OPTIMAL PORTFOLIO SELECTION: A NOTE (Draft) Ignacio Vélez-Pareja ivelez@javeriana.edu.co Department of Management Universidad Javeriana Bogotá, Colombia Working Paper N 8 First version: August 23, 2000 This version: August 8, 2001 OPTIMAL PORTFOLIO SELECTION: A NOTE (Draft) Abstract Usually in financial textbooks and courses the theory of portfolio selection is taught in a strictly theoretical way. There is a model (Markowitz) that stipulates that an investor has preferences and that she will choose the best portfolio, given her preference curves and an efficient frontier. On the other hand, the Capital Asset Pricing Model (CAPM) is presented as it is: a genial idea that served to simplify and to make operative the Markowitz setup. Most students and practitioners conclude that those models are just inapplicable theory. This is the most rational behavior one can expect. What can an investor do with the textbook recipes to configure an optimal portfolio? Very little. My purpose with this note is to rescue a simple procedure presented by Black (1972), Merton (1973) and later by Levy and Sarnat (1982), Elton and Gruber (1995) and Benninga (1997). They just propose that the optimal portfolio can be found maximizing the slope of the line that joins the point of risk-free return and the efficient frontier. When this maximum tangent is reached, that line is the capital market line (CML) (it is tangent to the efficient frontier). This is a simple procedure that does not require one to calculate the efficient frontier and is an easy task with Excel Solver. It is just one point of the efficient frontier. An example is presented. Keywords CAPM, efficient frontier, porfolio selection, capital market line, optimal portfolio JEL Classification: G11,G12 OPTIMAL PORTFOLIO SELECTION: A NOTE (Draft) Usually in financial textbooks and courses, portfolio selection is taught in a strictly theoretical way. There is a model (Markowitz) that stipulates that an investor has preferences and that she will choose the best portfolio, given her preference curves and an efficient frontier. On the other hand, the Capital Asset Pricing Model (CAPM) is presented as it is: an ingenious idea that served to simplify and to make operative the Markowitz setup. Most students and practitioners conclude that those models are just inapplicable theory. This is the most rational behavior one can expect. What can an investor do with the textbook recipes to configure an optimal portfolio? Very little. My purpose with this note is to rescue a simple procedure presented by Black (1972), Merton (1973) and later by Levy and Sarnat (1982), Elton and Gruber (1995) and Benninga (1997). They just propose that the optimal portfolio can be found maximizing the slope of the line that joins the point of risk-free return and the efficient frontier. When this maximum tangent is reached, that line is the capital market line (CML) (it is tangent to the efficient frontier). This is a simple procedure that does not require one to calculate the efficient frontier and is an easy task with Excel Solver. It is just one point of the efficient frontier. An example is presented. How to calculate the optimal portfolio What is an optimal portfolio? It is, according to the CAPM theory, a portfolio that lies in the efficient frontier and combined with certain proportion of risk free investment, and given a desired risk level, maximizes the return of the combined portfolio. This definition is valid even if the desired risk level is less than the minimum defined by the efficient frontier. Given the risk free rate of return, how to determine the optimal risky portfolio? That optimal risky portfolio is just the point of tangency between the Capital Market Line and the efficient frontier. As this optimal portfolio has to lie along the efficient frontier, then the point of tangency is located at the line with the maximum tangent between that line and the horizontal line. This solution is very good because it is not easy to determine the indifference curves for each decision maker. However, as it was said above, it is not necessary to generate the indifference curves nor even the efficient frontier given the Separation Theorem posited by Tobin. According to the CAPM theory, the investor will prefer a position in the “market portfolio” either levered or unlevered. Then, the optimal portfolio is given by this optimization problem. 