflation are quite high, as is often the case in many emerging markets. In these markets, the cross-term can be significant and thus should not be disregarded. weaker economic conditions generally lead to declining inflation, which, in turn, results in lower long-term rates.11 Now let’s consider the yield curve for corporate bonds. Recall that corporate bonds include a default risk premium (DRP) and a liquidity premium (LP). Therefore, the yield on a corporate bond that matures in t years can be expressed as follows: Corporate bond yield ¼ rt þ IPt þ MRPt þ DRPt þ LPt Corporate bonds’ default and liquidity risks are affected by their maturities. For example, the default risk on Coca-Cola’s short-term debt is very small since 11 Note that yield curves tend to rise or fall relatively sharply over the first 5 to 10 years and then flatten out. One reason this occurs is that when forecasting future interest rates, people often predict relatively high or low inflation for the next few years, after which they assume an average long-run inflation rate. Consequently, the short end of the yield curve tends to have more curvature and the long end of the yield curve tends to be more stable. Chapter 6 Interest Rates there is almost no chance that Coca-Cola will go bankrupt over the next few years. However, Coke has some bonds that have a maturity of almost 100 years; and while the odds of Coke defaulting on those bonds might not be very high, there is still a higher probability of default risk on Coke’s long-term bonds than its shortterm bonds. Longer-term corporate bonds also tend to be less liquid than shorter-term bonds. Since short-term debt has less default risk, someone can buy a short-term bond without doing as much credit checking as would be necessary for a longterm bond. Thus, people can move in and out of short-term corporate debt relatively rapidly. As a result, a corporation’s short-term bonds are typically more liquid and thus have lower liquidity premiums than its long-term bonds. Figure 6-6 shows yield curves for two hypothetical corporate bonds—an AA-rated bond with minimal default risk and a BBB-rated bond with more default risk— along with the yield curve for Treasury securities taken from Panel a of Figure 6-5. Here we assume that inflation is expected to increase, so the Treasury yield curve is upward-sloping. Because of their additional default and liquidity risk, corporate bonds yield more than Treasury bonds with the same maturity and BBB-rated bonds yield more than AA-rated bonds. Finally, note that the yield spread between corporate and Treasury bonds is larger the longer the maturity. This occurs because Illustrative Corporate and Treasury Yield Curves FIGURE 6-6 Interest Rate (%) 12 BBB-Rated Bond 10 AA-Rated Bond 8 Treasury Bond 6 4 2 0 10 20 30 Years to Maturity INTEREST RATE Term to Maturity 1 year 5 years 10 years 20 years 30 years Treasury Bond AA-Rated Bond BBB-Rated Bond 5.5% 6.1 6.8 7.4 7.7 6.7% 7.4 8.2 9.2 9.8 7.4% 8.1 9.1 10.2 11.1 179 180 Part 3 Financial Assets SE longer-term corporate bonds have more default and liquidity risk than shorter-term bonds, and both of these premiums are absent in Treasury bonds. LF TEST How do maturity risk premiums affect the yield curve? If the inflation rate is expected to increase, would this increase or decrease the slope of the yield curve? If the inflation rate is expected to remain constant at the current level in the future, would the yield curve slope up, slope down, or be horizontal? Consider all factors that affect the yield curve, not just inflation. Explain why corporate bonds’ default and liquidity premiums are likely to increase with their maturity. Explain why corporate bonds always yield more than Treasury bonds and why BBB-rated bonds always yield more than AA-rated bonds. 6-6 USING THE YIELD CURVE TO ESTIMATE FUTURE INTEREST RATES12 Pure Expectations Theory A theory that states that the shape of the yield curve depends on investors’ expectations about future interest rates. In the last section, we saw that the slope of the yield curve depends primarily on two factors: (1) expectations about future inflation and (2) effects of maturity on bonds’ risk. We also saw how to calculate the yield curve, given inflation and maturity-related risks. Note, though, that people can reverse the process: They can look at the yield curve and use information embedded in it to estimate the market’s expectations regarding future inflation, risk, and short-term interest rates. For example, suppose a company is in the midst of a 5-year expansion program and the treasurer knows that she will need to borrow short-term funds a year from now. She knows the current cost of 1-year money, read from the yield curve, but she wants to know the cost of 1-year money next year. That information can be “backed out” by analyzing the current yield curve, as will be discussed. The estimation process is straightforward provided we (1) focus on Treasury bonds and (2) assume that Treasury bonds contain no maturity risk premiums.13 This position has been called the pure expectations theory of the term structure of interest rates, often simply referred to as the “expectations theory.” The expectations theory assumes that bond traders establish bond prices and interest rates strictly on the basis of expectations for future interest rates and that they are indifferent to maturity because they do not view long-term bonds as being riskier than short-term bonds. If this were true, the maturity risk premium (MRP) would be zero and long-term interest rates would simply be a weighted average of current and expected future short-term interest rates. To illustrate the pure expectations theory, assume that a 1-year Treasury bond currently yields 5.00% while a 2-year bond yields 5.50%. Investors who want to invest for a 2-year horizon have two primary options: Option 1: Buy a two-year security and hold it for 2 years. 12 This section is relatively technical, but instructors can omit it without loss of continuity. Although most evidence suggests that there is a positive maturity risk premium, some academics and practitioners contend that this second assumption is reasonable, at least as an approximation. They argue that the market is dominated by large bond traders who buy and sell securities of different maturities each day, that these traders focus only on short-term returns, and that they are not concerned with maturity risk. According to this view, a bond trader is just as willing to buy a 20-year bond to pick up a short-term profit as he or she is to buy a 3month security. Proponents of this view argue that the shape of the Treasury yield curve is therefore determined only by market expectations about future interest rates. Later we show what happens when we include the effects of maturity risk premiums. 13 Chapter 6 Interest Rates Option 2: Buy a 1-year security; hold it for 1 year; and then at the end of the year, reinvest the proceeds in another 1-year security. If they select Option 1, for every dollar they invest today, they will have accumulated $1.113025 by the end of Year 2: Funds at end of Year 2 ¼ $1  ð1:055Þ2 ¼ $1:113025 If they select Option 2, they should end up with the same amount; but this equation is used to find the ending amount: Funds at end of Year 2 ¼ $1  ð1:05Þ  ð1 þ XÞ Here X is the expected interest rate on a 1-year Treasury security 1 year from now. If the expectations theory is correct, each option must provide the same amount of cash at the end of 2 years, which implies the following: ð1:05Þð1 þ XÞ ¼ ð1:055Þ2 We can rearrange this equation and then solve for X: 1 þ X ¼ ð1:055Þ2 =1:05 X ¼ ð1:055Þ2 =1:05  1 ¼ 0:0600238 ¼ 6:00238% Therefore, X, the 1-year rate 1 year from today, must be 6.00238%; otherwise, one option will be better than the other and the market will not be in equilibrium. However, if the market is not in equilibrium, buying and selling will quickly bring about equilibrium. For example, suppose investors expect the 1-year Treasury rate to be 6.00238% a year from now but a 2-year bond now yields 5.25%, not the 5.50% rate required for equilibrium. Bond traders could earn a profit by adopting the following strategy: 1. Borrow money for 2 years at the 2-year rate, 5.25% per year. 2. Invest the money in a series of 1-year securities, expecting to earn 5.00% this year and 6.00238% next year, for an overall expected return over the 2 years of [(1.05) × (1.0600238)]1/2– 1 ¼ 5.50%. Borrowing at 5.25% and investing to earn 5.50% is a good deal, so bond traders would rush to borrow money (demand funds) in the 2-year market and invest (or supply funds) in the 1-year market. Recall from Figure 6-1 that a decline in the supply of funds raises interest rates, while an increase in the supply lowers rates. Likewise, an increase in the demand for funds raises rates, while a decline in demand lowers rates. Therefore, bond traders would push up the 2-year yield and simultaneously lower the yield on 1-year bonds. This buying and selling would cease when the 2-year rate becomes a weighted average of expected future 1-year rates.14 The preceding analysis was based on the assumption that the maturity risk premium is zero. However, most evidence suggests that a positive maturity risk premium exists. For example, assume once again that 1- and 2-year maturities yield 5.00% and 5.50%, respectively; so we have a rising yield curve. However, 14 In our calculations, we used the geometric average of the current and expected 1-year rates: [(1.05) × (1.0600238)]1/2 – 1 ¼ 0.055 or 5.50%. The arithmetic average of the two rates is (5% þ 6.00238%)/2 ¼ 5.50119%. The geometric average is theoretically correct, but the difference is only 0.00119%. With interest rates at the levels they have been in the United States and most other nations in recent years, the geometric and arithmetic averages are so close that many people use the arithmetic average, especially given the other assumptions that underlie the estimation of future 1-year rates. 181 Part 3 Financial Assets now assume that the maturity risk premium on the 2-year bond is 0.20% versus zero for the 1-year bond. This premium means that in equilibrium, the expected annual return on a 2-year bond (5.50%) must be 0.20% higher than the expected return on a series of two 1-year bonds (5.00% and X%). Therefore, the expected return on the series must be 5.50% – 0.20% = 5.30%: Expected return on 2-year series ¼ Rate on 2-year bond  MRP ¼ 0:055  0:002 ¼ 0:053 ¼ 5:30% Now recall that the annual expected return from the series of two 1-year bonds can be expressed as follows, where X is the 1-year rate next year: ð1:05Þð1 þ XÞ ¼ ð1 þ Expected return on 2-year seriesÞ2 ¼ ð1:053Þ2 1:05X ¼ ð1:053Þ2  1:05 0:0588090 X ¼ ¼ 0:0560086 ¼ 5:60086% 1:05 Under these conditions, equilibrium requires that market participants expect the 1-year rate next year to be 5.60086%. Note that the rate read from the yield curve rises by 0.50% when the years to maturity increase from one to two: 5.50% – 5.00% = 0.50%. Of this 0.50% increase, 0.20% is attributable to the MRP and the remaining 0.30% is due to the increase in expected 1-year rates next year. Putting all of this together, we see that one can use the yield curve to estimate what the market expects the short-term rate to be next year. However, this requires an estimate of the maturity risk premium; and if our estimated MRP is incorrect, then so will our yield-curve-based interest rate forecast. Thus, while the yield curve can be used to obtain insights into what the market thinks future interest rates will be, we calculate out these expectations with precision unless the pure expectations theory holds or we know with certainty the exact maturity risk premium. Since neither of these conditions holds, it is difficult to know for sure what the market is forecasting. Note too that even if we could determine the market’s consensus forecast for future rates, the market is not always right. So a forecast of next year’s rate based on the yield curve could be wrong. Therefore, obtaining an accurate forecast of rates for next year—or even for next month—is extremely difficult. SE 182 LF TEST What key assumption underlies the pure expectations theory? Assuming that the pure expectations theory is correct, how are expected short-term rates used to calculate expected long-term rates? According to the pure expectations theory, what would happen if long-term rates were not an average of expected short-term rates? Most evidence suggests that a positive maturity risk premium exists. How would this affect your calculations when determining interest rates? Assume that the interest rate on a 1-year T-bond is currently 7% and the rate on a 2-year bond is 9%. If the maturity risk premium is zero, what is a reasonable forecast of the rate on a 1-year bond next year? What would the forecast be if the maturity risk premium on the 2-year bond was 0.5% versus zero for the 1-year bond? (11.04%; 10.02%) Chapter 6 Interest Rates 183 6-7 MACROECONOMIC FACTORS THAT INFLUENCE INTEREST RATE LEVELS We described how key components such as expected inflation, default risk, maturity risk, and liquidity concerns influence the level of interest rates over time and across different markets. On a day-to-day basis, a variety of macroeconomic factors may influence one or more of these components; hence, macroeconomic factors have an important effect on both the general level of interest rates and the shape of the yield curve. The primary factors are (1) Federal Reserve policy; (2) the federal budget deficit or surplus; (3) international factors, including the foreign trade balance and interest rates in other countries; and (4) the level of business activity. 6-7a Federal Reserve Policy As you probably learned in your economics courses, (1) the money supply has a significant effect on the level of economic activity, inflation, and interest rates, and (2) in the United States, the Federal Reserve Board controls the money supply. If the Fed wants to stimulate the economy, it increases the money supply. The Fed buys and sells short-term securities, so the initial effect of a monetary easing would be to cause short-term rates to decline. However, a larger money supply might lead to an increase in expected future inflation, which would cause longterm rates to rise even as short-term rates fell. The reverse holds if the Fed tightens the money supply. As you can see from Figure 6-2, interest rates in recent years have been relatively low, with short-term rates especially low in 2003 and 2004. Those low rates enabled mortgage banks to write adjustable rate mortgage loans with very favorable rates, and that helped stimulate a huge housing boom along with growth of the economy. The Fed became concerned that the economy would overheat; so from 2004 to 2006, it raised its target rate 17 times, going from 2.0% to 5.25% in 2006. Long-term rates remained relatively stable during those years. The Fed left its target rate unchanged from June 2006 to September 2007, but the subprime credit crunch that began in 2007 caused increasing concerns about a possible recession. Those fears led the Fed to cut rates five times from September 2007 to February 2008, taking the target rate down from 5.25% to 3.00%. The Fed also signaled that more cuts were likely in the coming few months. Actions that lower short-term rates won’t necessarily lower long-term rates. This point was made in the following quote from the online edition of Investors’ Business Daily on February 15, 2008: U.S. government debt prices ended mostly lower Thursday, led by long-dated issues, as traders turned their focus to potential inflation risks resulting from additional interest rate cuts signaled by the Federal Reserve. It was a rough day for the Treasuries market, as traders concluded that more Fed rate cuts and the government’s fiscal stimulus program would come at the expense of higher long-term inflation. “Fiscal and monetary stimuli are focused on the current strain in the financial markets and its effect on the economy, but there are fears about what these actions may do to inflation down the road,” said Tom Sapio, a managing director at Cantor Fitzgerald in New York. Lower rates could also cause foreigners to sell their holdings of U.S. bonds. These investors would be paid with dollars, which they would then sell to buy their own currencies. The sale of dollars and the purchase of other currencies would lower the value of the dollar relative to other currencies, which would The home page for the Board of Governors of the Federal Reserve System can be found at www.federalreserve.gov. You can access general information about the Federal Reserve, including press releases, speeches, and monetary policy. 184 Part 3 Financial Assets make U.S. goods less expensive, which would help manufacturers and thus lower the trade deficit. Note also that during periods when the Fed is actively intervening in the markets, the yield curve may be temporarily distorted. Short-term rates may be driven below the long-run equilibrium level if the Fed is easing credit and above the equilibrium rate if the Fed is tightening credit. Long-term rates are not affected as much by Fed intervention. 6-7b Federal Budget Deficits or Surpluses If the federal government spends more than it takes in as taxes, it runs a deficit; and that deficit must be covered by additional borrowing (selling more Treasury bonds) or by printing money. If the government borrows, this increases the demand for funds and thus pushes up interest rates. If the government prints money, investors recognize that with “more money chasing a given amount of goods,” the result will be increased inflation, which will also increase interest rates. So the larger the federal deficit, other things held constant, the higher the level of interest rates. Over the past several decades, the federal government has generally run large budget deficits. There were some surpluses in the late 1990s; but the September 11, 2001, terrorist attacks, the subsequent recession, and the Iraq war all boosted government spending and caused the deficits to return. It is difficult to tell where fiscal policy will go and consequently what effect it will have on interest rates. 6-7c International Factors Foreign Trade Deficit The situation that exists when a country imports more than it exports. Businesses and individuals in the United States buy from and sell to people and firms all around the globe. If they buy more than they sell (that is, if there are more imports than exports), they are said to be running a foreign trade deficit. When trade deficits occur, they must be financed; and this generally means borrowing from nations with export surpluses. Thus, if the United States imported $200 billion of goods but exported only $100 billion, it would run a trade deficit of $100 billion while other countries would have a $100 billion trade surplus. The United States would probably borrow the $100 billion from the surplus nations.15 At any rate, the larger the trade deficit, the higher the tendency to borrow. Note that foreigners will hold U.S. debt if and only if the rates on U.S. securities are competitive with rates in other countries. This causes U.S. interest rates to be highly dependent on rates in other parts of the world. All this interdependency limits the ability of the Federal Reserve to use monetary policy to control economic activity in the United States. For example, if the Fed attempts to lower U.S. interest rates and this causes rates to fall below rates abroad, foreigners will begin selling U.S. bonds. Those sales will depress bond prices, which will push up rates in the United States. Thus, the large U.S. trade deficit (and foreigners’ holdings of U.S. debt that resulted from many years of deficits) hinders the Fed’s ability to combat a recession by lowering interest rates. For about 25 years following World War II, the United States ran large trade surpluses and the rest of the world owed it many billions of dollars. However, the situation changed, and the United States has been running trade deficits since the mid-1970s. The cumulative effect of these deficits has been to change the United States from being the largest creditor nation to being the largest debtor nation of 15 The deficit could also be financed by selling assets, including gold, corporate stocks, entire companies, and real estate. The United States has financed its massive trade deficits by all of these means in recent years. Although the primary method has been by borrowing from foreigners, in recent years, there has been a sharp increase in foreign purchases of U.S. assets, especially oil exporters’ purchases of U.S. businesses. Chapter 6 Interest Rates all time. As a result, interest rates are very much influenced by interest rates in other countries—higher or lower rates abroad lead to higher or lower U.S. rates. Because of all of this, U.S. corporate treasurers and everyone else who is affected by interest rates should keep up with developments in the world economy. 6-7d Business Activity SE You can examine Figure 6-2 to see how business conditions influence interest rates. Here are the key points revealed by the graph: 1. Because inflation increased from 1972 to 1981, the general tendency during that period was toward higher interest rates. However, since the 1981 peak, the trend has generally been downward. 2. The shaded areas in the graph represent recessions, during which (a) the demand for money and the rate of inflation tended to fall and (b) the Federal Reserve tended to increase the money supply in an effort to stimulate the economy. As a result, there is a tendency for interest rates to decline during recessions. For example, the economy began to slow down in 2000, and the country entered a mild recession in 2001. In response, the Federal Reserve cut interest rates. In 2004, the economy began to rebound; so the Fed began to raise rates. However, the subprime debacle hit in 2007; so the Fed began lowering rates in September 2007. By February, the Fed’s target rate had fallen from 5.25% to 3.00%, with indications that more reductions were likely. 3. During recessions, short-term rates decline more sharply than long-term rates. This occurs for two reasons: (a) The Fed operates mainly in the short-term sector, so its intervention has the strongest effect there. (b) Long-term rates reflect the average expected inflation rate over the next 20 to 30 years; and this expectation generally does not change much, even when the current inflation rate is low because of a recession or high because of a boom. So short-term rates are more volatile than long-term rates. Taking another look at Figure 6-2, we see that short-term rates did decline recently by much more than long-term rates. LF TEST Identify some macroeconomic factors that influence interest rates and explain the effects of each. How does the Fed stimulate the economy? How does the Fed affect interest rates? Does the Fed have complete control over U.S. interest rates? That is, can it set rates at any level it chooses? Why or why not? 6-8 INTEREST RATES AND BUSINESS DECISIONS The yield curve for January 2008 shown earlier in Figure 6-4 indicates how much the U.S. government had to pay in January 2008 to borrow money for 1 year, 5 years, 10 years, and so forth. A business borrower would have paid somewhat more, but assume for the moment that it is January 2008 and the yield curve shown for that year applies to your company. Now suppose you decide to build a new plant with a 30-year life that will cost $1 million and you will raise the $1 million by borrowing rather than by issuing new stock. If you borrowed in January 2008 on a short-term basis—say for 1 year—your annual interest cost would be only 2.7%, or $27,000. On the other hand, if you used long-term financing, your annual cost would be 4.3%, or $43,000. Therefore, at first glance, it would seem that you should use short-term debt. 185 186 Part 3 Financial Assets However, this could prove to be a horrible mistake. If you use short-term debt, you will have to renew your loan every 6 months; and the rate charged on each new loan will reflect the then-current short-term rate. Interest rates could return to their previous highs, in which case you would be paying 14%, or $140,000, per year. Those high interest payments would cut into and perhaps eliminate your profits. Your reduced profitability could increase your firm’s risk to the point where your bond rating was lowered, causing lenders to increase the risk premium built into your interest rate. That would further increase your interest payments, which would further reduce your profitability, worry lenders still more, and make them reluctant to renew your loan. If your lenders refused to renew the loan and demanded its repayment, as they would have every right to do, you might have to sell assets at a loss, which could result in bankruptcy. On the other hand, if you used long-term financing in 2008, your interest costs would remain constant at $43,000 per year; so an increase in interest rates in the economy would not hurt you. You might even be able to acquire some of your bankrupt competitors at bargain prices—bankruptcies increase dramatically when interest rates rise, primarily because many firms use so much short-term debt. Does all of this suggest that firms should avoid short-term debt? Not at all. If inflation falls over the next few years, so will interest rates. If you had borrowed on a long-term basis for 4.3% in January 2008, your company would be at a disadvantage if it was locked into 4.3% debt while its competitors (who used short-term debt in 2008) had a borrowing cost of only 2.7%. Financing decisions would be easy if we could make accurate forecasts of future interest rates. Unfortunately, predicting interest rates with consistent accuracy is nearly impossible. However, although it is difficult to predict future interest rate levels, it is easy to predict that interest rates will fluctuate—they always have, and they always will. That being the case, sound financial policy calls for using a mix of long- and short-term debt as well as equity to position the firm so that it can survive in any interest rate environment. Further, the optimal financial policy depends in an important way on the nature of the firm’s assets—the easier it is to sell off assets to generate cash, the more feasible it is to use more short-term debt. This makes it logical for a firm to finance current assets such as inventories and receivables with short-term debt and to finance fixed assets such as buildings and equipment with long-term debt. We will return to this issue later in the book when we discuss capital structure and financing policy. Changes in interest rates also have implications for savers. For example, if you had a 401(k) plan—and someday most of you will—you would probably want to invest some of your money in a bond mutual fund. You could choose a fund that had an average maturity of 25 years, 20 years, on down to only a few months (a money market fund). How would your choice affect your investment results and hence your retirement income? First, your decision would affect your annual interest income. For example, if the yield curve was upward- sloping, as it normally is, you would earn more interest if you chose a fund that held longterm bonds. Note, though, that if you chose a long-term fund and interest rates then rose, the market value of your fund would decline. For example, as we will see in Chapter 7, if you had $100,000 in a fund whose average bond had a maturity of 25 years and a coupon rate of 6% and if interest rates then rose from 6% to 10%, the market value of your fund would decline from $100,000 to about $63,500. On the other hand, if rates declined, your fund would increase in value. If you invested in a short-term fund, its value would be stable, but it would probably provide less interest per year. In any event, your choice of maturity would have a major effect on your investment performance and hence on your future income. SE Chapter 6 Interest Rates LF TEST If short-term interest rates are lower than long-term rates, why might a borrower still choose to finance with long-term debt? Explain the following statement: The optimal financial policy depends in an important way on the nature of the firm’s assets. TYING IT ALL TOGETHER In this chapter, we discussed the way interest rates are determined, the term structure of interest rates, and some of the ways interest rates affect business decisions. We saw that the interest rate on a given bond, r, is based on this equation: r ¼ r þ IP þ DRP þ LP þ MRP Here r* is the real risk-free rate, IP is the premium for expected inflation, DRP is the premium for potential default risk, LP is the premium for lack of liquidity, and MRP is the premium to compensate for the risk inherent in bonds with long maturities. Both r* and the various premiums can and do change over time depending on economic conditions, Federal Reserve actions, and the like. Since changes in these factors are difficult to predict, it is hard to forecast the future direction of interest rates. The yield curve, which relates bonds’ interest rates to their maturities, usually has an upward slope; but it can slope up or down, and both its slope and level change over time. The main determinants of the slope of the curve are expectations for future inflation and the MRP. We can analyze yield curve data to estimate what market participants think future interest rates are likely to be. We will use the insights gained from this chapter in later chapters, when we analyze the values of bonds and stocks and when we examine various corporate investment and financing decisions. SELF-TEST QUESTIONS AND PROBLEMS (Solutions Appear in Appendix A) ST-1 KEY TERMS Define each of the following terms: a. Production opportunities; time preferences for consumption; risk; inflation b. Real risk-free rate of interest, r*; nominal (quoted) risk-free rate of interest, rRF c. Inflation premium (IP) d. Default risk premium (DRP) e. Liquidity premium (LP); maturity risk premium (MRP) f. Interest rate risk; reinvestment rate risk g. Term structure of interest rates; yield curve h. “Normal” yield curve; inverted (“abnormal”) yield curve; humped yield curve i. Pure expectations theory 187 188 Part 3 Financial Assets ST-2 INFLATION AND INTEREST RATES The real risk-free rate of interest, r*, is 3%; and it is expected to remain constant over time. Inflation is expected to be 2% per year for the next 3 years and 4% per year for the next 5 years. The maturity risk premium is equal to 0.1 (t – 1)%, where t = the bond’s maturity. The default risk premium for a BBB-rated bond is 1.3%. a. What is the average expected inflation rate over the next 4 years? b. What is the yield on a 4-year Treasury bond? c. What is the yield on a 4-year BBB-rated corporate bond with a liquidity premium of 0.5%? What is the yield on an 8-year Treasury bond? d. e. f. ST-3 What is the yield on an 8-year BBB-rated corporate bond with a liquidity premium of 0.5%? If the yield on a 9-year Treasury bond is 7.3%, what does that imply about expected inflation in 9 years? PURE EXPECTATIONS THEORY The yield on 1-year Treasury securities is 6%, 2-year securities yield 6.2%, and 3-year securities yield 6.3%. There is no maturity risk premium. Using expectations theory, forecast the yields on the following securities: a. A 1-year security, 1 year from now b. A 1-year security, 2 years from now c. A 2-year security, 1 year from now QUESTIONS 6-1 6-2 6-3 6-4 Suppose interest rates on residential mortgages of equal risk are 5.5% in California and 7.0% in New York. Could this differential persist? What forces might tend to equalize rates? Would differentials in borrowing costs for businesses of equal risk located in California and New York be more or less likely to exist than differentials in residential mortgage rates? Would differentials in the cost of money for New York and California firms be more likely to exist if the firms being compared were very large or if they were very small? What are the implications of all of this with respect to nationwide branching? Which fluctuate more—long-term or short-term interest rates? Why? Suppose you believe that the economy is just entering a recession. Your firm must raise capital immediately, and debt will be used. Should you borrow on a long-term or a shortterm basis? Why? Suppose the population of Area Y is relatively young and the population of Area O is relatively old but everything else about the two areas is the same. a. Would interest rates likely be the same or different in the two areas? Explain. b. Would a trend toward nationwide branching by banks and the development of nationwide diversified financial corporations affect your answer to part a? Explain. 6-5 Suppose a new process was developed that could be used to make oil out of seawater. The equipment required is quite expensive; but it would, in time, lead to low prices for gasoline, electricity, and other types of energy. What effect would this have on interest rates? 6-6 Suppose a new and more liberal Congress and administration are elected. Their first order of business is to take away the independence of the Federal Reserve System and to force the Fed to greatly expand the money supply. What effect will this have: a. On the level and slope of the yield curve immediately after the announcement? b. 6-7 On the level and slope of the yield curve that would exist two or three years in the future? It is a fact that the federal government (1) encouraged the development of the savings and loan industry, (2) virtually forced the industry to make long-term fixed-interest-rate mortgages, and (3) forced the savings and loans to obtain most of their capital as deposits that were withdrawable on demand. a. Would the savings and loans have higher profits in a world with a “normal” or an inverted yield curve? Chapter 6 Interest Rates b. Would the savings and loan industry be better off if the individual institutions sold their mortgages to federal agencies and then collected servicing fees or if the institutions held the mortgages that they originated? 6-8 Suppose interest rates on Treasury bonds rose from 5% to 9% as a result of higher interest rates in Europe. What effect would this have on the price of an average company’s common stock? 6-9 What does it mean when it is said that the United States is running a trade deficit? What impact will a trade deficit have on interest rates? PROBLEMS Easy Problems 1–7 6-1 YIELD CURVES The following yields on U.S. Treasury securities were taken from a recent financial publication: a. 6-2 Term Rate 6 months 1 year 2 years 3 years 4 years 5 years 10 years 20 years 30 years 5.1% 5.5 5.6 5.7 5.8 6.0 6.1 6.5 6.3 Plot a yield curve based on these data. b. What type of yield curve is shown? c. What information does this graph tell you? d. Based on this yield curve, if you needed to borrow money for longer than 1 year, would it make sense for you to borrow short-term and renew the loan or borrow long-term? Explain. REAL RISK-FREE RATE You read in The Wall Street Journal that 30-day T-bills are currently yielding 5.5%. Your brother-in-law, a broker at Safe and Sound Securities, has given you the following estimates of current interest rate premiums: l l l l Inflation premium ¼ 3.25% Liquidity premium ¼ 0.6% Maturity risk premium ¼ 1.8% Default risk premium ¼ 2.15% On the basis of these data, what is the real risk-free rate of return? 6-3 EXPECTED INTEREST RATE The real risk-free rate is 3%. Inflation is expected to be 2% this year and 4% during the next 2 years. Assume that the maturity risk premium is zero. What is the yield on 2-year Treasury securities? What is the yield on 3-year Treasury securities? 6-4 DEFAULT RISK PREMIUM A Treasury bond that matures in 10 years has a yield of 6%. A 10-year corporate bond has a yield of 8%. Assume that the liquidity premium on the corporate bond is 0.5%. What is the default risk premium on the corporate bond? 6-5 MATURITY RISK PREMIUM The real risk-free rate is 3%, and inflation is expected to be 3% for the next 2 years. A 2-year Treasury security yields 6.2%. What is the maturity risk premium for the 2-year security? 189 190 Part 3 Financial Assets Intermediate Problems 8–16 6-6 INFLATION CROSS-PRODUCT An analyst is evaluating securities in a developing nation where the inflation rate is very high. As a result, the analyst has been warned not to ignore the cross-product between the real rate and inflation. If the real risk-free rate is 5% and inflation is expected to be 16% each of the next 4 years, what is the yield on a 4-year security with no maturity, default, or liquidity risk? (Hint: Refer to “The Links between Expected Inflation and Interest Rates: A Closer Look” on Page 178.) 6-7 EXPECTATIONS THEORY One-year Treasury securities yield 5%. The market anticipates that 1 year from now, 1-year Treasury securities will yield 6%. If the pure expectations theory is correct, what is the yield today for 2-year Treasury securities? 6-8 EXPECTATIONS THEORY Interest rates on 4-year Treasury securities are currently 7%, while 6-year Treasury securities yield 7.5%. If the pure expectations theory is correct, what does the market believe that 2-year securities will be yielding 4 years from now? 6-9 EXPECTED INTEREST RATE The real risk-free rate is 3%. Inflation is expected to be 3% this year, 4% next year, and 3.5% thereafter. The maturity risk premium is estimated to be 0.05 × (t – 1)%, where t ¼ number of years to maturity. What is the yield on a 7-year Treasury note? 6-10 INFLATION Due to a recession, expected inflation this year is only 3%. However, the inflation rate in Year 2 and thereafter is expected to be constant at some level above 3%. Assume that the expectations theory holds and the real risk-free rate is r* = 2%. If the yield on 3-year Treasury bonds equals the 1-year yield plus 2%, what inflation rate is expected after Year 1? 6-11 DEFAULT RISK PREMIUM A company’s 5-year bonds are yielding 7.75% per year. Treasury bonds with the same maturity are yielding 5.2% per year, and the real risk-free rate (r*) is 2.3%. The average inflation premium is 2.5%; and the maturity risk premium is estimated to be 0.1 × (t – 1)%, where t ¼ number of years to maturity. If the liquidity premium is 1%, what is the default risk premium on the corporate bonds? 6-12 MATURITY RISK PREMIUM An investor in Treasury securities expects inflation to be 2.5% in Year 1, 3.2% in Year 2, and 3.6% each year thereafter. Assume that the real risk-free rate is 2.75% and that this rate will remain constant. Three-year Treasury securities yield 6.25%, while 5-year Treasury securities yield 6.80%. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is MRP5 – MRP3? 6-13 DEFAULT RISK PREMIUM The real risk-free rate, r*, is 2.5%. Inflation is expected to average 2.8% a year for the next 4 years, after which time inflation is expected to average 3.75% a year. Assume that there is no maturity risk premium. An 8-year corporate bond has a yield of 8.3%, which includes a liquidity premium of 0.75%. What is its default risk premium? 6-14 EXPECTATIONS THEORY AND INFLATION Suppose 2-year Treasury bonds yield 4.5%, while 1-year bonds yield 3%. r* is 1%, and the maturity risk premium is zero. a. Using the expectations theory, what is the yield on a 1-year bond 1 year from now? b. Challenging Problems 17−19 What is the expected inflation rate in Year 1? Year 2? 6-15 EXPECTATIONS THEORY Assume that the real risk-free rate is 2% and that the maturity risk premium is zero. If the 1-year bond yield is 5% and a 2-year bond (of similar risk) yields 7%, what is the 1-year interest rate that is expected for Year 2? What inflation rate is expected during Year 2? Comment on why the average interest rate during the 2-year period differs from the 1-year interest rate expected for Year 2. 6-16 INFLATION CROSS-PRODUCT An analyst is evaluating securities in a developing nation where the inflation rate is very high. As a result, the analyst has been warned not to ignore the cross-product between the real rate and inflation. A 6-year security with no maturity, default, or liquidity risk has a yield of 20.84%. If the real risk-free rate is 6%, what average rate of inflation is expected in this country over the next 6 years? (Hint: Refer to “The Links between Expected Inflation and Interest Rates: A Closer Look” on Page 178.) 6-17 INTEREST RATE PREMIUMS A 5-year Treasury bond has a 5.2% yield. A 10-year Treasury bond yields 6.4%, and a 10-year corporate bond yields 8.4%. The market expects that inflation will average 2.5% over the next 10 years (IP10 ¼ 2.5%). Assume that there is no maturity risk premium (MRP ¼ 0) and that the annual real risk-free rate, r*, will remain constant over the next 10 years. (Hint: Remember that the default risk premium and the liquidity premium are zero for Treasury securities: DRP ¼ LP ¼ 0.) A 5-year corporate bond Chapter 6 Interest Rates 6-18 6-19 has the same default risk premium and liquidity premium as the 10-year corporate bond described. What is the yield on this 5-year corporate bond? YIELD CURVES Suppose the inflation rate is expected to be 7% next year, 5% the following year, and 3% thereafter. Assume that the real risk-free rate, r*, will remain at 2% and that maturity risk premiums on Treasury securities rise from zero on very short-term bonds (those that mature in a few days) to 0.2% for 1-year securities. Furthermore, maturity risk premiums increase 0.2% for each year to maturity, up to a limit of 1.0% on 5-year or longerterm T-bonds. a. Calculate the interest rate on 1-, 2-, 3-, 4-, 5-, 10-, and 20-year Treasury securities and plot the yield curve. b. Suppose a AAA-rated company (which is the highest bond rating a firm can have) had bonds with the same maturities as the Treasury bonds. Estimate and plot what you believe a AAA-rated company’s yield curve would look like on the same graph with the Treasury bond yield curve. (Hint: Think about the default risk premium on its longterm versus its short-term bonds.) c. On the same graph, plot the approximate yield curve of a much riskier lower-rated company with a much higher risk of defaulting on its bonds. INFLATION AND INTEREST RATES In late 1980, the U.S. Commerce Department released new data showing inflation was 15%. At the time, the prime rate of interest was 21%, a record high. However, many investors expected the new Reagan administration to be more effective in controlling inflation than the Carter administration had been. Moreover, many observers believed that the extremely high interest rates and generally tight credit, which resulted from the Federal Reserve System’s attempts to curb the inflation rate, would lead to a recession, which, in turn, would lead to a decline in inflation and interest rates. Assume that at the beginning of 1981, the expected inflation rate for 1981 was 13%; for 1982, 9%; for 1983, 7%; and for 1984 and thereafter, 6%. a. b. c. d. e. What was the average expected inflation rate over the 5-year period 1981–1985? (Use the arithmetic average.) Over the 5-year period, what average nominal interest rate would be expected to produce a 2% real risk-free return on 5-year Treasury securities? Assume MRP ¼ 0. Assuming a real risk-free rate of 2% and a maturity risk premium that equals 0.1 × (t)%, where t is the number of years to maturity, estimate the interest rate in January 1981 on bonds that mature in 1, 2, 5, 10, and 20 years. Draw a yield curve based on these data. Describe the general economic conditions that could lead to an upward-sloping yield curve. If investors in early 1981 expected the inflation rate for every future year to be 10% (that is, It ¼ Itþ1 ¼ 10% for t ¼ 1 to ∞), what would the yield curve have looked like? Consider all the factors that are likely to affect the curve. Does your answer here make you question the yield curve you drew in part c? COMPREHENSIVE/SPREADSHEET PROBLEM 6-20 INTEREST RATE DETERMINATION AND YIELD CURVES a. What effect would each of the following events likely have on the level of nominal interest rates? (1) Households dramatically increase their savings rate. (2) Corporations increase their demand for funds following an increase in investment opportunities. (3) The government runs a larger-than-expected budget deficit. (4) There is an increase in expected inflation. b. Suppose you are considering two possible investment opportunities: a 12-year Treasury bond and a 7-year, A-rated corporate bond. The current real risk-free rate is 4%; and inflation is expected to be 2% for the next 2 years, 3% for the following 4 years, and 4% thereafter. The maturity risk premium is estimated by this formula: MRP ¼ 0.1 (t – 1)%. The liquidity premium for the corporate bond is estimated to be 0.7%. Finally, 191 192 Part 3 Financial Assets c. d. e. f. you may determine the default risk premium, given the company’s bond rating, from the default risk premium table in the text. What yield would you predict for each of these two investments? Given the following Treasury bond yield information from a recent financial publication, construct a graph of the yield curve. Maturity Yield 1 year 2 years 3 years 4 years 5 years 10 years 20 years 30 years 5.37% 5.47 5.65 5.71 5.64 5.75 6.33 5.94 Based on the information about the corporate bond provided in part b, calculate yields and then construct a new yield curve graph that shows both the Treasury and the corporate bonds. Which part of the yield curve (the left side or right side) is likely to be most volatile over time? Using the Treasury yield information in part c, calculate the following rates: (1) The 1-year rate 1 year from now (2) The 5-year rate 5 years from now (3) The 10-year rate 10 years from now (4) The 10-year rate 20 years from now INTEGRATED CASE MORTON HANDLEY & COMPANY 6-21 INTEREST RATE DETERMINATION Maria Juarez is a professional tennis player, and your firm manages her money. She has asked you to give her information about what determines the level of various interest rates. Your boss has prepared some questions for you to consider. a. b. c. d. What are the four most fundamental factors that affect the cost of money, or the general level of interest rates, in the economy? What is the real risk-free rate of interest (r*) and the nominal risk-free rate (rRF)? How are these two rates measured? Define the terms inflation premium (IP), default risk premium (DRP), liquidity premium (LP), and maturity risk premium (MRP). Which of these premiums is included in determining the interest rate on (1) short-term U.S. Treasury securities, (2) long-term U.S. Treasury securities, (3) short-term corporate securities, and (4) longterm corporate securities? Explain how the premiums would vary over time and among the different securities listed. What is the term structure of interest rates? What is a yield curve? e. Suppose most investors expect the inflation rate to be 5% next year, 6% the following year, and 8% thereafter. The real risk-free rate is 3%. The maturity risk premium is zero for bonds that mature in 1 year or less and 0.1% for 2-year bonds; then the MRP increases by 0.1% per year thereafter for 20 years, after which it is stable. What is the interest rate on 1-, 10-, and 20-year Treasury bonds? Draw a yield curve with these data. What factors can explain why this constructed yield curve is upward-sloping? f. At any given time, how would the yield curve facing a AAA-rated company compare with the yield curve for U.S. Treasury securities? At any given time, how would the yield curve facing a BB-rated company compare with the yield curve for U.S. Treasury securities? Draw a graph to illustrate your answer. Chapter 6 Interest Rates g. What is the pure expectations theory? What does the pure expectations theory imply about the term structure of interest rates? h. Suppose you observe the following term structure for Treasury securities: i. Maturity Yield 1 2 3 4 5 6.0% 6.2 6.4 6.5 6.5 year years years years years 193 Assume that the pure expectations theory of the term structure is correct. (This implies that you can use the yield curve provided to “back out” the market’s expectations about future interest rates.) What does the market expect will be the interest rate on 1-year securities 1 year from now? What does the market expect will be the interest rate on 3-year securities 2 years from now? ª JOHN CLARK, 2008/USED UNDER LICENSE FROM SHUTTERSTOCK.COM CHAPTER 7 Bonds and Their Valuation Sizing Up Risk in the Bond Market Many people view Treasury securities as a lackluster but ultra-safe investment. From a default standpoint, Treasuries are indeed our safest investments; but their prices can still decline in any given year if interest rates increase. This is especially true for long-term bonds, which lost nearly 9% in 1999. However, bonds can perform well—in fact, they outgained stocks in 5 of the 8 years between 2000 and 2007. All bonds aren’t alike, and they don’t necessarily all move in the same direction. For example, corporate bonds are callable and they can default, whereas Treasury bonds are not exposed to these risks. This results in higher nominal yields on corporates, but the spread between corporate and Treasury yields differs widely depending on the risk of the particular corporate bond. Moreover, yield spreads vary substantially over time, especially for lower-rated securities. For example, as information about WorldCom’s deteriorating condition began coming out in 2002, the spread on its 5-year bonds jumped from 1.67% to over 20% in mid-2002. These 194 bonds subsequently defaulted, so greedy people who bought them expecting a high return ended up with a large loss. When the economy is strong, corporate bonds generally produce higher returns than Treasuries— their promised returns are higher, and most make their promised payments because few go into default. However, when the economy weakens, concerns about defaults rise, which leads to declines in corporate bond prices. For example, from the beginning of 2000 to the end of 2002, a sluggish economy and a string of accounting scandals led to some major corporate defaults, which worried investors. All corporate bond prices then declined relative to Treasuries, and the result was an increase in yield spreads. As the economy rebounded in 2003, yield spreads declined to their former levels, which resulted in good gains in corporate bond prices. The situation is once again worrisome in 2008. The subprime mortgage crisis has led to fears of recession; and this has caused spreads to rise dramatically, especially for lower-rated bonds. Chapter 7 Bonds and Their Valuation For example, the spread on junk bonds over Treasuries rose from 2.4% to 7.5% in the 6 months from mid-2007 to January 2008. Bond investors are rightly worried today. If a recession does occur, this will lead to increased defaults on corporate bonds. A recession might benefit investors in Treasury bonds. However, because there have already been several rounds of Federal Reserve rate cuts, Treasury rates may not have much room to fall. Also, there is concern that recent Fed easing is sowing the seeds for higher inflation down the road, which would lead to higher rates and lower bond prices. 