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Mean–variance portfolio optimal problem under concave transaction cost
Applied Mathematics and Computation 174 (2006) 1–12 www.elsevier.com/locate/amc
Mean–variance portfolio optimal problem under concave transaction cost Hong-Gang Xue
a,*
q
, Cheng-Xian Xu b, Zong-Xian Feng
c
a
c
School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710049, China b Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China JinHe Center for Economic Research, Xi’an Jiaotong University, Xi’an 710049, China
Abstract In this paper, the classical mean–variance portfolio model is modified for calculating a globally optimal portfolio under concave transaction costs. A non-decreasing concave function is employed to approximate origin transaction cost function. The resulting model is a D-C (difference of two convex functions) programming and a branch and bound algorithm is designed to solve the problem. A series of numerical experiments on the model is presented. The history data of nine stocks in Shan Xi province is used in experiments, and efficient frontiers generated from the resulting model with different limitations on investments are presented to show the effect of the model and the efficiency of the algorithm solving the model. 2005 Elsevier Inc. All rights reserved. Keywords: Mean–variance; Concave transaction cost; Globally optimal portfolio; Branch and bound algorithm; Efficient frontier
q
This work is supported by National Natural Key Product Foundations of China 10231060. Corresponding author. Address: Faculty of Science, XiÕan Jiaotong University, XiÕan 710049, China. E-mail address: [email protected] (H.-G. Xue). *
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.05.005
2
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
1. Introduction The mean variance model for portfolio selection was first proposed by Markowitz in 1952, and has been used by many companies and investors in financial market. Assume that there exist n risky assets with expected return rate ri of ith asset, i = 1, 2, . . . ,n. Then the model can be expressed as [8] min
l T x Vx ð1 lÞRT x 2
s.t.
x 2 D \ S;
ðM-VÞ
where x = (x1, x2, . . . , xn)T denotes the investment weight vector, R ¼ T ðr1 ; r2 ; . . . ; rn Þ the expected return rate vector and V = (rij)n·n the variance– covariance matrix of the return distributions of these n assets. Parameter l 2 (0, 1] reflects the investorÕs attitude towards risk (which is measured by the variance of the portfolio). The set D is a polytope that is generally formulated by a series of linear P inequalities such as Ax 6 b. One typical case is the capital budget constraint ni¼1 xi ¼ 1. The set S = {li 6 xi 6 ui, i = 1, 2, . . . , n} is a rectangle and gives the low bound l = (l1, l2, . . . , ln)T and the upper bound u = (u1, u2, . . . , un)T on the vector x. In general, the variance–covariance matrix V is symmetric and positive definite, and problem (M-V) is a convex quadratic programming. Effective methods are available for the solution of problem (M-V) [9]. The transaction cost does not considered in problem (M-V). However, recent researches show that the transaction cost is considerable in some investment actions. Konno and Wijayake [5] studied a portfolio optimization problem under concave transaction cost. The absolute deviation (AD) of the portfolioÕs return distribution is used to measure the portfolioÕ risk, and the resulting problem is a Mean–AD (MAD) model under concave transaction cost. A branch-and-bound algorithm is proposed to solve the MAD model, and linear approximations are used to result in linear sub-programming [5]. Krokhmal, Palmquist and Uryasev [7] also took the transaction cost into account in their portfolio optimization problem in which the CVaR is used to measure the portfolioÕs risk and transaction cost is expressed as a linear function. In this paper, we consider the M–V model (M-V), but the transaction will included into the model. Since the concave transaction cost is employed, the resulting model is a D-C (difference of two convex functions) programming, and it is difficult to find itÕs global solution. However, the transaction cost funcPn tion C(x) is separable, i.e., CðxÞ ¼ i¼1 C i ðxi Þ, a branch and bound algorithm that is similar to [1,5,10,13,14] is proposed to solve the problem by using its structure. Konno [5,6] proposed a branch and bound algorithm based on linear underestimation to the concave transaction cost function to solve large scale portfolio optimal problems with MAD model and concave transaction cost.