3 Max tnθ = Rm − r m m ∑ ∑α α σ k j kj k=1 i=1 s.t. m ∑α i =1 i=1 where αi is the is weight of stock i in the portfolio, σkj is the covariance between stocks k and j, Rm is the portfolio return, r is the risk-free rate of return and m is the number of stocks in the analysis. The restriction that the α’s be positive might be included. In that case there is no short position. This idea can be seen in the next figure Figure 1. Capital Market Line, Efficient frontier and optimal portfolio θ Return Capital Market Line Optimal portfolio m Standard Deviation The solution to his optimization problem produces the αj‘s and hence the optimal portfolio return. Done this, the investor will select the desired risk level (for instance, she will reduce the risk beyond the minimum defined by the efficient frontier) combining the optimal portfolio with an appropriate portion of risk free investment. I have been examining the solution for historical data from the Bolsa de Bogota (Bogota Stock Exchange), and the resulting optimal portfolio has been composed for a few stocks (in some cases the optimal solution is comprised of only one stock). This apparently contradicts the basic idea behind portfolio selection: a high degree of diversification. However, when contrasted with the real practice performed by stock traders, they intuitively form very small portfolios with predominance of one or two stocks. The preliminary results of this work, performed by Professor Irina 4 Dubova and four Business Administration senior students show that the optimal solutions for 42 portfolios (one for every quarter and redefined quarterly) tend to confirm this assertion. An example Suppose four stocks with the following returns: Month 1 2 3 4 5 6 7 8 9 10 11 12 Average Variance Standard Deviation Weight Stock 1 18.25% 2.03% -11.15% -14.50% 21.37% 14.28% 11.50% -6.11% -2.81% -14.23% -10.71% 15.15% 1.92% 1.68% 12.95% Stock 2 19.13% 2.14% -5.27% -6.27% 4.01% 4.00% 9.40% -4.04% -1.07% -8.56% -8.83% 10.22% 1.24% 0.68% 8.23% Stock 3 18.87% 13.42% 2.32% -10.81% -21.58% 12.57% 15.42% -6.32% 5.71% 4.00% 2.12% -24.44% 0.94% 1.83% 13.54% Stock 4 7.59% -7.83% -25.11% -10.08% 11.54% -13.72% -18.73% 11.50% 18.72% 9.25% 20.02% 15.72% 1.57% 2.26% 15.03% 25% 25% 25% 25% The excess return matrix is: Month Stock 1 Stock 2 Stock 3 Stock 4 1 16.33% 17.89% 17.93% 6.02% 2 0.11% 0.90% 12.48% -9.40% 3 -13.07% -6.51% 1.38% -26.68% 4 -16.42% -7.51% -11.75% -11.65% 5 19.45% 2.77% -22.52% 9.97% 6 12.36% 2.76% 11.63% -15.29% 7 9.58% 8.16% 14.48% -20.30% 8 -8.03% -5.28% -7.26% 9.93% 9 -4.73% -2.31% 4.77% 17.15% 10 -16.15% -9.80% 3.06% 7.68% 11 -12.63% -10.07% 1.18% 18.45% 12 13.23% 8.98% -25.38% 14.15% The transpose matrix is found by the Excel Search and Reference function, =TRANSPOSE(Matrix). 5 Month 1 2 3 4 5 6 7 8 9 10 11 12 Stock 1 16.33% 0.11% -13.07% -16.42% 19.45% 12.36% 9.58% -8.03% -4.73% -16.15% -12.63% 13.23% Stock 2 17.89% 0.90% -6.51% -7.51% 8.16% -5.28% -2.31% -9.80% -10.07% Stock 3 17.93% 12.48% Stock 4 2.77% 2.76% 1.38% -11.75% -22.52% 11.63% 14.48% -7.26% 6.02% -9.40% -26.68% -11.65% 4.77% 9.97% -15.29% -20.30% 9.93% 17.15% 3.06% 8.98% 1.18% -25.38% 7.68% 18.45% 14.15% By matrix multiplication and dividing by the number of observations (n=12), then the covariance matrix is: Stock 1 Stock 2 Stock 3 Stock 4 Stock 1 Stock 2 Stock 3 Stock 4 0.01676831 0.00936991 -0.00042371 0.00097519 0.00936991 0.00677348 0.00213887 -0.00038209 -0.00042371 0.00213887 0.01833034 -0.00758114 0.00097519 -0.00038209 -0.00758114 0.02258129 1. Define a weight vector for each stock Stock 1 25% Stock 2 25% Stock 3 25% Stock 4 25% 2. Calculate the average return for the portfolio. It is the scalar product of the weight vector times the return vector (the return vector is the average return for the stocks in the first table). In Excel use SUMPRODUCT. For this level of participation the average return of the portfolio is 1.42%. 3. Multiply the weight vector by the covariance matrix (you will obtain a vector). Use the Excel function for matrix multiplication. In the example. Stock 1 Weights vector x covariance matrix Stock 2 Stock 3 Stock 4 0.00667243 0.004475046 0.00311609 0.00389831 4. The portfolio variance is calculated as the scalar product of the participation or weight vector times the vector obtained in 3. In this example the portfolio variance is 0.00454047. The portfolio standard deviation is the square root of the variance. 5. Assume that the risk free rate is 1.5%, then construct the tangent: tn θ = Rm − r m m ∑∑α α σ k j kj k=1 i=1 6. Use Solver to maximize the tangent subject to the conditions that the sum of the weights is one and the weights are nonnegative. 6 7. As an Excel operation (the image is in Spanish, but Excel users will recognize the correspondent words and functions) as in figures 2 and 3 Figure 2. Setup for the optimal portfolio problem. Figure 3. Solver for optimal portfolio. 8. When you press the Resolver (Solve) button, you get the optimal portfolio composition. Stock 1 Weights Stock 2 92.18% 0.00% 7 Stock 3 0.00% Stock 4 7.82% This produces a portfolio return of 1.90% and a portfolio variance 0.12052486. Bibliographic References Benninga, Simon Z. (1997),Financial Modeling, MIT Press. Black, F. (1972). "Capital Market Equilibrium with Restricted Borrowing." Journal of Business 45 (July), 444-455. Levy, Haim and Marshall Sarnat (1982), Capital Investment and Financial Decisions, 2 nd Ed. Prentice Hall. Merton, R. C. (1973), "An Analytic Derivation of the Efficient Portfolio Frontier." Journal of Financial and Quantitative Analysis 7 (September), 1851-1872. Elton Edwin J. and Martin Jay Gruber (1995), Modern portfolio theory and investment analysis, Wiley. 8 View publication stats