195 In the face of similar risks in 2001, a BusinessWeek Online article gave investors the following advice, which is still applicable today: Take the same diversified approach to bonds as you do with stocks. Blend in U.S. government, corporate— both high-quality and high-yield—and perhaps even some foreign government debt. If you’re investing taxable dollars, consider tax-exempt municipal bonds. And it doesn’t hurt to layer in some inflation-indexed bonds. Sources: Scott Patterson, “Ahead of the Tape: Junk Yields Flashing Back to ’01 Slump,” The Wall Street Journal, January 30, 2008, p. C1; Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Chicago: Morningstar, Inc., 2008); and Susan Scherreik, “Getting the Most Bang Out of Your Bonds,” BusinessWeek Online, November 12, 2001. PUTTING THINGS IN PERSPECTIVE In previous chapters, we noted that companies raise capital in two main forms: debt and equity. In this chapter, we examine the characteristics of bonds and discuss the various factors that influence bond prices. In Chapter 9, we will turn our attention to stocks and their valuation. If you skim through The Wall Street Journal, you will see references to a wide variety of bonds. This variety may seem confusing; but in actuality, only a few characteristics distinguish the various types of bonds. When you finish this chapter, you should be able to: Identify the different features of corporate and government bonds. Discuss how bond prices are determined in the market, what the relationship is between interest rates and bond prices, and how a bond’s price changes over time as it approaches maturity. Calculate a bond’s yield to maturity and its yield to call if it is callable and determine the “true” yield. Explain the different types of risk that bond investors and issuers face and the way a bond’s terms and collateral can be changed to affect its interest rate. l l l l 7-1 WHO ISSUES BONDS? A bond is a long-term contract under which a borrower agrees to make payments of interest and principal on specific dates to the holders of the bond. Bonds are issued by corporations and government agencies that are looking for long-term debt capital. For example, on January 3, 2009, Allied Food Products borrowed $50 million by issuing $50 million of bonds. For convenience, we assume that Allied sold 50,000 individual bonds for $1,000 each. Actually, it could have sold one $50 million bond, 10 bonds each with a $5 million face value, or any other combination that totaled $50 million. In any event, Allied received the $50 million; and in exchange, it promised to make annual interest payments and to repay the $50 million on a specified maturity date. Bond A long-term debt instrument. 196 Part 3 Financial Assets Until the 1970s, most bonds were beautifully engraved pieces of paper and their key terms, including their face values, were spelled out on the bonds. Today, though, virtually all bonds are represented by electronic data stored in secure computers, much like the “money” in a bank checking account. Bonds are grouped in several ways. One grouping is based on the issuer: the U.S. Treasury, corporations, state and local governments, and foreigners. Each bond differs with respect to risk and consequently its expected return. Treasury bonds, generally called Treasuries and sometimes referred to as government bonds, are issued by the federal government.1 It is reasonable to assume that the U.S. government will make good on its promised payments, so Treasuries have no default risk. However, these bonds’ prices do decline when interest rates rise; so they are not completely riskless. Corporate bonds are issued by business firms. Unlike Treasuries, corporates are exposed to default risk—if the issuing company gets into trouble, it may be unable to make the promised interest and principal payments and bondholders may suffer losses. Different corporate bonds have different levels of default risk depending on the issuing company’s characteristics and the terms of the specific bond. Default risk is often referred to as “credit risk”; and as we saw in Chapter 6, the larger this risk, the higher the interest rate investors demand. Municipal bonds, or munis, is the term given to bonds issued by state and local governments. Like corporates, munis are exposed to some default risk; but they have one major advantage over all other bonds: As we discussed in Chapter 3, the interest earned on most munis is exempt from federal taxes and from state taxes if the holder is a resident of the issuing state. Consequently, the market interest rate on a muni is considerably lower than on a corporate of equivalent risk. Foreign bonds are issued by a foreign government or a foreign corporation. All foreign corporate bonds are exposed to default risk, as are some foreign government bonds. An additional risk exists when the bonds are denominated in a currency other than that of the investor’s home currency. Consider, for example, a U.S. investor who purchases a corporate bond denominated in Japanese yen. At some point, the investor will want to close out his investment and convert the yen back to U.S. dollars. If the Japanese yen unexpectedly falls relative to the dollar, the investor will have fewer dollars than he originally expected to receive. Consequently, the investor could still lose money even if the bond does not default. Treasury Bonds Bonds issued by the federal government, sometimes referred to as government bonds. Corporate Bonds Bonds issued by corporations. Municipal Bonds Bonds issued by state and local governments. SE Foreign Bonds Bonds issued by foreign governments or by foreign corporations. LF TEST What is a bond? What are the four main issuers of bonds? Why are U.S. Treasury bonds not completely riskless? In addition to default risk, what key risk do investors in foreign bonds face? 7-2 KEY CHARACTERISTICS OF BONDS Although all bonds have some common characteristics, different bonds can have different contractual features. For example, most corporate bonds have provisions that allow the issuer to pay them off early (“call” features), but the specific call 1 The U.S. Treasury actually calls its debt “bills,” “notes,” or “bonds.” T-bills generally have maturities of 1 year or less at the time of issue, notes generally have original maturities of 2 to 7 years, and bonds originally mature in 8 to 30 years. There are technical differences between bills, notes, and bonds; but they are not important for our purposes. So we generally call all Treasury securities “bonds.” Note too that a 30-year T-bond at the time of issue becomes a 29-year bond the next year, and it is a 1-year bond after 29 years. Chapter 7 Bonds and Their Valuation 197 provisions vary widely among different bonds. Similarly, some bonds are backed by specific assets that must be turned over to the bondholders if the issuer defaults, while other bonds have no such collateral backup. Differences in contractual provisions (and in the fundamental underlying financial strength of the companies backing the bonds) lead to differences in bonds’ risks, prices, and expected returns. To understand bonds, it is essential that you understand the following terms. 7-2a Par Value The par value is the stated face value of the bond; for illustrative purposes, we generally assume a par value of $1,000, although any multiple of $1,000 (e.g., $5,000 or $5 million) can be used. The par value generally represents the amount of money the firm borrows and promises to repay on the maturity date. Par Value The face value of a bond. 7-2b Coupon Interest Rate Allied Food Products’ bonds require the company to pay a fixed number of dollars of interest each year. This payment, generally referred to as the coupon payment, is set at the time the bond is issued and remains in force during the bond’s life.2 Typically, at the time a bond is issued, its coupon payment is set at a level that will induce investors to buy the bond at or near its par value. Most of the examples and problems throughout this text will focus on bonds with fixed coupon rates. When this annual coupon payment is divided by the par value, the result is the coupon interest rate. For example, Allied’s bonds have a $1,000 par value, and they pay $100 in interest each year. The bond’s coupon payment is $100, so its coupon interest rate is $100/$1,000 ¼ 10%. In this regard, the $100 is the annual income that an investor receives when he or she invests in the bond. Allied’s bonds are fixed-rate bonds because the coupon rate is fixed for the life of the bond. In some cases, however, a bond’s coupon payment is allowed to vary over time. These floating-rate bonds work as follows: The coupon rate is set for an initial period, often 6 months, after which it is adjusted every 6 months based on some open market rate. For example, the bond’s rate may be adjusted so as to equal the 10-year Treasury bond rate plus a “spread” of 1.5 percentage points. Other provisions can be included in corporate bonds. For example, some can be converted at the holders’ option into fixed-rate debt, and some floaters have upper limits (caps) and lower limits (floors) on how high or low the rate can go. Some bonds pay no coupons at all, but are offered at a discount below their par values and hence provide capital appreciation rather than interest income. These securities are called zero coupon bonds (zeros). Other bonds pay some coupon interest, but not enough to induce investors to buy them at par. In general, any bond originally offered at a price significantly below its par value is called an original issue discount (OID) bond. Some of the details associated with issuing or investing in zero coupon bonds are discussed more fully in Web Appendix 7A. 7-2c Maturity Date Bonds generally have a specified maturity date on which the par value must be repaid. Allied’s bonds, which were issued on January 3, 2009, will mature on 2 Back when bonds were ornate, they were engraved pieces of paper rather than electronic information stored on a computer. Each bond had a number of small (1/2- by 2-inch) dated coupons attached to them; and on each interest payment date, the owner would “clip the coupon” for that date, send it to the company’s paying agent, and receive a check for the interest. A 30-year semiannual bond would start with 60 coupons, whereas a 5-year annual payment bond would start with only 5 coupons. Today no physical coupons are involved, and interest checks are mailed or deposited automatically to the bonds’ registered owners on the payment date. Even so, people continue to use the terms coupon and coupon interest rate when discussing bonds. You can think of the coupon interest rate as the promised rate. Coupon Payment The specified number of dollars of interest paid each year. Coupon Interest Rate The stated annual interest rate on a bond. Fixed-Rate Bond A bond whose interest rate is fixed for its entire life. Floating-Rate Bond A bond whose interest rate fluctuates with shifts in the general level of interest rates. Zero Coupon Bond A bond that pays no annual interest but is sold at a discount below par, thus compensating investors in the form of capital appreciation. Original Issue Discount (OID) Bond Any bond originally offered at a price below its par value. Maturity Date A specified date on which the par value of a bond must be repaid. 198 Part 3 Financial Assets Original Maturity The number of years to maturity at the time a bond is issued. January 2, 2024; thus, they had a 15-year maturity at the time they were issued. Most bonds have original maturities (the maturity at the time the bond is issued) ranging from 10 to 40 years, but any maturity is legally permissible.3 Of course, the effective maturity of a bond declines each year after it has been issued. Thus, Allied’s bonds had a 15-year original maturity. But in 2010, a year later, they will have a 14-year maturity; a year after that, they will have a 13-year maturity; and so forth. 7-2d Call Provisions Call Provision A provision in a bond contract that gives the issuer the right to redeem the bonds under specified terms prior to the normal maturity date. Most corporate and municipal bonds, but not Treasuries, contain a call provision that gives the issuer the right to call the bonds for redemption.4 The call provision generally states that the issuer must pay the bondholders an amount greater than the par value if they are called. The additional sum, which is termed a call premium, is often equal to one year’s interest. For example, the call premium on a 10-year bond with a 10% annual coupon and a par value of $1,000 might be $100, which means that the issuer would have to pay investors $1,100 (the par value plus the call premium) if it wanted to call the bonds. In most cases, the provisions in the bond contract are set so that the call premium declines over time as the bonds approach maturity. Also, while some bonds are immediately callable, in most cases, bonds are often not callable until several years after issue, generally 5 to 10 years. This is known as a deferred call, and such bonds are said to have call protection. Companies are not likely to call bonds unless interest rates have declined significantly since the bonds were issued. Suppose a company sold bonds when interest rates were relatively high. Provided the issue is callable, the company could sell a new issue of low-yielding securities if and when interest rates drop, use the proceeds of the new issue to retire the high-rate issue, and thus reduce its interest expense. This process is called a refunding operation. Thus, the call privilege is valuable to the firm but detrimental to long-term investors, who will need to reinvest the funds they receive at the new and lower rates. Accordingly, the interest rate on a new issue of callable bonds will exceed that on the company’s new noncallable bonds. For example, on February 29, 2008, Pacific Timber Company sold a bond issue yielding 8% that was callable immediately. On the same day, Northwest Milling Company sold an issue with similar risk and maturity that yielded only 7.5%; but its bonds were noncallable for 10 years. Investors were willing to accept a 0.5% lower coupon interest rate on Northwest’s bonds for the assurance that the 7.5% interest rate would be earned for at least 10 years. Pacific, on the other hand, had to incur a 0.5% higher annual interest rate for the option of calling the bonds in the event of a decline in rates. Note that the refunding operation is similar to a homeowner refinancing his or her home mortgage after a decline in rates. Consider, for example, a homeowner with an outstanding mortgage at 8%. If mortgage rates have fallen to 5%, the homeowner will probably find it beneficial to refinance the mortgage. There may 3 In July 1993, The Walt Disney Company, attempting to lock in a low interest rate, stretched the meaning of “longterm bond” by issuing the first 100-year bonds sold by any borrower in modern times. Soon after, Coca-Cola became the second company to sell 100-year bonds. A number of other companies have followed. 4 The number of new corporate issues with call provisions has declined somewhat in recent years. In the 1980s, nearly 80% of new issues contained call provisions; but in recent years, this number has fallen to about 35%. The use of call provisions also varies with credit quality. Roughly 25% of investment-grade bonds in recent years have call provisions versus about 75% of non-investment-grade bonds. Interest rates were historically high in the 1980s, so issuers wanted to be able to refund their debt if and when rates fell. Similarly, companies with low ratings hoped their ratings would rise, lowering their market rates and giving them an opportunity to refund. For more information on the use of callable bonds, see Levent Güntay, N. R. Prabhala, and Haluk Unal, “Callable Bonds, Interest-Rate Risk, and the Supply Side of Hedging,” May 2005, a Wharton Financial Institutions Center working paper. Chapter 7 Bonds and Their Valuation 199 be some fees involved in the refinancing, but the lower rate may be more than enough to offset those fees. The analysis required is essentially the same for homeowners and corporations. 7-2e Sinking Funds Some bonds include a sinking fund provision that facilitates the orderly retirement of the bond issue. Years ago firms were required to deposit money with a trustee, which invested the funds and then used the accumulated sum to retire the bonds when they matured. Today, though, sinking fund provisions require the issuer to buy back a specified percentage of the issue each year. A failure to meet the sinking fund requirement constitutes a default, which may throw the company into bankruptcy. Therefore, a sinking fund is a mandatory payment. Suppose a company issued $100 million of 20-year bonds and it is required to call 5% of the issue, or $5 million of bonds, each year. In most cases, the issuer can handle the sinking fund requirement in either of two ways: 1. It can call in for redemption, at par value, the required $5 million of bonds. The bonds are numbered serially, and those called for redemption would be determined by a lottery administered by the trustee. 2. The company can buy the required number of bonds on the open market. Sinking Fund Provision A provision in a bond contract that requires the issuer to retire a portion of the bond issue each year. The firm will choose the least-cost method. If interest rates have fallen since the bond was issued, the bond will sell for more than its par value. In this case, the firm will use the call option. However, if interest rates have risen, the bonds will sell at a price below par; so the firm can and will buy $5 million par value of bonds in the open market for less than $5 million. Note that a call for sinking fund purposes is generally different from a refunding call because most sinking fund calls require no call premium. However, only a small percentage of the issue is normally callable in a given year. Although sinking funds are designed to protect investors by ensuring that the bonds are retired in an orderly fashion, these funds work to the detriment of bondholders if the bond’s coupon rate is higher than the current market rate. For example, suppose the bond has a 10% coupon but similar bonds now yield only 7.5%. A sinking fund call at par would require a long-term investor to give up a bond that pays $100 of interest and then to reinvest in a bond that pays only $75 per year. This is an obvious disadvantage to those bondholders whose bonds are called. On balance, however, bonds that have a sinking fund are regarded as being safer than those without such a provision; so at the time they are issued, sinking fund bonds have lower coupon rates than otherwise similar bonds without sinking funds. 7-2f Other Features Several other types of bonds are used sufficiently often to warrant mention.5 First, convertible bonds are bonds that are exchangeable into shares of common stock at a fixed price at the option of the bondholder. Convertibles offer investors the chance for capital gains if the stock increases, but that feature enables the issuing company to set a lower coupon rate than on nonconvertible debt with similar credit risk. Bonds issued with warrants are similar to convertibles; but instead of giving the investor an option to exchange the bonds for stock, warrants give the holder an option to buy stock for a stated price, thereby providing a capital gain if the stock’s price rises. Because of this factor, bonds issued with warrants, like convertibles, carry lower coupon rates than otherwise similar nonconvertible bonds. 5 A recent article by John D. Finnerty and Douglas R. Emery reviews new types of debt (and other) securities that have been created in recent years. See “Corporate Securities Innovations: An Update,” Journal of Applied Finance: Theory, Practice, Education, Vol. 12, no. 1 (Spring/Summer 2002), pp. 21–47. Convertible Bond A bond that is exchangeable at the option of the holder for the issuing firm’s common stock. Warrant A long-term option to buy a stated number of shares of common stock at a specified price. 200 Part 3 Financial Assets Whereas callable bonds give the issuer the right to retire the debt prior to maturity, putable bonds allow investors to require the company to pay in advance. If interest rates rise, investors will put the bonds back to the company and reinvest in higher coupon bonds. Yet another type of bond is the income bond, which pays interest only if the issuer has earned enough money to pay the interest. Thus, income bonds cannot bankrupt a company; but from an investor’s standpoint, they are riskier than “regular” bonds. Yet another bond is the indexed, or purchasing power, bond. The interest rate is based on an inflation index such as the consumer price index; so the interest paid rises automatically when the inflation rate rises, thus protecting bondholders against inflation. As we mentioned in Chapter 6, the U.S. Treasury is the main issuer of indexed bonds. Recall that these Treasury Inflation Protected Securities (TIPS) generally pay a real return varying from 1% to 3%, plus the rate of inflation during the past year. Putable Bond A bond with a provision that allows its investors to sell it back to the company prior to maturity at a prearranged price. Income Bond A bond that pays interest only if it is earned. SE Indexed (Purchasing Power) Bond A bond that has interest payments based on an inflation index so as to protect the holder from inflation. Define floating-rate bonds, zero coupon bonds, callable bonds, putable bonds, income bonds, convertible bonds, and inflation-indexed bonds (TIPS). LF TEST Which is riskier to an investor, other things held constant—a callable bond or a putable bond? In general, how is the rate on a floating-rate bond determined? What are the two ways sinking funds can be handled? Which alternative will be used if interest rates have risen? if interest rates have fallen? 7-3 BOND VALUATION The value of any financial asset—a stock, a bond, a lease, or even a physical asset such as an apartment building or a piece of machinery—is the present value of the cash flows the asset is expected to produce. The cash flows for a standard coupon-bearing bond, like those of Allied Foods, consist of interest payments during the bond’s 15-year life plus the amount borrowed (generally the par value) when the bond matures. In the case of a floating-rate bond, the interest payments vary over time. For zero coupon bonds, there are no interest payments; so the only cash flow is the face amount when the bond matures. For a “regular” bond with a fixed coupon, like Allied’s, here is the situation: 0 Bond’s value rd% 1 2 3 N INT INT INT INT M Here rd ¼ the market rate of interest on the bond, 10%. This is the discount rate used to calculate the present value of the cash flows, which is also the bond’s price. In Chapter 6, we discussed in detail the various factors that determine market interest rates. Note that rd is not the coupon interest rate. However, rd is equal to the coupon rate at times, especially the day the bond is issued; and when the two rates are equal, as in this case, the bond sells at par. Chapter 7 Bonds and Their Valuation N ¼ the number of years before the bond matures ¼ 15. N declines over time after the bond has been issued; so a bond that had a maturity of 15 years when it was issued (original maturity ¼ 15) will have N ¼ 14 after 1 year, N ¼ 13 after 2 years, and so forth. At this point, we assume that the bond pays interest once a year, or annually; so N is measured in years. Later on we will analyze semiannual payment bonds, which pay interest every 6 months. INT ¼ dollars of interest paid each year ¼ Coupon rate  Par value ¼ 0.10 ($1,000) ¼ $100. In calculator terminology, INT ¼ PMT ¼ 100. If the bond had been a semiannual payment bond, the payment would have been $50 every 6 months. The payment would have been zero if Allied had issued zero coupon bonds, and it would have varied over time if the bond had been a “floater.” M ¼ the par, or maturity, value of the bond ¼ $1,000. This amount must be paid at maturity. Back in the 1970s and before, when paper bonds with paper coupons were used, most bonds had a $1,000 value. Now with computer-entry bonds, the par amount purchased can vary; but we use $1,000 for simplicity. We can now redraw the time line to show the numerical values for all variables except the bond’s value (and price, assuming an equilibrium exists), VB: 0 1 2 3 100 100 100 10% Bond’s value 15 100 1,000 1,100 The following general equation can be solved to find the value of any bond: INT INT INT M þ þ þ  þ ð1 þ rd Þ1 ð1 þ rd Þ2 ð1 þ rd ÞN ð1 þ rd ÞN N X INT M ¼ t þ ð1 þ r Þ ð1 þ rd ÞN d t¼1 Bond0 s value ¼ VB ¼ 7-1 Inserting values for the Allied bond, we have VB ¼ 15 X t¼1 $100 $1,000 t þ ð1:10Þ ð1:10Þ15 The cash flows consist of an annuity of N years plus a lump sum payment at the end of Year N, and this fact is reflected in Equation 7-1. We could simply discount each cash flow back to the present and sum those PVs to find the bond’s value; see Figure 7-1 for an example. However, this procedure is not very efficient, especially when the bond has many years to maturity. Therefore, we use a financial calculator to solve the problem. Here is the setup: Inputs: Output: 15 10 N I/YR PV 100 1000 PMT FV = –1,000 Simply input N ¼ 15, rd ¼ I/YR ¼ 10, INT ¼ PMT ¼ 100, and M ¼ FV ¼ 1000; then press the PV key to find the bond’s value, $1,000.6 Since the PV is an outflow to the 6 Spreadsheets can also be used to solve for the bond’s value, as we show in the Excel model for this chapter. 201 202 Part 3 Financial Assets Time Line for Allied Food Products’ Bonds, 10% Interest Rate FIGURE 7-1 1/3/10 1/11 Payments 100 100 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 1/20 1/21 1/22 1/23 1/2/2024 100 100 100 100 100 100 100 100 100 100 100 100 100 ⫹ 1,000 90.91 82.64 75.13 68.30 62.09 56.45 51.32 46.65 42.41 38.55 35.05 31.86 28.97 26.33 23.94 239.39 Present Value ⫽ 1,000.00 when rd ⫽ 10% investor, it is shown with a negative sign. The calculator is programmed to solve Equation 7-1: It finds the PV of an annuity of $100 per year for 15 years discounted at 10%; then it finds the PV of the $1,000 maturity payment; then it adds those two PVs to find the bond’s value. In this example, the bond is selling at a price equal to its par value. Whenever the bond’s market, or going, rate, rd, is equal to its coupon rate, a fixed-rate bond will sell at its par value. Normally, the coupon rate is set at the going rate in the market the day a bond is issued, causing it to sell at par initially. The coupon rate remains fixed after the bond is issued, but interest rates in the market move up and down. Looking at Equation 7-1, we see that an increase in the market interest rate (rd) causes the price of an outstanding bond to fall, whereas a decrease in the rate causes the bond’s price to rise. For example, if the market interest rate on Allied’s bond increased to 15% immediately after it was issued, we would recalculate the price with the new market interest rate as follows: Inputs: Output: Discount Bond A bond that sells below its par value; occurs whenever the going rate of interest is above the coupon rate. 15 15 N I/YR PV 100 1000 PMT FV = –707.63 The bond’s price would fall to $707.63, well below par, as a result of the increase in interest rates. Whenever the going rate of interest rises above the coupon rate, a fixed-rate bond’s price will fall below its par value; this type of bond is called a discount bond. Chapter 7 Bonds and Their Valuation 203 On the other hand, bond prices rise when market interest rates fall. For example, if the market interest rate on Allied’s bond decreased to 5% immediately after it was issued, we would once again recalculate its price as follows: Inputs: 15 5 N I/YR Output: PV 100 1000 PMT FV = –1,518.98 In this case, the price rises to $1,518.98. In general, whenever the going interest rate falls below the coupon rate, a fixed-rate bond’s price will rise above its par value; this type of bond is called a premium bond. To summarize, here is the situation: SE rd ¼ coupon rate, fixed-rate bond sells at par; hence, it is a par bond rd > coupon rate, fixed-rate bond sells below par; hence, it is a discount bond rd < coupon rate, fixed-rate bond sells above par; hence, it is a premium bond LF TEST Premium Bond A bond that sells above its par value; occurs whenever the going rate of interest is below the coupon rate. A bond that matures in 8 years has a par value of $1,000 and an annual coupon payment of $70; its market interest rate is 9%. What is its price? ($889.30) A bond that matures in 12 years has a par value of $1,000 and an annual coupon of 10%; the market interest rate is 8%. What is its price? ($1,150.72) Which of those two bonds is a discount bond, and which is a premium bond? 7-4 BOND YIELDS If you examine the bond market table of The Wall Street Journal or a price sheet put out by a bond dealer, you will typically see information regarding each bond’s maturity date, price, and coupon interest rate. You will also see a reported yield. Unlike the coupon interest rate, which is fixed, the bond’s yield varies from day to day depending on current market conditions. To be most useful, the bond’s yield should give us an estimate of the rate of return we would earn if we bought the bond today and held it over its remaining life. If the bond is not callable, its remaining life is its years to maturity. If it is callable, its remaining life is the years to maturity if it is not called or the years to the call if it is called. In the following sections, we explain how to calculate those two possible yields and which one is likely to occur. 7-4a Yield to Maturity Suppose you were offered a 14-year, 10% annual coupon, $1,000 par value bond at a price of $1,494.93. What rate of interest would you earn on your investment if you bought the bond, held it to maturity, and received the promised interest and maturity payments? This rate is called the bond’s yield to maturity (YTM), and it is the interest rate generally discussed by investors when they talk about rates of return and the rate reported by The Wall Street Journal and other Yield to Maturity (YTM) The rate of return earned on a bond if it is held to maturity. 204 Part 3 Financial Assets publications. To find the YTM, all you need to do is solve Equation 7-1 for rd as follows: VB ¼ $1,494:93 ¼ INT INT INT M þ þ  þ þ ð1 þ rd Þ1 ð1 þ rd Þ2 ð1 þ rd ÞN ð1 þ rd ÞN $100 $100 $1,000 þ  þ þ ð1 þ rd Þ1 ð1 þ rd Þ14 ð1 þ rd Þ14 You can substitute values for rd until you find a value that “works” and force the sum of the PVs in the equation to equal $1,494.93. However, finding rd ¼ YTM by trial and error would be a tedious, time-consuming process. However, as you might guess, the calculation is easy with a financial calculator.7 Here is the setup: Inputs: 14 N Output: I/YR –1494.93 100 1000 PV PMT FV =5 Simply enter N ¼ 14, PV ¼ –1494.93, PMT ¼ 100, and FV ¼ 1000; then press the I/YR key. The answer, 5%, will appear. The yield to maturity can also be viewed as the bond’s promised rate of return, which is the return that investors will receive if all of the promised payments are made. However, the yield to maturity equals the expected rate of return only when (1) the probability of default is zero and (2) the bond cannot be called. If there is some default risk or the bond may be called, there is some chance that the promised payments to maturity will not be received, in which case the calculated yield to maturity will exceed the expected return. Note also that a bond’s calculated yield to maturity changes whenever interest rates in the economy change, which is almost daily. An investor who purchases a bond and holds it until it matures will receive the YTM that existed on the purchase date, but the bond’s calculated YTM will change frequently between the purchase date and the maturity date. 7-4b Yield to Call Yield to Call (YTC) The rate of return earned on a bond when it is called before its maturity date. If you purchase a bond that is callable and the company calls it, you do not have the option of holding it to maturity. Therefore, the yield to maturity would not be earned. For example, if Allied’s 10% coupon bonds were callable and if interest rates fell from 10% to 5%, the company could call in the 10% bonds, replace them with 5% bonds, and save $100 – $50 ¼ $50 interest per bond per year. This would be beneficial to the company but not to its bondholders. If current interest rates are well below an outstanding bond’s coupon rate, a callable bond is likely to be called; and investors will estimate its most likely rate of return as the yield to call (YTC) rather than the yield to maturity. To calculate the YTC, we modify Equation 7-1, using years to call as N and the call price rather than the maturity value as the ending payment. Here’s the modified equation: 7-2 Price of bond ¼ N X t¼1 INT Call price t þ ð1 þ rd Þ ð1 þ rd ÞN Here N is the number of years until the company can call the bond; call price is the price the company must pay in order to call the bond (it is often set equal to the par value plus one year’s interest); and rd is the YTC. 7 You can also find the YTM with a spreadsheet. In Excel, you use the Rate function, inputting Nper ¼ 14, Pmt ¼ 100, Pv ¼ –1494.93, Fv ¼ 1000, and 0 for Type and leaving Guess blank. Chapter 7 Bonds and Their Valuation To illustrate, suppose Allied’s bonds had a provision that permitted the company, if it desired, to call them 10 years after their issue date at a price of $1,100. Suppose further that interest rates had fallen and that 1 year after issuance, the going interest rate had declined, causing their price to rise to $1,494.93. Here is the time line and the setup for finding the bonds’ YTC with a financial calculator: 0 YTC = ? ⫺1,494.93 Inputs: 1 2 8 100 100 100 9 N Output: I/YR 9 100 1,100 –1494.93 100 1100 PV PMT FV 4.21 = YTC SE The YTC is 4.21%—this is the return you would earn if you bought an Allied bond at a price of $1,494.93 and it was called 9 years from today. (It could not be called until 10 years after issuance. One year has gone by, so there are 9 years left until the first call date.) Do you think Allied will call its 10% bonds when they become callable? Allied’s action will depend on what the going interest rate is when they become callable. If the going rate remains at rd ¼ 5%, Allied could save 10% – 5% ¼ 5%, or $50 per bond per year; so it would call the 10% bonds and replace them with a new 5% issue. There would be some cost to the company to refund the bonds; but because the interest savings would most likely be worth the cost, Allied would probably refund them. Therefore, you should expect to earn the YTC ¼ 4.21% rather than the YTM ¼ 5% if you bought the bond under the indicated conditions. In the balance of this chapter, we assume that bonds are not callable unless otherwise noted. However, some of the end-of-chapter problems deal with yield to call.8 LF TEST Explain the difference between yield to maturity and yield to call. Halley Enterprises’ bonds currently sell for $975. They have a 7-year maturity, an annual coupon of $90, and a par value of $1,000. What is their yield to maturity? (9.51%) The Henderson Company’s bonds currently sell for $1,275. They pay a $120 annual coupon and have a 20-year maturity, but they can be called in 5 years at $1,120. What are their YTM and their YTC, and which is “more relevant” in the sense that investors should expect to earn it? (8.99%; 7.31%; YTC) 8 Brokerage houses occasionally report a bond’s current yield, defined as the annual interest payment divided by the current price. For example, if Allied’s 10% coupon bonds were selling for $985, the current yield would be $100/$985 ¼ 10.15%. Unlike the YTM or YTC, the current yield does not represent the actual return that investors should expect because it does not account for the capital gain or loss that will be realized if the bond is held until it matures or is called. The current yield was popular before calculators and computers came along because it was easy to calculate. However, it can be misleading, and now it’s easy enough to calculate the YTM and YTC. 205 206 Part 3 Financial Assets 7-5 CHANGES IN BOND VALUES OVER TIME When a coupon bond is issued, the coupon is generally set at a level that causes the bond’s market price to equal its par value. If a lower coupon were set, investors would not be willing to pay $1,000 for the bond; but if a higher coupon were set, investors would clamor for it and bid its price up over $1,000. Investment bankers can judge quite precisely the coupon rate that will cause a bond to sell at its $1,000 par value. A bond that has just been issued is known as a new issue. Once it has been issued, it is an outstanding bond, also called a seasoned issue. Newly issued bonds generally sell at prices very close to par, but the prices of outstanding bonds can vary widely from par. Except for floating-rate bonds, coupon payments are constant; so when economic conditions change, a bond with a $100 coupon that sold at its $1,000 par value when it was issued will sell for more or less than $1,000 thereafter. Among its outstanding bonds, Allied currently has three equally risky issues that will mature in 15 years: Allied’s just-issued 15-year bonds have a 10% annual coupon. They were issued at par, which means that the market interest rate on their issue date was also 10%. Because the coupon rate equals the market interest rate, these bonds are trading at par, or $1,000. Five years ago Allied issued 20-year bonds with a 7% annual coupon. These bonds currently have 15 years remaining until maturity. They were originally issued at par, which means that 5 years ago the market interest rate was 7%. Currently, this bond’s coupon rate is less than the 10% market rate, so they sell at a discount. Using a financial calculator or spreadsheet, we can quickly find that they have a price of $771.82. (Set N ¼ 15, I/YR = 10, PMT ¼ 70, and FV ¼ 1000 and solve for the PV to get the price.) Ten years ago Allied issued 25-year bonds with a 13% coupon rate. These bonds currently have 15 years remaining until maturity. They were originally issued at par, which means that 10 years ago the market interest rate must have been 13%. Because their coupon rate is greater than the current market rate, they sell at a premium. Using a financial calculator or spreadsheet, we can find that their price is $1,228.18. (Set N ¼ 15, I/YR ¼ 10, PMT ¼ 130, and FV ¼ 1000 and solve for the PV to get the price.) l l l Each of these three bonds has a 15-year maturity; each has the same credit risk; and thus each has the same market interest rate, 10%. However, the bonds have different prices because of their different coupon rates. Now let’s consider what would happen to the prices of these three bonds over the 15 years until they mature, assuming that market interest rates remain constant at 10% and Allied does not default on its payments. Table 7-1 demonstrates how the prices of each of these bonds will change over time if market interest rates remain at 10%. One year from now each bond will have a maturity of 14 years— that is, N ¼ 14. With a financial calculator, override N ¼ 15 with N ¼ 14 and press the PV key; that gives you the value of each bond 1 year from now. Continuing, set N ¼ 13, N ¼ 12, and so forth, to see how the prices change over time. Table 7-1 also shows the current yield (which is the coupon interest divided by the bond’s price), the capital gains yield, and the total return over time. For any given year, the capital gains yield is calculated as the bond’s annual change in price divided by the beginning-of-year price. For example, if a bond was selling for $1,000 at the beginning of the year and $1,035 at the end of the year, its capital gains yield for the year would be $35/$1,000 ¼ 3.5%. (If the bond was selling at a premium, its price would decline over time. Then the capital gains yield would be negative, but it would be offset by a high current yield.) A bond’s total return is Table 7-1 Calculation of Current Yields, Capital Gains Yields, and Total Returns for 7%, 10%, and 13% Coupon Bonds When the Market Rate Remains Constant at 10% 7% COUPON BOND Pricea 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 $ 771.82 779.00 786.90 795.59 805.15 815.66 827.23 839.95 853.95 869.34 886.28 904.90 925.39 947.93 972.73 1,000.00 Expected Current Yieldb Expected Capital Gains Yieldc Expected Total Returnd 9.1% 9.0 8.9 8.8 8.7 8.6 8.5 8.3 8.2 8.1 7.9 7.7 7.6 7.4 7.2 0.9% 1.0 1.1 1.2 1.3 1.4 1.5 1.7 1.8 1.9 2.1 2.3 2.4 2.6 2.8 10.0% 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 Pricea $1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00 13% COUPON BOND Expected Current Yieldb Expected Capital Gains Yieldc Expected Total Returnd 10.0% 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 0.0% 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10.0% 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 Pricea $1,228.18 1,221.00 1,213.10 1,204.41 1,194.85 1,184.34 1,172.77 1,160.05 1,146.05 1,130.66 1,113.72 1,095.10 1,074.61 1,052.07 1,027.27 1,000.00 Expected Current Yieldb Expected Capital Gains Yieldc Expected Total Returnd 10.6% 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.5 11.7 11.9 12.1 12.4 12.7 0.6% 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.5 1.7 1.9 2.1 2.4 2.7 10.0% 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 Notes: Using a financial calculator, the price of each bond is calculated by entering the data for N, I/YR, PMT, and FV, then solving for PV ¼ the bond’s value. b The expected current yield is calculated as the annual interest divided by the price of the bond. c The expected capital gains yield is calculated as the differece between the end-of-year bond price and the beginning-of-year bond price divided by the beginning-of-year price. d The expected total return is the sum of the expected current yield and the expected capital gains yield. a Chapter 7 Bonds and Their Valuation Number of Years Until Maturity 10% COUPON BOND 207 208 Part 3 Financial Assets equal to the current yield plus the capital gains yield. In the absence of default risk and assuming market equilibrium, the total return is also equal to YTM and the market interest rate, which in our example is 10%. Figure 7-2 plots the three bonds’ predicted prices as calculated in Table 7-1. Notice that the bonds have very different price paths over time but that at maturity, all three will sell at their par value of $1,000. Here are some points about the prices of the bonds over time: The price of the 10% coupon bond trading at par will remain at $1,000 if the market interest rate remains at 10%. Therefore, its current yield will remain at 10% and its capital gains yield will be zero each year. The 7% bond trades at a discount; but at maturity, it must sell at par because that is the amount the company will give to its holders. Therefore, its price must rise over time. The 13% coupon bond trades at a premium. However, its price must be equal to its par value at maturity; so the price must decline over time. l l l While the prices of the 7% and 13% coupon bonds move in opposite directions over time, each bond provides investors with the same total return, 10%, which is also the total return on the 10% coupon bond that sells at par. The discount bond has a low coupon rate (and therefore a low current yield), but it provides a capital gain each year. In contrast, the premium bond has a high current yield, but it has an expected capital loss each year.9 Time Paths of 7%, 10%, and 13% Coupon Bonds When the Market Rate Remains Constant at 10% FIGURE 7-2 Bond Value ($) 1,500 Coupon = 13% 1,250 Coupon = 10% 1,000 750 Coupon = 7% 500 0 15 12 9 6 3 0 Years Remaining until Maturity 9 In this example (and throughout the text), we ignore the tax effects associated with purchasing different types of bonds. For coupon bonds, under the current Tax Code, coupon payments are taxed as ordinary income, whereas capital gains are taxed at the capital gains tax rate. As we mentioned in Chapter 3, for most investors, the capital gains tax rate is lower than the personal tax rate. Moreover, while coupon payments are taxed each year, capital gains taxes are deferred until the bond is sold or matures. Consequently, all else equal, investors end up paying lower taxes on discount bonds because a greater percentage of their total return comes in the form of capital gains. For details on the tax treatment of zero coupon bonds, see Web Appendix 7A. SE Chapter 7 Bonds and Their Valuation LF TEST What is meant by the terms new issue and seasoned issue? Last year a firm issued 20-year, 8% annual coupon bonds at a par value of $1,000. (1) Suppose that one year later the going rate drops to 6%. What is the new price of the bonds assuming they now have 19 years to maturity? ($1,223.16) (2) Suppose that one year after issue, the going interest rate is 10% (rather than 6%). What would the price have been? ($832.70) Why do the prices of fixed-rate bonds fall if expectations for inflation rise? 7-6 BONDS WITH SEMIANNUAL COUPONS Although some bonds pay interest annually, the vast majority actually make payments semiannually. To evaluate semiannual bonds, we must modify the valuation model (Equation 7-1) as follows: 1. Divide the annual coupon interest payment by 2 to determine the dollars of interest paid each six months. 2. Multiply the years to maturity, N, by 2 to determine the number of semiannual periods. 3. Divide the nominal (quoted) interest rate, rd, by 2 to determine the periodic (semiannual) interest rate. On a time line, there would be twice as many payments, but each would be half as large as with an annual payment bond. Making the indicated changes results in the following equation for finding a semiannual bond’s value: VB ¼ 2N X t¼1 INT=2 M þ ð1 þ rd =2Þt ð1 þ rd =2Þ2N 7-1a To illustrate, assume that Allied Food’s 15-year bonds as discussed in Section 7-3 pay $50 of interest each 6 months rather than $100 at the end of each year. Thus, each interest payment is only half as large but there are twice as many of them. We would describe the coupon rate as “10% with semiannual payments.”10 When the going (nominal) rate is rd ¼ 5% with semiannual compounding, the value of a 15-year, 10% semiannual coupon bond that pays $50 interest every 6 months is found as follows: Inputs: Output: 30 2.5 N I/YR PV 50 1000 PMT FV = –1,523.26 Enter N ¼ 30, rd ¼ I/YR ¼ 2.5, PMT ¼ 50, and FV ¼ 1000; then press the PV key to obtain the bond’s value, $1,523.26. The value with semiannual interest payments is slightly larger than $1,518.98, the value when interest is paid annually as we 10 In this situation, the coupon rate of “10% paid semiannually” is the rate that bond dealers, corporate treasurers, and investors generally discuss. Of course, if this bond were issued at par, its effective annual rate would be higher than 10%.    rNOM M 0:10 2 EAR ¼ EFF% ¼ 1 þ 1 ¼ ð1:05Þ2  1 ¼ 10:25% 1 ¼ 1 þ M 2 Since 10% with annual payments is quite different from 10% with semiannual payments, we have assumed a change in effective rates in this section from the situation in Section 7-3, where we assumed 10% with annual payments. 209 210 Part 3 Financial Assets calculated in Section 7-3. This higher value occurs because each interest payment is received somewhat faster under semiannual compounding. Alternatively, when we know the price of a semiannual bond, we can easily back out the bond’s nominal yield to maturity. In the previous example, if you were told that a 15-year bond with a 10% semiannual coupon was selling for $1,523.26, you could solve for the bond’s periodic interest rate as follows: Inputs: 30 N Output: I/YR –1,523.26 50 1000 PV PMT FV = 2.5 SE In this case, enter N ¼ 30, PV ¼ 1523.26, PMT ¼ 50, and FV ¼ 1000; then press the I/YR key to obtain the interest rate per semiannual period, 2.5%. Multiplying by 2, we calculate the bond’s nominal yield to maturity to be 5%.11 LF TEST Describe how the annual payment bond valuation formula is changed to evaluate semiannual coupon bonds and write the revised formula. Hartwell Corporation’s bonds have a 20-year maturity, an 8% semiannual coupon, and a face value of $1,000. The going interest rate (rd) is 7% based on semiannual compounding. What is the bond’s price? ($1,106.78) 7-7 ASSESSING A BOND’S RISKINESS In this section, we identify and explain the two key factors that impact a bond’s riskiness. Once those factors are identified, we differentiate between them and discuss how you can minimize these risks. 7-7a Interest Rate Risk Interest Rate (Price) Risk The risk of a decline in a bond’s price due to an increase in interest rates. As we saw in Chapter 6, interest rates fluctuate over time and when they rise, the value of outstanding bonds decline. This risk of a decline in bond values due to an increase in interest rates is called interest rate risk (or interest rate price risk). To illustrate, refer back to Allied’s bonds; assume once more that they have a 10% annual coupon; and assume that you bought one of these bonds at its par value, $1,000. Shortly after your purchase, the going interest rate rises from 10 to 15%.12 As we saw in Section 7-3, this interest rate increase would cause the bond’s price to fall from $1,000 to $707.63; so you would have a loss of $292.37 on the bond.13 Since interest rates can and do rise, rising rates cause losses to bondholders; people or firms who invest in bonds are exposed to risk from increasing interest rates. 