Portfolio Optimization Under Liquidity Costs

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 39 No. 2 2007, 217-233 PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS Rudi Zagst1 § , Dieter Kalin2 1 HVB-Institute of Mathematical Finance Munich University of Technology Boltzmannstr. 3, Garching, D-85748, GERMANY e-mai: zagst@ma.tum.de 2 Insitute of Optimization and Operations Research University of Ulm Helmholtzstr. 18, Ulm, D-89069, GERMANY e-mail: dieter.kalin@uni-ulm.de Abstract: In this paper we examine the problem of optimally structuring a portfolio of assets with respect to transaction costs and liquidity effects. We claim that the intention of the portfolio manager is to maximize the expected net return of his portfolio, i.e. the expected return after costs, under a given limit for the portfolio risk. We show how this problem can be characterized by a convex optimization problem and that it can be solved by an equivalent quadratic optimization problem minimizing the portfolio risk under a given minimum level for the expected net return. The liquidity cost is estimated using intraday data of the German stock market. A case study shows how the results can be applied to practical trading problems. AMS Subject Classification: 91B28 Key Words: portfolio optimization, transaction costs, liquidity effects 1. Introduction The process of performing an optimal asset allocation basically deals with the problem of finding a portfolio that maximizes the expected utility of the investor or portfolio manager. As long as it is supposed that the returns of the portfolio assets follow a normal distribution, the return distribution of any portfolio considered will also be normal. In this case, as is done throughout the Received: May 14, 2007 § Correspondence author c 2007, Academic Publications Ltd. 218 R. Zagst, D. Kalin traditional portfolio theory introduced by Markowitz [22] and Sharpe [26], the problem of finding an expected utility-maximizing portfolio for a risk-averse investor, represented by a concave utility function, can be restricted to finding an optimal combination of the two parameters mean and variance. This dramatically simplifies the whole asset allocation process and is known as meanvariance analysis. It is the aim of the portfolio manager to find a portfolio that maximizes his expected return under a given level of risk or a portfolio that minimizes his risk under a given return level. Risk in this case is measured by the variance of the portfolio return. Unfortunately, a portfolio manager or trader also faces transaction costs reducing the net return of his portfolio. Placing a trade with a broker for execution, the portfolio manager must pay a direct cost of trading no matter if he buys or sells the position. This cost is due to broker commissions, custodial fees, etc. and is also called the explicit cost (EC). Within an optimzation program, EC can be considered by introducing a portfolio turnover constraint, a cost factor that is proportional to the amount of assets bought or sold (see, e.g., Best and Hlouskova [3], Cornuejols [8] and Pogue [24]), a fixed transaction cost (see, e.g., Brennan [5]), a fixed plus a proportional cost, or a proportional cost with lower limit (see, e.g., Lobo et al [19], Maringer [20] and Maringer [21]). Mitchell and Braun [23], Isaenko [11], and Isaenko [12] consider convex (linear, piecewise linear and quadratic) transaction costs with a special focus on illiquid stocks. However, the total cost of the trade also depends on the size of the trade and the broker’s ability to place the required trading volume in the market. If the trading volume is too high, i.e. above a critical trading level, the price of the share may rise (fall) between the investment decision and the complete trade execution if the share is to be bought (sold). Thus, this additional cost occurs because the transaction itself may change the market price of the traded asset and is implied by the actual liquidity situation in the market or the broker’s ability to trade. The difference between the transaction price and what the market price would have been in the absence of the transaction is termed the market impact cost (MIC), per share, of the transaction. The vast amount of data generated by electronic trading raised quite some research both, in academic and industrial communities to understand and quantify the MIC. Representative for many others, BARRA [2] analyzes MIC based on the assets of the S&P 500, Bikker et al [4] on equity trades by a Dutch pension fund, Chan and Lakonishok [6] and [7] examine the price impact of single institutional trades and trade packages performed by large institutional money management firms, Gallagher and Looi [9] examine the MIC incurred by active Australian equity managers, Almgren et al [1] examine MIC based on US PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS 219 stock trades executed by Citigroup equity trading desk. The major findings are that MIC is not negligible