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
3
Numerical tests on 200 assets and 60 simulation sceneries showed that the algorithm can solve large scale problems effectively. Subdivision processes play an important role in proving the convergence of a branch and bound method. The concept of ‘‘normal rectangular subdivision’’ introduced in [3,4,11] will be used in branch and bound algorithms for the solution of the problem with set S. The class of normal subdivision includes wellknow exhaustive bisection, x-subdivision, adaptive bisection. Xue and Xu [13,14] proposed a branch and bound algorithm based on the largest distance bisection, and it is proved that the largest distance bisection is also a normal rectangular subdivision. Solve these problems from 20 to 160 variables effectively. Some evidence showed that the x-subdivision is better than the exhaustive bisection and the adaptive bisection [10], while numerical tests in [13,14] showed that the largest distance bisection is more efficient than the x-subdivision. Hence, we will use the largest distance bisection in the proposed branchand-bound algorithm for the solution of the resulting D-C programming problems. The rest of the paper is organized as follows. In Section 2, we will construct the mean–variance model under concave transaction cost and describe the new branch and bound algorithm for solving it. In Section 3, we will disrupt the detail process of the largest distance bisection method. In Section 4, we will compare the case that the transaction cost exist with the other case that the transaction cost exist, and we will analysis the effective of the investment upper bound and low bound for efficient frontier by numerical tests. Conclusions are given in Section 5.
2. The M–V portfolio problem under concave transaction cost In this section, we will give the mean–variance portfolio problem under concave transaction costs. The transaction cost associated with a portfolio x = (x1, x2, . . . , xn)T is usually defined as the sum of individual transaction cost on each asset n X C i ðxi Þ; CðxÞ ¼ i¼1
where Ci(xi) is the individual cost on the ith asset. Ci(xi) generally is a nondecreasing concave function up to certain point ^xi . It is assumed that Ci(xi), i = 1, 2, . . . ,n are smooth enough. Then the first-order derivative of the transaction cost function has the property C 0i ðxi Þ P 0 and the second-order derivative of Ci(x) satisfies C 00i ðxi Þ < 0 for all i = 1, 2, ,n. Such properties of the cost function coincide with the rule of margin cost decreasing, and can be described by the curve in Fig. 1.
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H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
Fig. 1. concave transaction cost.
However, the unit transaction cost increases beyond the point ^xj , due to the ‘‘ill-liquidity’’ effects. Konno and Wijayanayake explained this economic phenomena in [5], and made an assumption that the amount of investment on the jth asset is below the critical point ^xj . The value of ^xj can be obtained by finding the inflexion point of original transaction cost function. Under such an assumption, the transaction cost is a well specified concave function and can be calculated by the transaction cost table of the agent. Under the concave transaction cost, the portfolioÕs net return is RTx C(x), and the portfolio optimal problem can be expressed as follows: min
f ðxÞ
s.t.
x 2 D \ S;
ðQ0 Þ
where the objective function f ðxÞ ¼ pðxÞ þ uðxÞ; l pðxÞ ¼ xT Vx 2 is the convex part which denotes the portfolioÕs risk, and n X ui ðxi Þ; ui ðxi Þ ¼ ð1 lÞðC i ðxÞ ri xi Þ; i ¼ 1; 2; . . . ; n uðxÞ ¼
ð1Þ
ð2Þ
i¼1
is the portfolioÕs net return. Problem (Q0) is a D-C programming. The constraints in (Q0) is the same as in problem (M-V). In general, 1 6 ui 6 ^xi . If short sells are not permitted, then li P 0, else li 6 0. If we set n 1 xP weight vector x 2 S, and i ¼ n for all i = 1, 2, . . . , n, then the 1 investment n 1 T x ¼ 1, so x 2 D. That is, x ¼ ; . . . ; is a feasible point, and hence, i i¼1 n n the feasible set is nonempty. Theorem 2.1. Suppose the concave transaction cost functions Ci(xi) i = 1, 2, . . . , n are continues, then the global solution of the problem (Q0) exists.
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
5
Proof. It is clear that the feasible set D \ S is nonempty, closed and bounded. While the assumption implies that the objective function is continues on the feasible region. Therefore, the conclusion comes from the fact that the minimum exists for a continuous function on a closed and bounded nonempty set. h
3. The branch and bound algorithm In this section, we will present the proposed branch and bound method in which the largest distance rectangular subdivision process will be used to divide problems into sub-problems. 3.1. The description of the algorithm Let S0 = {li 6 xi 6 ui, i = 1, 2, . . . ,n} be the initial rectangle. We replace the individual concave functions ui(xi) in u(x) by underestimated linear functions w0i ðxi Þ over S0 (see Fig. 2) w0i ðxi Þ ¼ di xi þ gi ;
i ¼ 1; 2; . . . ; n;
ð3Þ
where di ¼
ui ðui Þ ui ðli Þ ; ui li
gi ¼ ui ðli Þ di li ;
i ¼ 1; 2; . . . ; n.