11 We can use a similar process to calculate the nominal yield to call for a semiannual bond. The only difference would be that N should represent the number of semiannual periods until the bond is callable and FV should be the bond’s call price rather than its par value. 12 An immediate increase in rates from 10% to 15% would be quite unusual, and it would occur only if something quite bad were revealed about the company or happened in the economy. Smaller but still significant rate increases that adversely affect bondholders do occur fairly often. 13 You would have an accounting (and tax) loss only if you sold the bond; if you held it to maturity, you would not have such a loss. However, even if you did not sell, you would still have suffered a real economic loss in an opportunity cost sense because you would have lost the opportunity to invest at 15% and would be stuck with a 10% bond in a 15% market. In an economic sense, “paper losses” are just as bad as realized accounting losses. Chapter 7 Bonds and Their Valuation Interest rate risk is higher on bonds that have long maturities than on bonds that will mature in the near future.14 This follows because the longer the maturity, the longer before the bond will be paid off and the bondholder can replace it with another bond with a higher coupon. This point can be demonstrated by showing how the value of a 1-year bond with a 10% annual coupon fluctuates with changes in rd and then comparing those changes with changes on a 15-year bond. The 1-year bond’s values at different interest rates are shown here: Value of a 1-year bond at rd = 5%: Inputs: 1 5 N I/YR Output (Bond Value): rd = 10%: Inputs: Inputs: Output (Bond Value): 1000 PMT FV 100 1000 PMT FV 100 1000 PMT FV –1,047.62 1 10 N I/YR PV –1,000.00 Output (Bond Value): rd = 15%: PV 100 1 15 N I/YR PV –956.52 You would obtain the first value with a financial calculator by entering N ¼ 1, I/YR ¼ 5, PMT ¼ 100, and FV ¼ 1000 and then pressing PV to get $1,047.62. With everything still in your calculator, enter I/YR ¼ 10 to override the old I/YR ¼ 5 and press PV to find the bond’s value at a 10% rate; it drops to $1,000. Then enter I/YR ¼ 15 and press the PV key to find the last bond value, $956.52. The effects of increasing rates on the 15-year bond as found earlier can be compared with the just-calculated effects for the 1-year bond. This comparison is shown in Figure 7-3, where we show bond prices at several rates and then plot those prices on the graph. Compared to the 1-year bond, the 15-year bond is far more sensitive to changes in rates. At a 10% interest rate, both the 15-year and 1-year bonds are valued at $1,000. When rates rise to 15%, the 15-year bond falls to $707.63, but the 1-year bond falls only to $956.52. The price decline for the 1-year bond is only 4.35%, while that for the 15-year bond is 29.24%. For bonds with similar coupons, this differential interest rate sensitivity always holds true—the longer its maturity, the more its price changes in response to a given change in interest rates. Thus, even if the risk of default on two bonds is exactly the same, the 14 Actually, a bond’s maturity and coupon rate both affect interest rate risk. Low coupons mean that most of the bond’s return will come from repayment of principal, whereas on a high-coupon bond with the same maturity, more of the cash flows will come in during the early years due to the relatively large coupon payments. A measurement called duration, which finds the average number of years the bond’s PV of cash flows remain outstanding, has been developed to combine maturity and coupons. A zero coupon bond, which has no interest payments and whose payments all come at maturity, has a duration equal to its maturity. All coupon bonds have durations that are shorter than their maturity; and the higher the coupon rate, the shorter the duration. Bonds with longer duration are exposed to more interest rate risk. A discussion of duration would go beyond the scope of this book, but see any investments text for a discussion of the concept. 211 212 Part 3 Financial Assets Values of Long- and Short-Term 10% Annual Coupon Bonds at Different Market Interest Rates FIGURE 7-3 Bond Value ($) 2,500 2,000 1,500 15-Year Bond 1,000 1-Year Bond 500 0 5 10 15 20 25 Interest Rate (%) VALUE OF Current Market Interest Rate, rd 5% 10 15 20 25 1-Year Bond 15-Year Bond $1,047.62 1,000.00 956.52 916.67 880.00 $1,518.98 1,000.00 707.63 532.45 421.11 Note: Bond values were calculated using a financial calculator assuming annual, or once-a-year, compounding. one with the longer maturity is typically exposed to more risk from a rise in interest rates.15 The logical explanation for this difference in interest rate risk is simple. Suppose you bought a 15-year bond that yielded 10%, or $100 a year. Now suppose interest rates on comparable-risk bonds rose to 15%. You would be stuck with only $100 of interest for the next 15 years. On the other hand, had you bought a 1-year bond, you would have had a low return for only 1 year. At the end of the year, you would have received your $1,000 back; then you could have reinvested it and earned 15%, or $150 per year, for the next 14 years. 15 If a 10-year bond were plotted on the graph in Figure 7-3, its curve would lie between those of the 15-year and the 1-year bonds. The curve of a 1-month bond would be almost horizontal, indicating that its price would change very little in response to an interest rate change; but a 100-year bond would have a very steep slope, and the slope of a perpetuity would be even steeper. Also, a zero coupon bond’s price is quite sensitive to interest rate changes; and the longer its maturity, the greater its price sensitivity. Therefore, a 30-year zero coupon bond would have a huge amount of interest rate risk. Chapter 7 Bonds and Their Valuation 213 7-7b Reinvestment Rate Risk As we saw in the preceding section, an increase in interest rates hurts bondholders because it leads to a decline in the current value of a bond portfolio. But can a decrease in interest rates also hurt bondholders? Actually, the answer is yes because if interest rates fall, long-term investors will suffer a reduction in income. For example, consider a retiree who has a bond portfolio and lives off the income it produces. The bonds in the portfolio, on average, have coupon rates of 10%. Now suppose interest rates decline to 5%. Many of the bonds will mature or be called; as this occurs, the bondholder will have to replace 10% bonds with 5% bonds. Thus, the retiree will suffer a reduction of income. The risk of an income decline due to a drop in interest rates is called reinvestment rate risk, and its importance has been demonstrated to all bondholders in recent years as a result of the sharp drop in rates since the mid-1980s. Reinvestment rate risk is obviously high on callable bonds. It is also high on shortterm bonds because the shorter the bond’s maturity, the fewer the years before the relatively high old-coupon bonds will be replaced with the new low-coupon issues. Thus, retirees whose primary holdings are short-term bonds or other debt securities will be hurt badly by a decline in rates, but holders of noncallable longterm bonds will continue to enjoy the old high rates. Reinvestment Rate Risk The risk that a decline in interest rates will lead to a decline in income from a bond portfolio. 7-7c Comparing Interest Rate and Reinvestment Rate Risk Note that interest rate risk relates to the current market value of the bond portfolio, while reinvestment rate risk relates to the income the portfolio produces. If you hold long-term bonds, you will face significant interest rate price risk because the value of your portfolio will decline if interest rates rise, but you will not face much reinvestment rate risk because your income will be stable. On the other hand, if you hold short-term bonds, you will not be exposed to much interest rate price risk, but you will be exposed to significant reinvestment rate risk. Which type of risk is “more relevant” to a given investor depends critically on how long the investor plans to hold the bonds—this is often referred to as his or her investment horizon. To illustrate, consider an investor who has a relatively short 1-year investment horizon—say, the investor plans to go to graduate school a year from now and needs money for tuition and expenses. Reinvestment rate risk is of minimal concern to this investor because there is little time for reinvestment. The investor could eliminate interest rate risk by buying a 1-year Treasury security since he would be assured of receiving the face value of the bond 1 year from now (the investment horizon). However, if this investor were to buy a long-term Treasury security, he would bear a considerable amount of interest rate risk because, as we have seen, long-term bond prices decline when interest rates rise. Consequently, investors with shorter investment horizons should view long-term bonds as being more risky than short-term bonds. By contrast, the reinvestment risk inherent in short-term bonds is especially relevant to investors with longer investment horizons. Consider a retiree who is living on income from her portfolio. If this investor buys 1-year bonds, she will have to “roll them over” every year; and if rates fall, her income in subsequent years will likewise decline. A younger couple saving for their retirement or their children’s college costs, for example, would be affected similarly because if they buy short-term bonds, they too will have to roll over their portfolio at possibly much lower rates. Since there is uncertainty today about the rates that will be earned on these reinvested cash flows, long-term investors should be especially concerned about the reinvestment rate risk inherent in short-term bonds. Investment Horizon The period of time an investor plans to hold a particular investment. Part 3 Financial Assets One way to manage both interest rate and reinvestment rate risk is to buy a zero coupon Treasury bond with a maturity that matches the investor’s investment horizon. For example, assume your investment horizon is 10 years. If you buy a 10-year zero, you will receive a guaranteed payment in 10 years equal to the bond’s face value.16 Moreover, as there are no coupons to reinvest, there is no reinvestment rate risk. This explains why investors with specific goals often invest in zero coupon bonds.17 Recall from Chapter 6 that maturity risk premiums are generally positive. Moreover, a positive maturity risk premium implies that investors, on average, regard longer-term bonds as being riskier than shorter-term bonds. That, in turn, suggests that the average investor is most concerned with interest rate price risk. Still, it is appropriate for each investor to consider his or her own situation, to recognize the risks inherent in bonds with different maturities, and to construct a portfolio that deals best with the investor’s most relevant risk. SE 214 LF TEST Differentiate between interest rate risk and reinvestment rate risk. To which type of risk are holders of long-term bonds more exposed? shortterm bondholders? What type of security can be used to minimize both interest rate and reinvestment rate risk for an investor with a fixed investment horizon? 7-8 DEFAULT RISK Potential default is another important risk that bondholders face. If the issuer defaults, investors will receive less than the promised return. Recall from Chapter 6 that the quoted interest rate includes a default risk premium—the higher the probability of default, the higher the premium and thus the yield to maturity. Default risk on Treasuries is zero, but this risk is substantial for lower-grade corporate and municipal bonds. To illustrate, suppose two bonds have the same promised cash flows—their coupon rates, maturities, liquidity, and inflation exposures are identical; but one has more default risk than the other. Investors will naturally pay more for the one with less chance of default. As a result, bonds with higher default risk have higher market rates: rd ¼ r* þ IP þ DRP þ LP þ MRP. If a bond’s default risk changes, rd and thus the price will be affected. Thus, if the default risk on Allied’s bonds increases, their price will fall and the yield to maturity (YTM ¼ rd) will increase. 16 Note that in this example, the 10-year zero technically has a considerable amount of interest rate risk since its current price is highly sensitive to changes in interest rates. However, the year-to year movements in price should not be of great concern to an investor with a 10-year horizon. The reason is that the investor knows that regardless of what happens to interest rates, the bond’s price will still be $1,000 when it matures. 17 Two words of caution about zeros are in order. First, as we show in Web Appendix 7A, investors in zeros must pay taxes each year on their accrued gain in value even though the bonds don’t pay any cash until they mature. Second, buying a zero coupon with a maturity equal to your investment horizon enables you to lock in a nominal cash payoff, but the real value of that payment still depends on what happens to inflation during your investment horizon. What we need is an inflation-indexed zero coupon Treasury bond; but to date, no such bond exists. Also, the fact that maturity risk premiums are positive suggests that most investors have relatively short investment horizons, or at least worry about short-term changes in their net worth. See Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Chicago: Morningstar, Inc., 2008), which finds that the maturity risk premium for long-term bonds has averaged 1.4% over the past 82 years. Chapter 7 Bonds and Their Valuation 215 7-8a Various Types of Corporate Bonds Default risk is influenced by the financial strength of the issuer and the terms of the bond contract, including whether collateral has been pledged to secure the bond. The characteristics of some key types of bonds are described in this section. Mortgage Bonds Under a mortgage bond, the corporation pledges specific assets as security for the bond. To illustrate, in 2008, Billingham Corporation needed $10 million to build a regional distribution center. Bonds in the amount of $4 million, secured by a first mortgage on the property, were issued. (The remaining $6 million was financed with equity capital.) If Billingham defaults on the bonds, the bondholders can foreclose on the property and sell it to satisfy their claims. If Billingham had chosen to, it could have issued second mortgage bonds secured by the same $10 million of assets. In the event of liquidation, the holders of the second mortgage bonds would have a claim against the property, but only after the first mortgage bondholders had been paid off in full. Thus, second mortgages are sometimes called junior mortgages because they are junior in priority to the claims of senior mortgages, or first mortgage bonds. All mortgage bonds are subject to an indenture, which is a legal document that spells out in detail the rights of the bondholders and the corporation. The indentures of many major corporations were written 20, 30, 40, or more years ago. These indentures are generally “open-ended,” meaning that new bonds can be issued from time to time under the same indenture. However, the amount of new bonds that can be issued is usually limited to a specified percentage of the firm’s total “bondable property,” which generally includes all land, plant, and equipment. And, of course, the coupon interest rate on the newly issued bonds changes over time, along with the market rate on the older bonds. Mortgage Bond A bond backed by fixed assets. First mortgage bonds are senior in priority to claims of second mortgage bonds. Indenture A formal agreement between the issuer and the bondholders. Debentures A debenture is an unsecured bond; and as such, it provides no specific collateral as security for the obligation. Therefore, debenture holders are general creditors whose claims are protected by property not otherwise pledged. In practice, the use of debentures depends on the nature of the firm’s assets and on its general credit strength. Extremely strong companies such as General Electric and ExxonMobil can use debentures because they do not need to put up property as security for their debt. Debentures are also issued by weak companies that have already pledged most of their assets as collateral for mortgage loans. In this case, the debentures are quite risky and that risk will be reflected in their interest rates. Debenture A long-term bond that is not secured by a mortgage on specific property. Subordinated Debentures The term subordinate means “below” or “inferior to”; and in the event of bankruptcy, subordinated debt has a claim on assets only after senior debt has been paid in full. Subordinated debentures may be subordinated to designated notes payable (usually bank loans) or to all other debt. In the event of liquidation or reorganization, holders of subordinated debentures receive nothing until all senior debt, as named in the debentures’ indenture, has been paid. Precisely how subordination works and how it strengthens the position of senior debtholders are explained in detail in Web Appendix 7B. 7-8b Bond Ratings Since the early 1900s, bonds have been assigned quality ratings that reflect their probability of going into default. The three major rating agencies are Moody’s Investors Service (Moody’s), Standard & Poor’s Corporation (S&P), and Fitch Subordinated Debenture A bond having a claim on assets only after the senior debt has been paid off in the event of liquidation. 216 Part 3 Financial Assets Investment-Grade Bond Bonds rated triple-B or higher; many banks and other institutional investors are permitted by law to hold only investmentgrade bonds. Junk Bond A high-risk, high-yield bond. Table 7-2 Investor’s Service. Moody’s and S&P’s rating designations are shown in Table 7-2.18 The triple- and double-A bonds are extremely safe. Single-A and triple-B bonds are also strong enough to be called investment-grade bonds, and they are the lowestrated bonds that many banks and other institutional investors are permitted by law to hold. Double-B and lower bonds are speculative, or junk, bonds; and they have a significant probability of going into default. Bond Rating Criteria Bond ratings are based on financial ratios such as those discussed in Chapter 4 and on various qualitative factors. The ratios, especially the debt and interest coverage ratios, are generally the most important ratings determinants; but at times, other factors that are expected to affect the ratios in the future take center stage. In 2008, firms’ exposures to subprime mortgages are leading to downgrades of firms whose ratios still look “reasonable.” Published ratios are, of course, historical— they show the firm’s condition in the past, whereas bond investors are more interested in the firm’s condition in the future. The qualitative factors can be divided into two groups: factors that are related to the bond contract and all other factors. Following is an outline of the determinants of bond ratings: 1. Financial Ratios. All of the ratios are potentially important, but the debt and interest coverage ratios are key. The rating agencies’ analysts go through a financial analysis along the lines discussed in Chapter 4 and forecast future ratios along the lines described in the financial planning and forecasting chapter. For the forecasts, the qualitative factors discussed next are important. 2. Qualitative Factors: Bond Contract Terms. Every bond is covered by a contract, often called an indenture, between the issuer and the bondholders. The indenture spells out all the terms related to the bond. Included in the indenture are the maturity, the coupon interest rate, a statement of whether the bond is secured by a mortgage on specific assets, any sinking fund provisions, and a statement of whether the bond is guaranteed by some other party with a high credit ranking. Other provisions might include restrictive covenants such as requirements that the firm not let its debt ratio exceed a stated level and that it keep its times-interestearned ratio at or above a given level. Some bond indentures are hundreds of pages long, while others are quite short and cover just the terms of the loan. 3. Miscellaneous Qualitative Factors. Included here are issues like the sensitivity of the firm’s earnings to the strength of the economy, the way it is affected by inflation, a statement of whether it is having or likely to have labor problems, the extent of its international operations (including the stability of the countries in which it operates), potential environmental problems, and potential antitrust problems. Today the most important factor is exposure to subprime loans, including the difficulty to determine the extent of this exposure as a result of the complexity of the assets backed by such loans. Moody’s and S&P Bond Ratings INVESTMENT GRADE Moody’s S&P Aaa AAA Aa AA A A JUNK BONDS Baa BBB Ba BB B B Caa CCC C C Note: Both Moody’s and S&P use “modifiers” for bonds rated below triple A. S&P uses a plus and minus system. Thus, Aþ designates the strongest A-rated bonds; A-, the weakest. Moody’s uses a 1, 2, or 3 designation, with 1 denoting the strongest and 3 denoting the weakest; thus, within the double-A category, Aa1 is the best, Aa2 is average, and Aa3 is the weakest. 18 In the discussion to follow, reference to the S&P rating is intended to imply the Moody’s and Fitch’s ratings as well. Thus, triple-B bonds mean both BBB and Baa bonds; double-B bonds mean both BB and Ba bonds; and so forth. Chapter 7 Bonds and Their Valuation We see that bond ratings are determined by a great many factors, some quantitative and some qualitative (or subjective). Also, the rating process is dynamic—at times, one factor is of primary importance; at other times, some other factor is key. Nevertheless, as we can see from Table 7-3, there is a strong correlation between bond ratings and many of the ratios that we described in Chapter 4. Not surprisingly, companies with lower debt ratios, higher free cash flow to debt, higher returns on invested capital, higher EBITDA coverage ratios, and higher TIE ratios typically have higher bond ratings. Importance of Bond Ratings Bond ratings are important to both firms and investors. First, because a bond’s rating is an indicator of its default risk, the rating has a direct, measurable influence on the bond’s interest rate and the firm’s cost of debt. Second, most bonds are purchased by institutional investors rather than individuals and many institutions are restricted to investment-grade securities. Thus, if a firm’s bonds fall below BBB, it will have a difficult time selling new bonds because many potential purchasers will not be allowed to buy them. As a result of their higher risk and more restricted market, lower-grade bonds have higher required rates of return, rd, than high-grade bonds. Figure 7-4 illustrates this point. In each of the years shown on the graph, U.S. government bonds have had the lowest yields, AAA bonds have been next, and BBB bonds have had the highest yields. The figure also shows that the gaps between yields on the three types of bonds vary over time, indicating that the cost differentials, or yield spreads, fluctuate from year to year. This point is highlighted in Figure 7-5, which gives the yields on the three types of bonds and the yield spreads for AAA and BBB bonds over Treasuries in January 1994 and January 2008.19 Note first from Table 7-3 Times interest earned (EBIT/Interest) EBITDA interest coverage (EBITDA/Interest) Net cash flow/Total debt Free cash flow/Total debt Return on capital Total debt/EBITDA Total debt/Total capital Bond Rating Criteria: Three-Year (2002–2004) Median Financial Ratios for Different Bond Rating Classifications of Industrial Companiesa AAA AA 23.8 19.5 25.5 203.3% 127.6 27.6 0.4 12.4 24.6 79.9% 44.5 27.0 0.9 28.3 A BBB BB B CCC 8.0 4.7 2.5 1.2 0.4 10.2 48.0% 25.0 17.5 1.6 37.5 6.5 35.9% 17.3 13.4 2.2 42.5 3.5 22.4% 8.3 11.3 3.5 53.7 1.9 11.5% 2.8 8.7 5.3 75.9 0.9 5.0% (2.1) 3.2 7.9 113.5 a Somewhat different criteria are applied to firms in different industries, such as utilities and financial corporations. This table pertains to industrial companies, which include manufacturers, retailers, and service firms. Source: Adapted from “CreditStats Adjusted Key Industrial Financial Ratios,” Standard & Poor’s 2006 Corporate Ratings Criteria, September 10, 2007, p. 43. 19 A yield spread is related to but not identical to risk premiums on corporate bonds. The true risk premium reflects only the difference in expected (and required) returns between two securities that results from differences in their risk. However, yield spreads reflect (1) a true risk premium; (2) a liquidity premium, which reflects the fact that U.S. Treasury bonds are more readily marketable than most corporate bonds; (3) a call premium because most Treasury bonds are not callable whereas corporate bonds are; and (4) an expected loss differential, which reflects the probability of loss on the corporate bonds. As an example of the last point, suppose the yield to maturity on a BBB bond was 6.0% versus 4.8% on government bonds but there was a 5% probability of total default loss on the corporate bond. In this case, the expected return on the BBB bond would be 0.95(6.0%) þ 0.05(0%) ¼ 5.7% and the yield spread would be 0.9%, not the full 1.2 percentage points difference in “promised” yields to maturity. 217 218 Part 3 Financial Assets Yields on Selected Long-Term Bonds, 1994–2008 FIGURE 7-4 Yield (%) Narrow Spread 10 Corporate BBB 8 6 Wide Spread U.S. Government 4 Corporate AAA 2 0 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 Years Source: Federal Reserve Statistical Release, Selected Interest Rates (Historical Data), www.federalreserve.gov/releases/H15/data.htm. Figure 7-5 that the risk-free rate, or vertical axis intercept, was lower in January 2008 than it was in January 1994, primarily reflecting the decline in both rates and expected inflation over the past few years. Second, the slope of the line has increased, indicating an increase in investors’ risk aversion largely due to the subprime mortgage problem and growing fears of a possible recession. Thus, the penalty for having a low credit rating varies over time. Occasionally, as in 2008, the penalty is quite large; but at times, as in 1994 (shown in Figures 7-4 and 7-5), it is small. These spread differences reflect investors’ risk aversion and their optimism or pessimism regarding the economy and corporate profits. In 2008, as more and more homeowners default on their loans and poor economic news continues, investors were both pessimistic and risk-averse; so spreads were quite high. Changes in Ratings Changes in a firm’s bond rating affect its ability to borrow funds capital and its cost of that capital. Rating agencies review outstanding bonds on a periodic basis, occasionally upgrading or downgrading a bond as a result of its issuer’s changed circumstances. For example, on March 4, 2008, S&P upgraded Reliant Energy’s secured debt facilities from B to BB–; however, the firm’s “B” corporate credit rating remained unchanged. The secured debt’s upgrade was due to the firm’s refinancing the secured debt with unsecured debt, reducing the size of its secured revolving loan, and paying down the senior secured notes. On the other hand, on March 6, 2008, S&P downgraded Airborne Health Inc.’s corporate credit rating from B– to CCC+. The downgrade was largely due to S&P’s concern about the company’s future sales following negative publicity from its recent settlement of a class action lawsuit. (The lawsuit came about from the company’s claims that its product helped prevent the common cold, a fact that was proved to be untrue.) Chapter 7 Bonds and Their Valuation FIGURE 7-5 Relationship between Bond Ratings and Bond Yields, 1994 and 2008 Yield (%) 9 Yield SpreadBBB = 1.3% 8 Yield SpreadAAA = 0.5% 7 January 1994 January 2008 6 5 4 Yield SpreadBBB = 2.2% Yield SpreadAAA = 1.0% 3 2 1 0 Treasury AAA Long-Term Government Bonds (Default-Free) (1) January 1994 January 2008 6.4% 4.3 BBB AAA Corporate Bonds (2) 6.9% 5.3 BBB Corporate Bonds (3) 7.7% 6.5 YIELD SPREADS AAA (4) ¼ (2)  (1) 0.5% 1.0 BBB (5) ¼ (3)  (1) 1.3% 2.2 Source: Federal Reserve Statistical Release, Selected Interest Rates (Historical Data), www.federalreserve.gov/releases/H15/data.htm. Over the long run, rating agencies have done a reasonably good job of measuring the average credit risk of bonds and of changing ratings whenever there is a significant change in credit quality. However, it is important to understand that ratings do not adjust immediately to changes in credit quality; and in some cases, there can be a considerable lag between a change in credit quality and a change in rating. For example, Enron’s bonds still carried an investment-grade rating on a Friday in December 2001, but the company declared bankruptcy 2 days later, on Sunday. Many other abrupt downgrades occurred in 2007 and 2008, leading to calls by Congress and the SEC for changes in rating agencies and the way they rate bonds. Improvements can clearly be made, but there will always be surprises when we learn that supposedly strong bonds were in fact quite weak. 7-8c Bankruptcy and Reorganization When a business becomes insolvent, it doesn’t have enough cash to meet its interest and principal payments. A decision must then be made whether to dissolve the firm through liquidation or to permit it to reorganize and thus continue to operate. These issues are addressed in Chapter 7 and Chapter 11 of the federal bankruptcy statutes, and the final decision is made by a federal bankruptcy court judge. The decision to force a firm to liquidate versus permitting it to reorganize depends on whether the value of the reorganized business is likely to be greater than the value of its assets if they were sold off piecemeal. In a reorganization, the firm’s creditors negotiate with management on the terms of a potential reorganization. The reorganization plan may call for restructuring the debt, in which case the interest rate may be reduced, the term to maturity lengthened, or some of the debt exchanged for equity. The point of the restructuring is to reduce the financial 219 Part 3 Financial Assets charges to a level that is supportable by the firm’s projected cash flows. Of course, the common stockholders also have to “take a haircut”—they generally see their position diluted as a result of additional shares being given to debtholders in exchange for accepting a reduced amount of debt principal and interest. A trustee may be appointed by the court to oversee the reorganization, but the existing management generally is allowed to retain control. Liquidation occurs if the company is deemed to be worth more “dead” than “alive.” If the bankruptcy court orders a liquidation, assets are auctioned off and the cash obtained is distributed as specified in Chapter 7 of the Bankruptcy Act. Web Appendix 7B provides an illustration of how a firm’s assets are distributed after liquidation. For now, you should know that (1) the federal bankruptcy statutes govern reorganization and liquidation, (2) bankruptcies occur frequently, (3) a priority of the specified claims must be followed when the assets of a liquidated firm are distributed, (4) bondholders’ treatment depends on the terms of the bond, and (5) stockholders generally receive little in reorganizations and nothing in liquidations because the assets are usually worth less than the amount of debt outstanding. SE 220 LF TEST Differentiate between mortgage bonds and debentures. Name the major rating agencies and list some factors that affect bond ratings. Why are bond ratings important to firms and investors? Do bond ratings adjust immediately to changes in credit quality? Explain. Differentiate between Chapter 7 liquidations and Chapter 11 reorganizations. In general, when should each be used? 7-9 BOND MARKETS Corporate bonds are traded primarily in the over-the-counter market. Most bonds are owned by and traded among large financial institutions (for example, life insurance companies, mutual funds, hedge funds, and pension funds, all of which deal in very large blocks of securities), and it is relatively easy for over-the-counter bond dealers to arrange the transfer of large blocks of bonds among the relatively few holders of the bonds. It would be more difficult to conduct similar operations in the stock market among the literally millions of large and small stockholders, so a higher percentage of stock trades occur on the exchanges. The Wall Street Journal routinely reports key developments in the Treasury, corporate, and municipal bond markets. The online edition of The Wall Street Journal also lists for each trading day the most actively traded investment-grade bonds, high-yield bonds, and convertible bonds. Table 7-4 reprints portions of the online edition’s “Corporate Bonds Data” section which shows the most active issues that traded on March 6, 2008, in descending order of sales volume. Looking at Table 7-4, you will see the coupon rate, maturity date, bond rating, high and low prices for the day, closing (last) price, change in price, and yield to maturity. The table assumes that each bond has a par value of $100. Not surprisingly, the high-yield bonds have much higher yields to maturity because of their higher default risk and the convertible bonds have much lower yields because investors are willing to accept lower yields in return for the option to convert their bonds to common stock. If you examine the table closely, you will also see that the bonds with a yield to maturity above their coupon rate trade at a discount, whereas bonds with a 221 Chapter 7 Bonds and Their Valuation Table 7-4 Most Active Investment-Grade, High-Yield, and Convertible Corporate Bonds, March 6, 2008 CORPORATE BONDS Last updated: 3/6/2008 at 6:35 PM ET Market Breadth All Issues Total Issues Traded Advances Declines Unchanged 52 Week High 52 Week Low Dollar Volume * 3,774 1,457 1,873 126 170 344 15,640 Investment High Yield Convertibles Grade 2,587 1,083 1,187 53 161 191 7,350 942 299 532 68 8 117 4,989 245 75 154 5 1 36 3,301 About This Information: End of Day data. Activity as reported to FINRA TRACE (Trade Reporting and Compliance Engine). The Market breadth information represents activity in all TRACE eligible publicly traded securities. The most active information represents the most active fixedcoupon bonds (ranked by par value traded). Inclusion in Investment Grade or High Yield tables based on TRACE dissemination criteria. ”C” indicates yield is unavailable because of issues call criteria. * Par value in millions. Most Active Investment Grade Bonds Issuer Name Symbol Coupon Maturity Rating Moody’s/S&P/ Fitch MERRlLL LYNCH BANK OF AMERICA CORP JPMORGAN CHASE & CO SPRINT CAPITAL GOLDMAN SACHS GP GENERAL ELECTRIC CAPITAL SPRINT CAPITAL TELECOM ITALIA CAPITAL UNITED PARCEL SERVICE SPRINT CAPITAL MER.GDW BAC.HBM JPM.JPF S.GJ GS.YL GE.HEE S.HK Tl.GK UPS.QE S.GM 4.125% 5.750% 6.000% 6.875% 5.950% 5.250% 8.750% 5.250% 4.500% 6.900% Jan 2009 Dec 2017 Jan 2018 Nov 2028 Jan 2018 Dec 2017 Mar 2032 Nov 2013 Jan 2013 May 2019 A1/A+/A+ Aa1/AA/AA Aa2/AA–/AA– Baa3/BBB–/BB+ Aa3/AA–/AA– Aaa/AAA/-Baa3/BBB–/BB+ Baa2/BBB+/BBB+ Aa2/AA–/-Baa3/BBB–/BB+ Symbol Coupon Maturity Rating Moody’s/S&P/ Fitch TMA.GB GM.HB ET.GF CHTR.HM BBI.GB CYH.GI F.GRY GMA.HE NMGA.GD INTEL.GR 8.000% 8.375% 8.000% 11.000% 9.000% 8.875% 8.875% 6.875% 9.000% 9.250% May 2013 Jul 2033 Jun 2011 Oct 2015 Sep 2012 Jul 2015 Jan 2014 Sep 2011 Oct 2015 Jun 2016 Caa2/CCC+/CCC– Caa1/B–/B– Ba3/B/-Caa2/CCC/CCC Caa2/CCC/CC B3/B–/CCC+ B1/B/BB– B1/B+/BB B2/B/B– B3/B–/BB– Symbol Coupon Maturity Rating Moody’s/S&P/ Fitch High Low Last AMGN.GM SNDK.GC NBR.GP PDLI.GF AMGN.GN 0.125% 1.000% 0.940% 2.000% 0.375% Feb 2011 May 2013 May 2011 Feb 2012 Feb 2013 A2/--/---/BB–/---/BBB+/A– --/--/-A2/--/-- 92.438 74.000 100.500 80.608 88.467 91.813 72.690 96.000 78.882 87.000 91.883 74.000 100.000 79.443 87.587 High Low 100.886 99.500 103.143 99.280 104.566 100.632 74.000 69.000 100.516 95.956 101.750 97.678 81.120 76.063 100.834 95.908 103.964 103.617 76.313 73.950 Last 100.886 99.280 101.587 72.563 98.520 98.770 80.000 95.908 103.734 76.313 Change Yield % 0.910 –1.339 –0.413 0.063 0.576 –0.335 0.000 –1.949 1.053 0.563 3.051 5.847 5.784 10.048 6.151 5.413 11.159 6.112 3.651 10.565 Most Active High Yield Bonds Issuer Name THORNBURG MORTGAGE GENERAL MOTORS E TRADE FINANClAL CCH I BLOCKBUSTER COMMUNITY HEALTH SYSTEMS HERTZ CORP GENERAL MOTORS ACCEPTANCE NEIMAN MARCUS GP INTELSAT(BERMUDA) High Low 49.000 35.500 79.750 74.000 86.000 85.000 70.125 69.688 87.500 83.000 99.500 98.750 99.000 94.750 81.710 79.000 98.250 97.000 101.250 100.875 Last Change Yield % 40.000 –23.750 75.938 –1.063 86.000 0.500 70.070 –0.430 83.500 1.625 98.750 –0.750 97.086 0.586 80.516 –0.484 97.250 –0.688 100.875 0.000 32.807 11.262 13.429 18.517 14.096 9.108 9.535 14.099 9.511 9.093 Most Active Convertible Bonds Issuer Name AMGEN SANDISK CORP NABORS INDUSTRlES PROTEIN DESIGN LABS AMGEN Change Yield % –0.745 –0.116 –0.750 0.203 –0.663 Source: FINRA TRACE data. Reference information from Reuters DataScope Data. Credit ratings from Moody’s, Standard & Poor’s, and Fitch Ratings. Source: http://online.wsj.com, “Corporate Bonds,” The Wall Street Journal Online, March 7, 2008. 3.083 7.086 0.940 8.231 3.133 Part 3 Financial Assets yield below their coupon rate trade at a premium above par. We see that the large majority of high-yield bonds trade at a discount to par, which suggests that because of increased default risk, most of these bonds now trade at higher yields relative to when they were issued. (Recall that most bonds are issued at par, so the coupon rate tells us what the bond’s yield was at the time it was issued.) You should also note that when bonds with similar ratings are compared, bonds with longer maturities tend to have higher yields, which is consistent with the upwardsloping yield curve during this time period. SE 222 LF TEST Why do most bond trades occur in the over-the-counter market? If a bond issue is to be sold at par, at what rate must its coupon rate be set? Explain. TYING IT ALL TOGETHER This chapter described the different types of bonds governments and corporations issue, explained how bond prices are established, and discussed how investors estimate rates of return on bonds. It also discussed various types of risks that investors face when they purchase bonds. When an investor purchases a company’s bonds, the investor is providing the company with capital. Moreover, when a firm issues bonds, the return that investors require on the bonds represents the cost of debt capital to the firm. This point is extended in Chapter 10, where the ideas developed in this chapter are used to help determine a company’s overall cost of capital, which is a basic component of the capital budgeting process. In recent years, many companies have used zero coupon bonds to raise billions of dollars, while bankruptcy is an important consideration for companies that issue debt and for investors. Therefore, these two related issues are discussed in detail in Web Appendixes 7A and 7B. Go to the textbook’s web site to access these appendixes. SELF-TEST QUESTIONS AND PROBLEMS (Solutions Appear in Appendix A) ST-1 KEY TERMS a. Define each of the following terms: Bond; treasury bond; corporate bond; municipal bond; foreign bond b. Par value; maturity date; original maturity c. Coupon payment; coupon interest rate d. Fixed-rate bond; floating-rate bond; zero coupon bond; original issue discount (OID) bond Call provision; sinking fund provision e. f. Convertible bond; warrant; putable bond; income bond; indexed, or purchasing power, bond Chapter 7 Bonds and Their Valuation g. Discount bond; premium bond h. Yield to maturity (YTM); yield to call (YTC); total return; yield spread i. Interest rate risk; reinvestment rate risk; investment horizon; default risk j. Mortgage bond; indenture; debenture; subordinated debenture k. ST-2 BOND VALUATION The Pennington Corporation issued a new series of bonds on January 1, 1985. The bonds were sold at par ($1,000); had a 12% coupon; and mature in 30 years, on December 31, 2014. Coupon payments are made semiannually (on June 30 and December 31). a. What was the YTM on January 1, 1985? b. What was the price of the bonds on January 1, 1990, 5 years later, assuming that interest rates had fallen to 10%? Find the current yield, capital gains yield, and total return on January 1, 1990, given the price as determined in Part b. c. d. e. ST-3 Investment-grade bond; junk bond On July 1, 2008, 6½ years before maturity, Pennington’s bonds sold for $916.42. What were the YTM, the current yield, the capital gains yield, and the total return at that time? Now assume that you plan to purchase an outstanding Pennington bond on March 1, 2008, when the going rate of interest given its risk was 15.5%. How large a check must you write to complete the transaction? This is a difficult question. SINKING FUND The Vancouver Development Company (VDC) is planning to sell a $100 million, 10-year, 12%, semiannual payment bond issue. Provisions for a sinking fund to retire the issue over its life will be included in the indenture. Sinking fund payments will be made at the end of each year, and each payment must be sufficient to retire 10% of the original amount of the issue. The last sinking fund payment will retire the last of the bonds. The bonds to be retired each period can be purchased on the open market or obtained by calling up to 5% of the original issue at par, at VDC’s option. a. How large must each sinking fund payment be if the company (1) uses the option to call bonds at par or (2) decides to buy bonds on the open market? For Part (2), you can only answer in words. b. What will happen to debt service requirements per year associated with this issue over its 10-year life? Now consider an alternative plan where VDC sets up its sinking fund so that equal annual amounts are paid into a sinking fund trust held by a bank, with the proceeds being used to buy government bonds that are expected to pay 7% annual interest. The payments, plus accumulated interest, must total $100 million at the end of 10 years, when the proceeds will be used to retire the issue. How large must the annual sinking fund payments be? Is this amount known with certainty, or might it be higher or lower? What are the annual cash requirements for covering bond service costs under the trusteeship arrangement described in Part c? (Note: Interest must be paid on Vancouver’s outstanding bonds but not on bonds that have been retired.) Assume level interest rates for purposes of answering this question. What would have to happen to interest rates to cause the company to buy bonds on the open market rather than call them under the plan where some bonds are retired each year? c. d. e. QUESTIONS 7-1 A sinking fund can be set up in one of two ways: a. b. The corporation makes annual payments to the trustee, who invests the proceeds in securities (frequently government bonds) and uses the accumulated total to retire the bond issue at maturity. The trustee uses the annual payments to retire a portion of the issue each year, calling a given percentage of the issue by a lottery and paying a specified price per bond or buying bonds on the open market, whichever is cheaper. What are the advantages and disadvantages of each procedure from the viewpoint of (a) the firm and (b) the bondholders? 223 224 Part 3 Financial Assets 7-2 Is it true that the following equation can be used to find the value of a bond with N years to maturity that pays interest once a year? Assume that the bond was issued several years ago. VB ¼ 7-3 7-4 7-5 7-6 7-7 The values of outstanding bonds change whenever the going rate of interest changes. In general, short-term interest rates are more volatile than long-term interest rates. Therefore, short-term bond prices are more sensitive to interest rate changes than are long-term bond prices. Is that statement true or false? Explain. (Hint: Make up a “reasonable” example based on a 1-year and a 20-year bond to help answer the question.) If interest rates rise after a bond issue, what will happen to the bond’s price and YTM? Does the time to maturity affect the extent to which interest rate changes affect the bond’s price? (Again, an example might help you answer this question.) If you buy a callable bond and interest rates decline, will the value of your bond rise by as much as it would have risen if the bond had not been callable? Explain. Assume that you have a short investment horizon (less than 1 year). You are considering two investments: a 1-year Treasury security and a 20-year Treasury security. Which of the two investments would you view as being riskier? Explain. Indicate whether each of the following actions will increase or decrease a bond’s yield to maturity: a. The bond is downgraded by the rating agencies. c. A change in the bankruptcy code makes it more difficult for bondholders to receive payments in the event the firm declares bankruptcy. The economy seems to be shifting from a boom to a recession. Discuss the effects of the firm’s credit strength in your answer. Investors learn that the bonds are subordinated to another debt issue. e. 7-9 7-10 7-11 7-12 7-13 7-14 The bond’s price increases. b. d. 7-8 N X Annual interest Par value þ t Þ ð1 þ r ð1 þ rd Þ N d t¼1 Why is a call provision advantageous to a bond issuer? When would the issuer be likely to initiate a refunding call? Are securities that provide for a sinking fund more or less risky from the bondholder’s perspective than those without this type of provision? Explain. What’s the difference between a call for sinking fund purposes and a refunding call? Why are convertibles and bonds with warrants typically offered with lower coupons than similarly rated straight bonds? Explain whether the following statement is true or false: Only weak companies issue debentures. Would the yield spread on a corporate bond over a Treasury bond with the same maturity tend to become wider or narrower if the economy appeared to be heading toward a recession? Would the change in the spread for a given company be affected by the firm’s credit strength? Explain. A bond’s expected return is sometimes estimated by its YTM and sometimes by its YTC. Under what conditions would the YTM provide a better estimate, and when would the YTC be better? PROBLEMS Easy Problems 1–4 7-1 7-2 BOND VALUATION Callaghan Motors’ bonds have 10 years remaining to maturity. Interest is paid annually, they have a $1,000 par value, the coupon interest rate is 8%, and the yield to maturity is 9%. What is the bond’s current market price? YIELD TO MATURITY AND FUTURE PRICE A bond has a $1,000 par value, 10 years to maturity, and a 7% annual coupon and sells for $985. a. What is its yield to maturity (YTM)? b. Assume that the yield to maturity remains constant for the next 3 years. What will the price be 3 years from today? Chapter 7 Bonds and Their Valuation 7-3 7-4 Intermediate Problems 5–14 7-5 BOND VALUATION Nungesser Corporation’s outstanding bonds have a $1,000 par value, a 9% semiannual coupon, 8 years to maturity, and an 8.5% YTM. What is the bond’s price? YIELD TO MATURITY A firm’s bonds have a maturity of 10 years with a $1,000 face value, have an 8% semiannual coupon, are callable in 5 years at $1,050, and currently sell at a price of $1,100. What are their nominal yield to maturity and their nominal yield to call? What return should investors expect to earn on these bonds? BOND VALUATION An investor has two bonds in his portfolio that have a face value of $1,000 and pay a 10% annual coupon. Bond L matures in 15 years, while Bond S matures in 1 year. a. b. 7-6 What will the value of each bond be if the going interest rate is 5%, 8%, and 12%? Assume that only one more interest payment is to be made on Bond S at its maturity and that 15 more payments are to be made on Bond L. Why does the longer-term bond’s price vary more than the price of the shorter-term bond when interest rates change? BOND VALUATION An investor has two bonds in her portfolio, Bond C and Bond Z. Each bond matures in 4 years, has a face value of $1,000, and has a yield to maturity of 9.6%. Bond C pays a 10% annual coupon, while Bond Z is a zero coupon bond. a. Assuming that the yield to maturity of each bond remains at 9.6% over the next 4 years, calculate the price of the bonds at each of the following years to maturity: Years to Maturity Price of Bond C Price of Bond Z 4 3 2 1 0 b. 7-7 Plot the time path of prices for each bond. INTEREST RATE SENSITIVITY An investor purchased the following 5 bonds. Each bond had a par value of $1,000 and an 8% yield to maturity on the purchase day. Immediately after the investor purchased them, interest rates fell and each then had a new YTM of 7%. What is the percentage change in price for each bond after the decline in interest rates? Fill in the following table: Price @ 8% Price @ 7% Percentage Change 10-year, 10% annual coupon 10-year zero 5-year zero 30-year zero $100 perpetuity 7-8 YIELD TO CALL Six years ago the Singleton Company issued 20-year bonds with a 14% annual coupon rate at their $1,000 par value. The bonds had a 9% call premium, with 5 years of call protection. Today Singleton called the bonds. Compute the realized rate of return for an investor who purchased the bonds when they were issued and held them until they were called. Explain why the investor should or should not be happy that Singleton called them. 7-9 YIELD TO MATURITY Heymann Company bonds have 4 years left to maturity. Interest is paid annually, and the bonds have a $1,000 par value and a coupon rate of 9%. 