and can make up a large proportion of the execution costs and that, beside others, MIC is influenced by the stock price volatility, the size of the trade, the average daily trading volume, the relative package size, and the investment style. Dependence on the investment style of the portfolio manager was also stated in Keim and Madhavan [13] and Keim and Madhavan [14]. However, information about the investment style of a portfolio manager or trader is usually not available in the market and will therefore be neglected here. Furthermore, for the sake of simplicity, we assume that we are dealing within one currency, i.e. that all costs and share prices are already reported in local currency. Despite all the work on MIC and on portfolio optimization with (explicit) transaction cost, very little studies concentrated on the combination of the both, i.e. the optimization of portfolios under the special consideration of MIC. Konno and Wijayanayake [15], [16] and [17] consider non-convex transaction costs and minimal transaction unit constraints. They also allow for the unit price to increase beyond a certain point due to illiquidity and market impact effects. They propose a branch-and-bound algorithm for the resulting d.c. (difference of two convex functions) optimization program and linear programming subproblems for an efficient solution. It is the focus of this paper to propose an easy quadratic optimization model to derive portfolios under the consideration of explicit and market impact costs. In Section 2 we introduce the notion of explicit and implicit transaction cost and define what we understand under the market impact cost. We will show how the market impact cost can be estimated using intraday data of the German stock market in Section 3. In Section 4 we introduce a portfolio optimization problem to find a portfolio with maximum expected net return including explicit and implicit cost under a given maximum level of risk. It is also shown that we can always define an equivalent quadratic optimization problem minimizing risk under a given minimum level of expected net return and hence find an efficient frontier according to the well-known mean-variance theory. We conclude with a practical case study in Section 5. 2. Transaction Costs As already stated, we decompose the total transaction costs into explicit and implicit costs. Explicit costs are directly observable such as broker commissions or custodial fees. Implicit costs are implied by market or liquidity restrictions 220 R. Zagst, D. Kalin and defined as the deviation of the transaction price from the “unperturbed price” that would have prevailed if the trade had not occurred. In other words, market impact cost is the additional price an investor pays for immediate execution. This cost is difficult to measure because the unperturbed price is not observable. Here, the corresponding quote just prior to the transaction is chosen as a proxy for the unperturbed price Si > 0 per share i = 1, . . . , n. Within the problem of portfolio optimization we will assume, for the sake of simplicity, that each share is traded at it is mid price. Therefore, we neglect the cost implied by the bid-ask spread. Hence, the market impact cost is the change in the stock price that only occurs when the number of stocks an investor desires to buy or sell exceeds the number other market participants are willing − to buy (x+ i,min ≥ 0) or sell (xi,min ≥ 0) at that price. Typically, market impact cost would decrease over time because a trader with more time can split up the transaction into smaller transactions that individually exert little or no price pressure. We assume that there is a maximum execution time from which on the market impact cost vanishes. On the other hand, waiting for the complete execution may lead to a loss in oppotunities related to changing market prices or a decaying value of the information responsible for the original portfolio decision. This so-called opportunity cost tends to increase over time. The reflection principle introduced by Hafner [10] states that there is a trade-off between market impact and opportunity cost. It holds under the main assumptions that the liquidity demander and the liquidity provider have the same risk aversion and that liquidity is priced efficiently. For an immediate execution only market impact cost will occur as we have no opportunity cost. As time increases market impact cost will vanish leaving us with opportunity cost only. Due to the reflection principle there is an exact shift from market impact to opportunity cost, i.e. market impact cost for an immediate execution is equal to the opportunity cost for the maximum execution time. Therefore, market impact cost will include a factor for the time the broker needs to execute the position and a factor for the risk of the unknown asset price at which the position can be executed. The first factor will increase with the volume to be traded and decrease with the average (tick or daily) trade size xi > 0 as a measure for the liquidity of the share, the second factor is usually measured in terms of the share’s intraday volatility σi,intraday > 0, i = 1, . . . , n. We assume immediate execution and a market impact cost function of cM I (x∗i ) = λi · Si · σi,intraday ·  + x∗i − x∗i,min xi = ki · Si · x∗i − x∗i,min
+ PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS 221 with ∗ ∈ {+, −}, x∗i denoting the number of shares to be bought (+) or sold (−), and liquidity cost (factor) ki = λi · σi,intraday · 1 . xi We hereby assume that the market impact cost per share i = 1, . . . , n is proportional to the excess traded volume above the critical trade size x∗i,min as well as to the inverse of the   average trade size and hence to the average time for + execution x∗i − x∗i,min /xi . Furthermore, it is proportional to the volatility σi,intraday as well as to a factor λi ≥ 0 which we call the price of liquidity risk for share i and which may depend on the (excess) traded volume or the average trade size. Each choice of λi leads to a different model of liquidity risk. However, we have chosen λi to be constant for the sake of simplicity here. This approach corresponds to the empirical findings stated above and follows the economic approaches in BARRA [2] and Hafner [10]. 3. Estimating the Cost of Liquidity The sample data for estimating the model parameters consists of cleaned tick data from the German stock market between 17-th April 2001 and 5-th June 2001 as it was used by Hafner [10]. Each data record includes date, time (accurate to seconds), bid, ask and last price as well as the corresponding (critical) trade sizes. Only trades during normal market hours, i.e. after 9:00 a.m. and before 8:00 p.m. are considered. To be a trade the cumulative traded volume of the day must have changed. If the trade price is above (below) the latest mid price, the trade is considered as buyer- (seller-) initiated. By definition, the unperturbed price is the latest quote prior to the trade. To calculate the volatility of the stocks in a consistent way we had to consider that the data may be nonsynchronous because some shares were more frequently traded than others. Therefore, the time unit ∆t is chosen such that each stock is traded at least once in each time interval. Given the different trades and their volumes in a specific time interval, the synthetic trade price of the corresponding stock is set to the volume-weighted average trade price and the trading volume to the average trade size in this interval. We then use this synthetic empirical price data to calculate the log-returns and their empirical variance assuming that the expected log-return equals zero. Dividing the variance by ∆t and taking the square root we end up with the stock’s volatility. Given the trade considered is buyer- (seller-) initiated and the trading volume exceeds the ask (bid) size, the 222 R. Zagst, D. Kalin market impact cost is defined to be the absolute difference between the trade price and the ask (bid) price just before the trade multiplied with the liquidity cost (factor). Thus, the only parameter missing is the price of liquidity risk λi , i = 1, . . . , n. This parameter can now be determined using an OLS regression. The results are summarized in Table 1. 4. The Optimization Problem In this section we state the optimization problem which maximizes the net profit over a given planning horizon T under a given maximum level
of risk − ≥ 0 denote ≥ 0 x σmax > 0 for the portfolio return. As above, let x+ i i the number of stocks from asset i = 1, . . . , n which are to be bought (sold) for an optimal portfolio decision. The number of stocks x = (x1 , . . . , xn )′ in − the portfolio is then given by xi = x+ i − xi , i = 1, . . . , n. Furthermore, let c = (c1 , . . . , cn )′ ≥ 0 denote the proportional explicit cost per share, i.e. the explicit cost for a number of x± i shares bought or sold at an unperturbed price Si > 0, i = 1, . . . , n, is given by
cE x± = ci · Si · x± i i . We assume that the portfolio decision is for an immediate execution resulting in an additional market impact cost if the optimal number of stocks to be bought or sold exceeds the critical trade size x± i,min . The prices at the end of the planning horizon are given by the random vector S (T ) = (S1 (T ) , . . . , Sn (T ))′ resulting in a corresponding vector R = (R1 , . . . , Rn )′ for the rate of return with Si (T ) − Si , i = 1, . . . , n. Ri = Si The expected rate of return is denoted by µ = (µ1 , . . . , µn )′ with µi = E [Ri ], i = 1, . . . , n, and the covariance matrix is given by C = (σij )i,j=1,...,n with σij = Cov [Ri , Rj ] and σi2 := σii > 0, i, j = 1, . . . , n. It is assumed that C is positive definit and that the total budget or trading volume is restricted to a cash amount of B > 0 where the part of the budget which is not used for a stock investment can be allocated at a constant rate of return r > 0. Hence, the total cost T C (x, x+ , x− ) of the portfolio is limited by n
X