ð4Þ
Let g0 ðxÞ ¼ pðxÞ þ w0 ðxÞ; then g0(x) Pnis the convex envelope of the function f(x) over the set S0, where w0 ðxÞ ¼ i¼1 w0i ðxi Þ. We solve the quadratic underestimated approximation to (Q0),
Fig. 2. The underestimated linear function.
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minfg0 ðxÞ ¼ pðxÞ þ w0 ðxÞjx 2 D; l 6 x 6 ug.
ðQ0 Þ
(Q0 ) is a convex quadratic programming problem. The active set method in [2,12] can be used to effectively solve ðQ0 Þ. Let x0 be an optimal solution ðQ0 Þ, then we obtain an lower bound g0 ðx0 Þ and a upper bound f ðx0 Þ of the optimal value f(x*) of the problem (Q0) according to the following theorem. Theorem 3.1. Let x0 be an optimal solution of problem ðQ0 Þ and x* be a global optimal solution of problem (Q0). Then the following relations hold g0 ðx0 Þ 6 f 6 f ðx0 Þ;
ð5Þ
where f* = f(x*). Proof. It follows from g0(x) 6 f(x) "x 2 [l, u], that: g0 ðx0 Þ ¼ minfg0 ðxÞjx 2 D; l 6 x 6 ug 6 minff ðxÞjx 2 D; l 6 x 6 ug ¼ f 6 f ðx0 Þ. This gives the conclusion.
h
Theorem 3.1 indicates that if uðx0 Þ w0 ðx0 Þ 6
ð6Þ
is satisfied with a given tolerance , then x0 is an approximate solution of (Q0) with error less than . If (6) does not hold, we will use an NRS process to divide the problem (Q0) into two subproblems: minff ðxÞjx 2 D; x 2 S 1 g
ðQ1 Þ
minff ðxÞjx 2 D; x 2 S 2 g;
ðQ2 Þ
and
where the sub-rectangles S1 and S2 are generated from S0 S 1 ¼ fx j ls 6 xs 6 hs ; lj 6 xj 6 uj ; j ¼ 1; 2; . . . ; n; j 6¼ sg;
ð7Þ
S 2 ¼ fxjhs 6 xs 6 us ; lj 6 xj 6 uj ; j ¼ 1; 2; . . . ; n; j 6¼ sg:
ð8Þ
Using a similar way, we can get two underestimated quadratic programming programs to the branched subproblems (Q1) and (Q2) by replacing the function u(x) with new underestimated linear functions w1(x) and w2(x), where X 0 wi ðxi Þ þ w1s ðxs Þ ð9Þ w1 ðxÞ ¼ i6¼s
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
and w2 ðxÞ ¼
X
w0i ðxi Þ þ w2s ðxs Þ.
7
ð10Þ
i6¼s
The following is the underestimated quadratic programming to problem (Q1): minfg1 ðxÞ ¼ pðxÞ þ w1 ðxÞjx 2 D; x 2 S 1 g: ðQ1 Þ If problem (Q1 ) is infeasible, then problem (Q1) is also infeasible, and we will delete problem (Q1). Otherwise, let x1 be an optimal solution of (Q1 ), then we obtain an low bound g1 ðx1 Þ and a upper bound f ðx1 Þ for the optimal value f(x1) of problem (Q1), where x1 is an optimal solution of (Q1). If g1 ðx1 Þ > f ðx0 Þ, then f ðx1 Þ P g1 ðx1 Þ > f ðx0 Þ (according to Theorem 3.1), and problem (Q1) will be deleted from further consideration. Otherwise, if f ðx1 Þ g1 ðx1 Þ < , then problem (Q1) is solved with x1 being an approximate solution, if f ðx1 Þ < f ðx0 Þ, x1 will replace x0 as an approximation to the optimal solution. If f ðx1 Þ g1 ðx1 Þ P , the set S1 will be further divided into two subsets to generate two new subproblems. Repeat this process until no subproblems exists. Now we give the detailed description of the proposed branch-and-bound algorithm. Algorithm 1 (The branch-and-bound algorithm). Step 0. Let l0 = l, u0 = u, and give tolerance > 0. Solve ðQ0 Þ to obtain an optimal solution x0 , and set x0 ¼ x0 ; a0 ¼ f ðx0 Þ; b0 ¼ bðQ0 Þ ¼ g0 ðx0 Þ, Q = {Q0} and k = 0. Step 1. Delete all Qi from Q with b(Qi) > ak . Let Q be the set of remaining subproblems. If Q = /, terminate and xk is an global optimal solution of (Q0). Step 2. Select a problem (Qk) from Q minff ðxÞjx 2 V ; x 2 S k g
ðQk Þ
such that bk ¼ bðQk Þ ¼ minfbðQj Þ; Qj 2 Qg and divide Sk into Sk,1 and Sk,2 using a NRS process, generate two subproblems Qk,1 and Qk,2, Set Q = Qn(Qk). Step 3. For i = 1, 2, solve quadratic underestimate problem Qk;i to obtain optimal solutions xk;i . If Qk;i is infeasible, then Q = Q, else Q = Q [ {Qk,i, and set bk;i ¼ gk;i ðxk;i Þ; ak;i ¼ f ðxk;i Þ. Step 4. Set ak+1 = min{ak, ak,1, ak,2}, xk+1 = argmin{ak+1}, k = k + 1 and then go to Step 1. The convergence of the algorithm is given by the following theorem. Since the proof process is similar to one of Theorem 2.2 in [14] and hence is omitted here.