7-10 a. What is the yield to maturity at a current market price of (1) $829 and (2) $1,104? b. Would you pay $829 for each bond if you thought that a “fair” market interest rate for such bonds was 12%—that is, if rd ¼ 12%? Explain your answer. CURRENT YIELD, CAPITAL GAINS YIELD, AND YIELD TO MATURITY Hooper Printing Inc. has bonds outstanding with 9 years left to maturity. The bonds have an 8% annual coupon rate and were issued 1 year ago at their par value of $1,000. However, due to changes in 225 226 Part 3 Financial Assets interest rates, the bond’s market price has fallen to $901.40. The capital gains yield last year was –9.86%. a. What is the yield to maturity? b. For the coming year, what are the expected current and capital gains yields? (Hint: Refer to Footnote 8 for the definition of the current yield and to Table 7-1.) Will the actual realized yields be equal to the expected yields if interest rates change? If not, how will they differ? c. 7-11 BOND YIELDS Last year Clark Company issued a 10-year, 12% semiannual coupon bond at its par value of $1,000. Currently, the bond can be called in 4 years at a price of $1,060 and it sells for $1,100. a. b. c. 7-12 7-13 7-14 Challenging Problems 15–19 7-15 7-16 7-17 7-18 What are the bond’s nominal yield to maturity and its nominal yield to call? Would an investor be more likely to earn the YTM or the YTC? What is the current yield? Is this yield affected by whether the bond is likely to be called? (Hint: Refer to Footnote 8 for the definition of the current yield and to Table 7-1.) What is the expected capital gains (or loss) yield for the coming year? Is this yield dependent on whether the bond is expected to be called? YIELD TO CALL It is now January 1, 2009, and you are considering the purchase of an outstanding bond that was issued on January 1, 2007. It has a 9.5% annual coupon and had a 30-year original maturity. (It matures on December 31, 2036.) There is 5 years of call protection (until December 31, 2011), after which time it can be called at 109—that is, at 109% of par, or $1,090. Interest rates have declined since it was issued; and it is now selling at 116.575% of par, or $1,165.75. a. What is the yield to maturity? What is the yield to call? b. If you bought this bond, which return would you actually earn? Explain your reasoning. c. Suppose the bond had been selling at a discount rather than a premium. Would the yield to maturity have been the most likely return, or would the yield to call have been most likely? PRICE AND YIELD An 8% semiannual coupon bond matures in 5 years. The bond has a face value of $1,000 and a current yield of 8.21%. What are the bond’s price and YTM? (Hint: Refer to Footnote 8 for the definition of the current yield and to Table 7-1.) EXPECTED INTEREST RATE Lloyd Corporation’s 14% coupon rate, semiannual payment, $1,000 par value bonds, which mature in 30 years, are callable 5 years from today at $1,050. They sell at a price of $1,353.54, and the yield curve is flat. Assume that interest rates are expected to remain at their current level. a. What is the best estimate of these bonds’ remaining life? b. If Lloyd plans to raise additional capital and wants to use debt financing, what coupon rate would it have to set in order to issue new bonds at par? BOND VALUATION Bond X is noncallable and has 20 years to maturity, a 9% annual coupon, and a $1,000 par value. Your required return on Bond X is 10%; and if you buy it, you plan to hold it for 5 years. You (and the market) have expectations that in 5 years, the yield to maturity on a 15-year bond with similar risk will be 8.5%. How much should you be willing to pay for Bond X today? (Hint: You will need to know how much the bond will be worth at the end of 5 years.) BOND VALUATION You are considering a 10-year, $1,000 par value bond. Its coupon rate is 9%, and interest is paid semiannually. If you require an “effective” annual interest rate (not a nominal rate) of 8.16%, how much should you be willing to pay for the bond? BOND RETURNS Last year Joan purchased a $1,000 face value corporate bond with an 11% annual coupon rate and a 10-year maturity. At the time of the purchase, it had an expected yield to maturity of 9.79%. If Joan sold the bond today for $1,060.49, what rate of return would she have earned for the past year? BOND REPORTING Look back at Table 7-4 and examine United Parcel Service and Telecom Italia Capital bonds that mature in 2013. a. b. If these companies were to sell new $1,000 par value long-term bonds, approximately what coupon interest rate would they have to set if they wanted to bring them out at par? If you had $10,000 and wanted to invest in United Parcel Service bonds, what return would you expect to earn? What about Telecom Italia Capital bonds? Based just on the data in the table, would you have more confidence about earning your expected rate of return if you bought United Parcel Service or Telecom Italia Capital bonds? Explain. Chapter 7 Bonds and Their Valuation 7-19 YIELD TO MATURITY AND YIELD TO CALL Kaufman Enterprises has bonds outstanding with a $1,000 face value and 10 years left until maturity. They have an 11% annual coupon payment, and their current price is $1,175. The bonds may be called in 5 years at 109% of face value (Call price ¼ $1,090). a. What is the yield to maturity? b. What is the yield to call if they are called in 5 years? c. Which yield might investors expect to earn on these bonds? Why? d. The bond’s indenture indicates that the call provision gives the firm the right to call the bonds at the end of each year beginning in Year 5. In Year 5, the bonds may be called at 109% of face value; but in each of the next 4 years, the call percentage will decline by 1%. Thus, in Year 6, they may be called at 108% of face value; in Year 7, they may be called at 107% of face value; and so forth. If the yield curve is horizontal and interest rates remain at their current level, when is the latest that investors might expect the firm to call the bonds? COMPREHENSIVE/SPREADSHEET PROBLEM 7-20 BOND VALUATION Clifford Clark is a recent retiree who is interested in investing some of his savings in corporate bonds. His financial planner has suggested the following bonds: l Bond A has a 7% annual coupon, matures in 12 years, and has a $1,000 face value. l Bond B has a 9% annual coupon, matures in 12 years, and has a $1,000 face value. l Bond C has an 11% annual coupon, matures in 12 years, and has a $1,000 face value. Each bond has a yield to maturity of 9%. a. b. Before calculating the prices of the bonds, indicate whether each bond is trading at a premium, at a discount, or at par. Calculate the price of each of the three bonds. c. Calculate the current yield for each of the three bonds. (Hint: Refer to Footnote 8 for the definition of the current yield and to Table 7-1.) d. If the yield to maturity for each bond remains at 9%, what will be the price of each bond 1 year from now? What is the expected capital gains yield for each bond? What is the expected total return for each bond? Mr. Clark is considering another bond, Bond D. It has an 8% semiannual coupon and a $1,000 face value (i.e., it pays a $40 coupon every 6 months). Bond D is scheduled to mature in 9 years and has a price of $1,150. It is also callable in 5 years at a call price of $1,040. e. (1) f. g. What is the bond’s nominal yield to maturity? (2) What is the bond’s nominal yield to call? (3) If Mr. Clark were to purchase this bond, would he be more likely to receive the yield to maturity or yield to call? Explain your answer. Explain briefly the difference between interest rate (or price) risk and reinvestment rate risk. Which of the following bonds has the most interest rate risk? l A 5-year bond with a 9% annual coupon l A 5-year bond with a zero coupon l A 10-year bond with a 9% annual coupon l A 10-year bond with a zero coupon Only do this part if you are using a spreadsheet. Calculate the price of each bond (A, B, and C) at the end of each year until maturity, assuming interest rates remain constant. Create a graph showing the time path of each bond’s value similar to Figure 7-2. (1) What is the expected interest yield for each bond in each year? (2) What is the expected capital gains yield for each bond in each year? (3) What is the total return for each bond in each year? 227 228 Part 3 Financial Assets INTEGRATED CASE WESTERN MONEY MANAGEMENT INC. 7-21 BOND VALUATION Robert Black and Carol Alvarez are vice presidents of Western Money Management and codirectors of the company’s pension fund management division. A major new client, the California League of Cities, has requested that Western present an investment seminar to the mayors of the represented cities. Black and Alvarez, who will make the presentation, have asked you to help them by answering the following questions. a. What are a bond’s key features? b. What are call provisions and sinking fund provisions? Do these provisions make bonds more or less risky? c. How is the value of any asset whose value is based on expected future cash flows determined? d. How is a bond’s value determined? What is the value of a 10-year, $1,000 par value bond with a 10% annual coupon if its required return is 10%? (1) What is the value of a 13% coupon bond that is otherwise identical to the bond described in Part d? Would we now have a discount or a premium bond? e. f. g. h. i. j. k. l. m. (2) What is the value of a 7% coupon bond with these characteristics? Would we now have a discount or premium bond? (3) What would happen to the values of the 7%, 10%, and 13% coupon bonds over time if the required return remained at 10%? [Hint: With a financial calculator, enter PMT, I/YR, FV, and N; then change (override) N to see what happens to the PV as it approaches maturity.] (1) What is the yield to maturity on a 10-year, 9%, annual coupon, $1,000 par value bond that sells for $887.00? that sells for $1,134.20? What does the fact that it sells at a discount or at a premium tell you about the relationship between rd and the coupon rate? (2) What are the total return, the current yield, and the capital gains yield for the discount bond? Assume that it is held to maturity and the company does not default on it. (Hint: Refer to Footnote 8 for the definition of the current yield and to Table 7-1.) What is interest rate (or price) risk? Which has more interest rate risk, an annual payment 1-year bond or a 10-year bond? Why? What is reinvestment rate risk? Which has more reinvestment rate risk, a 1-year bond or a 10-year bond? How does the equation for valuing a bond change if semiannual payments are made? Find the value of a 10-year, semiannual payment, 10% coupon bond if nominal rd ¼ 13%. Suppose for $1,000 you could buy a 10%, 10-year, annual payment bond or a 10%, 10-year, semiannual payment bond. They are equally risky. Which would you prefer? If $1,000 is the proper price for the semiannual bond, what is the equilibrium price for the annual payment bond? Suppose a 10-year, 10%, semiannual coupon bond with a par value of $1,000 is currently selling for $1,135.90, producing a nominal yield to maturity of 8%. However, it can be called after 4 years for $1,050. (1) What is the bond’s nominal yield to call (YTC)? (2) If you bought this bond, would you be more likely to earn the YTM or the YTC? Why? Does the yield to maturity represent the promised or expected return on the bond? Explain. These bonds were rated AA- by S&P. Would you consider them investment-grade or junk bonds? n. What factors determine a company’s bond rating? o. If this firm were to default on the bonds, would the company be immediately liquidated? Would the bondholders be assured of receiving all of their promised payments? Explain. ª FRANK SITEMAN/PHOTOLIBRARY CHAPTER 8 Risk and Rates of Return A Tale of Three Markets—or Is It Four? The purpose of this vignette is to give you some additional perspective on the stock market. Refer to Figure 8-1 on page 232 as you read the following paragraphs. Market 1: 1975–2000. These were great years, especially the last five. Only 3 years saw losses; and toward the end of the run, most investors and money managers had never experienced a really bad market and acted as though bad markets had been banished and would never reappear again. However, Alan Greenspan, Chairman of the Federal Reserve Board at that time, knew the wild ride couldn’t continue. In 1995, he stated that investors were exhibiting “irrational exuberance”; but the market ignored him and kept roaring ahead. Market 2: 2000–2003. Greenspan was right. In 2000, the bubble started to leak and the market fell by 10%. Then in 2001, the 9/11 terrorist attacks on the World Trade Center knocked stocks down another 14%. Finally, in 2002, fears of another attack in addition to a recession led to a gut-wrenching 24% decline. Those 3 years cost the average investor almost 50% of his or her beginning-of-2000 market value. People planning to retire rich and young had to rethink those plans. Market 3: 2003–2007. Investors had overreacted; so in 2003, the market rebounded, rising by just over 25%. The market remained strong through 2007—the economy was robust, profits were rising rapidly, and the Federal Reserve encouraged a bull market by cutting interest rates 11 times. In 2007, the Dow Jones and other stock averages hit all-time highs. But the debt markets were suffering from the subprime mortgage debacle, and institutions such as Merrill Lynch and Citigroup were writing off tens of billions of dollars of bad loans. Oil prices hit $100 per barrel, gasoline prices hit new highs, and unemployment rates were creeping up. With this backdrop, some observers wondered if we were again suffering from irrational exuberance. 229 230 Part 3 Financial Assets Market 4: 2008 and Thereafter: Bull or Bear? In early 2008, the big question is this: Will the bull market continue; or are we entering another bear market? It turned out that the bears were right—by October 2008, the market had fallen nearly 30% from its high earlier in the year in the aftermath of a credit crisis on Wall Street, the collapse of several leading financial firms, and fears of a sharp economic decline. In response, Congress passed an unprecedented $700 billion plan to rescue the financial system. What's next? Will the market stabilize or will it continue to see further declines? We wish we knew! By the time you read this, you will know, but it will be too late to profit from that knowledge. PUTTING THINGS IN PERSPECTIVE We start this chapter from the basic premise that investors like returns and dislike risk; hence, they will invest in risky assets only if those assets offer higher expected returns. We define what risk means as it relates to investments, examine procedures that are used to measure risk, and discuss the relationship between risk and return. Investors should understand these concepts, as should corporate managers as they develop the plans that will shape their firms’ futures. Risk can be measured in different ways, and different conclusions about an asset’s riskiness can be reached depending on the measure used. Risk analysis can be confusing, but it will help if you keep the following points in mind: 1. All business assets are expected to produce cash flows, and the riskiness of an asset is based on the riskiness of its cash flows. The riskier the cash flows, the riskier the asset. 2. Assets can be categorized as financial assets, especially stocks and bonds, and as real assets, such as trucks, machines, and whole businesses. In theory, risk analysis for all types of assets is similar and the same fundamental concepts apply to all assets. However, in practice, differences in the types of available data lead to different procedures for stocks, bonds, and real assets. Our focus in this chapter is on financial assets, especially stocks. We considered bonds in Chapter 7; and we take up real assets in the capital budgeting chapters, especially Chapter 12. 3. A stock’s risk can be considered in two ways: (a) on a stand-alone, or single-stock, basis, or (b) in a portfolio context, where a number of stocks are combined and their consolidated cash flows are analyzed.1 There is an important difference between stand-alone and portfolio risk, and a stock that has a great deal of risk held by itself may be much less risky when held as part of a larger portfolio. 4. In a portfolio context, a stock’s risk can be divided into two components: (a) diversifiable risk, which can be diversified away and is thus of little concern to diversified investors, and (b) market risk, which reflects the risk of a general stock market decline and cannot be eliminated by diversification (hence, does concern investors). Only market risk is relevant to rational investors because diversifiable risk can and will be eliminated. 5. A stock with high market risk must offer a relatively high expected rate of return to attract investors. Investors in general are averse to risk, so they will not buy risky assets unless they are compensated with high expected returns. 6. If investors, on average, think a stock’s expected return is too low to compensate for its risk, they will start selling it, driving down its price and boosting its expected 1 A portfolio is a collection of investment securities. If you owned stock in General Motors, ExxonMobil, and IBM, you would be holding a three-stock portfolio. Because diversification lowers risk without sacrificing much if any expected return, most stocks are held in portfolios. Chapter 8 Risk and Rates of Return return. Conversely, if the expected return on a stock is more than enough to compensate for the risk, people will start buying it, raising its price and thus lowering its expected return. The stock will be in equilibrium, with neither buying nor selling pressure, when its expected return is exactly sufficient to compensate for its risk. 7. Stand-alone risk, the topic of Section 8-2, is important in stock analysis primarily as a lead-in to portfolio risk analysis. However, stand-alone risk is extremely important when analyzing real assets such as capital budgeting projects. When you finish this chapter, you should be able to: Explain the difference between stand-alone risk and risk in a portfolio context. Explain how risk aversion affects a stock’s required rate of return. Discuss the difference between diversifiable risk and market risk, and explain how each type of risk affects well-diversified investors. Explain what the CAPM is and how it can be used to estimate a stock’s required rate of return. Discuss how changes in the general stock and the bond markets could lead to changes in the required rate of return on a firm’s stock. Discuss how changes in a firm’s operations might lead to changes in the required rate of return on the firm’s stock. l l l l l l 8-1 STOCK PRICES OVER THE LAST 20 YEARS Figure 8-1 gives you an idea about how stocks have performed over the period from 1988 through 2007.2 The top graph compares General Electric (GE), the broad stock market as measured by the S&P 500, and General Motors (GM). GE illustrates companies that have done well, GM illustrates those that have not done well, and the S&P 500 shows how an average company has performed. Most stocks climbed sharply until 2000 (Market 1 in the vignette), then dropped equally sharply during Market 2, then rose nicely through most of Market 3. Since there are thousands of stocks, we could have shown many different pictures, with some rising much faster than GE and others falling much faster than GM—with some going to zero and vanishing. Most of the indexes rise and fall together; but if we had shown the Nasdaq index, it would have looked a great deal like GE, rising much faster than the S&P but then falling faster later on. Also note that the beginning and ending dates can lead to totally different “pictures” of stocks’ performances. If we had started in 1990 and ended in 2000, it would have looked as though stocks were wonderful investments. On the other hand, if we had started in 2000 and ended in 2003, it would have looked as though stocks were a terrible place to put our money. It would be great if we knew when to get in and out of the market. The lower graph shows GE’s P/E ratio. The P/E ratio depends on a number of factors, including fundamental factors such as interest rates and earnings growth rates; but it also reflects investors’ optimism or pessimism—or in Alan Greenspan’s words, their “irrational exuberance” or pessimism. Security analysts and investors forecast the future, but they seem to be overly optimistic at certain times and overly pessimistic at other times. Looking back, we can see that they were overly optimistic in 2000. But what about in 1997? There had been a sharp run-up to that time, and some “experts” thought the market was at a top and recommended getting out. Those experts turned out to be wrong, and they “left a lot of money on the table.” 2 The graph reflects stock prices; dividends are not included. If dividends were included, the percentage gains would be somewhat higher. 231 232 Part 3 Financial Assets FIGURE 8-1 Stock Performance, 1988–2007 GE Daily SP500 GM 6/18/07 +1,600% +1,400% +1,200% +1,000% +800% +600% +400% +200% +0% 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 –200% 60 P/E Ratio 50 40 30 20 10 0 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 © BigCharts.com Source: http://online.wsj.com, The Wall Street Journal Online, January 12, 2008. Note also that if you had bought and held GE stock you would have done quite well; but if you had bought GM stock, you wouldn’t have done well at all. If you had formed a portfolio with some GM and some GE stocks, you would have had “average” performance. The portfolio would have limited your potential gain but also would have limited your low-end returns. We will have more to say about portfolios later, but keep this in mind as you go through the chapter. 8-2 STAND-ALONE RISK Risk The chance that some unfavorable event will occur. Risk is defined by Webster as “a hazard; a peril; exposure to loss or injury.” Thus, risk refers to the chance that some unfavorable event will occur. If you engage in skydiving, you are taking a chance with your life—skydiving is risky. If you bet on the horses, you are risking your money. As we saw in previous chapters, individuals and firms invest funds today with the expectation of receiving additional funds in the future. Bonds offer relatively low returns, but with relatively little risk—at least if you stick to Treasury and high-grade corporate bonds. Stocks offer the chance of higher returns, but stocks are generally riskier than bonds. If you invest in speculative stocks (or, really, any stock), you are taking a significant risk in the hope of making an appreciable return. Chapter 8 Risk and Rates of Return An asset’s risk can be analyzed in two ways: (1) on a stand-alone basis, where the asset is considered by itself, and (2) on a portfolio basis, where the asset is held as one of a number of assets in a portfolio. Thus, an asset’s stand-alone risk is the risk an investor would face if he or she held only this one asset. Most financial assets, and stocks in particular, are held in portfolios; but it is necessary to understand stand-alone risk to understand risk in a portfolio context. To illustrate stand-alone risk, suppose an investor buys $100,000 of short-term Treasury bills with an expected return of 5%. In this case, the investment’s return, 5%, can be estimated quite precisely; and the investment is defined as being essentially risk-free. This same investor could also invest the $100,000 in the stock of a company just being organized to prospect for oil in the mid-Atlantic. Returns on the stock would be much harder to predict. In the worst case, the company would go bankrupt and the investor would lose all of his or her money, in which case the return would be 100%. In the best-case scenario, the company would discover huge amounts of oil and the investor would receive a 1,000% return. When evaluating this investment, the investor might analyze the situation and conclude that the expected rate of return, in a statistical sense, is 20%; but the actual rate of return could range from, say, þ1,000% to 100%. Because there is a significant danger of earning much less than the expected return, such a stock would be relatively risky. No investment should be undertaken unless the expected rate of return is high enough to compensate for the perceived risk. In our example, it is clear that few if any investors would be willing to buy the oil exploration stock if its expected return didn’t exceed that of the T-bill. This is an extreme example. Generally, things are much less obvious; and we need to measure risk in order to decide whether a potential investment should be undertaken. Therefore, we need to define risk more precisely. As you will see, the risk of an asset is different when the asset is held by itself versus when it is held as a part of a group, or portfolio, of assets. We look at standalone risk in this section, then at portfolio risk in later sections. It’s necessary to know something about stand-alone risk in order to understand portfolio risk. Also, stand-alone risk is important to the owners of small businesses and in our examination of physical assets in the capital budgeting chapters. For stocks and most financial assets, though, it is portfolio risk that is most important. Still, you need to understand the key elements of both types of risk. 233 Stand-Alone Risk The risk an investor would face if he or she held only one asset. 8-2a Statistical Measures of Stand-Alone Risk This is not a statistics book, and we won’t spend a great deal of time on statistics. However, you do need an intuitive understanding of the relatively simple statistics presented in this section. All of the calculations can be done easily with a calculator or with Excel; and while we show pictures of the Excel setup, Excel is not needed for the calculations. Here are the five key items that are covered: Probability distributions Expected rates of return, ^ r (“r hat”) Historical, or past realized, rates of return, r (“r bar”) Standard deviation,  (sigma) Coefficient of variation (CV) l l l l l Table 8-1 gives the probability distributions for Martin Products, which makes engines for long-haul trucks (18-wheelers), and for U.S. Water, which supplies an essential product and thus has very stable sales and profits. Three possible states of the economy are shown in Column 1; and the probabilities of these outcomes, expressed as decimals rather than percentages, are given in Column 2 and then repeated in Column 5. There is a 30% chance of a strong economy and thus strong demand, a 40% probability of normal demand, and a 30% probability of weak demand. Probability Distribution A listing of possible outcomes or events with a probability (chance of occurrence) assigned to each outcome. 234 Part 3 Financial Assets Table 8-1 A 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Economy, Which Affects Demand (1) Strong Normal Weak Expected Rate of Return, ^r The rate of return expected to be realized from an investment; the weighted average of the probability distribution of possible results. Probability Distributions and Expected Returns B C D Martin Products Rate of Probability Return of This If This Demand Demand Occurring Occurs (2) (3) 80% 0.30 10 0.40 -60 0.30 Expected return = 1.00 Product (2)⫻(3) (4) 24% 4 -18 10% E Probability of This Demand Occurring (5) 0.30 0.40 0.30 1.00 F G U.S. Water Rate of Return If This Demand Product Occurs (5)⫻(6) (6) (7) 4.5% 15% 4.0 10 1.5 5 Expected return = 10.0% Columns 3 and 6 show the returns for the two companies under each state of the economy. Returns are relatively high when demand is strong and low when demand is weak. Notice, though, that Martin’s rate of return could vary far more widely than U.S. Water’s. Indeed, there is a fairly high probability that Martin’s stock will suffer a 60% loss, while at worst, U.S. Water should have a 5% return.3 Columns 4 and 7 show the products of the probabilities times the returns under the different demand levels. When we sum these products, we obtain the expected rates of return, ^ r “r-hat,” for the stocks. Both stocks have an expected return of 10%.4 We can graph the data in Table 8-1 as we do in Figure 8-2. The height of each bar indicates the probability that a given outcome will occur. The range of possible returns for Martin is from 60% to þ80%, and the expected return is 10%. The expected return for U.S. Water is also 10%, but its possible range (and thus maximum loss) is much narrower. In Figure 8-2, we assumed that only three economic states could occur: strong, normal, and weak. Actually, the economy can range from a deep depression to a fantastic boom; and there are an unlimited number of possibilities in between. Suppose we had the time and patience to assign a probability to each possible level of demand (with the sum of the probabilities still equaling 1.0) and to assign a rate of return to each stock for each level of demand. We would have a table similar to Table 8-1 except that it would have many more demand levels. This table could be used to calculate expected rates of return as shown previously, and the probabilities and outcomes could be represented by continuous curves such as those shown in Figure 8-3. Here we changed the assumptions so that there is essentially no chance that Martin’s return will be less than 60% or more than 80% or that 3 It is completely unrealistic to think that any stock has no chance of a loss. Only in hypothetical examples could this occur. To illustrate, the price of Countrywide Financial’s stock dropped from $45.26 to $4.43 in the 12 months ending January 2008. 4 The expected return can also be calculated with an equation that does the same thing as the table: Expected rate of return ¼ ^r ¼ P1 r1 þ P2 r2 þ    þ PN rN N X ^r ¼ Pi ri 8-1 i¼1 The second form of the equation is a shorthand expression in which sigma (∑) means “sum up,” or add the values of n factors. If i ¼ 1, then Piri ¼ P1r1; if i ¼ 2, then Piri ¼ P2r2; and so forth; until i ¼ N, the last possible outcome. The symbol N X simply says, “Go through the following process: First, let i ¼ 1 and find the first product; then let i¼1 i ¼ 2 and find the second product; then continue until each individual product up to N has been found. Add these individual products to find the expected rate of return.” Chapter 8 Risk and Rates of Return Probability Distributions of Martin Products’ and U.S. Water’s Rates of Return FIGURE 8-2 b. U.S. Water a. Martin Products Probability of Occurrence 0.4 Probability of Occurrence 0.4 –60 0.3 0.3 0.2 0.2 0.1 0.1 0 235 10 80 Rate of Return (%) 0 5 10 15 Rate of Return (%) Expected Rate of Return Expected Rate of Return Continuous Probability Distributions of Martin Products’ and U.S. Water’s Rates of Return FIGURE 8-3 Probability Density U.S. Water Martin Products –60 0 10 80 Rate of Return (%) Expected Rate of Return Note: The assumptions regarding the probabilities of various outcomes have been changed from those in Figure 8-2. There the probability of obtaining exactly 10% was 40%; here it is much smaller because there are many possible outcomes instead of just three. With continuous distributions, it is more appropriate to ask what the probability is of obtaining at least some specified rate of return than to ask what the probability is of obtaining exactly that rate. This topic is covered in detail in statistics courses. 236 Part 3 Financial Assets U.S. Water’s return will be less than 5% or more than 15%. However, virtually any return within these limits is possible. The tighter (or more peaked) the probability distributions shown in Figure 8-3, the more likely the actual outcome will be close to the expected value and, consequently, the less likely the actual return will end up far below the expected return. Thus, the tighter the probability distribution, the lower the risk. Since U.S. Water has a relatively tight distribution, its actual return is likely to be closer to its 10% expected return than is true for Martin; so U.S. Water is less risky.5 8-2b Measuring Stand-Alone Risk: The Standard Deviation6 It is useful to measure risk for comparative purposes, but risk can be defined and measured in several ways. A common definition that is simple and is satisfactory for our purpose is based on probability distributions such as those shown in Figure 8-3: The tighter the probability distribution of expected future returns, the smaller the risk of a given investment. According to this definition, U.S. Water is less risky than Martin Products because there is a smaller chance that the actual return of U.S. Water will end up far below its expected return. We can use the standard deviation (, pronounced “sigma”) to quantify the tightness of the probability distribution.7 The smaller the standard deviation, the tighter the probability distribution and, accordingly, the lower the risk. We calculate Martin’s in Table 8-2. We picked up Columns 1, 2, and 3 from Table 8-1. Then in Column 4, we find the deviation of the return in each demand state from the expected return: Actual return – Expected 10% return. The deviations are squared and shown in Column 5. Each squared deviation is then multiplied by the relevant probability and shown in Column 6. The sum of the products in Column 6 is the variance of the distribution. Finally, we find the square root of the variance—this is Table 8-2 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Calculating Martin Products’ Standard Deviation A B Economy, Which Affects Demand (1) Strong Normal Weak Probability of This Demand Occurring (2) 0.30 0.40 0.30 1.00 5 C D Rate of Return If This Demand Occurs (3) 80% 10 -60 Deviation: Actual 10% Expected Return (4) 70% 0 -70 E F Squared Deviation Deviation Squared x Prob. (5) (6) 0.4900 0.1470 0.0000 0.0000 0.1470 0.4900 ⌺ = Variance: 0.2940 Standard deviation = square root of variance: ␴ = 0.5422 Standard deviation expressed as a percentage: ␴ = 54.22% In this example, we implicitly assume that the state of the economy is the only factor that affects returns. In reality, many factors, including labor, materials, and development costs, influence returns. This is discussed at greater length in the chapters on capital budgeting. 6 This section is relatively technical, but it can be omitted without loss of continuity. 7 There are actually two types of standard deviations, one for complete distributions and one for situations that involve only a sample. Different formulas and notations are used. Also, the standard deviation should be modified if the distribution is not normal, or bell-shaped. Since our purpose is simply to get the general idea across, we leave the refinements to advanced finance and statistics courses. 237 Chapter 8 Risk and Rates of Return the standard deviation, and it is shown at the bottom of Column 6 as a fraction and a percentage.8 The standard deviation is a measure of how far the actual return is likely to deviate from the expected return. Martin’s standard deviation is 54.2%, so its actual return is likely to be quite different from the expected 10%.9 U.S. Water’s standard deviation is 3.9%, so its actual return should be much closer to the expected return of 10%. The average publicly traded firm’s has been in the range of 20% to 30% in recent years; so Martin is more risky than most stocks, while U.S. Water is less risky. Standard Deviation, σ (sigma) A statistical measure of the variability of a set of observations. 8-2c Using Historical Data to Measure Risk 10 In the last section, we found the mean and standard deviation based on a subjective probability distribution. If we had actual historical data instead, the standard deviation of returns could be found as shown in Table 8-3.11 Because past results are often repeated in the future, the historical  is often used as an estimate of future risk.12 A key question that arises when historical data is used to forecast the future is how far back in time we should go. Unfortunately, there is no simple answer. Using a longer historical time series has the benefit of giving more information, but some of that information may be misleading if you believe that the level of risk in the future is likely to be very different than the level of risk in the past. Table 8-3 A 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Year (1) 2005 2006 2007 2008 Average Finding  Based On Historical Data B C D E Deviation from Average (3) 19.8% -20.3 -29.3 29.8 Return (2) 30.0% -10.0 -19.0 40.0 10.3% F Squared Deviation (4) 3.9% 4.1 8.6 8.9 Variance = ⌺: 25.4% Variance/(N–1) = Variance/3: Standard deviation = Square root of variance: ␴ = 8 This formula summarizes what we did in Table 8-2: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX Standard deviation ¼  ¼ t ðri  ^ r Þ2 Pi 8-2 i¼1 9 With a normal (bell-shaped) distribution, the actual return should be within one  about 68% of the time. Again, this section is relatively technical, but it can be omitted without loss of continuity. 11 The 4 years of historical data are considered to be a “sample” of the full (but unknown) set of data, and the procedure used to find the standard deviation is different from the one used for probabilistic data. Here is the equation for sample data, and it is the basis for Table 8-3: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP u N ðr  r Þ2 t Avg t 8-2a Estimated  ¼ t¼1 N1 10 Here rt (“r bar t”) denotes the past realized rate of return in Period t, and rAvg is the average annual return earned over the last N years. 12 The average return for the past period (10.3% in our example) may also be used as an estimate of future returns, but this is problematic because the average historical return varies widely depending on the period examined. In our example, if we went from 2005 to 2007, we would get a different average from the 10.3%. The average historical return stabilizes with more years of data, but that brings into question whether data from many years ago is still relevant today. 8.5% 29.1% 238 Part 3 Financial Assets All financial calculators (and Excel) have easy-to-use functions for finding  based on historical data.13 Simply enter the rates of return and press the key marked S (or Sx) to obtain the standard deviation. However, neither calculators nor Excel have a built-in formula for finding  where probabilistic data are involved. In those cases, you must go through the process outlined in Table 8-2. 8-2d Measuring Stand-Alone Risk: The Coefficient of Variation Coefficient of Variation (CV) The standardized measure of the risk per unit of return; calculated as the standard deviation divided by the expected return. If a choice has to be made between two investments that have the same expected returns but different standard deviations, most people would choose the one with the lower standard deviation and, therefore, the lower risk. Similarly, given a choice between two investments with the same risk (standard deviation) but different expected returns, investors would generally prefer the investment with the higher expected return. To most people, this is common sense—return is “good” and risk is “bad”; consequently, investors want as much return and as little risk as possible. But how do we choose between two investments if one has the higher expected return but the other has the lower standard deviation? To help answer that question, we use another measure of risk, the coefficient of variation (CV), which is the standard deviation divided by the expected return: 8-3 Coefficient of variation ¼ CV ¼  ^r The coefficient of variation shows the risk per unit of return, and it provides a more meaningful risk measure when the expected returns on two alternatives are not the same. Since U.S. Water and Martin Products have the same expected return, the coefficient of variation is not necessary in this case. Here the firm with the larger standard deviation, Martin, must have the larger coefficient of variation. In fact, the coefficient of variation for Martin is 54.22/10 ¼ 5.42 and the coefficient of variation for U.S. Water is 3.87/10 ¼ 0.39. Thus, Martin is about 14 times riskier than U.S. Water on the basis of this criterion. 8-2e Risk Aversion and Required Returns Suppose you inherited $1 million, which you plan to invest and then retire on the income. You can buy a 5% U.S. Treasury bill, and you will be sure of earning $50,000 of interest. Alternatively, you can buy stock in R&D Enterprises. If R&D’s research programs are successful, your stock will increase to $2.1 million. However, if the research is a failure, the value of your stock will be zero and you will be penniless. You regard R&D’s chances of success or failure as 50-50, so the expected value of the stock a year from now is 0.5($0) + 0.5($2,100,000) ¼ $1,050,000. Subtracting the $1 million cost leaves an expected $50,000 profit and a 5% rate of return, the same as for the T-bill: Expected ending value  Cost Cost $1,050,000  $1,000,000 ¼ $1,000,000 $50,000 ¼ ¼ 5% $1,000,000 Expected rate of return ¼ Given the choice of the sure $50,000 profit (and 5% rate of return) and the risky expected $50,000 profit and 5% return, which one would you choose? If you choose the 13 See our tutorials on the text’s web site (http://academic.cengage.com/finance/brigham) or your calculator manual for instructions on calculating historical standard deviations. Chapter 8 Risk and Rates of Return less risky investment, you are risk-averse. Most investors are risk-averse, and certainly the average investor is with regard to his or her “serious money.” Because this is a well-documented fact, we assume risk aversion in our discussions throughout the remainder of the book. What are the implications of risk aversion for security prices and rates of return? The answer is that, other things held constant, the higher a security’s risk, the higher its required return; and if this situation does not hold, prices will change to bring about the required condition. To illustrate this point, look back at Figure 8-3 and consider again the U.S. Water and Martin Products stocks. Suppose each stock sells for $100 per share and each has an expected rate of return of 10%. Investors are averse to risk; so under those conditions, there would be a general preference for U.S. Water. People with money to invest would bid for U.S. Water, and Martin’s stockholders would want to sell and use the money to buy U.S. Water. Buying pressure would quickly drive U.S. Water’s stock up, and selling pressure would simultaneously cause Martin’s price to fall. These price changes, in turn, would change the expected returns of the two securities. Suppose, for example, that U.S. Water’s stock was bid up from $100 to $125 and Martin’s stock declined from $100 to $77. These price changes would cause U.S. Water’s expected return to fall to 8% and Martin’s return to rise to 13%.14 The difference in returns, 13% – 8% ¼ 5%, would be a risk premium (RP), which represents the additional compensation investors require for bearing Martin’s higher risk. This example demonstrates a very important principle: In a market dominated by risk-averse investors, riskier securities compared to less risky securities must have higher expected returns as estimated by the marginal investor. If this situation does not exist, buying and selling will occur until it does exist. Later in the chapter we will consider the question of how much higher the returns on risky securities must be, after we see how diversification affects the way risk should be measured. Text not available due to copyright restrictions 14 We assume that each stock is expected to pay shareholders $10 a year in perpetuity. The price of this perpetuity can be found by dividing the annual cash flow by the stock’s return. Thus, if the stock’s expected return is 10%, the price must be $10/0.10 ¼ $100. Likewise, an 8% expected return would be consistent with a $125 stock price ($10/0.08 ¼ $125) and a 13% return with a $77 stock price ($10/0.13 ¼ $77). 239 Risk Aversion Risk-averse investors dislike risk and require higher rates of return as an inducement to buy riskier securities. Risk Premium (RP) The difference between the expected rate of return on a given risky asset and that on a less risky asset. Part 3 Financial Assets SE 240 LF TEST What does investment risk mean? Set up an illustrative probability distribution table for an investment with probabilities for different conditions, returns under those conditions, and the expected return. Which of the two stocks graphed in Figure 8-3 is less risky? Why? Explain why you agree or disagree with this statement: Most investors are risk-averse. How does risk aversion affect rates of return? An investment has a 50% chance of producing a 20% return, a 25% chance of producing an 8% return, and a 25% chance of producing a 12% return. What is its expected return? (9%) 8-3 RISK IN A PORTFOLIO CONTEXT: THE CAPM Capital Asset Pricing Model (CAPM) A model based on the proposition that any stock’s required rate of return is equal to the riskfree rate of return plus a risk premium that reflects only the risk remaining after diversification. In this section, we discuss the risk of stocks when they are held in portfolios rather than as stand-alone assets. Our discussion is based on an extremely important theory, the Capital Asset Pricing Model, or CAPM, that was developed in the 1960s.15 We do not attempt to cover the CAPM in detail—rather, we simply use its intuition to explain how risk should be considered in a world where stocks and other assets are held in portfolios. If you go on to take a course in investments, you will cover the CAPM in detail. Thus far in the chapter we have considered the riskiness of assets when they are held in isolation. This is generally appropriate for small businesses, many real estate investments, and capital budgeting projects. However, the risk of a stock held in a portfolio is typically lower than the stock’s risk when it is held alone. Since investors dislike risk and since risk can be reduced by holding portfolios, most stocks are held in portfolios. Banks, pension funds, insurance companies, mutual funds, and other financial institutions are required by law to hold diversified portfolios. Most individual investors—at least those whose security holdings constitute a significant part of their total wealth—also hold portfolios. Therefore, the fact that one particular stock’s price goes up or down is not important—what is important is the return on the portfolio and the portfolio’s risk. Logically, then, the risk and return of an individual stock should be analyzed in terms of how the security affects the risk and return of the portfolio in which it is held. To illustrate, Pay Up Inc. is a collection agency that operates nationwide through 37 offices. The company is not well known, its stock is not very liquid, and its earnings have experienced sharp fluctuations in the past. This suggests that Pay Up is risky and that its required rate of return, r, should be relatively high. However, Pay Up’s required return in 2008 (and all other years) was quite low in comparison to most other companies. This indicates that investors think Pay Up is a low-risk company in spite of its uncertain profits. This counterintuitive finding has to do with diversification and its effect on risk. Pay Up’s earnings rise during recessions, whereas most other companies’ earnings decline when the economy slumps. Thus, Pay Up’s stock is like insurance—it pays off when other things go bad—so adding Pay Up to a portfolio of “regular” stocks stabilizes the portfolio’s returns and makes it less risky. 15 The CAPM was originated by Professor William F. Sharpe in his article “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk,” Journal of Finance, 1964. Literally thousands of articles exploring various aspects of the CAPM have been published subsequently, and it is very widely used in investment analysis. 241 Chapter 8 Risk and Rates of Return 8-3a Expected Portfolio Returns, ^rp The expected return on a portfolio, ^r p, is the weighted average of the expected returns of the individual assets in the portfolio, with the weights being the percentage of the total portfolio invested in each asset : ^r p ¼ w1^r 1 þ w2^r 2 þ    þ wN^r N N X ¼ wi^r i Expected Return on a Portfolio, ^rp The weighted average of the expected returns on the assets held in the portfolio. 8-4 i¼1 Here ^r i is the expected return on the ith stock; the wi’s are the stocks’ weights, or the percentage of the total value of the portfolio invested in each stock; and N is the number of stocks in the portfolio. Table 8-4 can be used to implement the equation. Here we assume that an analyst estimated returns on the four stocks shown in Column 1 for the coming year, as shown in Column 2. Suppose further that you had $100,000 and you planned to invest $25,000, or 25% of the total, in each stock. You could multiply each stock’s percentage weight as shown in Column 4 by its expected return; get the product terms in Column 5; and then sum Column 5 to get the expected portfolio return, 10.75%. If you added a fifth stock with a higher expected return, the portfolio’s expected return would increase, and vice versa if you added a stock with a lower expected return. The key point to remember is that the expected return on a portfolio is a weighted average of expected returns on the stocks in the portfolio. Several additional points should be made: 1. The expected returns in Column 2 would be based on a study of some type, but they would still be essentially subjective and judgmental because different analysts could look at the same data and reach different conclusions. Therefore, this type of analysis must be viewed with a critical eye. Nevertheless, it is useful, indeed necessary, if one is to make intelligent investment decisions. 2. If we added companies such as Delta Airlines and Ford, which are generally considered to be relatively risky, their expected returns as estimated by the marginal investor would be relatively high; otherwise, investors would sell them, drive down their prices, and force the expected returns above the returns on safer stocks. 