− + − T C x, x+ , x− = xi · Si + cE x+ ≤ B, i + xi + cM I xi + cM I xi i=1 PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS Share ALLIANZ TELEKOM MUNICH RE DAIMLERCHRYSLER SIEMENS SAP DEUTSCHE BANK E.ON BASF RWE BAYER BMW HYPOVEREINSBANK VOLKSWAGEN INFINEON METRO COMMERZBANK SCHERING HENKEL DEUTSCHE POST THYSSEN KRUPP FRESENIUS DT LUFTHANSA PREUSSAG DEGUSSA LINDE MAN ADIDAS SALOMON EPCOS Last Price 321,45 19,10 328,10 52,50 59,90 159,91 74,70 60,60 44,92 47,75 36,10 38,60 49,41 52,22 24,99 44,80 26,83 59,39 73,25 17,96 15,57 88,60 18,70 34,60 29,50 47,50 26,63 75,15 49,74 Volatility 0,0035% 0,0124% 0,0054% 0,0061% 0,0110% 0,0052% 0,0087% 0,0067% 0,0044% 0,0059% 0,0097% 0,0059% 0,0089% 0,0061% 0,0230% 0,0040% 0,0092% 0,0065% 0,0074% 0,0036% 0,0045% 0,0038% 0,0061% 0,0080% 0,0026% 0,0051% 0,0037% 0,0052% 0,0120% Aver. daily trade size 782040 22696585 455819 2717134 4537083 1657061 3308851 1668693 2599873 1073789 8246891 1916431 986790 1113849 4168153 802733 2465530 441379 393766 710135 971003 205062 1224505 411352 262865 186740 427172 205732 373249 223 Liquidity cost 0,03% 0,07% 0,03% 0,05% 0,07% 0,05% 0,08% 0,01% 0,04% 0,06% 0,01% 0,03% 0,10% 0,04% 0,11% 0,04% 0,07% 0,06% 0,10% 0,01% 0,04% 0,09% 0,07% 0,14% 0,05% 0,15% 0,06% 0,06% 0,20% Table 1: Liquidity cost and market information or equivalently with