3. After the fact and a year later, the actual realized rates of return, ri, on the individual stocks—the r i, or “r-bar,” values—would almost certainly be different from the initial expected values. That would cause the portfolio’s actual return, r p, to differ from the expected return, ^r p ¼ 10.75%. For example, Microsoft’s price might double and thus provide a return of þ100%, whereas IBM might have a terrible year, fall sharply, and have a return of 75%. Note, though, that those two events would be offsetting; so the portfolio’s return still might be close to its expected return even though the returns on the individual stocks were far from their expected values. Table 8-4 52 53 54 55 56 57 58 59 60 61 62 Realized Rate of Return, r The return that was actually earned during some past period. The actual return (r) usually turns out to be different from the expected return (r^) except for riskless assets. Expected Return on a Portfolio, ^ rp A B C D E F Stock (1) Microsoft IBM GE Exxon Expected Return (2) 12.00% 11.50 10.00 9.50 10.75% Dollars Invested (3) $ 25,000 25,000 25,000 25,000 $100,000 Percent of Total (wi) (4) 25.0% 25.0 25.0 25.0 100.0% Product: (2)⫻(4) (5) 3.000% 2.875 2.500 2.375 10.750% = Expected rp 242 Part 3 Financial Assets 8-3b Portfolio Risk Although the expected return on a portfolio is simply the weighted average of the expected returns on its individual stocks, the portfolio’s risk, p, is not the weighted average of the individual stocks’ standard deviations. The portfolio’s risk is generally smaller than the average of the stocks’ s because diversification lowers the portfolio’s risk. To illustrate this point, consider the situation in Figure 8-4. The bottom section gives data on Stocks W and M individually and data on a portfolio with 50% in each stock. The left graph plots the data in a time series format, and it shows that the returns on the individual stocks vary widely from year to year. Therefore, the individual stocks are risky. However, the portfolio’s returns are constant at 15%, indicating that it is not risky at all. The probability distribution graphs to the right show the same thing—the two stocks would be quite risky if they were held in isolation; but when they are combined to form Portfolio WM, they have no risk whatsoever. If you invested all of your money in Stock W, you would have an expected return of 15%, but you would face a great deal of risk. The same thing would hold if you invested entirely in Stock M. However, if you invested 50% in each stock, you would have the same expected return of 15%, but with no risk whatsoever. Being rational and averse to risk, you and all other rational investors would choose to hold the portfolio, not the stocks individually. Returns With Perfect Negative Correlation, r ¼ 1.0 FIGURE 8-4 A 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 B C D Rate of Return 45% F G Stocks W and M, held separately M W E 30% -20% WM 15% 40% Rate of Return (%) 15% Portfolio WM 0% -15% 2004 Year 2004 2005 2006 2007 2008 Avg return = Estimated ␴ = -20% 2005 2006 2007 2008 Stock W Stock M Portfolio WM 40.00% -10.00% 15.00% -10.00 40.00 15.00 40.00 -10.00 15.00 -10.00 40.00 15.00 15.00 15.00 15.00 15.00% 15.00% 15.00% 25.00% 25.00% 0.00% Correlation coefficient = -1.00 15% 40% Rate of Return (%) Chapter 8 Risk and Rates of Return Stocks W and M can be combined to form a riskless portfolio because their returns move countercyclically to each other—when W’s fall, M’s rise, and vice versa. The tendency of two variables to move together is called correlation, and the correlation coefficient, r (pronounced “rho”), measures this tendency.16 In statistical terms, we say that the returns on Stocks W and M are perfectly negatively correlated, with r ¼ 1.0. The opposite of perfect negative correlation is perfect positive correlation, with r ¼ þ1.0. If returns are not related to one another at all, they are said to be independent and r ¼ 0. The returns on two perfectly positively correlated stocks with the same expected return would move up and down together, and a portfolio consisting of these stocks would be exactly as risky as the individual stocks. If we drew a graph like Figure 8-4, we would see just one line because the two stocks and the portfolio would have the same return at each point in time. Thus, diversification is completely useless for reducing risk if the stocks in the portfolio are perfectly positively correlated. We see then that when stocks are perfectly negatively correlated (r ¼ 1.0), all risk can be diversified away; but when stocks are perfectly positively correlated (r ¼ þ1.0), diversification does no good. In reality, most stocks are positively correlated but not perfectly so. Past studies have estimated that on average, the correlation coefficient between the returns of two randomly selected stocks is about 0.30.17 Under this condition, combining stocks into portfolios reduces risk but does not completely eliminate it.18 Figure 8-5 illustrates this point using two stocks whose correlation coefficient is r ¼ þ0.35. The portfolio’s average return is 15%, which is the same as the average return for the two stocks; but its standard deviation is 18.62%, which is below the stocks’ standard deviations and their average . Again, a rational, risk-averse investor would be better off holding the portfolio rather than just one of the individual stocks. In our examples, we considered portfolios with only two stocks. What would happen if we increased the number of stocks in the portfolio? 243 Correlation The tendency of two variables to move together. Correlation Coefficient, r A measure of the degree of relationship between two variables. As a rule, portfolio risk declines as the number of stocks in a portfolio increases. If we added enough partially correlated stocks, could we completely eliminate risk? In general, the answer is no. For an illustration, see Figure 8-5 on page 244 which shows that a portfolio’s risk declines as stocks are added. Here are some points to keep in mind about the figure: 1. The portfolio’s risk declines as stocks are added, but at a decreasing rate; and once 40 to 50 stocks are in the portfolio, additional stocks do little to reduce risk. 2. The portfolio’s total risk can be divided into two parts, diversifiable risk and market risk.19 Diversifiable risk is the risk that is eliminated by adding stocks. Market risk is the risk that remains even if the portfolio holds every stock in 16 The correlation coefficient, r, can range from +1.0, denoting that the two variables move up and down in perfect synchronization, to 1.0, denoting that the variables move in exactly opposite directions. A correlation coefficient of zero indicates that the two variables are not related to each other—that is, changes in one variable are independent of changes in the other. It is easy to calculate correlation coefficients with a financial calculator. Simply enter the returns on the two stocks and press a key labeled “r.” For W and M, r ¼ 1.0. See our tutorial on the text’s web site or your calculator manual for the exact steps. Also note that the correlation coefficient is often denoted by the term r. We use r here to avoid confusion with r used to denote the rate of return. 17 A study by Chan, Karceski, and Lakonishok (1999) estimated that the average correlation coefficient between two randomly selected stocks was 0.28, while the average correlation coefficient between two large-company stocks was 0.33. The time period of their sample was 1968 to 1998. See Louis K. C. Chan, Jason Karceski, and Josef Lakonishok, “On Portfolio Optimization: Forecasting Covariance and Choosing the Risk Model,” The Review of Financial Studies, Vol. 12, no. 5 (Winter 1999), pp. 937–974. 18 If we combined a large number of stocks with r ¼ 0, we could form a riskless portfolio. However, there are not many stocks with r ¼ 0—stocks’ returns tend to move together, not to be independent of one another. 19 Diversifiable risk is also known as company-specific, or unsystematic, risk. Market risk is also known as nondiversifiable or systematic or beta risk; it is the risk that remains in the portfolio after diversification has eliminated all company-specific risk. Diversifiable Risk That part of a security’s risk associated with random events; it can be eliminated by proper diversification. This risk is also known as companyspecific, or unsystematic, risk. Market Risk The risk that remains in a portfolio after diversification has eliminated all company-specific risk. This risk is also known as nondiversifiable or systematic or beta risk. 244 Part 3 Financial Assets Returns with Partial Correlation, r ¼ þ 0.35 FIGURE 8-5 A 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 B C D Rate of Return 45% E F G Stocks W and Y, held separately W 30% -20% 15% 40% Rate of Return (%) WY 15% Portfolio WY Y 0% -15% 2004 Year 2004 2005 2006 2007 2008 Avg return = Estimated ␴ = -20% 2005 2006 2008 15% 40% Rate of Return (%) Stock W Stock Y Portfolio WY 40.00% 40.00% 40.00% -10.00 15.00 2.50 35.00 -5.00 15.00 -5.00 -10.00 -7.50 15.00 35.00 25.00 15.00% 15.00% 15.00% 22.64% 22.64% 18.62% Correlation coefficient = 0.35 3. 4. 5. Market Portfolio A portfolio consisting of all stocks. 2007 the market. Market risk is the risk that we discussed in the opening vignette and in our discussion of Figure 8-1. Diversifiable risk is caused by such random, unsystematic events as lawsuits, strikes, successful and unsuccessful marketing and R&D programs, the winning or losing of a major contract, and other events that are unique to the particular firm. Because these events are random, their effects on a portfolio can be eliminated by diversification—bad events for one firm will be offset by good events for another. Market risk, on the other hand, stems from factors that systematically affect most firms: war, inflation, recessions, high interest rates, and other macro factors. Because most stocks are affected by macro factors, market risk cannot be eliminated by diversification. If we carefully selected the stocks included in the portfolio rather than adding them randomly, the graph would change. In particular, if we chose stocks with low correlations with one another and with low stand-alone risk, the portfolio’s risk would decline faster than if random stocks were added. The reverse would hold if we added stocks with high correlations and high s. Most investors are rational in the sense that they dislike risk, other things held constant. That being the case, why would an investor ever hold one (or a few) stocks? Why not hold a market portfolio consisting of all stocks? There are several reasons. First, high administrative costs and commissions would more than offset Chapter 8 Risk and Rates of Return 6. 245 the benefits for individual investors. Second, index funds can diversify for investors, and many individuals can and do get broad diversification through these funds. Third, some people think that they can pick stocks that will “beat the market”; so they buy them rather than the broad market. And fourth, some people can, through superior analysis, beat the market; so they find and buy undervalued stocks and sell overvalued ones and, in the process, cause most stocks to be properly valued, with their expected returns consistent with their risks. One key question remains: How should the risk of an individual stock be measured? The standard deviation of expected returns, , is not appropriate because it includes risk that can be eliminated by holding the stock in a portfolio. How then should we measure a stock’s risk in a world where most people hold portfolios? That’s the subject of the next section. 8-3c Risk in a Portfolio Context: The Beta Coefficient When a stock is held by itself, its risk can be measured by the standard deviation of its expected returns. However,  is not appropriate when the stock is held in a portfolio, as stocks generally are. So how do we measure a stock’s relevant risk in a portfolio context? First, note that all risk except that related to broad market movements can and will be diversified away by most investors—rational investors will hold enough stocks to move down the risk curve in Figure 8-6 to the point where only market risk remains in their portfolios. The risk that remains once a stock is in a diversified portfolio is its contribution to the portfolio’s market risk, and that risk can be measured by the extent to which the stock moves up or down with the market. The tendency of a stock to move with the market is measured by its beta coefficient, b. Ideally, when estimating a stock’s beta, we would like to have a crystal ball that tells us how the stock is going to move relative to the overall stock market in the future. But since we can’t look into the future, we often use historical data and assume that the stock’s historical beta will give us a reasonable estimate of how the stock will move relative to the market in the future. To illustrate the use of historical data, consider Figure 8-7, which shows the historical returns on three stocks and a market index. In Year 1, “the market,” as defined by a portfolio containing all stocks, had a total return (dividend yield plus capital gains yield) of 10%, as did the three individual stocks. In Year 2, the market went up sharply and its return was 20%. Stocks H (for high) soared by 30%; A (for average) returned 20%, the same as the market; and L (for low) returned 15%. In Year 3, the market dropped sharply; its return was 10%. The three stocks’ returns also fell—H’s return was 30%, A’s was 10%, and L broke even with a 0% return. In Years 4 and 5, the market returned 0% and 5%, respectively, and the three stocks’ returns were as shown in the figure. A plot of the data shows that the three stocks moved up or down with the market but that H was twice as volatile as the market, A was exactly as volatile as the market, and L had only half the market’s volatility. It is apparent that the steeper a stock’s line, the greater its volatility and thus the larger its loss in a down market. The slopes of the lines are the stocks’ beta coefficients. We see in the figure that the slope coefficient for H is 2.0; for A, it is 1.0; and for L, it is 0.5.20 Thus, beta measures a given stock’s volatility relative to the market, and an average stock’s beta, bA ¼ 1.0. 20 For more on calculating betas, see Brigham and Daves, Intermediate Financial Management, 9th ed., (Mason, OH: Thomson/South-Western, 2007), pp. 55–58 and pp. 89–94. Relevant Risk The risk that remains once a stock is in a diversified portfolio is its contribution to the portfolio’s market risk. It is measured by the extent to which the stock moves up or down with the market. Beta Coefficient, b A metric that shows the extent to which a given stock’s returns move up and down with the stock market. Beta thus measures market risk. Average Stock’s Beta, bA By definition, bA ¼ 1 because an average-risk stock is one that tends to move up and down in step with the general market. 246 Part 3 Financial Assets Effects of Portfolio Size on Risk for a Portfolio of Randomly Selected Stocks FIGURE 8-6 Portfolio Risk, σp (%) 35 30 Portfolio’s Risk, ␴p 25 σ M = 20.4 Portfolio’s Diversifiable Risk: Could Be Reduced by Adding More Stocks 15 Portfolio’s Total Risk: Declines as Stocks 10 Are Added Minimum Attainable Risk in a Portfolio of Average Stocks Portfolio’s Market Risk: Remains Constant 5 0 1 10 20 30 40 2,000+ Number of Stocks in the Portfolio Note: This graph assumes that stocks in the portfolio are randomly selected from the universe of large, publicly-traded stocks listed on the NYSE. Stock A is defined as an average-risk stock because it has a beta of b ¼ 1.0 and thus moves up and down in step with the general market. Thus, an average stock will, in general, move up by 10% when the market moves up by 10% and fall by 10% when the market falls by 10%. A large portfolio of such b ¼ 1.0 stocks would (1) have all of its diversifiable risk removed but (2) would still move up and down with the broad market averages and thus have a degree of risk. Stock H, which has b ¼ 2.0, is twice as volatile as an average stock, which means that it is twice as risky. The value of a portfolio consisting of b ¼ 2.0 stocks could double—or halve—in a short time; and if you held such a portfolio, you could quickly go from being a millionaire to being a pauper. Stock L, on the other hand, with b ¼ 0.5, is only half as volatile as the average stock, and a portfolio of such stocks would rise and fall only half as rapidly as the market. Thus, its risk would be half that of an average-risk portfolio with b ¼ 1.0. Betas for literally thousands of companies are calculated and published by Merrill Lynch, Value Line, Yahoo, Google, and numerous other organizations; and the beta coefficients of some well-known companies are shown in Table 8-5. Most stocks have betas in the range of 0.50 to 1.50; and the average beta for all stocks is 1.0, which indicates that the average stock moves in sync with the market.21 21 While fairly uncommon, it is possible for a stock to have a negative beta. In that case, the stock’s returns would tend to rise whenever the returns on other stocks fell. 247 Chapter 8 Risk and Rates of Return FIGURE 8-7 A 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 Betas: Relative Volatility of Stocks H, A, and L B C D E Return on Stocks 30.0% High: b = 2.0 20.0% Average: b = 1.0 F Low: b = 0.5 10.0% -20.0% 0 -10.0% 10.0% 20.0% 30.0% Return on Market -10.0% -20.0% -30.0% Year 1 2 3 4 5 rM 10.0% 20.0 -10.0 0.0 5.0 rH 10.0% 30.0 -30.0 -10.0 0.0 rA 10.0% 20.0 -10.0 0.0 5.0 rL 10.0% 15.0 0.0 5.0 7.5 Calculating beta: 1. Rise-Over-Run. Divide the vertical axis change that results from a given change on the horizontal axis (i.e., the change in the stock’s return divided by the changes in the market return). For Stock H, when the market rises from –10% to +20%, or by 30%, the stock’s return goes from –30% to +30%, or by 60%. Thus, beta H by the rise-over-run method is 60/30 = 2.0. In the same way, we find beta A to be 1.0 and beta L to be 0.5. This procedure is easy in our example because all of the points lie on a straight line; but if the points were scattered around the trend line, we could not calculate an exact beta. 2. Financial Calculator. Financial calculators have a built-in function that can be used to calculate beta. The procedure differs somewhat from calculator to calculator. See our tutorial on the text’s web site for instructions on several calculators. 3. Excel. Excel’s Slope function can be used to calculate betas. Here are the functions for our three stocks: BetaH 2.0 =SLOPE(C163:C164,B163:B164) BetaA 1.0 =SLOPE(D163:D164,B163:B164) BetaL 0.5 =SLOPE(E163:E164,B163:B164) If a stock whose beta is greater than 1.0 (say 1.5) is added to a bp ¼ 1.0 portfolio, the portfolio’s beta and consequently its risk will increase. Conversely, if a stock whose beta is less than 1.0 is added to a bp ¼ 1.0 portfolio, the portfolio’s beta and risk will decline. Thus, because a stock’s beta reflects its contribution to the riskiness of a portfolio, beta is the theoretically correct measure of the stock’s riskiness. 248 Part 3 Financial Assets Table 8-5 Illustrative List of Beta Coefficients Stock Beta Merrill Lynch Best Buy eBay General Electric Microsoft ExxonMobil Heinz Coca-Cola FPL Group Procter & Gamble 1.35 1.25 1.20 0.95 0.95 0.90 0.80 0.75 0.75 0.65 Source: Adapted from Value Line, February 2008. 1. 2. 3. 4. We can summarize our discussion up to this point as follows: A stock’s risk has two components, diversifiable risk and market risk. Diversifiable risk can be eliminated; and most investors do eliminate it, either by holding very large portfolios or by buying shares in a mutual fund. We are left, then, with market risk, which is caused by general movements in the stock market and reflects the fact that most stocks are systematically affected by events such as wars, recessions, and inflation. Market risk is the only risk that should matter to a rational, diversified investor. Investors must be compensated for bearing risk—the greater the risk of a stock, the higher its required return. However, compensation is required only for risk that cannot be eliminated by diversification. If risk premiums existed on a stock due to its diversifiable risk, that stock would be a bargain to welldiversified investors. They would start buying it and bid up its price, and the stock’s final (equilibrium) price would be consistent with an expected return that reflected only its market risk. To illustrate this point, suppose half of Stock B’s risk is market risk (it occurs because the stock moves up and down with the market), while the other half is diversifiable. You are thinking of buying Stock B and holding it in a one-stock portfolio, so you would be exposed to all of its risk. As compensation for bearing so much risk, you want a risk premium of 8% over the 6% T-bond rate; so your required return is rA ¼ 6% þ 8% ¼ 14%. But other investors, including your professor, are well diversified. They are also looking at Stock B; but they would hold it in diversified portfolios, eliminate its diversifiable risk, and thus be exposed to only half as much risk as you. Therefore, their required risk premium would be half as large as yours, and their required rate of return would be rB ¼ 6% þ 4% ¼ 10%. If the stock was priced to yield the 14% you require, those diversified investors, including your professor, would buy it, push its price up and its yield down, and prevent you from getting the stock at a price low enough to provide the 14% return. In the end, you would have to accept a 10% return or keep your money in the bank. The market risk of a stock is measured by its beta coefficient, which is an index of the stock’s relative volatility. Here are some benchmark betas: b ¼ 0.5: Stock is only half as volatile, or risky, as an average stock. b ¼ 1.0: Stock is of average risk. b ¼ 2.0: Stock is twice as risky as an average stock. Chapter 8 Risk and Rates of Return 5. A portfolio consisting of low-beta stocks will also have a low beta because the beta of a portfolio is a weighted average of its individual securities’ betas, found using this equation: bp ¼ w1 b1 þ w2 b2 þ    þ wN bN N X ¼ wi bi : 8-5 i¼1 Here bp is the beta of the portfolio, and it shows how volatile the portfolio is relative to the market; wi is the fraction of the portfolio invested in the ith stock; and bi is the beta coefficient of the ith stock. To illustrate, if an investor holds a $100,000 portfolio consisting of $33,333.33 invested in each of three stocks and if each of the stocks has a beta of 0.70, the portfolio’s beta will be bp ¼ 0.70: bp ¼ 0:333ð0:70Þ þ 0:333ð0:70Þ þ 0:333ð0:70Þ ¼ 0:70: Such a portfolio would be less risky than the market, so it should experience relatively narrow price swings and have relatively small rate-of-return fluctuations. In terms of Figure 8-7, the slope of its regression line would be 0.70, which is less than that for a portfolio of average stocks. Now suppose one of the existing stocks is sold and replaced by a stock with bi ¼ 2.00. This action will increase the portfolio’s beta from bp1 ¼ 0.70 to bp2 ¼ 1.13: bp2 ¼ 0:333ð0:70Þ þ 0:333ð0:70Þ þ 0:333ð2:00Þ ¼ 1:13: SE 6. Had a stock with bi ¼ 0.20 been added, the portfolio’s beta would have declined from 0.70 to 0.53. Adding a low-beta stock would therefore reduce the portfolio’s riskiness. Consequently, changing the stocks in a portfolio can change the riskiness of that portfolio. Because a stock’s beta coefficient determines how the stock affects the riskiness of a diversified portfolio, beta is, in theory, the most relevant measure of a stock’s risk. LF TEST Explain the following statement: An asset held as part of a portfolio is generally less risky than the same asset held in isolation. What is meant by perfect positive correlation, perfect negative correlation, and zero correlation? In general, can the riskiness of a portfolio be reduced to zero by increasing the number of stocks in the portfolio? Explain. What is an average-risk stock? What is the beta of such a stock? Why is it argued that beta is the best measure of a stock’s risk? If you plotted a particular stock’s returns versus those on the S&P 500 Index over the past five years, what would the slope of the regression line indicate about the stock’s risk? An investor has a two-stock portfolio with $25,000 invested in Stock X and $50,000 invested in Stock Y. X’s beta is 1.50, and Y’s beta is 0.60. What is the beta of the investor’s portfolio? (0.90) 249 250 Part 3 Financial Assets G LOBAL P ERSPECTIVES THE BENEFITS OF DIVERSIFYING OVERSEAS The increasing availability of international securities is making it possible to achieve a better risk-return trade-off than could be obtained by investing only in U.S. securities. So investing overseas might result in a portfolio with less risk but a higher expected return. This result occurs because of low correlations between the returns on U.S. and international securities, along with potentially high returns on overseas stocks. Figure 8-6, presented earlier, demonstrated that an investor can reduce the risk of his or her portfolio by holding a number of stocks. The figure that follows suggests that investors may be able to reduce risk even further by holding a portfolio of stocks from all around the world, given the fact that the returns on domestic and international stocks are not perfectly correlated. Even though foreign stocks represent roughly 60% of the worldwide equity market and despite the apparent benefits from investing overseas, the typical U.S. investor still puts less than 10% of his or her money in foreign stocks. One possible explanation for this reluctance to invest overseas is that investors prefer domestic stocks because of lower transactions costs. However, this explanation is questionable because recent studies reveal that investors buy and sell overseas stocks more frequently than they trade their domestic stocks. Other explanations for the domestic bias include the additional risks from investing overseas (for example, exchange rate risk) and the fact that the typical U.S. investor is uninformed about international investments and/or thinks that international investments are extremely risky. It has been argued that world capital markets have become more integrated, causing the correlation of returns between different countries to increase, which reduces the benefits from international diversification. In addition, U.S. corporations are investing more internationally, providing U.S. investors with international diversification even if they purchase only U.S. stocks. Whatever the reason for their relatively small holdings of international assets, our guess is that in the future U.S. investors will shift more of their assets to overseas investments. Portfolio Risk, σp (%) U.S. Stocks U.S. and International Stocks Number of Stocks in the Portfolio Source: For further reading, see also Kenneth Kasa, “Measuring the Gains from International Portfolio Diversification,” Federal Reserve Bank of San Francisco Weekly Letter, Number 94–14, April 8, 1994. Chapter 8 Risk and Rates of Return 251 8-4 THE RELATIONSHIP BETWEEN RISK AND RATES OF RETURN The preceding section demonstrated that under the CAPM theory, beta is the most appropriate measure of a stock’s relevant risk. The next issue is this: For a given level of risk as measured by beta, what rate of return is required to compensate investors for bearing that risk? To begin, let us define the following terms: ^ ri ¼ ri ¼ r ¼ rRF ¼ bi ¼ rM ¼ RPM ¼ RPi ¼ expected rate of return on the ith stock. required rate of return on the ith stock. Note that if ^ r i is less than ri, the typical investor will not purchase this stock or will sell it if he or she owns it. If ^ r i is greater than ri, the investor will purchase the stock because it looks like a bargain. Investors will be indifferent if ^ r i ¼ ri. Buying and selling by investors tends to force the expected return to equal the required return, although the two can differ from time to time before the adjustment is completed. realized, after-the-fact return. A person obviously does not know r at the time he or she is considering the purchase of a stock. risk-free rate of return. In this context, rRF is generally measured by the return on U.S. Treasury securities. Some analysts recommend that short-term T-bills be used; others recommend long-term T-bonds. We generally use T-bonds because their maturity is closer to the average investor’s holding period of stocks. beta coefficient of the ith stock. The beta of an average stock is bA ¼ 1.0. required rate of return on a portfolio consisting of all stocks, which is called the market portfolio. rM is also the required rate of return on an average (bA ¼ 1.0) stock. (rM – rRF) ¼ risk premium on “the market” and the premium on an average stock. This is the additional return over the risk-free rate required to compensate an average investor for assuming an average amount of risk. Average risk means a stock where bi ¼ bA ¼ 1.0. (rM – rRF)bi ¼ (RPM)bi ¼ risk premium on the ith stock. A stock’s risk premium will be less than, equal to, or greater than the premium on an average stock, RPM, depending on whether its beta is less than, equal to, or greater than 1.0. If bi ¼ bA ¼ 1.0, then RPi ¼ RPM. The market risk premium, RPM, shows the premium that investors require for bearing the risk of an average stock. The size of this premium depends on how risky investors think the stock market is and on their degree of risk aversion. Let us assume that at the current time, Treasury bonds yield rRF ¼ 6% and an average share of stock has a required rate of return of rM ¼ 11%. Therefore, the market risk premium is 5%, calculated as follows: RPM ¼ rM  rRF ¼ 11%  6% ¼ 5% It should be noted that the risk premium of an average stock, rM – rRF, is actually hard to measure because it is impossible to obtain a precise estimate of the expected future return of the market, rM.22 Given the difficulty of estimating future 22 This concept, as well as other aspects of the CAPM, is discussed in more detail in Chapter 3 of Eugene F. Brigham and Philip R. Daves, Intermediate Financial Management, 9th ed., (Mason, OH: Thomson/South-Western, 2007). That chapter also discusses the assumptions embodied in the CAPM framework. Some of those assumptions are unrealistic; and because of this, the theory does not hold exactly. Market Risk Premium, RPM The additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk. 252 Part 3 Financial Assets ESTIMATING THE MARKET RISK PREMIUM The Capital Asset Pricing Model (CAPM) is more than a theory describing the trade-off between risk and return—it is also widely used in practice. As we will see later, investors use the CAPM to determine the discount rate for valuing stocks and corporate managers use it to estimate the cost of equity capital. The market risk premium is a key component of the CAPM, and it should be the difference between the expected future return on the overall stock market and the expected future return on a riskless investment. However, we cannot obtain investors’ expectations; instead, academicians and practitioners often use a historical risk premium as a proxy for the expected risk premium. The historical premium is found by taking the difference between the actual return on the overall stock market and the risk-free rate during a number of different years and then averaging the annual results. Morningstar (through its recent purchase of Ibbotson Associates) may provide the most comprehensive estimates of historical risk premiums. It reports that the annual premiums have averaged 7.1% over the past 82 years. However, there are three potential problems with historical risk premiums. First, what is the proper number of years over which to compute the average? Morningstar goes back to 1926, when good data first became available; but that is an arbitrary choice, and the starting and ending points make a major difference in the calculated premium. Second, historical premiums are likely to be misleading at times when the market risk premium is changing. To illustrate, the stock market was very strong from 1995 through 1999, in part because investors were becoming less risk-averse, which means that they applied a lower risk premium when they valued stocks. The strong market resulted in stock returns of about 30% per year; and when bond yields were subtracted from the high stock returns, the calculated risk premiums averaged 22.3% a year. When those high numbers were added to data from prior years, they caused the long-run historical risk premium as reported by Morningstar to increase. Thus, a declining “true” risk premium led to very high stock returns, which, in turn, led to an increase in the calculated historical risk premium. That’s a worrisome result, to say the least. The third concern is that historical estimates may be biased upward because they include only the returns of firms that have survived—they do not reflect the losses incurred on investments in failed firms. Stephen Brown, William Goetzmann, and Stephen Ross discussed the implications of this “survivorship bias” in a 1995 Journal of Finance article. Putting these ideas into practice, Tim Koller, Marc Goedhart, and David Wessels recently suggested that survivorship bias increases historical returns by 1% to 2% a year. Therefore, they suggest that practitioners subtract 1% to 2% from the historical estimates to obtain the risk premium used in the CAPM. Sources: Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Chicago: Morningstar, Inc., 2008); Stephen J. Brown, William N. Goetzmann, and Stephen A. Ross, “Survival,” Journal of Finance, Vol. 50, no. 3 (July 1995), pp. 853–873; and Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the Value of Companies, 4th edition (New York: McKinsey & Company, 2005). market returns, analysts often look to historical data to estimate the market risk premium. Historical data suggest that the market risk premium varies somewhat from year to year due to changes in investors’ risk aversion but that it has generally ranged from 4% to 8%. While historical estimates might be a good starting point for estimating the market risk premium, those estimates would be misleading if investors’ attitudes toward risk changed considerably over time. (See “Estimating the Market Risk Premium” box above.) Indeed, many analysts have argued that the market risk premium has fallen in recent years. If this claim is correct, the market risk premium is considerably lower than one based on historical data. The risk premium on individual stocks varies in a systematic manner from the market risk premium. For example, if one stock is twice as risky as another stock as measured by their beta coefficients, its risk premium should be twice as high. Therefore, if we know the market risk premium, RPM, and the stock’s beta, bi, we can find its risk premium as the product (RPM)bi. For example, if beta for Stock L ¼ 0.5 and RPM ¼ 5%, RPL will be 2.5%: 8-6 Risk premium for Stock L ¼ RPi ¼ ðRPM Þbi ¼ ð5%Þð0:5Þ ¼ 2:5% Chapter 8 Risk and Rates of Return 253 As the discussion in Chapter 6 implied, the required return for any stock can be found as follows: Required return on a stock ¼ Risk-free return þ Premium for the stock 0 s risk Here the risk-free return includes a premium for expected inflation; and if we assume that the stocks under consideration have similar maturities and liquidity, the required return on Stock L can be found using the Security Market Line (SML) equation:    Required return ¼ Risk-free + Market risk Stock L0 s on Stock L premium beta return rL ¼ rRF þ ðrM  rRF ÞbL ¼ rRF þ ðRPM ÞbL 8-7 ¼ 6% þ ð11%  6%Þð0:5Þ ¼ 6% þ 2:5% ¼ 8:5% Stock H had bH ¼ 2.0, so its required rate of return is 16%: rH ¼ 6% þ ð5%Þ2:0 ¼ 16% An average stock, with b ¼ 1.0, would have a required return of 11%, the same as the market return: rA ¼ 6% þ ð5%Þ1:0 ¼ 11% ¼ rM The SML equation is plotted in Figure 8-8 using the data shown below the graph on Stocks L, A, and H and assuming that rRF ¼ 6% and rM ¼ 11%. Note the following points: 1. Required rates of return are shown on the vertical axis, while risk as measured by beta is shown on the horizontal axis. This graph is quite different from the one shown in Figure 8-7, where we calculated betas. In the earlier graph, the returns on individual stocks were plotted on the vertical axis and returns on the market index were shown on the horizontal axis. The betas found in Figure 8-7 were then plotted as points on the horizontal axis of Figure 8-8. 2. Riskless securities have bi ¼ 0; so the return on the riskless asset, rRF ¼ 6.0%, is shown as the vertical axis intercept in Figure 8-8. 3. The slope of the SML in Figure 8-8 can be found using the rise-over-run procedure. When beta goes from 0 to 1.0, the required return goes from 6% to 11%, or 5%; so the slope is 5%/1.0 ¼ 5%. Thus, a 1-unit increase in beta causes a 5% increase in the required rate of return. 4. The slope of the SML reflects the degree of risk aversion in the economy—the greater the average investor’s risk aversion, (a) the steeper the slope of the line and (b) the greater the risk premium for all stocks—hence, the higher the required rate of return on all stocks. Both the SML and a company’s position on it change over time due to changes in interest rates, investors’ risk aversion, and individual companies’ betas. Such changes are discussed in the following sections. 8-4a The Impact of Expected Inflation As we discussed in Chapter 6, interest amounts to “rent” on borrowed money, or the price of money. Thus, rRF is the price of money to a riskless borrower. We also Security Market Line (SML) Equation An equation that shows the relationship between risk as measured by beta and the required rates of return on individual securities. 254 Part 3 Financial Assets The Security Market Line (SML) FIGURE 8-8 A 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 B C D E F G H I Required Rate of Return (%) rH = 16.0% SML = rRF + RPM x bi rA = rM = 11.0% rL = 8.5% L’s Risk Premium rRF = 6.0% H’s Risk Premium Market Risk Premium, RPM. Also Stock A’s Risk Premium Risk-Free Return, rRF 0.0 0.5 1.0 1.5 2.0 2.5 Beta Coefficient ri = rRF + RPM(bi) Key Inputs rRF 6.0% rM 11.0 5.0 RPM = rM – rRF Riskless asset: Stock L: Stock A: Stock H: Beta 0.0 0.5 1.0 2.0 ri 6.00% 8.50 11.00 16.00 saw that the risk-free rate as measured by the rate on U.S. Treasury securities is called the nominal, or quoted, rate; and it consists of two elements: (1) a real inflation-free rate of return, r* and (2) an inflation premium, IP, equal to the anticipated rate of inflation.23 Thus, rRF ¼ r* + IP. The real rate on long-term Treasury bonds has historically ranged from 2% to 4%, with a mean of about 3%. Therefore, if no inflation were expected, long-term Treasury bonds would yield about 3%. However, as the expected rate of inflation increases, a premium must be added to the real risk-free rate of return to compensate investors for the loss of purchasing power that results from inflation. Therefore, the 6% rRF shown in Figure 8-8 might be thought of as consisting of a 3% real risk-free rate of return plus a 3% inflation premium: rRF ¼ r* + IP ¼ 3% + 3% ¼ 6%. 23 Long-term Treasury bonds also contain a maturity risk premium, MRP. We include the MRP in r* to simplify the discussion. Chapter 8 Risk and Rates of Return If the expected inflation rate rose by 2%, to 3% + 2% ¼ 5%, rRF would rise to 8%. Such a change is shown in Figure 8-9. Notice that the increase in rRF leads to an equal increase in the rates of return on all risky assets because the same inflation premium is built into required rates of return on both riskless and risky assets.24 Therefore, the rate of return on our illustrative average stock, rA, increases from 11% to 13%. Other risky securities’ returns also rise by two percentage points. 8-4b Changes in Risk Aversion The slope of the SML reflects the extent to which investors are averse to risk—the steeper the slope of the line, the more the average investor requires as compensation for bearing risk. Suppose investors were indifferent to risk; that is, they were not at all risk-averse. If rRF was 6%, risky assets would also have a required return of 6% because if there were no risk aversion, there would be no risk premium. In that case, the SML would plot as a horizontal line. However, because investors are risk-averse, there is a risk premium; and the greater the risk aversion, the steeper the slope of the SML. Shift in the SML Caused by an Increase in Expected Inflation FIGURE 8-9 Required Rate of Return (%) SML2 = 8% + 5%(bi) SML1 = 6% + 5%(bi) rA2 = rM2 = 13 rA1 = rM1 = 11 rRF2 = 8 Increase in Anticipated Inflation, ⌬IP = 2% rRF1 = 6 Original IP = 3% r* = 3 Real Risk-Free Rate of Return, r* 0 0.5 1.0 1.5 2.0 Risk, bi 24 Recall that the inflation premium for any asset is the average expected rate of inflation over the asset’s life. Thus, in this analysis, we must assume that all securities plotted on the SML graph have the same life or that the expected rate of future inflation is constant. It should also be noted that rRF in a CAPM analysis can be proxied by either a long-term rate (the T-bond rate) or a short-term rate (the T-bill rate). Traditionally, the T-bill rate was used; but in recent years, there has been a movement toward use of the T-bond rate because there is a closer relationship between T-bond yields and stocks’ returns than between T-bill yields and stocks’ returns. See Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2008 Yearbook (Chicago: Morningstar, Inc., 2008) for a discussion. 255 256 Part 3 Financial Assets Figure 8-10 illustrates an increase in risk aversion. The market risk premium rises from 5% to 7.5%, causing rM to rise from rM1 ¼ 11% to rM2 ¼ 13.5%. The returns on other risky assets also rise, and the effect of this shift in risk aversion is more pronounced on riskier securities. For example, the required return on Stock L with bA ¼ 0.5 increases by only 1.25 percentage points, from 8.5% to 9.75%, whereas the required return on a stock with a beta of 1.5 increases by 3.75 percentage points, from 13.5% to 17.25%. 8-4c Changes in a Stock’s Beta Coefficient As we will see later in the book, a firm can influence its market risk (hence, its beta) through changes in the composition of its assets and through changes in the amount of debt it uses. A company’s beta can also change as a result of external factors such as increased competition in its industry and expiration of basic patents. When such changes occur, the firm’s required rate of return also changes; and as we will see in Chapter 9, this change will affect its stock price. For example, consider Allied Food Products, with a beta of 1.48. Now suppose some action occurred that caused Allied’s beta to increase from 1.48 to 2.0. If the conditions depicted in Figure 8-8 held, Allied’s required rate of return would increase from 13.4% to 16%: r1 ¼ rRF þ ðrM  rRF Þbi ¼ 6% þ ð11%  6%Þ1:48 ¼ 13:4% to r2 ¼ 6% þ ð11%  6%Þ2:0 ¼ 16:0% Shift in the SML Caused by Increased Risk Aversion FIGURE 8-10 SML2 = 6% + 7.5%(bi) Required Rate of Return (%) 17.25 SML1 = 6% + 5%(bi) rA2 = rM2 = 13.5 rA1 = rM1 = 11 rL2 = 9.75 rL1 = 8.5 New Market Risk Premium, rM2 – rRF = RPM2 = RPA2 = 7.5% rRF = 6 Original Market Risk Premium, rM1 – rRF = 5% 0 0.5 1.0 1.5 2.0 Risk, b i Chapter 8 Risk and Rates of Return 257 SE As we will see in Chapter 9, this change would have a negative effect on Allied’s stock price.25 LF TEST Differentiate between a stock’s expected rate of return (^r ); required rate of return (r); and realized, after-the-fact historical return (r). Which would have to be larger to induce you to buy the stock, ^r or r? At a given point in time, would ^r , r, and r typically be the same or different? Explain. What are the differences between the relative volatility graph (Figure 8-7), where “betas are made,” and the SML graph (Figure 8-8), where “betas are used”? Explain how both graphs are constructed and what information they convey. What would happen to the SML graph in Figure 8-8 if expected inflation increased or decreased? What happens to the SML graph when risk aversion increases or decreases? What would the SML look like if investors were indifferent to risk, that is, if they had zero risk aversion? How can a firm influence the size of its beta? A stock has a beta of 1.2. Assume that the risk-free rate is 4.5% and the market risk premium is 5%. What is the stock’s required rate of return? (10.5%) 8-5 SOME CONCERNS ABOUT BETA AND THE CAPM The Capital Asset Pricing Model (CAPM) is more than just an abstract theory described in textbooks—it has great intuitive appeal and is widely used by analysts, investors, and corporations. However, a number of recent studies have raised concerns about its validity. For example, a study by Eugene Fama of the University of Chicago and Kenneth French of Dartmouth found no historical relationship between stocks’ returns and their market betas, confirming a position long held by some professors and stock market analysts.26 As an alternative to the traditional CAPM, researchers and practitioners are developing models with more explanatory variables than just beta. These multivariable models represent an attractive generalization of the traditional CAPM model’s insight that market risk—risk that cannot be diversified away—underlies the pricing of assets. In the multivariable models, risk is assumed to be caused by a number of different factors, whereas the CAPM gauges risk only relative to returns on the market portfolio. These multivariable models represent a potentially important step forward in finance theory; they also have some deficiencies 25 The concepts covered in this chapter are obviously important to investors, but they are also important for managers in two key ways. First, as we will see in the next chapter, the risk of a stock affects the required rate of return on equity capital, and that feeds directly into the important subject of capital budgeting. Second, and also related to capital budgeting, the “true” risk of individual projects is impacted by their correlation with the firm’s other projects and with other assets that the firm’s stockholders might hold. We will discuss these topics in later chapters. 26 See Eugene F. Fama and Kenneth R. French, “The Cross-Section of Expected Stock Returns,” Journal of Finance, Vol. 47 (1992), pp. 427–465; and Eugene F. Fama and Kenneth R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, Vol. 33 (1993), pp. 3–56. They found that stock returns are related to firm size and market/book ratios. Small firms and firms with low market/book ratios had higher returns; however, they found no relationship between returns and beta. Kenneth French’s web site http://mba.tuck.dartmouth. edu/pages/faculty/ken. french/index.html is an excellent resource for information regarding factors related to stock returns. Part 3 Financial Assets when applied in practice. As a result, the basic CAPM is still the most widely used method for estimating required rates of return on stocks. SE 258 LF TEST Have there been any studies that question the validity of the CAPM? Explain. 8-6 SOME CONCLUDING THOUGHTS: IMPLICATIONS FOR CORPORATE MANAGERS AND INVESTORS The connection between risk and return is an important concept, and it has numerous implications for both corporate managers and investors. As we will see in later chapters, corporate managers spend a great deal of time assessing the risk and returns on individual projects. Indeed, given their concerns about the risk of individual projects, it might be fair to ask why we spend so much time discussing the riskiness of stocks. Why not begin by looking at the riskiness of such business assets as plant and equipment? The reason is that for management whose primary goal is stock price maximization, the overriding consideration is the riskiness of the firm’s stock, and the relevant risk of any physical asset must be measured in terms of its effect on the stock’s risk as seen by investors. For example, suppose Goodyear, the tire company, is considering a major investment in a new product, recapped tires. Sales of recaps (hence, earnings on the new operation) are highly uncertain; so on a standalone basis, the new venture appears to be quite risky. However, suppose returns in the recap business are negatively correlated with Goodyear’s other operations— when times are good and people have plenty of money, they buy new cars with new tires; but when times are bad, they tend to keep their old cars and buy recaps for them. Therefore, returns would be high on regular operations and low on the recap division during good times, but the opposite would be true during recessions. The result might be a pattern like that shown earlier in Figure 8-4 for Stocks W and M. Thus, what appears to be a risky investment when viewed on a standalone basis might not be very risky when viewed within the context of the company as a whole. This analysis can be extended to the corporation’s stockholders. Because Goodyear’s stock is owned by diversified stockholders, the real issue each time management makes an investment decision is this: How will this investment affect the risk of our stockholders? Again, the stand-alone risk of an individual project may look quite high; however, viewed in the context of the project’s effect on stockholder risk, it may not be very large. We will address this issue again in Chapter 12, where we examine the effects of capital budgeting on companies’ beta coefficients and thus on stockholders’ risks. While these concepts are obviously important for individual investors, they are also important for corporate managers. We summarize some key ideas that all investors should consider: 1. There is a trade-off between risk and return. The average investor likes higher returns but dislikes risk. It follows that higher-risk investments need to offer investors higher expected returns. Put another way—if you are seeking higher returns, you must be willing to assume higher risks. Chapter 8 Risk and Rates of Return 2. 3. 