+
+ e′ x e + c′ x e+ + x e− + k′ x e+ − x e+ + k′ x e− − x e− ≤1 min min x± xi · Si ± x± i,min · Si ± i · Si ,x ei = , and x ei,min = , i = 1, . . . , n, x ei = B B B ′  ± ± + − , . . . , x e = x e and x e± n,min . Furthermore, the (net) return R(x, x , x ) 1,min 1 min 224 R. Zagst, D. Kalin of the portfolio is given by Pn + −
+ − i=1 xi · Si (T ) + (B − T C (x, x , x )) · (1 + r) − B R x, x , x = B

= R′ x e + r · 1 − e′ x e − (1 + r) · c x e, x e+ , x e− with e = (1, . . . , 1)′ and


+
+ c x e, x e+ , x e− = c′ x e+ + x e− + k′ x e+ − x e+ + k′ x e− − x e− . min min Consequently, the expected portfolio return is


µ x e, x e+ , x e− = µ′ x e + r · 1 − e′ x e − (1 + r) · c x e, x e+ , x e− and the variance of the portfolio return is given by σ 2 (e x) = x e′ C x e. Replacing x e+ = x e+x e− we consider the following optimization problem  ′ µ bx e + r · (1 − e′ x e) − b c′ x e− − b k′ (y + + y − ) → max ,    ′ 2  x e Cx e ≤ σmax ,   
(e + c)′ x e + 2 · c′ x e− + k′ (y + + y − ) ≤ 1 , 2 P1 σmax +  x e+x e− − x e+  min ≤ y ,  −  − −  x e −x emin ≤ y ,   − x e+x e ≥ 0, x e− ≥ 0, y + ≥ 0, y − ≥ 0 , b := (1 + r) · k. Let In denote with µ b := µ − (1 + r) · c, b c := 2 · (1 + r) · c, and k the n− dimensional identity matrix, 0n the n−dimensional matrix filled with zeros,      ′   ′  (e + c)′ 2 · c′ k k         I I −I 0 n n     n    n          0n    In   0n   −In         A1 =   −In  , A2 =  −In  , A3 =  0n  , A4 =  0n          0n    −In   0n   0n         0n 0n −In 0n  0n 0n 0n −In 
′
′ ′ ′ ′ ′ ′ − and b = 1, x e+ , x e . Then we can reformulate our optimin min , 0 , 0 , 0 , 0 mization problem to  b′ x e + r · (1 − e′ x e) − b c′ x e− − b k′ (y + + y − ) → max ,  µ
2 ′ 2 P1 σmax x e Cx e ≤ σmax ,  A1 x e + A2 x e− + A3 y + + A4 y − ≤ b . PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS 225 We generally assume that the expected excess rate of return after cost exceeds the interest we pay for financing the transaction cost, i.e. µi − r − ci − ki > r · (ci + ki ) for all i = 1, . . . , n, or equivalently µ bi > r + (1 + r) · ki for all i = 1, . . . , n. In the special case of no transaction cost this reduces to the well-known assumption that µi > r for all i = 1, . . . , n.
′ 2 Lemma 4.1. Let (e x′ , x e−′ , y +′ , y −′ ) be an optimal solution for P1 σmax . Then,

µ b′ x e + r · 1 − e′ x e −b c′ x e− − b k′ y + + y − > r and for each i ∈ {1, . . . , n} with ci > 0 we have + − x e+ e− i = yi = 0 or x i = yi = 0.
′ 2 Proof. Let (e x′ , x e−′ , y +′ , y −′ ) be an optimal solution for P1 σmax . Further′ ′ −′ ′ −′ more, let (x , x , x , x ) be defined by o n ( 1 , , if i = 1 min σmax σ1 1+c1 +k1 and x− ≡ 0. xi := 0 , if i 6= 1
′ 2 Then, (x′ , x−′ , x′ , x−′ ) is a feasible solution for P1 σmax with

µ b′ x + r · 1 − e′ x − b c′ x− − b k ′ x + x− = µ b1 · x1 + r − r · x1 − b k1 · x1   = r+ µ b1 − r − b k1 · x1 > r. |{z} | {z } >0 >0 ′ Due to the optimality of (e x′ , x e−′ , y +′ , y −′ ) we conclude that

r < µ b′ x + r · 1 − e′ x − b c′ x− − b k′ x + x−

≤ µ b′ x e + r · 1 − e′ x e −b c′ x e− − b k′ y + + y − . ′ Also due to the optimality of (e x′ , x e−′ , y +′ , y −′ ) it is straightforward that y ± = ± ± e+ e± − x e± max{e x± − x e± i > 0 min and y ≥ 0. Assume that x min ; 0} because y ≥ x ′ − − + − and x ei > 0 for some i ∈ {1, . . . , n}. Define (b x, x b , yb , yb ) by  + e− , if j = i, x e+ e− ei − x  x i , i ≥x i + + e− 0 , if j = i, x ei < x x bj := i ,  + x ej , if j 6= i , 226 R. Zagst, D. Kalin and  e− , if j = i, x e+  0 i , i ≥x − + + − − x e , if j = i, x e < x e x e x b− := i i i i , j  − x ej , if j 6= i , for j = 1, . . . , n and x b := x b+ − x b− . Then, x e=x b, x b± < x e± , x b+ + x b− < x e+ + x e−
′ 2 with and hence, (b x, x b− , y + , y − ) is a feasible solution for P1 σmax µ b′ x b + r · (1 − e′ x b) − b c′ x b− − b k′ (y + + y − ) > >µ b′ x e + r · (1 − e′ x e) − b c′ x e− − b k′ (y + + y − ) ′ which is a contradiction that (e x′ , x e−′ , y +′ , y −′ ) is an optimal
to the assumption + − 2 ei = 0 or x ei = 0 and consequently y + = solution for P1 σmax . Hence, x


+
+ + + x e+ − x e+ = −e x+ = 0 or y − = x e− − x e− = −e x− = 0.  min min min min According to the proof of Lemma 4.1 there is always an optimal solution
+ − 2 , σmax > 0, with x e+ e− x e for P1 σmax i = yi = 0 or x i = yi = 0 for each i ∈ {1, . . . , n}, even if the corresponding ci = 0. Lemma 4.1 genaralizes the results of Lemma 2.1 in Best and Hlouskova [3], who assume k ≡ 0, for the case k ≥ 0.
′ 2 Lemma 4.2. Let (e x′ , x e−′ , y +′ , y −′ ) be an optimal solution for P1 σmax . 2 . Then, x e′ C x e = σmax
′ 2 iff it is a Proof. (e x′ , x e−′ , y +′ , y −′ ) is an optimal solution for P1 σmax feasible solution and there are non-negative u1 , u e such that the following KuhnTucker conditions are satisfied: (1) (2) (3) (4) (5) (6) e = 0, µ b − r · e + 2 · u1 · C x e + A′1 u ′ −b c + A2 u e = 0, −b k + A′3 u e = 0, ′ b e = 0,
−k + A4 u ′ 2 u1 · x e Cx e − σmax = 0, ′ − u e (A1 x e + A2 x e + A3 y + + A4 y − − b) = 0 . Adding (5) and (6) we get
2 e+u e′ A2 x e− + u e′ A3 y + + u e′ A4 y − − u e′ b 0 = −u1 · σmax + u1 · x e′ C + u e′ A1 x and thus, using (1)-(4):
2 (7) −u1 · x e′ C x e + σmax − (b µ − r · e)′ x e+b c′ x e− + b k′ (y + + y − ) − u e′ b = 0. PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS 227 2 . Then, using (5), we get u = 0 and thus from (7): Assume that x e′ C x e < σmax 1
(b µ − r · e)′ x e−b c′ x e− − b k′ y + + y − + |{z} u e′ b = 0 , ≥0 which leads us to

µ b′ x e + r · 1 − e′ x e −b c′ x e− − b k′ y + + y − ≤ r. This is a contradiction to the statement in Lemma 4.1 and thus 2 x e′ C x e = σmax .  Let us now fix a minimum level µmin > r for the expected portfolio return and consider the quadratic optimization problem  ′ e Cx e → min ,  x P2 (µmin ) µ b′ x e + r · (1 − e′ x e) − b c′ x e− − b k′ (y + + y − ) ≥ µmin ,  A1 x e + A2 x e− + A3 y + + A4 y − ≤ b. Then we can proof the following analogon to Lemma 4.2. ′ Lemma 4.3. Let (b x′ , x b−′ , yb+′ , yb−′ ) be an optimal solution for P2 (µmin ). ′ Then, x b Cx b > 0 and

µ b′ x b + r · 1 − e′ x b −b c′ x b− − b k′ yb+ + yb− = µmin . ′ Proof. Let (b x′ , x b−′ , yb+′ , yb−′ ) be an optimal solution for P2 (µmin ) and as′ sume that x b Cx b = 0. Because C is positive definit this is equivalent to x b ≡ 0. Thus, !
′ ′ b′ yb+ + yb− ≤ r µmin ≤ µ bx b + r · 1 − |{z} c′ x b b− − k ex b − |{z} |{z} | {z } =0 =0 ≥0 ≥0 ′ which is a contradiction to our assumption µmin > r. Now, (b x′ , x b−′ , yb+′ , yb−′ ) is an optimal solution for P2 (µmin ) iff it is a feasible solution and there are non-negative v1 , ve such that the following Kuhn-Tucker conditions are satisfied: (1′ ) (2′ ) (3′ ) (4′ ) 2 · Cx b − v1 · (b µ − r · e) + A′1 ve = 0 , ′ v1 · b c + A2 ve = 0 , v1 · b k + A′3 ve = 0 , ′ v1 · b k  + A4 ve = 0 ,  (5′ ) v1 · µmin − µ b′ x b − r · (1 − e′ x b) + b c′ x b− + b k′ (b y + + yb− ) = 0 , (6′ ) ve′ (A1 x b + A2 x b− + A3 yb+ + A4 yb− − b) = 0 . 228 R. Zagst, D. Kalin Adding (5′ ) and (6′ ) we get