4. SE 5. Diversification is crucial. By diversifying wisely, investors can dramatically reduce risk without reducing their expected returns. Don’t put all of your money in one or two stocks or in one or two industries. A huge mistake that many people make is to invest a high percentage of their funds in their employer’s stock. If the company goes bankrupt, they not only lose their job but also their invested capital. While no stock is completely riskless, you can smooth out the bumps by holding a well-diversified portfolio. Real returns are what matters. All investors should understand the difference between nominal and real returns. When assessing performance, the real return (what you have left over after inflation) is what matters. It follows that as expected inflation increases, investors need to receive higher nominal returns. The risk of an investment often depends on how long you plan to hold the investment. Common stocks, for example, can be extremely risky for shortterm investors. However, over the long haul, the bumps tend to even out; thus, stocks are less risky when held as part of a long-term portfolio. Indeed, in his best-selling book Stocks for the Long Run, Jeremy Siegel of the University of Pennsylvania concludes that “[t]he safest long-term investment for the preservation of purchasing power has clearly been stocks, not bonds.” While the past gives us insights into the risk and returns on various investments, there is no guarantee that the future will repeat the past. Stocks that have performed well in recent years might tumble, while stocks that have struggled may rebound. The same thing may hold true for the stock market as a whole. Even Jeremy Siegel, who has preached that stocks have historically been good long-term investments, also has argued that there is no assurance that returns in the future will be as strong as they have been in the past. More importantly, when purchasing a stock, you always need to ask, “Is this stock fairly valued, or is it currently priced too high?” We discuss this issue more completely in the next chapter. LF TEST Explain the following statement: The stand-alone risk of an individual corporate project may be quite high; but viewed in the context of its effect on stockholders’ risk, the project’s true risk may not be very large. How does the correlation between returns on a project and returns on the firm’s other assets affect the project’s risk? What are some important concepts for individual investors to consider when evaluating the risk and returns of various investments? TYING IT ALL TOGETHER In this chapter, we described the relationship between risk and return. We discussed how to calculate risk and return for individual assets and for portfolios. In particular, we differentiated between stand-alone risk and risk in a portfolio context and we explained the benefits of diversification. We also discussed the CAPM, which describes how risk should be measured and how risk affects rates of return. In the chapters that follow, we will give you the tools needed to estimate the required rates of return on a firm’s common stock and explain how that return and the yield on its bonds are used to develop the firm’s cost of capital. As you will see, the cost of capital is a key element in the capital budgeting process. 259 260 Part 3 Financial Assets SELF-TEST QUESTIONS AND PROBLEMS (Solutions Appear in Appendix A) ST-1 KEY TERMS Define the following terms using graphs or equations to illustrate your answers whenever feasible: a. Risk; stand-alone risk; probability distribution b. Expected rate of return, ^r c. Standard deviation, ; coefficient of variation (CV) d. Risk aversion; risk premium (RP); realized rate of return, r e. Risk premium for Stock i, RPi; market risk premium, RPM f. Expected return on a portfolio, ^r p; market portfolio g. Correlation; correlation coefficient, r h. Market risk; diversifiable risk; relevant risk i. Capital Asset Pricing Model (CAPM) j. k. ST-2 Beta coefficient, b; average stock’s beta, bA Security Market Line (SML) equation REALIZED RATES OF RETURN a. b. c. d. Stocks A and B have the following historical returns: Year Stock A’s Returns, rA Stock B’s Returns, rB 2004 2005 2006 2007 2008 (24.25%) 18.50 38.67 14.33 39.13 5.50% 26.73 48.25 (4.50) 43.86 Calculate the average rate of return for each stock during the period 2004 through 2008. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would the realized rate of return on the portfolio have been in each year from 2004 through 2008? What would the average return on the portfolio have been during that period? Calculate the standard deviation of returns for each stock and for the portfolio. Use Equation 8-2a. Looking at the annual returns on the two stocks, would you guess that the correlation coefficient between the two stocks is closer to +0.8 or to –0.8? If more randomly selected stocks had been included in the portfolio, which of the following is the most accurate statement of what would have happened to p? (1) p would have remained constant. (2) p would have been in the vicinity of 20%. (3) p would have declined to zero if enough stocks had been included. ST-3 BETA AND THE REQUIRED RATE OF RETURN ECRI Corporation is a holding company with four main subsidiaries. The percentage of its capital invested in each of the subsidiaries (and their respective betas) are as follows: Subsidiary Electric utility Cable company Real estate development International/special projects Percentage of Capital Beta 60% 25 10 5 0.70 0.90 1.30 1.50 Chapter 8 Risk and Rates of Return a. What is the holding company’s beta? b. If the risk-free rate is 6% and the market risk premium is 5%, what is the holding company’s required rate of return? ECRI is considering a change in its strategic focus; it will reduce its reliance on the electric utility subsidiary, so the percentage of its capital in this subsidiary will be reduced to 50%. At the same time, it will increase its reliance on the international/special projects division, so the percentage of its capital in that subsidiary will rise to 15%. What will the company’s required rate of return be after these changes? c. QUESTIONS 8-1 Suppose you owned a portfolio consisting of $250,000 of long-term U.S. government bonds. a. Would your portfolio be riskless? Explain. b. c. 8-2 8-3 The probability distribution of a less risky expected return is more peaked than that of a riskier return. What shape would the probability distribution be for (a) completely certain returns and (b) completely uncertain returns? A life insurance policy is a financial asset, with the premiums paid representing the investment’s cost. a. How would you calculate the expected return on a 1-year life insurance policy? b. c. 8-4 8-5 8-6 Now suppose the portfolio consists of $250,000 of 30-day Treasury bills. Every 30 days your bills mature, and you will reinvest the principal ($250,000) in a new batch of bills. You plan to live on the investment income from your portfolio, and you want to maintain a constant standard of living. Is the T-bill portfolio truly riskless? Explain. What is the least risky security you can think of? Explain. Suppose the owner of a life insurance policy has no other financial assets—the person’s only other asset is “human capital,” or earnings capacity. What is the correlation coefficient between the return on the insurance policy and the return on the human capital? Life insurance companies must pay administrative costs and sales representatives’ commissions; hence, the expected rate of return on insurance premiums is generally low or even negative. Use portfolio concepts to explain why people buy life insurance in spite of low expected returns. Is it possible to construct a portfolio of real-world stocks that has an expected return equal to the risk-free rate? Stock A has an expected return of 7%, a standard deviation of expected returns of 35%, a correlation coefficient with the market of –0.3, and a beta coefficient of –0.5. Stock B has an expected return of 12%, a standard deviation of returns of 10%, a 0.7 correlation with the market, and a beta coefficient of 1.0. Which security is riskier? Why? A stock had a 12% return last year, a year when the overall stock market declined. Does this mean that the stock has a negative beta and thus very little risk if held in a portfolio? Explain. 8-7 If investors’ aversion to risk increased, would the risk premium on a high-beta stock increase by more or less than that on a low-beta stock? Explain. 8-8 8-9 If a company’s beta were to double, would its required return also double? In Chapter 7, we saw that if the market interest rate, rd, for a given bond increased, the price of the bond would decline. Applying this same logic to stocks, explain (a) how a decrease in risk aversion would affect stocks’ prices and earned rates of return, (b) how this would affect risk premiums as measured by the historical difference between returns on stocks and returns on bonds, and (c) what the implications of this would be for the use of historical risk premiums when applying the SML equation. 261 262 Part 3 Financial Assets PROBLEMS Easy Problems 1–5 8-1 EXPECTED RETURN A stock’s returns have the following distribution: Demand for the Company’s Products Probability of This Demand Occurring Weak Below average Average Above average Strong Rate of Return If This Demand Occurs 0.1 0.2 0.4 0.2 0.1 1.0 (50%) (5) 16 25 60 Calculate the stock’s expected return, standard deviation, and coefficient of variation. 8-2 8-3 8-4 8-5 Intermediate Problems 6–12 8-6 8-7 PORTFOLIO BETA An individual has $35,000 invested in a stock with a beta of 0.8 and another $40,000 invested in a stock with a beta of 1.4. If these are the only two investments in her portfolio, what is her portfolio’s beta? REQUIRED RATE OF RETURN Assume that the risk-free rate is 6% and the expected return on the market is 13%. What is the required rate of return on a stock with a beta of 0.7? EXPECTED AND REQUIRED RATES OF RETURN Assume that the risk-free rate is 5% and the market risk premium is 6%. What is the expected return for the overall stock market? What is the required rate of return on a stock with a beta of 1.2? BETA AND REQUIRED RATE OF RETURN A stock has a required return of 11%, the risk-free rate is 7%, and the market risk premium is 4%. a. What is the stock’s beta? b. If the market risk premium increased to 6%, what would happen to the stock’s required rate of return? Assume that the risk-free rate and the beta remain unchanged. EXPECTED RETURNS Stocks X and Y have the following probability distributions of expected future returns: Probability X Y 0.1 0.2 0.4 0.2 0.1 (10%) 2 12 20 38 (35%) 0 20 25 45 a. Calculate the expected rate of return, ^r Y, for Stock Y (^r X ¼ 12%). b. Calculate the standard deviation of expected returns, X, for Stock X (Y ¼ 20.35%). Now calculate the coefficient of variation for Stock Y. Is it possible that most investors will regard Stock Y as being less risky than Stock X? Explain. PORTFOLIO REQUIRED RETURN Suppose you are the money manager of a $4 million investment fund. The fund consists of four stocks with the following investments and betas: Stock Investment Beta A B C D $ 400,000 600,000 1,000,000 2,000,000 1.50 (0.50) 1.25 0.75 If the market’s required rate of return is 14% and the risk-free rate is 6%, what is the fund’s required rate of return? 8-8 BETA COEFFICIENT Given the following information, determine the beta coefficient for Stock J that is consistent with equilibrium: ^r J ¼ 12.5%; rRF ¼ 4.5%; rM ¼ 10.5%. Chapter 8 Risk and Rates of Return 8-9 8-10 8-11 8-12 REQUIRED RATE OF RETURN Stock R has a beta of 1.5, Stock S has a beta of 0.75, the expected rate of return on an average stock is 13%, and the risk-free rate of return is 7%. By how much does the required return on the riskier stock exceed the required return on the less risky stock? CAPM AND REQUIRED RETURN Bradford Manufacturing Company has a beta of 1.45, while Farley Industries has a beta of 0.85. The required return on an index fund that holds the entire stock market is 12.0%. The risk-free rate of interest is 5%. By how much does Bradford’s required return exceed Farley’s required return? CAPM AND REQUIRED RETURN Calculate the required rate of return for Manning Enterprises assuming that investors expect a 3.5% rate of inflation in the future. The real risk-free rate is 2.5%, and the market risk premium is 6.5%. Manning has a beta of 1.7, and its realized rate of return has averaged 13.5% over the past 5 years. REQUIRED RATE OF RETURN a. What is ri, the required rate of return on Stock i? b. Now suppose that rRF (1) increases to 10% or (2) decreases to 8%. The slope of the SML remains constant. How would this affect rM and ri? Now assume that rRF remains at 9% but rM (1) increases to 16% or (2) falls to 13%. The slope of the SML does not remain constant. How would these changes affect ri? c. Challenging Problems 13–21 8-13 Suppose rRF ¼ 9%, rM ¼ 14%, and bi ¼ 1.3. CAPM, PORTFOLIO RISK, AND RETURN Consider the following information for three stocks, Stocks X, Y, and Z. The returns on the three stocks are positively correlated, but they are not perfectly correlated. (That is, each of the correlation coefficients is between 0 and 1.) Stock Expected Return X Y Z 9.00% 10.75 12.50 Standard Deviation 15% 15 15 Beta 0.8 1.2 1.6 Fund Q has one-third of its funds invested in each of the three stocks. The risk-free rate is 5.5%, and the market is in equilibrium. (That is, required returns equal expected returns.) a. What is the market risk premium (rM – rRF)? b. What is the beta of Fund Q? c. What is the expected return of Fund Q? d. Would you expect the standard deviation of Fund Q to be less than 15%, equal to 15%, or greater than 15%? Explain. 8-14 PORTFOLIO BETA Suppose you held a diversified portfolio consisting of a $7,500 investment in each of 20 different common stocks. The portfolio’s beta is 1.12. Now suppose you decided to sell one of the stocks in your portfolio with a beta of 1.0 for $7,500 and use the proceeds to buy another stock with a beta of 1.75. What would your portfolio’s new beta be? 8-15 CAPM AND REQUIRED RETURN HR Industries (HRI) has a beta of 1.8, while LR Industries’ (LRI) beta is 0.6. The risk-free rate is 6%, and the required rate of return on an average stock is 13%. The expected rate of inflation built into rRF falls by 1.5 percentage points, the real risk-free rate remains constant, the required return on the market falls to 10.5%, and all betas remain constant. After all of these changes, what will be the difference in the required returns for HRI and LRI? 8-16 CAPM AND PORTFOLIO RETURN You have been managing a $5 million portfolio that has a beta of 1.25 and a required rate of return of 12%. The current risk-free rate is 5.25%. Assume that you receive another $500,000. If you invest the money in a stock with a beta of 0.75, what will be the required return on your $5.5 million portfolio? PORTFOLIO BETA A mutual fund manager has a $20 million portfolio with a beta of 1.5. The risk-free rate is 4.5%, and the market risk premium is 5.5%. The manager expects to receive an additional $5 million, which she plans to invest in a number of stocks. After investing the additional funds, she wants the fund’s required return to be 13%. What should be the average beta of the new stocks added to the portfolio? EXPECTED RETURNS Suppose you won the lottery and had two options: (1) receiving $0.5 million or (2) taking a gamble in which at the flip of a coin you receive $1 million if a head comes up but receive zero if a tail comes up. 8-17 8-18 a. What is the expected value of the gamble? b. Would you take the sure $0.5 million or the gamble? 263 264 Part 3 Financial Assets c. d. If you chose the sure $0.5 million, would that indicate that you are a risk averter or a risk seeker? Suppose the payoff was actually $0.5 million—that was the only choice. You now face the choice of investing it in a U.S. Treasury bond that will return $537,500 at the end of a year or a common stock that has a 50-50 chance of being worthless or worth $1,150,000 at the end of the year. (1) The expected profit on the T-bond investment is $37,500. What is the expected dollar profit on the stock investment? (2) The expected rate of return on the T-bond investment is 7.5%. What is the expected rate of return on the stock investment? (3) Would you invest in the bond or the stock? Why? 8-19 (4) Exactly how large would the expected profit (or the expected rate of return) have to be on the stock investment to make you invest in the stock, given the 7.5% return on the bond? (5) How might your decision be affected if, rather than buying one stock for $0.5 million, you could construct a portfolio consisting of 100 stocks with $5,000 invested in each? Each of these stocks has the same return characteristics as the one stock—that is, a 50-50 chance of being worth zero or $11,500 at year-end. Would the correlation between returns on these stocks matter? Explain. EVALUATING RISK AND RETURN Stock X has a 10% expected return, a beta coefficient of 0.9, and a 35% standard deviation of expected returns. Stock Y has a 12.5% expected return, a beta coefficient of 1.2, and a 25% standard deviation. The risk-free rate is 6%, and the market risk premium is 5%. a. Calculate each stock’s coefficient of variation. b. Which stock is riskier for a diversified investor? c. Calculate each stock’s required rate of return. d. On the basis of the two stocks’ expected and required returns, which stock would be more attractive to a diversified investor? Calculate the required return of a portfolio that has $7,500 invested in Stock X and $2,500 invested in Stock Y. If the market risk premium increased to 6%, which of the two stocks would have the larger increase in its required return? e. f. 8-20 REALIZED RATES OF RETURN Year Stock A’s Returns, rA Stock B’s Returns, rB 2004 2005 2006 2007 2008 (18.00%) 33.00 15.00 (0.50) 27.00 (14.50%) 21.80 30.50 (7.60) 26.30 a. Calculate the average rate of return for each stock during the period 2004 through 2008. b. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would the realized rate of return on the portfolio have been each year? What would the average return on the portfolio have been during this period? Calculate the standard deviation of returns for each stock and for the portfolio. c. 8-21 Stocks A and B have the following historical returns: d. Calculate the coefficient of variation for each stock and for the portfolio. e. Assuming you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why? SECURITY MARKET LINE You plan to invest in the Kish Hedge Fund, which has total capital of $500 million invested in five stocks: Stock Investment Stock’s Beta Coefficient A B C D E $160 million 120 million 80 million 80 million 60 million 0.5 1.2 1.8 1.0 1.6 Chapter 8 Risk and Rates of Return Kish’s beta coefficient can be found as a weighted average of its stocks’ betas. The risk-free rate is 6%, and you believe the following probability distribution for future market returns is realistic: a. b. c. Probability Market Return 0.1 0.2 0.4 0.2 0.1 28% 0 12 30 50 What is the equation for the Security Market Line (SML)? (Hint: First, determine the expected market return.) Calculate Kish’s required rate of return. Suppose Rick Kish, the president, receives a proposal from a company seeking new capital. The amount needed to take a position in the stock is $50 million, it has an expected return of 15%, and its estimated beta is 1.5. Should Kish invest in the new company? At what expected rate of return should Kish be indifferent to purchasing the stock? COMPREHENSIVE/SPREADSHEET PROBLEM 8-22 EVALUATING RISK AND RETURN Bartman Industries’ and Reynolds Inc.’s stock prices and dividends, along with the Winslow 5000 Index, are shown here for the period 2003–2008. The Winslow 5000 data are adjusted to include dividends. BARTMAN INDUSTRIES REYNOLDS INC. WINSLOW 5000 Year Stock Price Dividend Stock Price Dividend Includes Dividends 2008 2007 2006 2005 2004 2003 $17.250 14.750 16.500 10.750 11.375 7.625 $1.15 1.06 1.00 0.95 0.90 0.85 $48.750 52.300 48.750 57.250 60.000 55.750 $3.00 2.90 2.75 2.50 2.25 2.00 $11,663.98 8,785.70 8,679.98 6,434.03 5,602.28 4,705.97 a. b. c. d. e. f. g. Use the data to calculate annual rates of return for Bartman, Reynolds, and the Winslow 5000 Index. Then calculate each entity’s average return over the 5-year period. (Hint: Remember, returns are calculated by subtracting the beginning price from the ending price to get the capital gain or loss, adding the dividend to the capital gain or loss, and dividing the result by the beginning price. Assume that dividends are already included in the index. Also, you cannot calculate the rate of return for 2003 because you do not have 2002 data.) Calculate the standard deviations of the returns for Bartman, Reynolds, and the Winslow 5000. (Hint: Use the sample standard deviation formula, Equation 8-2a in this chapter, which corresponds to the STDEV function in Excel.) Calculate the coefficients of variation for Bartman, Reynolds, and the Winslow 5000. Construct a scatter diagram that shows Bartman’s and Reynolds’ returns on the vertical axis and the Winslow 5000 Index’s returns on the horizontal axis. Estimate Bartman’s and Reynolds’ betas by running regressions of their returns against the index’s returns. (Hint: Refer to Web Appendix 8A.) Are these betas consistent with your graph? Assume that the risk-free rate on long-term Treasury bonds is 6.04%. Assume also that the average annual return on the Winslow 5000 is not a good estimate of the market’s required return—it is too high. So use 11% as the expected return on the market. Use the SML equation to calculate the two companies’ required returns. If you formed a portfolio that consisted of 50% Bartman and 50% Reynolds, what would the portfolio’s beta and required return be? 265 266 Part 3 Financial Assets h. Suppose an investor wants to include Bartman Industries’ stock in his portfolio. Stocks A, B, and C are currently in the portfolio; and their betas are 0.769, 0.985, and 1.423, respectively. Calculate the new portfolio’s required return if it consists of 25% of Bartman, 15% of Stock A, 40% of Stock B, and 20% of Stock C. INTEGRATED CASE MERRILL FINCH INC. 8-23 RISK AND RETURN Assume that you recently graduated with a major in finance. You just landed a job as a financial planner with Merrill Finch Inc., a large financial services corporation. Your first assignment is to invest $100,000 for a client. Because the funds are to be invested in a business at the end of 1 year, you have been instructed to plan for a 1-year holding period. Further, your boss has restricted you to the investment alternatives in the following table, shown with their probabilities and associated outcomes. (For now, disregard the items at the bottom of the data; you will fill in the blanks later.) RETURNS ON ALTERNATIVE INVESTMENTS ESTIMATED RATE OF RETURN State of the Economy Recession Below average Average Above average Boom ^ r Probability T-Bills High Tech Collections U.S. Rubber Market Portfolio 2-Stock Portfolio 0.1 0.2 0.4 0.2 0.1 5.5% 5.5 5.5 5.5 5.5 (27.0%) (7.0) 15.0 30.0 45.0 27.0% 13.0 0.0 (11.0) (21.0) 1.0% 6.0%a (14.0) 3.0 41.0 26.0 9.8% (17.0%) (3.0) 10.0 25.0 38.0 10.5% 0.0% 13.2 13.2 0.87 18.8 1.9 0.88 15.2 1.4 0.0  CV b 7.5 12.0 3.4 0.5 a Note that the estimated returns of U.S. Rubber do not always move in the same direction as the overall economy. For example, when the economy is below average, consumers purchase fewer tires than they would if the economy were stronger. However, if the economy is in a flat-out recession, a large number of consumers who were planning to purchase a new car may choose to wait and instead purchase new tires for the car they currently own. Under these circumstances, we would expect U.S. Rubber’s stock price to be higher if there was a recession than if the economy was just below average. Merrill Finch’s economic forecasting staff has developed probability estimates for the state of the economy; and its security analysts have developed a sophisticated computer program, which was used to estimate the rate of return on each alternative under each state of the economy. High Tech Inc. is an electronics firm, Collections Inc. collects past-due debts, and U.S. Rubber manufactures tires and various other rubber and plastics products. Merrill Finch also maintains a “market portfolio” that owns a market-weighted fraction of all publicly traded stocks; you can invest in that portfolio and thus obtain average stock market results. Given the situation described, answer the following questions: a. (1) Why is the T-bill’s return independent of the state of the economy? Do T-bills promise a completely risk-free return? Explain. (2) b. c. Why are High Tech’s returns expected to move with the economy, whereas Collections’ are expected to move counter to the economy? Calculate the expected rate of return on each alternative and fill in the blanks on the row for ^ r in the previous table. You should recognize that basing a decision solely on expected returns is appropriate only for risk-neutral individuals. Because your client, like most people, is risk-averse, the riskiness of each alternative is an important aspect of the decision. One possible measure of risk is the standard deviation of returns. (1) Calculate this value for each alternative and fill in the blank on the row for  in the table. (2) What type of risk is measured by the standard deviation? Chapter 8 Risk and Rates of Return (3) d. e. f. g. h. Draw a graph that shows roughly the shape of the probability distributions for High Tech, U.S. Rubber, and T-bills. Suppose you suddenly remembered that the coefficient of variation (CV) is generally regarded as being a better measure of stand-alone risk than the standard deviation when the alternatives being considered have widely differing expected returns. Calculate the missing CVs and fill in the blanks on the row for CV in the table. Does the CV produce the same risk rankings as the standard deviation? Explain. Suppose you created a two-stock portfolio by investing $50,000 in High Tech and $50,000 in Collections. (1) Calculate the expected return ( ^ rp), the standard deviation (p), and the coefficient of variation (CVp) for this portfolio and fill in the appropriate blanks in the table. (2) How does the riskiness of this two-stock portfolio compare with the riskiness of the individual stocks if they were held in isolation? Suppose an investor starts with a portfolio consisting of one randomly selected stock. What would happen: (1) To the riskiness and to the expected return of the portfolio as more randomly selected stocks were added to the portfolio? (2) What is the implication for investors? Draw a graph of the two portfolios to illustrate your answer. (1) Should the effects of a portfolio impact the way investors think about the riskiness of individual stocks? (2) If you decided to hold a 1-stock portfolio (and consequently were exposed to more risk than diversified investors), could you expect to be compensated for all of your risk; that is, could you earn a risk premium on the part of your risk that you could have eliminated by diversifying? The expected rates of return and the beta coefficients of the alternatives supplied by Merrill Finch’s computer program are as follows: Security High Tech Market U.S. Rubber T-bills Collections i. j. 267 Return ( ^ r) Risk (Beta) 12.4% 10.5 9.8 5.5 1.0 1.32 1.00 0.88 0.00 (0.87) (1) What is a beta coefficient, and how are betas used in risk analysis? (2) Do the expected returns appear to be related to each alternative’s market risk? (3) Is it possible to choose among the alternatives on the basis of the information developed thus far? Use the data given at the start of the problem to construct a graph that shows how the T-bill’s, High Tech’s, and the market’s beta coefficients are calculated. Then discuss what betas measure and how they are used in risk analysis. The yield curve is currently flat; that is, long-term Treasury bonds also have a 5.5% yield. Consequently, Merrill Finch assumes that the risk-free rate is 5.5%. (1) Write out the Security Market Line (SML) equation, use it to calculate the required rate of return on each alternative, and graph the relationship between the expected and required rates of return. (2) How do the expected rates of return compare with the required rates of return? (3) Does the fact that Collections has an expected return that is less than the T-bill rate make any sense? Explain. (4) What would be the market risk and the required return of a 50-50 portfolio of High Tech and Collections? of High Tech and U.S. Rubber? (1) Suppose investors raised their inflation expectations by 3 percentage points over current estimates as reflected in the 5.5% risk-free rate. What effect would higher inflation have on the SML and on the returns required on highand low-risk securities? (2) Suppose instead that investors’ risk aversion increased enough to cause the market risk premium to increase by 3 percentage points. (Inflation remains constant.) What effect would this have on the SML and on returns of highand low-risk securities? 268 Part 3 Financial Assets Access the Thomson ONE problems through the CengageNOW™ web site. Use the Thomson ONE—Business School Edition online database to work this chapter’s questions. Using Past Information to Estimate Required Returns Chapter 8 discussed the basic trade-off between risk and return. In the Capital Asset Pricing Model (CAPM) discussion, beta was identified as the correct measure of risk for diversified shareholders. Recall that beta measures the extent to which the returns of a given stock move with the stock market. When using the CAPM to estimate required returns, we would like to know how the stock will move with the market in the future; but since we don’t have a crystal ball, we generally use historical data to estimate this relationship with beta. As mentioned in the Web Appendix for this chapter, beta can be estimated by regressing the individual stock’s returns against the returns of the overall market. As an alternative to running our own regressions, we can rely on reported betas from a variety of sources. These published sources make it easy for us to readily obtain beta estimates for most large publicly traded corporations. However, a word of caution is in order. Beta estimates can often be quite sensitive to the time period in which the data are estimated, the market index used, and the frequency of the data used. Therefore, it is not uncommon to find a wide range of beta estimates among the various published sources. Indeed, Thomson One reports multiple beta estimates. These multiple estimates reflect the fact that Thomson One puts together data from a variety of different sources. Discussion Questions 1. 2. 3. 4. 5. 6. 7. Begin by looking at the historical performance of the overall stock market. If you want to see, for example, the performance of the S&P 500, select “INDICES” and enter S&PCOMP. Click on “PERFORMANCE.” You will see a quick summary of the market’s performance in recent months and years. How has the market performed over the past year? the past 3 years? the past 5 years? the past 10 years? Now let’s take a closer look at the stocks of four companies: Colgate Palmolive (Ticker ¼ CL), Campbell Soup (CPB), Motorola (MOT), and Tiffany & Co (TIF). Before looking at the data, which of these companies would you expect to have a relatively high beta (greater than 1.0) and which of these companies would you expect to have a relatively low beta (less than 1.0)? Select one of the four stocks listed in Question 2 by selecting “COMPANY ANALYSIS,” entering the company’s ticker symbol in the blank companies box, and clicking “GO.” On the company overview page, you should see a chart that summarizes how the stock has done relative to the S&P 500 over the past 6 months. Has the stock outperformed or underperformed the overall market during this time period? If you scroll down the company overview page, you should see an estimate of the company’s beta. What is the company’s beta? What was the source of the estimated beta? Click on “PRICES” on the left-hand side of the screen. What is the company’s current dividend yield? What has been its total return to investors over the past 6 months? over the past year? over the past 3 years? (Remember that total return includes the dividend yield plus any capital gains or losses.) Assume that the risk-free rate is 5% and the market risk premium is 6%. What is the required return on the company’s stock? Repeat the same exercise for each of the 3 remaining companies. Do the reported betas confirm your earlier intuition? In general, do you find that the higher-beta stocks tend to do better in up markets and worse in down markets? Explain. ª SEBASTIAN KAULITZKI/SHUTTERSTOCK.COM CHAPTER 9 Stocks and Their Valuation Searching for the Right Stock A recent study by the securities industry found that roughly half of all U.S. households have invested in common stocks. As noted in Chapter 8, over the long run, returns in the U.S. stock market have been quite strong, averaging 12% per year. However, the market’s performance recently has been less than stellar. Trying to put things in perspective, Fortune magazine’s senior editor Allan Sloan offered the following comments about the market’s performance: When the greatest bull market in U.S. history started in the summer of 1982, only a relative handful of people owned stocks, which were cheap because they were considered highly risky. But by the time the Standard & Poor’s 500 peaked in March 2000 amid a fully inflated stock bubble, the masses were in the market. Stocks were magical, a supposedly can’tmiss way to pay for your kids’ college, save for retirement, enrich employees by giving them options, and regrow hair. (Just kidding about the hair. Alas.) Stocks might go down in any given year, the mantra went, but in the long term they’d produce double-digit returns. However, one of the lessons of the past eight years is that the long run can be . . . really long. As I write this in late February 2008, the U.S. market—which I’m defining as the Standard & Poor’s 500—is well below the high that it set on March 24, 2000. Even after you include dividends, which have run a bit below 2% a year, you’ve barely broken even, according to calculations for Fortune by Aronson & Johnson & Ortiz, a Philadelphia money manager. One month later in March 2008 the stock market fell further in the aftermath of the startling collapse of Wall Street giant Bear Stearns. While most experts believe the stock market will ultimately rebound, most doubt that investors will average double-digit returns from common stock returns in the years ahead. As we discussed in Chapter 8, the returns of individual stocks are more volatile than the 269 270 Part 3 Financial Assets returns of the overall market. For example, in 2007, the overall market (as measured by the S&P 500 Index) was up slightly (+5.49%). That same year some individual stocks realized huge gains while others declined sharply. On the plus side, Research in Motion was up 166%, Amazon.com rose 135%, and Apple Computer climbed 133%. On the down side, E*Trade Financial plummeted 84%; Circuit City, 78%; and Starbucks, 42%. This wide range in individual stocks’ returns shows, first, that diversification is important and, second, that when it comes to picking stocks, it is not enough to simply pick a good company—the stock must also be “fairly” priced. To determine whether a stock is fairly priced, you first need to estimate the stock’s true value, or “intrinsic value,” a concept first discussed in Chapter 1. With this objective in mind, in this chapter, we describe some models that analysts have used to estimate intrinsic values. As you will see, while it is difficult to predict stock prices, we are not completely in the dark. Indeed, after studying this chapter, you should have a reasonably good understanding of the factors that influence stock prices; and with that knowledge—plus a little luck—you should be able to successfully navigate the market’s often-treacherous ups and downs. Sources: Allan Sloan, “The Incredible Shrinking Bull,” Fortune, March 17, 2008, p. 24 and Alexandra Twin, “Best and Worst Stocks of 2007,” CNNMoney.com, December 31, 2007. PUTTING THINGS IN PERSPECTIVE Key trends in the securities industry are listed and explained at www.sifma.org/ research/statistics/ key_industry_trends.html. In Chapter 7, we examined bonds. We now turn to stocks, both common and preferred. Since the cash flows provided by bonds are set by contract, it is generally easy to predict their cash flows. Preferred stock returns are also set by contract, which makes them similar to bonds; and they are valued in much the same way. However, common stock returns are not contractual—they depend on the firm’s earnings, which in turn depend on many random factors, making their valuation more difficult. Two fairly straightforward models are used to estimate stocks’ intrinsic (or “true”) values: (1) the discounted dividend model and (2) the corporate valuation model. A stock should, of course, be bought if its price is less than its estimated intrinsic value and sold if its price exceeds its intrinsic value. By the time you finish this chapter, you should be able to: Discuss the legal rights of stockholders. Explain the distinction between a stock’s price and its intrinsic value. Identify the two models that can be used to estimate a stock’s intrinsic value: the discounted dividend model and the corporate model. List the key characteristics of preferred stock and explain how to estimate the value of preferred stock. Stock valuation is interesting in its own right; but you also need to understand valuation when you estimate the cost of capital for use in capital budgeting, which is probably a firm’s most important task. l l l l 9-1 LEGAL RIGHTS AND PRIVILEGES OF COMMON STOCKHOLDERS A corporation’s common stockholders are the owners of the corporation; and as such, they have certain rights and privileges, as discussed in this section. Chapter 9 Stocks and Their Valuation 271 9-1a Control of the Firm A firm’s common stockholders have the right to elect its directors, who, in turn, elect the officers who manage the business. In a small firm, usually the major stockholder is also the president and chair of the board of directors. In large, publicly owned firms, the managers typically have some stock, but their personal holdings are generally insufficient to give them voting control. Thus, the managements of most publicly owned firms can be removed by the stockholders if the management team is not effective. State and federal laws stipulate how stockholder control is to be exercised. First, corporations must hold elections of directors periodically, usually once a year, with the vote taken at the annual meeting. Each share of stock has one vote; thus, the owner of 1,000 shares has 1,000 votes for each director.1 Stockholders can appear at the annual meeting and vote in person, but typically they transfer their right to vote to another person by means of a proxy. Management always solicits stockholders’ proxies and usually receives them. However, if earnings are poor and stockholders are dissatisfied, an outside group may solicit the proxies in an effort to overthrow management and take control of the business. This is known as a proxy fight. The question of control has become a central issue in finance in recent years. The frequency of proxy fights has increased, as have attempts by one corporation to take over another by purchasing a majority of the outstanding stock. These actions are called takeovers. Some well-known examples of takeover battles in past years include KKR’s acquisition of RJR Nabisco, Chevron’s acquisition of Gulf Oil, and the QVC/Viacom fight to take over Paramount. More recently, in February 2008, Microsoft made an unsolicited offer for Yahoo; but thus far Yahoo’s management has resisted. Managers without more than 50% of their firms’ stock are very much concerned about proxy fights and takeovers, and many of them have attempted to obtain stockholder approval for changes in their corporate charters that would make takeovers more difficult. For example, a number of companies have gotten their stockholders to agree (1) to elect only one-third of the directors each year (rather than electing all directors each year), (2) to require 75% of the stockholders (rather than 50%) to approve a merger, and (3) to vote in a “poison pill” provision that would allow the stockholders of a firm that is taken over by another firm to buy shares in the second firm at a reduced price. The poison pill makes the acquisition unattractive and thus helps ward off hostile takeover attempts. Managers seeking such changes generally cite a fear that the firm will be picked up at a bargain price, but it often appears that the managers’ concern about their own positions is the primary consideration. Managers’ moves to make takeovers more difficult have been countered by stockholders, especially large institutional stockholders, who do not like barriers erected to protect incompetent managers. To illustrate, the California Public Employees Retirement System (CalPERS), which is one of the largest institutional investors, has led proxy fights with several corporations whose financial performances were poor in CalPERS’ judgment. CalPERS wants companies to increase outside (non-management) directors’ ability to force managers to be more responsive to stockholder complaints. 1 In the situation described, a 1,000-share stockholder could cast 1,000 votes for each of three directors if there were three contested seats on the board. An alternative procedure that may be prescribed in the corporate charter calls for cumulative voting. There the 1,000-share stockholder would get 3,000 votes if there were three vacancies, and he or she could cast all of them for one director. Cumulative voting helps small groups obtain representation on the board. Proxy A document giving one person the authority to act for another, typically the power to vote shares of common stock. Proxy Fight An attempt by a person or group to gain control of a firm by getting its stockholders to grant that person or group the authority to vote its shares to replace the current management. Takeover An action whereby a person or group succeeds in ousting a firm’s management and taking control of the company. 272 Part 3 Financial Assets Managers’ pay is another contentious issue. It has been asserted, with considerable support, that CEOs tend to pick other CEOs to serve on their boards, with “you-scratch-my-back-and-I’ll-scratch-yours” behavior resulting in excessive compensation packages to top managers. Boards have tried to conceal the facts by making it extremely difficult for stockholders to know what the top managers are being paid. Investors are galled to see CEOs such as Stan O’Neil of Merrill Lynch, who was fired because of his firm’s multibillion-dollar loss, walk away with stock and cash worth hundreds of millions. CalPERS and other institutional investors have weighed in on this issue, and most firms today have been forced to make their compensation packages more transparent. For many years, SEC rules prohibited large investors such as CalPERS from getting together to force corporate managers to institute policy changes. However, the SEC began changing its rules in 1993, and now large investors can work together to force management changes. These rulings have helped keep managers focused on stockholder concerns, which means the maximization of stock prices. 9-1b The Preemptive Right Preemptive Right A provision in the corporate charter or bylaws that gives common stockholders the right to purchase on a pro rata basis new issues of common stock (or convertible securities). SE Common stockholders often have the right, called the preemptive right, to purchase on a pro rata basis any additional shares sold by the firm. In some states, the preemptive right is automatically included in every corporate charter; in other states, it must be specifically inserted into the charter. The purpose of the preemptive right is twofold. First, it prevents the management of a corporation from issuing a large number of additional shares and purchasing those shares itself. Management could use this tactic to seize control of the corporation and frustrate the will of the current stockholders. The second, and far more important, reason for the preemptive right is to protect stockholders from a dilution of value. For example, suppose 1,000 shares of common stock, each with a price of $100, were outstanding, making the total market value of the firm $100,000. If an additional 1,000 shares were sold at $50 a share, or for $50,000, this would raise the firm’s total market value to $150,000. When the new total market value is divided by the 2,000 total shares now outstanding, a value of $75 a share is obtained. The old stockholders would thus lose $25 per share, and the new stockholders would have an instant profit of $25 per share. Thus, selling common stock at a price below the market value would dilute a firm’s price and transfer wealth from its present stockholders to those who were allowed to purchase the new shares. The preemptive right prevents this. LF TEST Identify some actions that companies have taken to make takeovers more difficult. What is the preemptive right, and what are the two primary reasons for its existence? 9-2 TYPES OF COMMON STOCK Classified Stock Common stock that is given a special designation such as Class A or Class B to meet special needs of the company. Although most firms have only one type of common stock, in some instances, classified stock is used to meet special needs. Generally, when special classifications are used, one type is designated Class A, another Class B, and so forth. Small, new companies seeking funds from outside sources frequently use different types of common stock. For example, when Google went public, it sold Class A stock to the public while its Class B stock was retained by the company’s insiders. Chapter 9 Stocks and Their Valuation SE The key difference is that the Class B stock has 10 votes per share while the Class A stock has 1 vote per share. Google’s Class B shares are predominantly held by the company’s two founders and its current CEO. The use of classified stock thus enables the company’s founders to maintain control over the company without having to own a majority of the common stock. For this reason, Class B stock of this type is sometimes called founders’ shares. Since dual-class share structures of this type give special voting privileges to key insiders, these structures are sometimes criticized because they may enable insiders to make decisions that are counter to the interests of the majority of stockholders. Note that “Class A,” “Class B,” and so forth, have no standard meanings. Most firms have no classified shares; but a firm that does could designate its Class B shares as founders’ shares and its Class A shares as those sold to the public, while another could reverse those designations. Still other firms could use stock classifications for entirely different purposes. For example, when General Motors acquired Hughes Aircraft for $5 billion, it paid in part with a new Class H common, GMH, which had limited voting rights and whose dividends were tied to Hughes’s performance as a GM subsidiary. The reasons for the new stock were that (1) GM wanted to limit voting privileges on the new classified stock because of management’s concern about a possible takeover and (2) Hughes’s employees wanted to be rewarded more directly on Hughes’s own performance than would have been possible through regular GM stock. These Class H shares disappeared in 2003 when GM decided to sell off the Hughes unit. LF TEST What are some reasons a company might use classified stock? 9-3 STOCK PRICE VS. INTRINSIC VALUE We saw in Chapter 1 that a manager should seek to maximize the value of his or her firm’s stock. In that chapter, we also emphasized the difference between stock price and intrinsic value. The stock price is simply the current market price, and it is easily observed for publicly traded companies. By contrast, intrinsic value, which represents the “true” value of the company’s stock, cannot be directly observed and must instead be estimated. Figure 9-1 illustrates once again the connection between stock price and intrinsic value. As the figure suggests, market equilibrium occurs when the stock’s price equals its intrinsic value. If the stock market is reasonably efficient, gaps between the stock price and intrinsic value should not be very large and they should not persist for very long. However, in some cases, an individual stock price may be much higher or lower than its intrinsic value. During several years leading up to the credit crunch of 2007–2008, most of the large investment banks were reporting record profits and selling at record prices. However, much of those earnings were illusory in that they did not reflect the huge risks that existed in the mortgagebacked securities they were buying. So with hindsight, we now know that the market prices of most financial firms’ stocks exceeded their intrinsic values just prior to 2007. Then when the market realized what was happening, those stock prices crashed. Citigroup, Merrill Lynch, and others lost over 60% of their value in a few short months; and Bear Stearns, the fifth largest investment bank, saw its 273 Founders’ Shares Stock owned by the firm’s founders that has sole voting rights but restricted dividends for a specified number of years. 274 Part 3 Financial Assets Determinants of Intrinsic Values and Stock Prices FIGURE 9-1 Managerial Actions, the Economic Environment, Taxes, and the Political Climate “True” Investor Returns “True” Risk “Perceived” Investor Returns Stock’s Intrinsic Value “Perceived” Risk Stock’s Market Price Market Equilibrium: Intrinsic Value = Stock Price stock price drop from $171 in 2007 to $2 in mid-March 2008. It clearly pays to question market prices at times! 9-3a Why Do Investors and Companies Care About Intrinsic Value? The remainder of this chapter focuses primarily on different approaches for estimating a stock’s intrinsic value. Before these approaches are described, it is worth asking why it is important for investors and companies to understand how to calculate intrinsic value. When investing in common stocks, one’s goal is to purchase stocks that are undervalued (i.e., the price is below the stock’s intrinsic value) and avoid stocks that are overvalued. Consequently, Wall Street analysts, institutional investors who control mutual funds and pension funds, and many individual investors are interested in finding reliable models that help predict intrinsic value. Investors obviously care about intrinsic value, but managers also need to understand how intrinsic value is estimated. First, managers need to know how alternative actions are likely to affect stock prices; and the models of intrinsic value that we cover help demonstrate the connection between managerial decisions and firm value. Second, managers should consider whether their stock is significantly undervalued or overvalued before making certain decisions. For example, firms should consider carefully the decision to issue new shares if they believe their stock is undervalued; and an estimate of their stock’s intrinsic value is the key to such decisions. Two basic models are used to estimate intrinsic values: the discounted dividend model and the corporate valuation model. The dividend model focuses on dividends, while the corporate model goes beyond dividends and focuses on sales, costs, and free cash flows. In the following sections, we describe these approaches in more detail. SE Chapter 9 Stocks and Their Valuation LF TEST 275 What is the difference between a stock’s price and its intrinsic value? Why do investors and managers need to understand how to estimate a firm’s intrinsic value? What are two commonly used approaches for estimating a stock’s intrinsic value? 9-4 THE DISCOUNTED DIVIDEND MODEL The value of a share of common stock depends on the cash flows it is expected to provide, and those flows consist of two elements: (1) the dividends the investor receives each year while he or she holds the stock and (2) the price received when the stock is sold. The final price includes the original price paid plus an expected capital gain. Keep in mind that there are many different investors in the market and thus many different sets of expectations. Therefore, different investors will have different opinions about a stock’s true intrinsic value and thus proper price. The analysis as performed by the marginal investor, whose actions actually determine the equilibrium stock price, is critical; but every investor, marginal or not, implicitly goes through the same type of analysis. The following terms are used in our analysis:2 Marginal investor ¼ the investor (or group of investors with similar views) who is at the margin and would be willing to buy if the stock price was slightly lower or to sell if the price was slightly higher. It is this investor’s expectations about dividends, growth, and risk that are key in the valuation process. Other investors ¼ all except the marginal investor. Some will be more optimistic than the marginal investor; others, more pessimistic. These investors will place new buy or sell orders if events occur to cause them to change their current expectations. Dt ¼ the dividend a stockholder expects to receive at the end of each Year t. D0 is the last dividend the company paid. Since it has already been paid, a buyer of the stock will not receive D0. The first dividend a new buyer will receive is D1, which is paid at the end of Year 1. D2 is the dividend expected at the end of Year 2; D3, at the end of Year 3; and so forth. D0 is known with certainty; but D1, D2, and all other future dividends are expected values; and different investors can have different expectations.3 Our primary concern is with Dt as forecasted by the marginal investor. P0 ¼ actual market price of the stock today. P0 is known with certainty, but predicted future prices are subject to uncertainty. 2 Many terms are described here, and students sometimes get concerned about having to memorize all of them. We tell our students that we will provide formula sheets for use on exams, so they don’t have to try to memorize everything. With their minds thus eased, they end up learning what the terms are rather than memorizing them. 3 Stocks generally pay dividends quarterly, so theoretically we should evaluate them on a quarterly basis. However, most analysts actually work with annual data because forecasted stock data are not precise enough to warrant the use of a quarterly model. For additional information on the quarterly model, see Charles M. Linke and J. Kenton Zumwalt, “Estimation Biases in Discounted Cash Flow Analysis of Equity Capital Costs in Rate Regulation,” Financial Management, Autumn 1984, pp. 15–21. Marginal Investor A representative investor whose actions reflect the beliefs of those people who are currently trading a stock. It is the marginal investor who determines a stock’s price. Market Price, P0 The price at which a stock sells in the market. 276 Part 3 Financial Assets Growth Rate, g The expected rate of growth in dividends per share. Required Rate of Return, rs The minimum rate of return on a common stock that a stockholder considers acceptable. Expected Rate of Return, ^rs The rate of return on a common stock that a stockholder expects to receive in the future. Actual (Realized) Rate of Return, rs The rate of return on a common stock actually received by stockholders in some past period. rs may be greater or less than ^rs and/or rs. Dividend Yield The expected dividend divided by the current price of a share of stock. Capital Gains Yield The capital gain during a given year divided by the beginning price. Expected Total Return The sum of the expected dividend yield and the expected capital gains yield. ^ t ¼ both the expected price and the expected intrinsic value of P the stock at the end of each Year t (pronounced “P hat t”) ^ t is based on as seen by the investor doing the analysis. P the investor’s estimates of the dividend stream and the riskiness of that stream. There are many investors in the ^ t . However, market, so there can be many estimates for P ^ for the marginal investor, P0 must equal P 0 . Otherwise, a disequilibrium would exist, and buying and selling in the ^ 0 as seen by the market would soon result in P0 equaling P marginal investor. g ¼ expected growth rate in dividends as predicted by an investor. If dividends are expected to grow at a constant rate, g should also equal the expected growth rate in earnings and the stock’s price. Different investors use different g’s to evaluate a firm’s stock; but the market price, P0, is based on g as estimated by the marginal investor. rs ¼ required, or minimum acceptable, rate of return on the stock considering its riskiness and the returns available on other investments. Different investors typically have different opinions, but the key is again the marginal investor. The determinants of rs include factors discussed in Chapter 8, including the real rate of return, expected inflation, and risk. ^r s ¼ expected rate of return (pronounced “r hat s”) that an investor believes the stock will provide in the future. The expected return can be above or below the required return; but a rational investor will buy the stock if ^r s exceeds rs, sell the stock if ^r s is less than rs, and simply hold the stock if these returns are equal. Again, the key is the marginal investor, whose views determine the actual stock price. r s ¼ actual, or realized, after-the-fact rate of return, pronounced “r bar s.” You can expect to obtain a return of r s ¼ 10% if you buy a stock today; but if the market goes down, you may end up with an actual realized return that is much lower, perhaps even negative. D1/P0 ¼ dividend yield expected during the coming year. If Company X’s stock is expected to pay a dividend of D1 ¼ $1 during the next 12 months and if X’s current price is P0 ¼ $20, the expected dividend yield will be $1/$20 ¼ 0.05 ¼ 5%. Different investors could have different expectations for D1; but again, the marginal investor is the key. ^ 1  P0 Þ=P0 ¼ expected capital gains yield on the stock during the ðP coming year. If the stock sells for $20.00 today and if it is expected to rise to $21.00 by the end of the year, the ^ 1  P0 ¼ $21.00 – $20.00 ¼ expected capital gain will be P $1.00 and the expected capital gains yield will be $1.00/$20.00 ¼ 0.05 = 5%. Different investors can have ^ 1 , but the marginal investor is key. different expectations for P Expected total return ¼ ^r s ¼ expected dividend yield (D1/P0) plus expected ^ 1 – P0)/P0]. In our example, the capital gains yield [(P expected total return ¼ ^r s ¼ 5% þ 5% ¼ 10%. Chapter 9 Stocks and Their Valuation All active investors hope to be better than average—they hope to identify stocks whose intrinsic values exceed their current prices and whose expected returns (expected by this investor) exceed the required rate of return. Note, though, that about half of all investors are likely to be disappointed. A good understanding of the points made in this chapter can help you avoid being disappointed. 9-4a Expected Dividends as the Basis for Stock Values In our discussion of bonds, we used Equation 7-1 to find the value of a bond; the equation is the present value of interest payments over the bond’s life plus the present value of its maturity (or par) value: VB ¼ INT INT INT M þ þ  þ þ ð1 þ rd Þ1 ð1 þ rd Þ2 ð1 þ rd ÞN ð1 þ rd ÞN Stock prices are likewise determined as the present value of a stream of cash flows, and the basic stock valuation equation is similar to the one for bonds. What are the cash flows that a corporation will provide to its stockholders? To answer that question, think of yourself as an investor who buys the stock of a company that is expected to go on indefinitely (for example, GE). You intend to hold it (in your family) forever. In this case, all you (and your heirs) will receive is a stream of dividends; and the value of the stock today can be calculated as the present value of an infinite stream of dividends: ^ 0 ¼ PV of expected future dividends Value of stock ¼ P D1 D2 D1 þ þ  þ ¼ ð1 þ rs Þ1 ð1 þ rs Þ1 ð1 þ rs Þ2 1 X Dt ¼ ð1 þ rs Þt t¼1 9-1 What about the more typical case, where you expect to hold the stock for a finite ^0 in this case? Unless the period and then sell it—what will be the value of P company is likely to be liquidated or sold and thus disappears, the value of the stock is again determined by Equation 9-1. To see this, recognize that for any individual investor, the expected cash flows consist of expected dividends plus the expected sale price of the stock. However, the sale price to the current investor depends on the dividends some future investor expects, and that investor’s expected sale price is also dependent on some future dividends, and so forth. Therefore, for all present and future investors in total, expected cash flows must be based on expected future dividends. Put another way, unless a firm is liquidated or sold to another concern, the cash flows it provides to its stockholders will consist only of a stream of dividends. Therefore, the value of a share of stock must be established as the present value of the stock’s expected dividend stream.4 4 The general validity of Equation 9-1 can also be confirmed by asking yourself the following question: Suppose I buy a stock and expect to hold it for 1 year. I will receive dividends during the year plus the value ^P1 when I sell it at the end of the year. But what will determine the value of ^P1? The answer is that it will be determined as the present value of the dividends expected during Year 2 plus the stock price at the end of that year, which, in turn, will be determined as the present value of another set of future dividends and an even more distant stock price. This process can be continued ad infinitum, and the ultimate result is Equation 9-1. We should note that investors periodically lose sight of the long-run nature of stocks as investments and forget that in order to sell a stock at a profit, one must find a buyer who will pay the higher price. If you analyze a stock’s value in accordance with Equation 9-1, conclude that the stock’s market price exceeds a reasonable value, and buy the stock anyway, you would be following the “bigger fool” theory of investment—you think you may be a fool to buy the stock at its excessive price; but you also believe that when you get ready to sell it, you can find someone who is an even bigger fool. The bigger fool theory was widely followed in the summer of 2000, just before the stock market crashed. 277 Part 3 Financial Assets SE 278 LF TEST Explain the following statement: Whereas a bond contains a promise to pay interest, a share of common stock typically provides an expectation of, but no promise of, dividends plus capital gains. What are the two parts of most stocks’ expected total return? If D1 ¼ $2.00, g ¼ 6%, and P0 ¼ $40.00, what are the stock’s expected dividend yield, capital gains yield, and total expected return for the coming year? (5%, 6%, 11%) Is it necessary for all investors to have the same expectations regarding a stock for the stock to be in equilibrium? (No, but explain.) What would happen to a stock’s price if the “marginal investor” examined a stock and concluded that its intrinsic value was greater than its current market price? (P0 would rise.) 9-5 CONSTANT GROWTH STOCKS Equation 9-1 is a generalized stock valuation model in the sense that the time pattern of Dt can be anything: Dt can be rising, falling, or fluctuating randomly; or it can be zero for several years. Equation 9-1 can be applied in any of these situations; and with a computer spreadsheet, we can easily use the equation to find a stock’s intrinsic value—provided we have an estimate of the future dividends. However, it is not easy to obtain accurate estimates of future dividends. Still, for many companies it is reasonable to predict that dividends will grow at a constant rate. In this case, Equation 9-1 may be rewritten as follows: 9-2 Constant Growth (Gordon) Model Used to find the value of a constant growth stock. 1 2 1 ^ 0 ¼ D0 ð1 þ gÞ þ D0 ð1 þ gÞ þ    þ D0 ð1 þ gÞ1 P 1 2 ð1 þ rs Þ ð1 þ rs Þ ð1 þ rs Þ D0 ð1 þ gÞ D1 ¼ ¼ rs  g rs  g The last term of Equation 9-2 is the constant growth model, or Gordon model, named after Myron J. Gordon, who did much to develop and popularize it.5 The term rs in Equation 9-2 is the required rate of return, which is a riskless rate plus a risk premium. However, we know that if the stock is in equilibrium, the required rate of return must equal the expected rate of return, which is the expected dividend yield plus an expected capital gains yield. So we can solve Equation 9-2 for rs, but now using the hat to indicate that we are dealing with an expected rate of return:6 9-3 Expected growth rate;or Expected Expected rate þ ¼ capital gains yield dividend yield of return D1 ^r s þ g ¼ P0 We illustrate Equations 9-2 and 9-3 in the following section. 5 The last term in Equation 9-2 is derived in the Web/CD Extension of Chapter 5 of Eugene F. Brigham and Phillip R. Daves, Intermediate Financial Management, 9th ed. (Mason, OH: Thomson/South-Western, 2007). In essence, Equation 9-2 is the sum of a geometric progression, and the final result is the solution value of the progression. 6 The rs value in Equation 9-2 is a required rate of return; but when we transform Equation 9-2 to obtain Equation 9-3, we are finding an expected rate of return. Obviously, the transformation requires that rs ¼ ^r s. This equality must hold if the stock is in equilibrium, as most normally are. 279 Chapter 9 Stocks and Their Valuation 9-5a Illustration of a Constant Growth Stock Table 9-1 presents an analysis of Allied Food Products’ stock as performed by a security analyst after a meeting for analysts and other investors presided over by Allied’s CFO. The table looks complicated, but it is really quite straightforward.7 Part I, in the upper left corner, provides some basic data. The last dividend, which was just paid, was $1.15; the stock’s last closing price was $23.06; and it is in equilibrium. Based on an analysis of Allied’s history and likely future, the analyst forecasts that earnings and dividends will grow at a constant rate of 8.3% per year and that the stock’s price will grow at this same rate. Moreover, the analyst believes that the most appropriate required rate of return is 13.7%. Different analysts might use different inputs; but we assume for now that since this analyst is widely followed, her results represent those of the marginal investor. Now look at Part IV, where we show the predicted stream of dividends and stock prices along with annual values for the dividend yield, the capital gains yield, and the expected total return. Notice that the total return shown in Column 6 is equal to the required rate of return shown in Part I. This indicates that the stock Table 9-1 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 7 Analysis of a Constant Growth Stock B C I. Basic Information $1.15 D0 = P0 $23.06 = g 8.30% = rs = 13.70% III. Examples: Col. 2 Col. 3 Col. 4 Col. 5 Col. 6 Col. 7 D E F G II. Formulas Used in the Analysis: Dividend in Year t, Dt, in Col. 2 Intrinsic value (and price) in Year t, Pt, in Col. 3 Dividend yield (constant), in Col. 4 Capital gains yield (constant), in Col. 5 Total return (constant), in Col. 6 PV of dividends, discounted at 13.7% Col. 7 H Dt–1(1 + g) Dt+1/(rs – g) Dt/Pt–1 (Pt – Pt–1)/Pt–1 Div. yield + CG yield Dt/(1 + rs)t D1 = $1.1500(1.083) P0 = $1.25/(0.137 – 0.083) Dividend yield, Year 1: $1.25/$23.06 Cap gains yield, Year 1: ($24.98 – $23.06)/$23.06 Total return, Year 1: 5.4% + 8.3% PV of D1 discounted at 13.7% IV. Table: Forecasted Results over Time At end Dividend of year: (2) (1) $1.15 2008 1.25 2009 1.35 2010 1.46 2011 1.58 2012 1.71 2013 1.86 2014 2.01 2015 2.18 2016 2017 2.36 2.55 2018 ⬁ Price (3) $23.06 24.98 27.05 29.30 31.73 34.36 37.21 40.30 43.65 47.27 51.19 I $1.25 $23.06 5.40% 8.30% 13.70% $1.10 Dividend yield (4) Capital gain yield (5) Total returns (6) PV of dividend at 13.7% (7) 5.40% 5.40 5.40 5.40 5.40 5.40 5.40 5.40 5.40 5.40 8.30% 8.30 8.30 8.30 8.30 8.30 8.30 8.30 8.30 8.30 13.70% 13.70 13.70 13.70 13.70 13.70 13.70 13.70 13.70 13.70 $1.10 1.04 0.99 0.95 0.90 0.86 0.82 0.78 0.74 0.71 Sum of PVs from 1 to ⬁ = P0 = $23.06 You may notice some minor “errors” in the table. These are not errors—they are simply differences caused by rounding. 280 Part 3 Financial Assets analyst thinks that the stock is fairly priced; hence, it is in equilibrium. She forecasted out 10 years, but she could have forecasted out to infinity. Part II shows the formulas used to calculate the data in Part IV, and Part III gives examples of the calculations. For example, D1, the first dividend a purchaser would receive, is forecasted to be D1 ¼ $1.15(1.083) ¼ $1.25, and the other forecasted dividends in Column 2 were calculated similarly. The estimated intrinsic values shown in Column 3 are based on Equation 9-2, the constant growth model: P0 ¼ D1/(rs – g) ¼ $1.25/(0.137 – 0.083) ¼ $23.06 (corrected for rounding), P1 ¼ $24.98, and so forth. Column 4 shows the dividend yield, which for 2009 is D1/P0 ¼ 5.40%; and this number is constant thereafter. The capital gain expected during 2009 is P1 – P0 ¼ $24.98 – $23.06 ¼ $1.92, which when divided by P0 gives the expected capital gains yield, $1.92/$23.06 ¼ 8.3%, again corrected for rounding. The total return is found as the dividend yield plus the capital gains yield, 13.7%; and it is both constant and equal to the required rate of return given in Part I. Finally, look at Column 7 in the table. Here we find the present value of each of the dividends shown in Column 2, discounted at the required rate of return. For example, the PV of D1 ¼ $1.25/(1.137)1 ¼ $1.10, the PV of D2 ¼ $1.35/(1.137)2 ¼ $1.04, and so forth. If you extended the table out to about 170 years (with Excel, this is easy), then summed the PVs of the dividends, you would get the same value as that found using Equation 9-2, $23.06.8 Figure 9-2 shows graphically what’s happening. We extended the table out 20 years and then plotted dividends from Column 2 in the upper step function curve and the PV of those dividends in the lower curve. The sum of the PVs is an estimate of the stock’s forecasted intrinsic value. Note that in Table 9-1, the forecasted intrinsic value is equal to the current stock price and the expected total return is equal to the required rate of return. In this situation, the analysis would call the stock a “Hold” and would recommend that investors not buy or sell it. However, if the analyst were somewhat more optimistic and thought the growth rate would be 10.0% rather than 8.3%, the forecasted intrinsic value would be (by Equation 9-2) $34.19 and the analyst would call it a “Buy.” At g ¼ 6%, the intrinsic value would be $15.83 and the stock would be a “Sell.” Changes in the required rate of return would produce similar changes in the forecasted intrinsic value and thus the equilibrium current price. 9-5b Dividends Versus Growth The discounted dividend model as expressed in Equation 9-2 shows that, other things held constant, a higher value for D1 increases a stock’s price. However, Equation 9-2 shows that a higher growth rate also increases the stock’s price. But now recognize the following: Dividends are paid out of earnings. Therefore, growth in dividends requires growth in earnings. Earnings growth in the long run occurs primarily because firms retain earnings and reinvest them in the business. Therefore, the higher the percentage of earnings retained, the higher the growth rate. l l l l To illustrate all this, suppose you inherit a business that has $1,000,000 of assets, no debt, and thus $1,000,000 of equity. The expected return on equity (ROE) 8 The dividends get quite large, but the discount rate exceeds the growth rate; so the PVs of the dividends become quite small. In theory, you would have to go out to infinity to find the exact price of a constant growth stock, but the difference between the Equation 9-2 value and the sum of the PVs can’t be seen out to 2 decimal places if you go out about 170 periods. Chapter 9 Stocks and Their Valuation Present Values of Dividends of a Constant Growth Stock where D0 ¼ $1.15, g ¼ 8.3%, rs ¼ 13.7% FIGURE 9-2 Dividend ($) Dollar Amount of Each Dividend = D0 (1 + g)t 1.15 PV D1 = 1.10 Pˆ 0 = 0 D0 (1 + g)t (1 + rs )t 8 PV of Each Dividend = Σ PV Dt = Area under PV Curve t=1 = $23.06 5 10 15 20 Years equals 10.0%, so its expected earnings for the coming year are (0.10)$1,000,000 ¼ $100,000. You could take out the entire $100,000 of earnings in dividends, or you could reinvest some or all of the $100,000 in the business. If you pay out all the earnings, you will have $100,000 of dividend income this year, but dividends will not grow because assets and therefore earnings will not grow. However, suppose you decide to have the firm pay out 40% and retain 60%. Now your dividend income in Year 1 will be $40,000; but assets will rise by $60,000, and earnings and dividends will likewise increase: Next year's earnings ¼ Prior earnings þ ROEðRetained earningsÞ ¼ $100,000 þ 0:1ð$60,000Þ ¼ $106,000 Next year's dividends ¼ 0:4ð$106,000Þ ¼ $42,400 Moreover, your dividend income will continue to grow by 6% per year thereafter: Growth rate ¼ ð1  Payout ratioÞROE ¼ ð1  0:4Þ10:0% 9-4 ¼ 0:6ð10:0%Þ ¼ 60% This demonstrates that in the long run, growth in dividends depends primarily on the firm’s payout ratio and its ROE. In our example, we assumed that other things remain constant. This is often but not always a logical assumption. For example, suppose the firm develops a 281 282 Part 3 Financial Assets successful new product or hires a better CEO or makes some other change that increased the ROE. Any of these actions could cause the ROE and thus the growth rate to increase. Also note that the earnings of new firms are often low or even negative for several years, then begin to rise rapidly; finally, growth levels off as the firm approaches maturity. Such a firm might pay no dividends for its first few years, then pay a low initial dividend but let it increase rapidly, and finally make regular payments that grow at a constant rate once earnings have stabilized. In any such situation, the nonconstant model as discussed in a later section must be used. 9-5c Which Is Better: Current Dividends or Growth? We saw in the preceding section that a firm can pay a higher current dividend by increasing its payout ratio, but that will lower its dividend growth rate. So the firm can provide a relatively high current dividend or a high growth rate but not both. This being the case, which would stockholders prefer? The answer is not clear. As we will see in the dividend chapter, some stockholders prefer current dividends while others prefer a lower payout ratio and future growth. Empirical studies have been unable to determine which strategy is optimal in the sense of maximizing a firm’s stock price. So dividend policy is an issue that management must decide on the basis of its judgment, not a mathematical formula. Logically, shareholders should prefer for the company to retain more earnings (hence pay less current dividends) if the firm has exceptionally good investment opportunities; however, shareholders should prefer a high payout if investment opportunities are poor. In spite of this, taxes and other factors complicate the situation. We will discuss all this in detail in the dividend chapter; but for now, just assume that the firm’s management has decided on a payout policy and uses that policy to determine the actual dividend. 9-5d Required Conditions for the Constant Growth Model Zero Growth Stock A common stock whose future dividends are not expected to grow at all; that is, g ¼ 0. Several conditions are necessary for Equation 9-2 to be used. First, the required rate of return, rs, must be greater than the long-run growth rate, g. If the equation is used in situations where g is greater than rs, the results will be wrong, meaningless, and misleading. For example, if the forecasted growth rate in our example was 15% and thus exceeded the 13.7% required rate of return, stock price as calculated by Equation 9-2 would be a negative $101.73. That would be nonsense—stocks can’t have negative prices. Moreover, in Table 9-1, the PV of each future dividend would exceed that of the prior year. If this situation was graphed in Figure 9-2, the step-function curve for the PV of dividends would be increasing, not decreasing; so the sum would be infinitely high, which would indicate an infinitely high stock price. Obviously, stock prices cannot be either infinite or negative, so Equation 9-2 cannot be used unless rs > g. Second, the constant growth model as expressed in Equation 9-2 is not appropriate unless a company’s growth rate is expected to remain constant in the future. This condition almost never holds for new start-up firms, but it does exist for many mature companies. Indeed, mature firms such as Allied and GE are generally expected to grow at about the same rate as nominal gross domestic product (that is, real GDP plus inflation). On this basis, one might expect the dividends of an average, or “normal,” company to grow at a rate of 5% to 8% a year. Note too that Equation 9-2 is sufficiently general to handle the case of a zero growth stock, where the dividend is expected to remain constant over time. If g ¼ 0, Equation 9-2 reduces to Equation 9-5: 9-5 ^0 ¼ D P rs Chapter 9 Stocks and Their Valuation 283 SE This is the same equation as the one we developed in Chapter 5 for a perpetuity, and it is simply the current dividend divided by the discount rate. Finally, as we discuss later in the chapter, most firms, even rapidly growing startups and others that pay no dividends at present, can be expected to pay dividends at some point in the future, at which time the constant growth model will be appropriate. For such firms, Equation 9-2 is used as one part of a more complicated valuation equation that we discuss next. LF TEST Write out and explain the valuation formula for a constant growth stock. Explain how the formula for a zero growth stock can be derived from that for a normal constant growth stock. Firm A is expected to pay a dividend of $1.00 at the end of the year. The required rate of return is rs ¼ 11%. Other things held constant, what would the stock’s price be if the growth rate was 5%? What if g was 0%? ($16.67; $9.09) Firm B has a 12% ROE. Other things held constant, what would its expected growth rate be if it paid out 25% of its earnings as dividends? 75%? (9%, 3%) If Firm B had a 75% payout ratio but then lowered it to 25%, causing its growth rate to rise from 3% to 9%, would that action necessarily increase the price of its stock? Why or why not? 9-6 VALUING NONCONSTANT GROWTH STOCKS For many companies, it is not appropriate to assume that dividends will grow at a constant rate. Indeed, most firms go through life cycles where they experience different growth rates during different parts of the cycle. In their early years, most firms grow much faster than the economy as a whole; then they match the economy’s growth; and finally they grow at a slower rate than the economy.9 Automobile manufacturers in the 1920s, computer software firms such as Microsoft in the 1990s, and Google in the 2000s are examples of firms in the early part of their cycle. These firms are defined as supernormal, or nonconstant growth, firms. Figure 9-3 illustrates nonconstant growth and compares it with normal growth, zero growth, and negative growth.10 In the figure, the dividends of the supernormal growth firm are expected to grow at a 30% rate for three years, after which the growth rate is expected to fall to 8%, the assumed average for the economy. The value of this firm’s stock, like any other asset, is the present value of its expected future dividends as determined by 9 The concept of life cycles could be broadened to product cycle, which would include both small start-up companies and large companies such as Microsoft and Procter & Gamble, which periodically introduce new products that give sales and earnings a boost. We should also mention business cycles, which alternately depress and boost sales and profits. The growth rate just after a major new product has been introduced (or just after a firm emerges from the depths of a recession) is likely to be much higher than the “expected long-run average growth rate,” which is the proper number for use in the discounted dividend model. 10 A negative growth rate indicates a declining company. A mining company whose profits are falling because of a declining ore body is an example. Someone buying such a company would expect its earnings (and consequently its dividends and stock price) to decline each year, which would lead to capital losses rather than capital gains. Obviously, a declining company’s stock price is relatively low, and its dividend yield must be high enough to offset the expected capital loss and still produce a competitive total return. Students sometimes argue that they would never be willing to buy a stock whose price was expected to decline. However, if the present value of the expected dividends exceeds the stock price, the stock is still a good investment that would provide a good return. Supernormal (Nonconstant) Growth The part of the firm’s life cycle in which it grows much faster than the economy as a whole. 284 Part 3 Financial Assets Illustrative Dividend Growth Rates FIGURE 9-3 Dividend ($) Normal Growth, 8% End of Supernormal Growth Period Supernormal Growth, 30% Normal Growth, 8% 1.15 Zero Growth, 0% Declining Growth, –8% 0 1 2 3 4 5 Years Terminal (Horizon) Date The date when the growth rate becomes constant. At this date, it is no longer necessary to forecast the individual dividends. Equation 9-1. When Dt is growing at a constant rate, we can simplify Equation 9-1 ^0 ¼ D1/(rs – g). In the supernormal case, however, the expected to Equation 9-2, P growth rate is not a constant. In our example, there are two distinctly different rates. Because Equation 9-2 requires a constant growth rate, we obviously cannot use it to value stocks that are not growing at a constant rate. However, assuming that a company currently enjoying supernormal growth will eventually slow down and become a constant growth stock, we can combine Equations 9-1 and 9-2 to construct a new formula, Equation 9-6, for valuing the stock. First, we assume that the dividend will grow at a nonconstant rate (generally a relatively high rate) for N periods, after which it will grow at a constant rate, g. N is often called the terminal, or horizon, date. Second, we can use the constant growth formula, Equation 9-2, to determine what the stock’s horizon, or terminal, value will be N periods from today: ^N ¼ Horizon Value ¼ P DNþ1 rs  g ^ 0 , is the present value of the dividends during The stock’s intrinsic value today, P the nonconstant growth period plus the present value of the horizon value: Horizon (Terminal) Value The value at the horizon date of all dividends expected thereafter. ^0 ¼ P D1 ð1 þ rs Þ1 ð1 þ rs Þ ð1 þ rs Þ ð1 þ rs Þ ð1 þ rs Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Horizon value ¼ PV of dividends PV of dividends during the nonconstant growth during the constant growth Period; t ¼ 1;    N Period; t ¼ N þ 1;    1 D1 1 þ D2 2 þ  þ DN N þ DNþ1 Nþ1 þ  þ Chapter 9 Stocks and Their Valuation ^N P D1 D2 DN þ 1þ 2 þ  þ N ð1 þ rs Þ ð1 þ rs Þ ð1 þ rs Þ ð1 þ rs ÞN |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} PV of dividends during the PV of horizon ^ N: value, P nonconstant growth period ½ðDNþ1 Þ=ðrs  gÞ t ¼ 1;    N ^0 ¼ P 9-6 ð1 þ rs ÞN To implement Equation 9-6, we go through the following three steps: 1. Find the PV of each dividend during the period of nonconstant growth and sum them. 2. Find the expected stock price at the end of the nonconstant growth period, at which point it has become a constant growth stock so it can be valued with the constant growth model, and discount this price back to the present. ^0. 3. Add these two components to find the stock’s intrinsic value, P Figure 9-4 illustrates the process for valuing nonconstant growth stocks. Here we use a new company, Firm M, and we assume that the following five facts exist: rs ¼ stockholders’ required rate of return ¼ 13.4%. This rate is used to discount the cash flows. N ¼ years of nonconstant growth ¼ 3. gs ¼ rate of growth in both earnings and dividends during the nonconstant growth period ¼ 30%. This rate is shown directly on the time line. (Note: The growth rate during the nonconstant growth period could vary from year to year. Also, there could be several different nonconstant growth periods—for example, 30% for three years, 20% for the next three years, and a constant 8% thereafter). gn ¼ rate of normal, constant growth after the nonconstant period ¼ 8.0%. This rate is also shown on the time line, after Period 3, when it is in effect. D0 ¼ last dividend the company paid ¼ $1.15. SE The valuation process diagrammed in Figure 9-4 is explained in the steps set forth below the time line. The value of the nonconstant growth stock is calculated to be $39.21. Note that in this example, we assumed a relatively short 3-year horizon to keep things simple. When evaluating stocks, most analysts use a longer horizon (for example, 5 years) to estimate intrinsic values. This requires a few more calculations; but because analysts use spreadsheets, the arithmetic is not a problem. In practice, the real limitation is obtaining reliable forecasts for future growth. LF TEST Explain how one would find the value of a nonconstant growth stock. Explain what is meant by terminal (horizon) date and horizon (terminal) value. 285 286 Part 3 Financial Assets Finding the Value of a Nonconstant Growth Stock FIGURE 9-4 0 1 gs = 30% 30% D1 = 1.4950 1.3183 1.5113 36.3838 2 30% D2 = 1.9435 13.4% 13.4% 13.4% 3 D3 = 2.5266 gn = 8% 4 D4 = 2.7287 P̂3 = 50.5310 53.0576 39.2134 = $39.21 = P̂0 Notes to Figure 9-4: Step 1. Calculate the dividends expected at the end of each year during the nonconstant growth period. Calculate the first dividend, D1 ¼ D0(1 + gs) ¼ $1.15(1.30) ¼ $1.4950. Here gs is the growth rate during the 3-year nonconstant growth period, 30%. Show the $1.4950 on the time line as the cash flow at Time 1. Calculate D2 ¼ D1(1 + gs) ¼ $1.4950(1.30) ¼ $1.9435, then D3 ¼ D2(1 + gs) ¼ $1.9435(1.30) ¼ $2.5266. Show these values on the time line as the cash flows at Times 2 and 3. Note that D0 is used only to calculate D1. Step 2. The price of the stock is the PV of dividends from Time 1 to infinity; so in theory, we could project each future dividend, with the normal growth rate, gn ¼ 8%, used to calculate D4 and subsequent dividends. However, we know that after D3 has been paid at Time 3, the stock becomes a constant growth stock. Therefore, we can use the constant growth formula to find ^P3, which is the PV of the dividends from Time 4 to infinity as evaluated at Time 3. First, we determine D4 ¼ $2.5266(1.08) ¼ $2.7287 for use in the formula; then we calculate ^P3 as follows: ^3 ¼ P D4 $2:7287 ¼ ¼ $50:5310 rs  gn 0:134  0:08 We show this $50.5310 on the time line as a second cash flow at Time 3. The $50.5310 is a Time 3 cash flow in the sense that the stockholder could sell the stock for $50.5310 at Time 3 and in the sense that $50.5310 is the present value of the dividend cash flows from Time 4 to infinity. Note that the total cash flow at Time 3 consists of the sum of D3 + ^P3 ¼ $2.5266 þ $50.5310 ¼ $53.0576. Step 3. Now that the cash flows have been placed on the time line, we can discount each cash flow at the required rate of return, rs ¼ 13.4%. We could discount each cash flow by dividing by (1.134)t, where t ¼ 1 for Time 1, t ¼ 2 for Time 2, and t ¼ 3 for Time 3. This produces the PVs shown to the left below the time line; and the sum of the PVs is the value of the nonconstant growth stock, $39.21. With a financial calculator, you can find the PV of the cash flows as shown on the time line with the cash flow (CFLO) register of your calculator. Enter 0 for CF0 because you receive no cash flow at Time 0, CF1 ¼ 1.495, CF2 ¼ 1.9435, and CF3 ¼ 2.5266 þ 50.5310 ¼ 53.0576. Then enter I/YR ¼ 13.4 and press the NPV key to find the value of the stock, $39.21. 9-7 VALUING THE ENTIRE CORPORATION11 Corporate Valuation Model A valuation model used as an alternative to the discounted dividend model to determine a firm’s value, especially one with no history of dividends, or the value of a division of a larger firm. The corporate model first calculates the firm’s free cash flows, then finds their present values to determine the firm’s value. Thus far we have discussed the discounted dividend model for valuing a firm’s common stock. This procedure is widely used, but it is based on the assumption that the analyst can forecast future dividends reasonably well. This is often true for mature companies that have a history of steadily growing dividends. However, dividends are dependent on earnings; so a really reliable dividend forecast must be based on an underlying forecast of the firm’s future sales, costs, and capital requirements. This recognition has led to an alternative stock valuation approach, the corporate valuation model. 11 The corporate valuation model presented in this section is widely used by analysts, and it is in many respects superior to the discounted dividend model. However, it is rather involved as it requires the estimation of sales, costs, and cash flows on out into the future before the discounting process is begun. Therefore, in the introductory course, some instructors may prefer to omit Section 9-7 and skip to Section 9-8. Chapter 9 Stocks and Their Valuation EVALUATING STOCKS THAT DON’T PAY DIVIDENDS The discounted dividend model assumes that the firm is currently paying a dividend. However, many firms, even highly profitable ones, including Google, Dell, and Apple, have never paid a dividend. If a firm is expected to begin paying dividends in the future, we can modify the equations presented in the chapter and use them to determine the value of the stock. A new business often expects to have low sales during its first few years of operation as it develops its product. Then if the product catches on, sales will grow rapidly for several years. Sales growth brings with it the need for additional assets—a firm cannot increase sales without also increasing its assets, and asset growth requires an increase in liability and/or equity accounts. Small firms can generally obtain some bank credit, but they must maintain a reasonable balance between debt and equity. Thus, additional bank borrowings require increases in equity, and getting the equity capital needed to support growth can be difficult for small firms. They have limited access to the capital markets; and even when they can sell common stock, their owners are reluctant to do so for fear of losing voting control. Therefore, the best source of equity for most small businesses is retained earnings; for this reason most small firms pay no dividends during their rapid growth years. Eventually, though, successful small firms do pay dividends, and those dividends generally grow rapidly at first but slow down to a sustainable constant rate once the firm reaches maturity. If a firm currently pays no dividends but is expected to pay future dividends, the value of its stock can be found as follows: 1. Estimate at what point dividends will be paid, the amount of the first dividend, the growth rate during the supernormal growth period, the length of the supernormal period, the long-run (constant) growth rate, and the rate of return required by investors. 2. Use the constant growth model to determine the price of the stock after the firm reaches a stable growth situation. 3. Set out on a time line the cash flows (dividends during the supernormal growth period and the stock price once the constant growth state is reached); then find the present value of these cash flows. That present value represents the value of the stock today. To illustrate this process, consider the situation for Marvel-Lure Inc., a company that was set up in 2007 to produce and market a new high-tech fishing lure. MarvelLure’s sales are currently growing at a rate of 200% per year. The company expects to experience a high but declining rate of growth in sales and earnings during the next 10 years, after which analysts estimate that it will grow at a steady 10% per year. The firm’s management has announced that it will pay no dividends for 5 years but that if earnings materialize as forecasted, it will pay a dividend of $0.20 per share at the end of Year 6, $0.30 in Year 7, $0.40 in Year 8, $0.45 in Year 9, and $0.50 in Year 10. After Year 10, current plans are to increase dividends by 10% per year. MarvelLure’s investment bankers estimate that investors require a 15% return on similar stocks. Therefore, we find the value of a share of MarvelLure’s stock as follows: P0 ¼ $0 $0 $0:20 $0:30 $0:40 þ  þ þ þ þ ð1:15Þ1 ð1:15Þ5 ð1:15Þ6 ð1:15Þ7 ð1:15Þ8 !   $0:45 $0:50 $0:50ð1:10Þ 1 þ þ þ 0:15  0:10 ð1:15Þ9 ð1:15Þ10 ð1:15Þ10 ¼ $3:30 The last term finds the expected stock price in Year 10 and then finds the present value of that price. Thus, we see that the discounted dividend model can be applied to firms that currently pay no dividends, provided we can estimate future dividends with a fair degree of confidence. However, in many cases, we can have more confidence in the forecasts of free cash flows; and in these situations, it is better to use the corporate valuation model. Rather than starting with a forecast of dividends, the corporate valuation model focuses on the firm’s future free cash flows. We discussed free cash flow (FCF) in Chapter 3, where we developed the following equation: 2 6 FCF ¼ 6 4EBITð1  TÞ þ 3 2 Depreciation 7 6 54 and amortizaton Capital expenditures þ 3