− 0 = v1 · (µmin − r) + ve′ A1 − v1 · (b µ − r · e)′ x b + ve′ A2 + v1 · b c′ x b     v ′ A4 + v1 · b k′ yb− − ve′ b + ve′ A3 + v1 · b k′ yb+ + e and thus, using (1′ )-(4′ ): (7′ ) v1 · (µmin − r) − 2 · x b′ C x b − ve′ b = 0. Assume that µmin < µ b′ x b + r · (1 − e′ x b) − b c′ x b− − b k′ (b y + + yb− ). Then, using (5′ ), we get v1 = 0 and thus from (7′ ): ve′ b = 0. 2 · |x b′{z Cx b} + |{z} ≥0 ≥0 Because C is positive definite, we conclude that x b ≡ 0 and thus !
k′ yb+ + yb− ≤ r c′ x b b− − b µmin ≤ µ b′ x b + r · 1 − |{z} e′ x b − |{z} |{z} | {z } =0 =0 ≥0 ≥0 in contradiction to our assumption µmin > r. Hence

µ b′ x b + r · 1 − e′ x b −b c′ x b− − b k′ yb+ + yb− = µmin . 
2 Theorem 4.4.
Let µ∗ σmax denote the maximum value of the objective 2 2 function in P1 σmax > 0. Furthermore, let σ ∗2 (µmin ) denote the with σmax minimum value of the objective function in P2 (µmin ) with µmin > r. Then,


2 2 µ∗ σ ∗2 (µmin ) = µmin and σ ∗2 µ∗ σmax = σmax .
′ Proof. Let (e x′ , x e−′ , y +′ , y −′ ) be an optimal solution for P1 σ ∗2 (µmin ) . Then, using Lemma 4.2, x e′ C x e = σ ∗2 (µmin ). Furthermore, let (b x′ , x b−′ , yb+′ , ′ −′ ′ ′ −′ +′ −′ yb ) be an optimal solution x ,x b , yb , yb ) is a feasible
for P2 (µmin ). Then, (b ∗2 solution for P1 σ (µmin ) and, using Lemma 4.3,


µ∗ σ ∗2 (µmin ) = µ b′ x e + r · 1 − e′ x e −b c′ x e− − b k′ y + + y −

b′ yb+ + yb− = µmin . ≥ µ b′ x b + r · 1 − e′ x b −b c′ x b− − k ′ Hence, (e x′ , x e−′ , y +′ , y −′ ) is a feasible solution for P2 (µmin ) with x e′ C x e = σ ∗2 (µmin ) and thus an optimal solution for P2 (µmin ). Therefore, again using Lemma 4.3,


µ∗ σ ∗2 (µmin ) = µ b′ x e + r · 1 − e′ x e −b c′ x e− − b k′ y + + y − = µmin . PORTFOLIO OPTIMIZATION UNDER LIQUIDITY COSTS 229

′ 2 . Then, Now, let (b x′ , x b−′ , yb+′ , yb−′ ) be an optimal solution for P2 µ∗ σmax using Lemma 4.3,


2 µ b′ x b + r · 1 − e′ x b −b c′ x b− − b k′ yb+ + yb− = µ∗ σmax .
′ 2 Furthermore, let (e x′ , x e−′ , y +′ , y −′ ) be an optimal solution for P1 σmax . Then,

′ ∗ 2 ′ −′ +′ −′ (e x ,x e , y , y ) is a feasible solution for P2 µ σmax and, using Lemma 4.2,