Net 7 working 5 capital EBIT is earnings before interest and taxes, and free cash flow represents the cash generated from current operations, less the cash that must be spent on investments in fixed assets and working capital to support future growth. Consider the case of Home Depot (HD). The first term in brackets in the preceding equation represents the amount of cash that HD is generating from its existing stores. The second term 287 288 Part 3 Financial Assets represents the amount of cash the company plans to spend this period to construct new stores. To open a new store, HD must spend cash to purchase the land and construct the building—these are the capital expenditures, and they lead to a corresponding increase in the firm’s fixed assets as shown on the balance sheet. But HD also needs to increase its working capital, especially inventory. Putting everything together, HD generates positive free cash flow for its investors if and only if the money from its existing stores exceeds the money required to build and equip its new stores. 9-7a The Corporate Valuation Model In Chapter 3, we explained that a firm’s value is determined by its ability to generate cash flow both now and in the future. Therefore, its market value can be expressed as follows: Market value of company 9-7 ¼ Vcompany ¼ PV of expected future free cash flows ¼ FCF1 ð1 þ WACCÞ 1 þ FCF2 ð1 þ WACCÞ 2 þ  þ FCF1 ð1 þ WACCÞ1 Here FCFt is the free cash flow in Year t; and the discount rate, the WACC, is the weighted average cost of all the firm’s capital. When thinking about the WACC, note these two points: 1. The firm finances with debt, preferred stock, and common equity. The WACC is the weighted average of these three types of capital, and we discuss it in detail in Chapter 10. 2. Free cash flow is the cash generated before any payments are made to any investors; so it must be used to compensate common stockholders, preferred stockholders, and bondholders. Moreover, each type of investor has a required rate of return; and the weighted average of those returns is the WACC, which is used to discount the free cash flows. Free cash flows are generally forecasted for 5 to 10 years, after which it is assumed that the final explicitly forecasted FCF will grow at some long-run constant rate. Once the company reaches its horizon date, when cash flows begin to grow at a constant rate, we can use the following formula to calculate the market value of the company as of that date: 9-8 Horizon value ¼ VCompany at t¼N ¼ FCFNþ1 =ðWACC  gFCF Þ The corporate model is applied internally by the firm’s financial staff and by outside security analysts. For illustrative purposes, we discuss an analysis conducted by Susan Buskirk, senior food analyst for the investment banking firm Morton Staley and Company. Her analysis is summarized in Table 9-2, which was reproduced from the chapter Excel model. Based on Allied’s history and Buskirk’s knowledge of the firm’s business plan, she estimated sales, costs, and cash flows on an annual basis for 5 years. Growth will vary during those years, but she assumes that things will stabilize and growth will be constant after the fifth year. She would have made explicit forecasts for more years if she thought it would take longer to reach a steadystate, constant growth situation. Buskirk next calculated the expected free cash flows (FCFs) for each of the 5 nonconstant growth years, and she found the PV of those cash flows discounted at the WACC. After Year 5, she assumed that FCF growth would be constant; hence, the constant growth model could be used to find Allied’s total market value at Year 5. This “horizon, or terminal, value” is the sum of the PVs of the FCFs from Year 6 on out into the future, discounted back to Year 5 at the WACC. It l l l 289 Chapter 9 Stocks and Their Valuation Table 9-2 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 l l l l A Part 1. Key Inputs Allied Food Products: Free Cash Flow Valuation B C D 2009 10.0% 87.0 8.0 6.0 40.0 10.0 6.0 Sales growth rate Operating costs as a % of sales Growth in operating capital Depr'n as a % of operating capital Tax rate WACC Long-run FCF growth, gLR E 2010 9.0% 87.0 8.0 8.0 Part 2. Forecast of Cash Flows During Period of Nonconstant Growth Historical 2008 2009 F Forecasted Years 2011 9.0% 86.0 8.0 7.0 Forecasted Years 2010 2011 G H 2012 9.0% 85.0 8.0 7.0 2013 8.0% 85.0 8.0 7.0 2012 2013 Sales Operating costs Depreciation EBIT EBIT ⫻ (1 - T) $3,000.0 2,616.2 100.0 $283.8 $170.3 $3,300.0 2,871.0 116.6 $312.4 $187.4 $3,597.0 3,129.4 168.0 $299.6 $179.8 $3,920.7 3,371.8 158.7 $390.2 $234.1 $4,273.6 3,632.6 171.4 $469.6 $281.8 $4,615.5 3,923.2 185.1 $507.2 $304.3 Total operating capital Net new operating cap Free Cash Flow, FCF PV of FCFs $1,800.0 280 -$109.7 N.A. $1,944.0 144.0 $43.4 $39.5 $2,099.5 155.5 $24.3 $20.1 $2,267.5 168.0 $66.1 $49.7 $2,448.9 181.4 $100.4 $68.6 $2,644.8 195.9 $108.4 $67.3 Part 3. Terminal Value and Intrinsic Value Estimation Estimated Value at the Horizon, 2013 $114.9 Free Cash Flow (2014) Terminal Value at 2013, TV $2,872.7 PV of the 2013 TV $1,783.7 Calculation of Firm's Intrinsic Value Sum of PVs of FCFs, 2009-2013 PV of 2013 TV Total corporate value Less: market value of debt and pfd Intrinsic value of common equity Shares outstanding (millions) Intrinsic Value Per Share FCF2013(1 + gLR) FCF2014 TV2013 = WACC - g TV / (1 + WACC)N $245.1 1,783.7 $2,028.8 860.0 $1,168.8 50.0 $23.38 follows that: Horizon Value at t ¼ 5 ¼ FCF6/(WACC – gFCF), where gFCF represents the long-run growth rate of free cash flow. Next, she discounted the Year 5 terminal value back to the present to find its PV at Year 0. She then summed all the PVs, the annual cash flows during the nonconstant period plus the PV of the horizon value, to find the firm’s estimated total market value. Then she subtracted the value of the debt and preferred stock to find the value of the common equity. Finally, she divided the equity value by the number of shares outstanding, and the result was her estimate of Allied’s intrinsic value per share. This value was quite close to the stock’s market price, so she concluded that Allied’s stock is priced at its equilibrium level. Consequently, she issued a “Hold” recommendation on the stock. If the estimated intrinsic value had been significantly below the market price, she would have issued a “Sell” recommendation; if the estimated intrinsic value had been well above the market price, she would have called the stock a “Buy.” 290 Part 3 Financial Assets OTHER APPROACHES TO VALUING COMMON STOCKS While the dividend growth and the corporate valuation models presented in this chapter are the most widely used methods for valuing common stocks, they are by no means the only approaches. Analysts often use a number of different techniques to value stocks. Two of these alternative approaches are described here. The P/E Multiple Approach Investors have long looked for simple rules of thumb to determine whether a stock is fairly valued. One such approach is to look at the stock’s price-to-earnings (P/E) ratio. Recall from Chapter 4 that a company’s P/E ratio shows how much investors are willing to pay for each dollar of reported earnings. As a starting point, you might conclude that stocks with low P/E ratios are undervalued since their price is “low” given current earnings, while stocks with high P/E ratios are overvalued. Unfortunately, however, valuing stocks is not that simple. We should not expect all companies to have the same P/E ratio. P/E ratios are affected by risk—investors discount the earnings of riskier stocks at a higher rate. Thus, all else equal, riskier stocks should have lower P/E ratios. In addition, when you buy a stock, you have a claim not only on current earnings but also on all future earnings. All else equal, companies with stronger growth opportunities will generate larger future earnings and thus should trade at higher P/E ratios. Therefore, eBay is not necessarily overvalued just because its P/E ratio is 121.2 at a time when the median firm has a P/E of 19.7. Investors believe that eBay’s growth potential is well above average. Whether the stock’s future prospects justify its P/E ratio remains to be seen; but in and of itself, a high P/E ratio does not mean that a stock is overvalued. Nevertheless, P/E ratios can provide a useful starting point in stock valuation. If a stock’s P/E ratio is well above its industry average and if the stock’s growth potential and risk are similar to other firms in the industry, the stock’s price may be too high. Likewise, if a company’s P/E ratio falls well below its historical average, the stock may be undervalued —particularly if the company’s growth prospects and risk are unchanged and if the overall P/E for the market has remained constant or increased. One obvious drawback of the P/E approach is that it depends on reported accounting earnings. For this reason, some analysts choose to rely on other multiples to value stocks. For example, some analysts look at a company’s price-to-cash-flow ratio, while others look at the price-tosales ratio. The EVA Approach In recent years, analysts have looked for more rigorous alternatives to the discounted dividend model. More than a quarter of all stocks listed on the NYSE pay no dividends. This proportion is even higher on Nasdaq. While the discounted dividend model can still be used for these stocks (see “Evaluating Stocks That Don’t Pay Dividends”), this approach requires that analysts forecast when the stock will begin paying dividends, what the dividend will be once it is established, and what the future dividend growth rate will be. In many cases, these forecasts contain considerable errors. An alternative approach is based on the concept of Economic Value Added (EVA), which we discussed in Chapter 4 in “Economic Value Added (EVA) versus Net Income,” that can be written as follows: EVA ¼ ðEquity capitalÞðROE  Cost of equity capitalÞ This equation suggests that companies can increase their EVA by investing in projects that provide shareholders with returns that are above their cost of equity capital, which is the return they could expect to earn on alternative investments with the same level of risk. When you purchase stock in a company, you receive more than just the book value of equity—you also receive a claim on all future value that is created by the firm’s managers (the present value of all future EVAs). It follows that a company’s market value of equity can be written as follows: Market value of equity ¼ Book value þ PV of all future EVAs We can find the “fundamental” value of the stock, P0, by simply dividing the preceding expression by the number of shares outstanding. As is the case with the discounted dividend model, we can simplify the expression by assuming that at some point in time, annual EVA becomes a perpetuity, or grows at some constant rate over time.a a What we have presented here is a simplified version of what is often referred to as the Edwards-Bell-Ohlson (EBO) model. For a more complete description of this technique and an excellent summary of how it can be used in practice, read the article “Measuring Wealth,” by Charles M. C. Lee, in CA Magazine, April 1996, pp. 32–37. 9-7b Comparing the Corporate Valuation and Discounted Dividend Models Analysts use both the discounted dividend model and the corporate valuation model when valuing mature, dividend-paying firms; and they generally use the corporate model when valuing divisions and firms that do not pay dividends. In Chapter 9 Stocks and Their Valuation SE principle, we should find the same intrinsic value using either model, but differences are often observed. When a conflict exists, the assumptions embedded in the corporate model can be reexamined; and once the analyst is convinced they are reasonable, the results of that model are used. In our Allied example, the estimates were extremely close—the discounted dividend model predicted a price of $23.06 per share versus $23.38 using the corporate model; both are essentially equal to Allied’s actual $23.06 price. In practice, intrinsic value estimates based on the two models normally deviate from one another and from actual stock prices, leading different analysts to reach different conclusions about the attractiveness of a given stock. The better the analyst, the more often his or her valuations turn out to be correct; but no one can make perfect predictions because too many things can change randomly and unpredictably in the future. Given all this, does it matter whether you use the corporate model or the discounted dividend model to value stocks? We would argue that it does. If we had to value, for example, 100 mature companies whose dividends were expected to grow steadily in the future, we would probably use the discounted dividend model. Here we would estimate only the growth rate in dividends, not the entire set of pro forma financial statements; hence, it would be more feasible to use the dividend model. However, if we were studying just one company or a few companies, especially companies still in the high-growth stage of their life cycles, we would want to project future financial statements before estimating future dividends. Because we would already have projected future financial statements, we would go ahead and apply the corporate model. Intel, which pays a dividend of $0.56 versus earnings of about $1.17, is an example of a company where either model could be used; but we think the corporate model is better. Now suppose you were trying to estimate the value of a company such as eBay that, to date (2008), has never paid a dividend or a new firm that is about to go public. In either situation, you would be better off using the corporate valuation model. Actually, even if a company is paying steady dividends, much can be learned from the corporate model; so analysts today use it for all types of valuations. The process of projecting future financial statements can reveal a great deal about a company’s operations and financing needs. Also, such an analysis can provide insights into actions that might be taken to increase the company’s value; and for this reason, it is integral to the planning and forecasting process, as we discuss in a later chapter. LF TEST Write out the equation for free cash flows and explain it. Why might someone use the corporate valuation model for companies that have a history of paying dividends? What steps are taken to find a stock price using the corporate model? Why might the calculated intrinsic value differ from the stock’s current market price? Which would be “correct,” and what does “correct” mean? 9-8 PREFERRED STOCK12 Preferred stock is a hybrid—it is similar to a bond in some respects and to common stock in others. This hybrid nature becomes apparent when we try to classify preferred stock in relation to bonds and common stock. Like bonds, preferred 12 Preferred stock is discussed in more detail in Chapter 20 of Fundamentals of Financial Management, 12th ed., (Mason, OH: Cengage Learning, 2010) and in Chapter 20 of Brigham & Daves, Intermediate Financial Management, 9th ed., (Mason, OH: Thomson/South-Western, 2007). 291 Part 3 Financial Assets stock has a par value and a fixed dividend that must be paid before dividends can be paid on the common stock. However, the directors can omit (or “pass”) the preferred dividend without throwing the company into bankruptcy. So although preferred stock calls for a fixed payment like bonds, skipping the payment will not lead to bankruptcy. As noted earlier, a preferred stock entitles its owners to regular, fixed dividend payments. If the payments last forever, the issue is a perpetuity whose value, Vp, is found as follows: Vp ¼ 9-9 Dp rp Vp is the value of the preferred stock, Dp is the preferred dividend, and rp is the required rate of return on the preferred. Allied Food has no preferred outstanding, but discussions about such an issue suggested that its preferred should pay a dividend of $10 per year. If its required return was 10.3%, the preferred’s value would be $97.09, found as follows: Vp ¼ $10:00 ¼ $97:09 0:103 In equilibrium, the expected return, ^r p , must be equal to the required return, rp. Thus, if we know the preferred’s current price and dividend, we can solve for the expected rate of return as follows: 9-9a ^r p ¼ Dp Vp Some preferreds have a stated maturity, often 50 years. Assume that our illustrative preferred matured in 50 years, paid a $10 annual dividend, and had a required return of 8%. We could then find its price as follows: Enter N ¼ 50, I/YR ¼ 8, PMT ¼ 10, and FV ¼ 100. Then press PV to find the price, Vp ¼ $124.47. If rp rose to 10%, change I/YR to 10, in which case Vp ¼ PV ¼ $100. If you know the price of a share of preferred stock, you can solve for I/YR to find the expected rate of return, ^r p . SE 292 LF TEST Explain the following statement: Preferred stock is a hybrid security. Is the equation used to value preferred stock more like the one used to evaluate a bond or the one used to evaluate a “normal” constant growth common stock? Explain. TYING IT ALL TOGETHER Corporate decisions should be analyzed in terms of how alternative courses of action are likely to affect a firm’s value. However, it is necessary to know how stock prices are established before attempting to measure how a given decision will affect a specific firm’s value. This chapter discussed the rights and privileges of common stockholders, showed how stock values are determined, and explained how investors estimate stocks’ intrinsic values and expected rates of return. Chapter 9 Stocks and Their Valuation Two types of stock valuation models were discussed: the discounted dividend model and the corporate valuation model. The discounted dividend model is useful for mature, stable companies. It is easier to use, but the corporate valuation model is more flexible and better for use with companies that do not pay dividends or whose dividends would be especially hard to predict. We also discussed preferred stock, which is a hybrid security that has some characteristics of a common stock and some of a bond. Preferreds are valued using models similar to those for perpetual and “regular” bonds. SELF-TEST QUESTIONS AND PROBLEMS (Solutions Appear in Appendix A) ST-1 KEY TERMS Define the following terms: a. b. Preemptive right c. Classified stock; founders’ shares d. ^ 0 ); market price (P0) Marginal investor; intrinsic value (P e. f. Required rate of return, rs; expected rate of return, ^r s ; actual (realized) rate of return, r s Capital gains yield; dividend yield; expected total return; growth rate, g g. Zero growth stock h. Constant growth (Gordon) model; supernormal (nonconstant) growth i. Corporate valuation model j. k. ST-2 Proxy; proxy fight; takeover Terminal (horizon) date; horizon (terminal) value Preferred stock STOCK GROWTH RATES AND VALUATION You are considering buying the stocks of two companies that operate in the same industry. They have very similar characteristics except for their dividend payout policies. Both companies are expected to earn $3 per share this year; but Company D (for “dividend”) is expected to pay out all of its earnings as dividends, while Company G (for “growth”) is expected to pay out only one-third of its earnings, or $1 per share. D’s stock price is $25. G and D are equally risky. Which of the following statements is most likely to be true? a. b. c. d. e. Company G will have a faster growth rate than Company D. Therefore, G’s stock price should be greater than $25. Although G’s growth rate should exceed D’s, D’s current dividend exceeds that of G, which should cause D’s price to exceed G’s. A long-term investor in Stock D will get his or her money back faster because D pays out more of its earnings as dividends. Thus, in a sense, D is like a short-term bond and G is like a long-term bond. Therefore, if economic shifts cause rd and rs to increase and if the expected dividend streams from D and G remain constant, both Stocks D and G will decline, but D’s price should decline further. D’s expected and required rate of return is ^rs¼ rs ¼ 12%. G’s expected return will be higher because of its higher expected growth rate. If we observe that G’s price is also $25, the best estimate of G’s growth rate is 8%. ST-3 CONSTANT GROWTH STOCK VALUATION Fletcher Company’s current stock price is $36.00, its last dividend was $2.40, and its required rate of return is 12%. If dividends are expected to grow at a constant rate, g, in the future and if rs is expected to remain at 12%, what is Fletcher’s expected stock price 5 years from now? ST-4 NONCONSTANT GROWTH STOCK VALUATION Snyder Computers Inc. is experiencing rapid growth. Earnings and dividends are expected to grow at a rate of 15% during the 293 294 Part 3 Financial Assets next 2 years, at 13% the following year, and at a constant rate of 6% during Year 4 and thereafter. Its last dividend was $1.15, and its required rate of return is 12%. b. a. Calculate the value of the stock today. ^ 1 and P ^ 2. Calculate P c. Calculate the dividend and capital gains yields for Years 1, 2, and 3. QUESTIONS 9-1 It is frequently stated that the one purpose of the preemptive right is to allow individuals to maintain their proportionate share of the ownership and control of a corporation. a. b. 9-2 How important do you suppose control is for the average stockholder of a firm whose shares are traded on the New York Stock Exchange? Is the control issue likely to be of more importance to stockholders of publicly owned or closely held (private) firms? Explain. Is the following equation correct for finding the value of a constant growth stock? Explain. ^0 ¼ P 9-3 9-4 9-5 D0 rs þ g If you bought a share of common stock, you would probably expect to receive dividends plus an eventual capital gain. Would the distribution between the dividend yield and the capital gains yield be influenced by the firm’s decision to pay more dividends rather than to retain and reinvest more of its earnings? Explain. Two investors are evaluating GE’s stock for possible purchase. They agree on the expected value of D1 and on the expected future dividend growth rate. Further, they agree on the riskiness of the stock. However, one investor normally holds stocks for 2 years, while the other holds stocks for 10 years. On the basis of the type of analysis done in this chapter, should they both be willing to pay the same price for GE’s stock? Explain. A bond that pays interest forever and has no maturity is a perpetual bond. In what respect is a perpetual bond similar to a no-growth common stock? Are there preferred stocks that are evaluated similarly to perpetual bonds and other preferred stocks that are more like bonds with finite lives? Explain. PROBLEMS Easy Problems 1–6 9-1 DPS CALCULATION Warr Corporation just paid a dividend of $1.50 a share (that is, D0 ¼ $1.50). The dividend is expected to grow 7% a year for the next 3 years and then at 5% a year thereafter. What is the expected dividend per share for each of the next 5 years? 9-2 CONSTANT GROWTH VALUATION Thomas Brothers is expected to pay a $0.50 per share dividend at the end of the year (that is, D1 ¼ $0.50). The dividend is expected to grow at a constant rate of 7% a year. The required rate of return on the stock, rs, is 15%. What is the stock’s current value per share? CONSTANT GROWTH VALUATION Harrison Clothiers’ stock currently sells for $20.00 a share. It just paid a dividend of $1.00 a share (that is, D0 ¼ $1.00). The dividend is expected to grow at a constant rate of 6% a year. What stock price is expected 1 year from now? What is the required rate of return? NONCONSTANT GROWTH VALUATION Hart Enterprises recently paid a dividend, D0, of $1.25. It expects to have nonconstant growth of 20% for 2 years followed by a constant rate of 5% thereafter. The firm’s required return is 10%. a. How far away is the terminal, or horizon, date? b. What is the firm’s horizon, or terminal, value? ^ 0? c. What is the firm’s intrinsic value today, P 9-3 9-4 9-5 CORPORATE VALUATION Smith Technologies is expected to generate $150 million in free cash flow next year, and FCF is expected to grow at a constant rate of 5% per year Chapter 9 Stocks and Their Valuation 9-6 Intermediate Problems 7–15 9-7 9-8 indefinitely. Smith has no debt or preferred stock, and its WACC is 10%. If Smith has 50 million shares of stock outstanding, what is the stock’s value per share? PREFERRED STOCK VALUATION Fee Founders has perpetual preferred stock outstanding that sells for $60 a share and pays a dividend of $5 at the end of each year. What is the required rate of return? PREFERRED STOCK RATE OF RETURN What will be the nominal rate of return on a perpetual preferred stock with a $100 par value, a stated dividend of 8% of par, and a current market price of (a) $60, (b) $80, (c) $100, and (d) $140? PREFERRED STOCK VALUATION Ezzell Corporation issued perpetual preferred stock with a 10% annual dividend. The stock currently yields 8%, and its par value is $100. a. What is the stock’s value? b. Suppose interest rates rise and pull the preferred stock’s yield up to 12%. What is its new market value? 9-9 PREFERRED STOCK RETURNS Bruner Aeronautics has perpetual preferred stock outstanding with a par value of $100. The stock pays a quarterly dividend of $2, and its current price is $80. a. What is its nominal annual rate of return? b. What is its effective annual rate of return? 9-10 VALUATION OF A DECLINING GROWTH STOCK Martell Mining Company’s ore reserves are being depleted, so its sales are falling. Also, because its pit is getting deeper each year, its costs are rising. As a result, the company’s earnings and dividends are declining at the constant rate of 5% per year. If D0 ¼ $5 and rs ¼ 15%, what is the value of Martell Mining’s stock? 9-11 VALUATION OF A CONSTANT GROWTH STOCK A stock is expected to pay a dividend of $0.50 at the end of the year (that is, D1 ¼ 0.50), and it should continue to grow at a constant rate of 7% a year. If its required return is 12%, what is the stock’s expected price 4 years from today? VALUATION OF A CONSTANT GROWTH STOCK Investors require a 15% rate of return on Levine Company’s stock (that is, rs ¼ 15%). a. What is its value if the previous dividend was D0 ¼ $2 and investors expect dividends to grow at a constant annual rate of (1) –5%, (2) 0%, (3) 5%, or (4) 10%? b. Using data from Part a, what would the Gordon (constant growth) model value be if the required rate of return was 15% and the expected growth rate was (1) 15% or (2) 20%? Are these reasonable results? Explain. c. Is it reasonable to think that a constant growth stock could have g > rs? Explain. 9-12 9-13 9-14 9-15 CONSTANT GROWTH You are considering an investment in Keller Corp’s stock, which is expected to pay a dividend of $2.00 a share at the end of the year (D1 ¼ $2.00) and has a beta of 0.9. The risk-free rate is 5.6%, and the market risk premium is 6%. Keller currently sells for $25.00 a share, and its dividend is expected to grow at some constant rate g. Assuming the market is in equilibrium, what does the market believe will be the stock price ^ 3 ?) at the end of 3 years? (That is, what is P NONCONSTANT GROWTH Microtech Corporation is expanding rapidly and currently needs to retain all of its earnings; hence, it does not pay dividends. However, investors expect Microtech to begin paying dividends, beginning with a dividend of $1.00 coming 3 years from today. The dividend should grow rapidly—at a rate of 50% per year—during Years 4 and 5; but after Year 5, growth should be a constant 8% per year. If the required return on Microtech is 15%, what is the value of the stock today? CORPORATE VALUATION Dozier Corporation is a fast-growing supplier of office products. Analysts project the following free cash flows (FCFs) during the next 3 years, after which FCF is expected to grow at a constant 7% rate. Dozier’s WACC is 13%. Year FCF ($ millions) a. b. c. 0 1 2 NA ⫺$20 $30 3 $40 What is Dozier’s terminal, or horizon, value? (Hint: Find the value of all free cash flows beyond Year 3 discounted back to Year 3.) What is the firm’s value today? Suppose Dozier has $100 million of debt and 10 million shares of stock outstanding. What is your estimate of the current price per share? 295 296 Part 3 Financial Assets Challenging Problems 16–21 9-16 9-17 NONCONSTANT GROWTH Mitts Cosmetics Co.’s stock price is $58.88, and it recently paid a $2.00 dividend. This dividend is expected to grow by 25% for the next 3 years, then grow forever at a constant rate, g; and rs ¼ 12%. At what constant rate is the stock expected to grow after Year 3? CONSTANT GROWTH Your broker offers to sell you some shares of Bahnsen & Co. common stock that paid a dividend of $2.00 yesterday. Bahnsen’s dividend is expected to grow at 5% per year for the next 3 years. If you buy the stock, you plan to hold it for 3 years and then sell it. The appropriate discount rate is 12%. a. Find the expected dividend for each of the next 3 years; that is, calculate D1, D2, and D3. Note that D0 ¼ $2.00. b. Given that the first dividend payment will occur 1 year from now, find the present value of the dividend stream; that is, calculate the PVs of D1, D2, and D3 and then sum these PVs. ^3 to c. You expect the price of the stock 3 years from now to be $34.73; that is, you expect P equal $34.73. Discounted at a 12% rate, what is the present value of this expected future stock price? In other words, calculate the PV of $34.73. d. If you plan to buy the stock, hold it for 3 years, and then sell it for $34.73, what is the most you should pay for it today? e. Use Equation 9-2 to calculate the present value of this stock. Assume that g ¼ 5% and that it is constant. f. Is the value of this stock dependent upon how long you plan to hold it? In other words, if your planned holding period was 2 years or 5 years rather than 3 years, would this ^0? Explain. affect the value of the stock today, P 9-18 NONCONSTANT GROWTH STOCK VALUATION Taussig Technologies Corporation (TTC) has been growing at a rate of 20% per year in recent years. This same growth rate is expected to last for another 2 years, then decline to gn ¼ 6%. a. If D0 ¼ $1.60 and rs ¼ 10%, what is TTC’s stock worth today? What are its expected dividend and capital gains yields at this time, that is, during Year 1? b. Now assume that TTC’s period of supernormal growth is to last for 5 years rather than 2 years. How would this affect the price, dividend yield, and capital gains yield? Answer in words only. c. What will TTC’s dividend and capital gains yields be once its period of supernormal growth ends? (Hint: These values will be the same regardless of whether you examine the case of 2 or 5 years of supernormal growth; the calculations are very easy.) d. Of what interest to investors is the changing relationship between dividend and capital gains yields over time? 9-19 CORPORATE VALUATION Barrett Industries invests a large sum of money in R&D; as a result, it retains and reinvests all of its earnings. In other words, Barrett does not pay any dividends and it has no plans to pay dividends in the near future. A major pension fund is interested in purchasing Barrett’s stock. The pension fund manager has estimated Barrett’s free cash flows for the next 4 years as follows: $3 million, $6 million, $10 million, and $15 million. After the fourth year, free cash flow is projected to grow at a constant 7%. Barrett’s WACC is 12%, its debt and preferred stock total $60 million, and it has 10 million shares of common stock outstanding. a. What is the present value of the free cash flows projected during the next 4 years? b. What is the firm’s terminal value? c. What is the firm’s total value today? d. What is an estimate of Barrett’s price per share? 9-20 CORPORATE VALUE MODEL Assume that today is December 31, 2008, and that the following information applies to Vermeil Airlines: l After-tax operating income [EBIT(1 – T)] for 2009 is expected to be $500 million. l The depreciation expense for 2009 is expected to be $100 million. l The capital expenditures for 2009 are expected to be $200 million. l No change is expected in net working capital. l The free cash flow is expected to grow at a constant rate of 6% per year. l The required return on equity is 14%. l The WACC is 10%. 297 Chapter 9 Stocks and Their Valuation l The market value of the company’s debt is $3 billion. l 200 million shares of stock are outstanding. Using the corporate valuation model approach, what should be the company’s stock price today? 9-21 NONCONSTANT GROWTH Assume that it is now January 1, 2009. Wayne-Martin Electric Inc. (WME) has developed a solar panel capable of generating 200% more electricity than any other solar panel currently on the market. As a result, WME is expected to experience a 15% annual growth rate for the next 5 years. Other firms will have developed comparable technology at the end of 5 years, and WME’s growth rate will slow to 5% per year indefinitely. Stockholders require a return of 12% on WME’s stock. The most recent annual dividend (D0), which was paid yesterday, was $1.75 per share. a. Calculate WME’s expected dividends for 2009, 2010, 2011, 2012, and 2013. ^0. Proceed by finding the present value of the b. Calculate the value of the stock today, P dividends expected at the end of 2009, 2010, 2011, 2012, and 2013 plus the present value of the stock price that should exist at the end of 2013. The year-end 2013 stock price can be found by using the constant growth equation. Notice that to find the December 31, 2013, price, you must use the dividend expected in 2014, which is 5% greater than the 2013 dividend. c. Calculate the expected dividend yield (D1/P0), capital gains yield, and total return ^0 ¼ P0 and (dividend yield plus capital gains yield) expected for 2009. (Assume that P recognize that the capital gains yield is equal to the total return minus the dividend yield.) Then calculate these same three yields for 2014. d. e. f. How might an investor’s tax situation affect his or her decision to purchase stocks of companies in the early stages of their lives, when they are growing rapidly, versus stocks of older, more mature firms? When does WME’s stock become “mature” for purposes of this question? Suppose your boss tells you she believes that WME’s annual growth rate will be only 12% during the next 5 years and that the firm’s long-run growth rate will be only 4%. Without doing any calculations, what general effect would these growth rate changes have on the price of WME’s stock? Suppose your boss also tells you that she regards WME as being quite risky and that she believes the required rate of return should be 14%, not 12%. Without doing any calculations, determine how the higher required rate of return would affect the price of the stock, the capital gains yield, and the dividend yield. Again, assume that the longrun growth rate is 4%. COMPREHENSIVE/SPREADSHEET PROBLEM 9-22 NONCONSTANT GROWTH AND CORPORATE VALUATION Rework Problem 9-18, Parts a, b, and c, using a spreadsheet model. For Part b, calculate the price, dividend yield, and capital gains yield as called for in the problem. After completing Parts a through c, answer the following additional question using the spreadsheet model. d. TTC recently introduced a new line of products that has been wildly successful. On the basis of this success and anticipated future success, the following free cash flows were projected: Year 1 2 3 4 5 6 7 8 9 10 FCF $5.5 $12.1 $23.8 $44.1 $69.0 $88.8 $107.5 $128.9 $147.1 $161.3 After the tenth year, TTC’s financial planners anticipate that its free cash flow will grow at a constant rate of 6%. Also, the firm concluded that the new product caused the WACC to fall to 9%. The market value of TTC’s debt is $1,200 million, it uses no preferred stock, and there are 20 million shares of common stock outstanding. Use the corporate valuation model approach to value the stock. 298 Part 3 Financial Assets INTEGRATED CASE MUTUAL OF CHICAGO INSURANCE COMPANY 9-23 STOCK VALUATION Robert Balik and Carol Kiefer are senior vice presidents of the Mutual of Chicago Insurance Company. They are codirectors of the company’s pension fund management division, with Balik having responsibility for fixed-income securities (primarily bonds) and Kiefer being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an investment seminar to the mayors of the represented cities; and Balik and Kiefer, who will make the actual presentation, have asked you to help them. To illustrate the common stock valuation process, Balik and Kiefer have asked you to analyze the Bon Temps Company, an employment agency that supplies word processor operators and computer programmers to businesses with temporarily heavy workloads. You are to answer the following questions: a. b. Describe briefly the legal rights and privileges of common stockholders. (1) Write a formula that can be used to value any stock, regardless of its dividend pattern. (2) What is a constant growth stock? How are constant growth stocks valued? (3) What are the implications if a company forecasts a constant g that exceeds its rs? Will many stocks have expected g > rs in the short run (that is, for the next few years)? in the long run (that is, forever)? c. d. Assume that Bon Temps has a beta coefficient of 1.2, that the risk-free rate (the yield on T-bonds) is 7%, and that the required rate of return on the market is 12%. What is Bon Temps’s required rate of return? Assume that Bon Temps is a constant growth company whose last dividend (D0, which was paid yesterday) was $2.00 and whose dividend is expected to grow indefinitely at a 6% rate. (1) What is the firm’s expected dividend stream over the next 3 years? (2) What is its current stock price? (3) What is the stock’s expected value 1 year from now? (4) What are the expected dividend yield, capital gains yield, and total return during the first year? e. Now assume that the stock is currently selling at $30.29. What is its expected rate of return? f. What would the stock price be if its dividends were expected to have zero growth? g. Now assume that Bon Temps is expected to experience nonconstant growth of 30% for the next 3 years, then return to its long-run constant growth rate of 6%. What is the stock’s value under these conditions? What are its expected dividend and capital gains yields in Year 1? Year 4? Suppose Bon Temps is expected to experience zero growth during the first 3 years and then resume its steady-state growth of 6% in the fourth year. What would be its value then? What would be its expected dividend and capital gains yields in Year 1? in Year 4? Finally, assume that Bon Temps’s earnings and dividends are expected to decline at a constant rate of 6% per year, that is, g ¼ –6%. Why would anyone be willing to buy such a stock, and at what price should it sell? What would be its dividend and capital gains yields in each year? Suppose Bon Temps embarked on an aggressive expansion that requires additional capital. Management decided to finance the expansion by borrowing $40 million and by halting dividend payments to increase retained earnings. Its WACC is now 10%, and the projected free cash flows for the next 3 years are –$5 million, $10 million, and $20 million. After Year 3, free cash flow is projected to grow at a constant 6%. What is Bon Temps’s total value? If it has 10 million shares of stock and $40 million of debt and preferred stock combined, what is the price per share? Suppose Bon Temps decided to issue preferred stock that would pay an annual dividend of $5.00 and that the issue price was $50.00 per share. What would be the stock’s expected return? Would the expected rate of return be the same if the preferred was a perpetual issue or if it had a 20-year maturity? h. i. j. k. Chapter 9 Stocks and Their Valuation Access the Thomson ONE problems through the CengageNOW™ web site. Use the Thomson ONE—Business School Edition online database to work this chapter’s questions. Estimating ExxonMobil’s Intrinsic Stock Value In this chapter, we described the various factors that influence stock prices and the approaches that analysts use to estimate a stock’s intrinsic value. By comparing these intrinsic value estimates to the current price, an investor can assess whether it makes sense to buy or sell a particular stock. Stocks trading at a price far below their estimated intrinsic values may be good candidates for purchase, whereas stocks trading at prices far in excess of their intrinsic value may be good stocks to avoid or sell. While estimating a stock’s intrinsic value is a complex exercise that requires reliable data and good judgment, we can use the data available in Thomson ONE to arrive at a quick “back-of-the-envelope” calculation of intrinsic value. Discussion Questions 1. 2. 3. 4. 5. 6. For purposes of this exercise, let’s take a closer look at the stock of ExxonMobil Corporation (XOM). Looking at the COMPANY ANALYSIS OVERVIEW, we can see the company’s current stock price and its performance relative to the overall market in recent months. What is ExxonMobil’s current stock price? How has the stock performed relative to the market over the past few months? Click on “NEWS & EVENTS” on the left-hand side of your screen to see the company’s recent news stories for the company. Have there been any recent events impacting the company’s stock price, or have things been relatively quiet? To provide a starting point for gauging a company’s relative valuation, analysts often look at a company’s price-to-earnings (P/E) ratio. Return to the COMPANY OVERVIEW page. Here you can see XOM’s forward P/E ratio, which uses XOM’s next 12-month estimate of earnings in the calculation. To see its current P/E ratio, click on “FINANCIALS” (on the left-hand side of your screen), scroll down to “WORLDSCOPE” (under Financial Ratios on the left-hand side of your screen), and click on “ANNUAL INCOME STATEMENT RATIOS.” The firm’s current P/E ratio is shown at the top right of your screen. What is the firm’s current P/E ratio? To put XOM’s P/E ratio in perspective, it is useful to see how this ratio has varied over time. Scroll down to the Stock Performance section of this screen. The first two lines of this section show the firm’s P/E ratio using the end-of-year closing price and the 5-year average over time. Is XOM’s current P/E ratio well above or well below its latest 5-year average? Do you have any explanation for why the current P/E deviates from its historical trend? Explain. On the basis of this information, does XOM’s current P/E suggest that the stock is undervalued or overvalued? Explain. To put the firm’s current P/E ratio in perspective, it is useful to compare this ratio with that of other companies in the same industry. To see how XOM’s P/E ratio stacks up to its peers, click on “COMPARABLES” (left-hand side of your screen). Select “KEY FINANCIAL RATIOS.” Toward the bottom of the table, you should see information on the P/E ratio in the section titled “Market Value Ratios.” For the most part, is XOM’s P/E ratio above or below that of its peers? In Chapter 4, we discussed the various factors that may influence P/E ratios. Can any of these factors explain why XOM’s P/E ratio differs from its peers? Explain. If you want to compare XOM to a different set of firms, click on “CLICK TO SELECT NEW PEER SET.” (This appears toward the top of the screen.) In the text, we discussed using the discounted dividend model to estimate a stock’s intrinsic value. To keep things as simple as possible, let’s assume at first that XOM’s dividend is expected to grow at some constant rate over time. If so, the intrinsic value equals D1/(rs – g), where D1 is the expected annual dividend 1 year from now, rs is the stock’s required rate of return, and g is the dividend’s constant growth rate. To estimate the dividend growth rate, it’s helpful to look at XOM’s dividend history. Go back to the COMPANY OVERVIEW page. Select “FINANCIALS”; and under “FINANCIAL RATIOS,” select “WORLDSCOPE” and “ANNUAL INCOME STATEMENT RATIOS.” On your screen at the bottom of the Per Share Data section, you should see 299 300 Part 3 Financial Assets 7. 8. 9. 10. the firm’s annual dividend over the past several years. On the basis of this information, what has been the average annual dividend growth rate? Another way to obtain estimates of dividend growth rates is to look at analysts’ forecasts for future dividends, which can be found under “ESTIMATES” (on the left-hand side of your screen). Near the top of your screen, you should see an area marked “Consensus Estimates.” Use the scroll bar to change from EPS estimates to DPS estimates. What is the median year-end dividend forecast? You can use this as an estimate of D1 in your measure of intrinsic value. Also notice that the last line of this area shows the long-term growth rate. What is the median forecast of the company’s long-term growth rate? You can use this as a forecast of the firm’s dividend growth rate, g. The required return on equity, rs, is the final input needed to estimate intrinsic value. For our purposes, you can assume a number (say, 8% or 9%) or you can use the CAPM to calculate an estimate of the cost of equity using the data available in Thomson ONE. (For more details, look at the Thomson ONE exercise for Chapter 8.) Having decided on your best estimates for D1, rs, and g, you can calculate XOM’s intrinsic value. How does this estimate compare with the current stock price? Does your preliminary analysis suggest that XOM is undervalued or overvalued? Explain. It is often useful to perform a sensitivity analysis, where you show how your estimate of intrinsic value varies according to different estimates of D1, rs, and g. To do so, recalculate your intrinsic value estimate for a range of different estimates for each of these key inputs. One convenient way to do this is to set up a simple data table in Excel. Refer to the Excel tutorial accessed through the CengageNOW™ web site for instructions on data tables. On the basis of this analysis, what inputs justify the current stock price? On the basis of the dividend history you uncovered in Question 6 and your assessment of XOM’s future dividend payout policies, do you think it is reasonable to assume that the constant growth model is a good proxy for intrinsic value? If not, how would you use the available data in Thomson ONE to estimate intrinsic value using the nonconstant growth model? Finally, you can also use the information in Thomson ONE to value the entire corporation. This approach requires that you estimate XOM’s annual free cash flows. Once you estimate the value of the entire corporation, you subtract the value of debt and preferred stock to arrive at an estimate of the company’s equity value. By dividing this number by the number of shares of common stock outstanding, you calculate an alternative estimate of the stock’s intrinsic value. While this approach may take additional time and involves more judgment concerning forecasts of future free cash flows, you can use the financial statements and growth forecasts in Thomson ONE as useful starting points. Go to Worldscope’s Cash Flow Ratios Report (which you find by clicking on “FINANCIALS, WORLDSCOPE”—under “FINANCIAL RATIOS”—and “ANNUAL CASH FLOW RATIOS”). There you will find an estimate of free cash flow per share. While this number is useful, Worldscope’s definition of free cash flow subtracts out dividends per share; therefore, to make it comparable to the measure in this text, you must add back dividends. To see Worldscope’s definition of free cash flow (or any term), go to the top of your screen and click on “GLOSSARY”. In the middle of your screen on the right-hand side, you will see a dialog box with terms. Use the down arrow to scroll through the terms, highlighting the term for which you would like to see a definition. Then click the SELECT button immediately below the dialog box. Chapter 9 Stocks and Their Valuation APPENDIX 9A Stock Market Equilibrium Recall that rX, the required return on Stock X, can be found using the Security Market Line (SML) equation from the Capital Asset Pricing Model (CAPM) as discussed in Chapter 8: rx ¼ rRF þ ðrM  rRF Þbx ¼ rRF þ ðRPM Þbx If the risk-free rate is 6%, the market risk premium is 5%, and Stock X has a beta of 2, the marginal investor will require a return of 16% on the stock: rx ¼ 6% þ ð5%Þ2:0 ¼ 16% This 16% required return is shown as the point on the SML in Figure 9A-1 associated with beta ¼ 2.0. A marginal investor will purchase Stock X if its expected return is more than 16%, will sell it if the expected return is less than 16%, and will be indifferent (will hold but not buy or sell) if the expected return is exactly 16%. Now suppose the investor’s portfolio contains Stock X; he or she analyzes its prospects and concludes that its earnings, dividends, and price can be expected to grow at a constant rate of 5% per year. The last dividend was D0 ¼ $2.8571, so the next expected dividend is as follows: D1 ¼ $2:8571ð1:05Þ ¼ $3 The investor observes that the present price of the stock, P0, is $30. Should he or she buy more of Stock X, sell the stock, or maintain the present position? The investor can calculate Stock X’s expected rate of return as follows: ^r x ¼ D1 $3 þ 5% ¼ 15% þg¼ $30 P0 Expected and Required Returns on Stock X FIGURE 9A-1 Rate of Return (%) SML: ri = rRF + (rM – rRF) bi rX = 16 rˆX = 15 X rM = 11 rRF = 6 0 1.0 2.0 Risk, bi 301 302 Part 3 Financial Assets Equilibrium The condition under which the expected return on a security is just equal to its required return, ^r ¼ r. Also, P^0 ¼ P0 , and the price is stable. This value is plotted on Figure 9A-1 as Point X, which is below the SML. Because the expected rate of return is less than the required return, he or she (and many other investors) would want to sell the stock. However, few people would want to buy at the $30 price; so the present owners would be unable to find buyers unless they cut the price of the stock. Thus, the price would decline, and the decline would continue until the price hit $27.27. At that point, the stock would be in equilibrium, defined as the price at which the expected rate of return, 16%, is equal to the required rate of return: ^r x ¼ $3:00 þ 5% ¼ 11% þ 5% ¼ 16% ¼ rx $27:27 Had the stock initially sold for less than $27.27 (say, $25), events would have been reversed. Investors would have wanted to purchase the stock because its expected rate of return would have exceeded its required rate of return, buy orders would have come in, and the stock’s price would have been driven up to $27.27. To summarize, in equilibrium, two related conditions must hold: 1. A stock’s expected rate of return as seen by the marginal investor must equal its required rate of return: ^r i ¼ ri : 2. The actual market price of the stock must equal its intrinsic value as estimated ^ 0: by the marginal investor: P0 ¼ P ^ 0 > P0 (hence, Of course, some individual investors may believe that ^r i > ri and P they would invest most of their funds in the stock), while other investors might have an opposite view and sell all of their shares. However, investors at the margin establish the actual market price; and for these investors, we must have ^ 0 ¼ P0 . If these conditions do not hold, trading will occur until they do. ^r i ¼ ri and P 9A-1 CHANGES IN EQUILIBRIUM STOCK PRICES Stock prices are not constant—they undergo violent changes at times. For example, on October 27, 1997, the Dow Jones Industrials fell 554 points, a 7.18% drop in value. Even worse, on October 19, 1987, the Dow lost 508 points, causing an average stock to lose 23% of its value on that one day, and some individual stocks lost more than 70%. To see what could cause such changes to occur, assume that Stock X is in equilibrium, selling at a price of $27.27 per share. If all expectations were met exactly, during the next year the price would gradually rise to $28.63, or by 5%. However, suppose conditions changed as indicated in the second column of the following table: VARIABLE VALUE Risk-free rate, rRF Market risk premium, rM – rRF Stock X’s beta coefficient, bX Stock X’s expected growth rate, gX D0 Price of Stock X Original New 6% 5% 2.0 5% $2.8571 $27.27 5% 4% 1.25 6% $2.8571 ? Now give yourself a test: How would the change in each variable, by itself, affect the price; and what new price would result? Every change, taken alone, would lead to an increase in the price. The first three changes together lower rX, which declines from 16% to 10%: Original rx ¼ 6% þ 5%ð2:0Þ ¼ 16% New rx ¼ 5% þ 4%ð1:25Þ ¼ 10% Chapter 9 Stocks and Their Valuation ^0 rises from $27.27 to Using these values, together with the new g, we find that P 1 $75.71, or by 178%: ^ Original P0 ¼ ^ New P0 ¼ $2:8571ð1:05Þ $3:00 ¼ ¼ $27:27 0:16  0:05 0:11 $2:8571ð1:06Þ $3:0285 ¼ ¼ $75:71 0:10  0:06 0:04 Note too that at the new price, the expected and required rates of return will be equal:2 ^r x ¼ $3:0285 þ 6% ¼ 10% ¼ rx $75:71 Evidence suggests that stocks, especially those of large companies, adjust rapidly when their fundamental positions change. Such stocks are followed closely by a number of security analysts; so as soon as things change, so does the stock price. Consequently, equilibrium ordinarily exists for any given stock, and required and expected returns are generally close to equal. Stock prices certainly change, sometimes violently and rapidly; but this simply reflects changing conditions and expectations. There are, of course, times when a stock will continue to react for several months to unfolding favorable or unfavorable developments. However, this does not signify a long adjustment period; rather, it simply indicates that as more new information about the situation becomes available, the market adjusts to it. QUESTIONS 9A-1 9A-2 For a stock to be in equilibrium, what two conditions must hold? If a stock is not in equilibrium, explain how financial markets adjust to bring it into equilibrium. PROBLEMS 9A-1 RATES OF RETURN AND EQUILIBRIUM Stock C’s beta coefficient is bC ¼ 0.4, while Stock D’s is bD ¼ –0.5. (Stock D’s beta is negative, indicating that its return rises when returns on most other stocks fall. There are very few negative beta stocks, although collection agency stocks are sometimes cited as an example.) a. b. 1 If the risk-free rate is 7% and the expected rate of return on an average stock is 11%, what are the required rates of return on Stocks C and D? For Stock C, suppose the current price, P0, is $25.00; the next expected dividend, D1, is $1.50; and the stock’s expected constant growth rate is 4%. Is the stock in equilibrium? Explain and describe what will happen if the stock is not in equilibrium. A price change of this magnitude is by no means rare. The prices of many stocks double or halve during a year. For example, during 2007, Amazon.com, a large online retailer of books, music, and videos, increased in value by 134.8%. On the other hand, E*Trade Financial, a discount brokerage firm, fell in value by 84.2%. 2 It should be obvious by now that actual realized rates of return are not necessarily equal to expected and required returns. Thus, an investor might have expected to receive a return of 15% if he or she had bought Amazon.com or E*Trade Financial stock in 2007; but after the fact, the realized return on Amazon.com was far above 15%, whereas the return on E*Trade Financial was far below. 303 304 Part 3 Financial Assets 9A-2 EQUILIBRIUM STOCK PRICE The risk-free rate of return, rRF, is 6%; the required rate of return on the market, rM, is 10%; and Upton Company’s stock has a beta coefficient of 1.5. a. If the dividend expected during the coming year, D1, is $2.25 and if g ¼ a constant 5%, at what price should Upton’s stock sell? b. Now suppose the Federal Reserve Board increases the money supply, causing the riskfree rate to drop to 5% and rM to fall to 9%. What would happen to Upton’s price? c. In addition to the change in Part b, suppose investors’ risk aversion declines and this, combined with the decline in rRF, causes rM to fall to 8%. Now what is Upton’s price? d. Suppose Upton has a change in management. The new group institutes policies that increase the expected constant growth rate from 5% to 6%. Also, the new management smoothes out fluctuations in sales and profits, causing beta to decline from 1.5 to 1.3. Assume that rRF and rM are equal to the values in Part c. After all these changes, what is its new equilibrium price? (Note: D1 is now $2.27.) 9A-3 BETA COEFFICIENTS Suppose Chance Chemical Company’s management conducted a study and concluded that if it expands its consumer products division (which is less risky than its primary business, industrial chemicals), its beta will decline from 1.2 to 0.9. However, consumer products have a somewhat lower profit margin, and this would cause its constant growth rate in earnings and dividends to fall from 6% to 4%. The following also apply: rM ¼ 9%, rRF ¼ 6%, and D0 ¼ $2.00. a. Should management expand the consumer products division? Explain. b. Assume all the facts given except the change in the beta coefficient. How low would the ^ 0 under the new beta have to fall to cause the expansion to be a good one? (Hint: Set P ^ 0 under the old one and find the new beta that will produce this policy equal to P equality.)