2 2 σ ∗2 µ∗ σmax . =x b′ C x b≤x e′ C x e = σmax
′ 2 Hence, (b x′ , x b−′ , yb+′ , yb−′ ) is a feasible solution for P1 σmax with µ b′ x b+r·
′ + − ∗ 2 ′ ′ − b y + yb ) = µ σmax and thus an optimal solution for (1 − e x b)
− b cx b − k (b 2 P1 σmax . Therefore, again using Lemma 4.2,

2 2 σ ∗2 µ∗ σmax .  =x b′ C x b = σmax Theorem 4.4 is consistent to Theorem 3 in Krokhmal et al [18]. However, we had to give a separate proof because our parameters u1 and v1 may not be positive. Theorem 4.5. µmin > r. The efficient frontier µmin → σ ∗ (µmin ) is convex for all ′ Proof. Let λ ∈ [0, 1], (b x′ , x b−′ , yb+′ , yb−′ ) be an optimal solution for P2 (µmin ) ′ ′ −′ +′ −′ and (x , x , y , y ) be an optimal solution for P2 (µmin ). Then,       x x (λ) x b  x− (λ)   −   b−    + (1 − λ) ·  x  =λ· x + +  y (λ)   y+   yb  y− y − (λ) yb− is a feasible solution for P2 (λ · µmin + (1 − λ) · µmin ) and thus, using the inequality of Cauchy-Schwartz, σ ∗2 (λ · µmin + (1 − λ) · µmin ) ≤ x (λ)′ Cx (λ) = λ2 · x b′ C x b + 2 · λ · (1 − λ) · x b′ Cx + (1 − λ)2 · x′ Cx √ √ b′ C x b · x′ Cx + (1 − λ)2 · x′ Cx = ≤ λ2 · x b′ C x b + 2 · λ · (1 − λ) · x 2  √ √ b′ C x b + (1 − λ) · x′ Cx = (λ · σ ∗ (µmin ) + (1 − λ) · σ ∗ (µmin ))2 .  λ· x Setting r = 0 we can easily see that the statements of Lemmas 4.1 and 4.2 as well as those of Theorems 4.4 and 4.5 also hold if there is no possibility of a riskless investment. 230 R. Zagst, D. Kalin 5. Case Study For studying the effect of market impact cost we use the same two-year time series of daily price data as Hafner [10] ending exactly at the same day for which the market impact cost was estimated, i.e. daily price data from 4-th June 1999 until 5-th June 2001. For the sake of simplicity we assume that the problem of the trader or portfolio manager is to decide on a portfolio consisting of the chemistry shares of BASF and BAYER and a riskless investment only. Given a maximum level for the volatility of 25% and a planning horizon of 1 year, the correlation matrix, the annualized standard deviation (STD), the expected rate of return (Exp. return) as well as the explicit cost (EC), the liquidity cost factor (Liq. cost), and the critical trade level (Crit. tr. level) are shown in Table 2. Correlation BASF BAYER STD Exp. return EC Liq. cost Crit. BASF 1,00 0,66 30,56% 8,45% 0 0,04% 5100 BAYER 0,66 1,00 28,69% 7,87% 0 0,01% 200 tr. level Table 2: Market information It is assumed that the critical trade level is the same, no matter if the stock is to be bought or sold. The riskless rate of return is 2% and the budget is increased fom 1000 EUR to 10 Mio. EUR by a factor of 10 for each step. If transaction cost is neglected, the structure of the optimal portfolio does not depend on the budget at all and is given by (xBASF ; xBAY ER ; xRiskless ) = (48, 38%; 44, 10%; 7, 52%) with an expected rate of return equal to 7, 71%. If we consider liquidity cost, the optimal portfolio changes with increasing budget. For a budget of 1.000 and 10.000 EUR there are no liquidity costs. For a budget of 100.000 EUR there is liquidity cost for BAYER only due to the lower critical trade level. Therefore, the BASF share is overweighted relative to the optimal portfolio without liquidity cost and the weigth for BAYER is reduced. However, if we increase the budget to 1 Mio. EUR, there is liquidity cost for half of the BASF shares and nearly all BAYER shares. Nevertheless, the higher liquidity cost for BASF becomes dominant and BAYER is now overweighted instead of BASF. 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