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Theorem 3.2. The sequence xk generated by the algorithm above converges to an optimal solution of (Q0) as k ! 1. 3.2. Normal rectangle subdivisions Various NRS processes for rectangle subdivision are available [10,13]. Numerical tests in [13] show that the the largest distance bisection (LDB) strategy is efficient for solving D-C programming [10,13,14], and hence the LDB strategy will be employed in Algorithm 1. Suppose that a rectangle S k ¼ fxjlki 6 xi 6 uki ; i ¼ 1; 2; ; ng is selected for further division in Algorithm 1. For simplicity, we will denote the two sub-rectangles obtained from a bisection method as S+1, S+2, that is S þ1 ¼ fxjlks 6 xks 6 hs ; lkj 6 xkj 6 ukj ; j ¼ 1; 2; . . . ; n; j 6¼ sg;
ð11Þ
S þ2 ¼ fxjhs 6 xks 6 uks ; lkj 6 xkj 6 ukj ; j ¼ 1; 2; . . . ; n; j 6¼ sg.
ð12Þ
Different choices in the index s and the point hks generate different rectangle subdivisions. The largest distance bisection choose the index s and the point hks in the following way. The largest distance bisection Algorithm 2 (Largest distance bisection (LDB)). Step 1. Calculate the slopes of lines of the underestimation functions wki ðxi Þ di ¼
ui ðuki Þ ui ðlki Þ ; uki lki
i ¼ 1; 2; . . . ; n:
Express the distance between ui(xi) and wki ðxi Þ for xi 2 ½lki ; uki d i ðxi Þ ¼ ui ðxi Þ wki ðxi Þ ¼ ui ðxi Þ di xi gi ;
i ¼ 1; 2; . . . ; n:
Step 2. Maximize the distance function di(xi) to get the solution hki . Let d i ðhki Þ be the maximum. Step 3. Determine the index s from d s ðhks Þ ¼ maxfd i ðhki Þ; i ¼ 1; 2; . . . ; ng:
ð13Þ
Step 4. Determine the bisection point hks . One method is to set hks ¼ xks , and another is to set hks as the point calculated at Step 2. It is proved in [13] that the LDB method is a NRS process that ensures the convergence of Algorithm 1.
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
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4. Numerical tests Numerical experiments on the proposed algorithm have been conducted on portfolio optimal problem (Q0) with nine stocks from Shan Xi province in China. Weekly prices for these stocks are used in numerical calculations. The program was coded using MatLab and tested on Pentium Pro 1794 MHZ with 256 Mbyte memory. The parameter value e = 105 is used in the algorithm to terminate the iteration for tests on both the subdivisions. Quadratic functions are used to fit transaction cost functions. In problem (Q0), we set polytop D = {xjx1 + x2 + + x9 = 1}. Fig. 3 presents the efficient frontiers generated from the the M–V portfolio models without transaction cost (the model (M–V)) and with transaction cost (the model (Q0). The low bound and upper bound on investment are li = 0 (i.e., short sell is forbidden), and ubi = 0.35, i = 1, 2, . . . , n. It can be observed that the efficient frontier without transaction is higher than the efficient frontier with transaction cost, and the difference between the returns with and without transaction cost under the same risk level increases as the risk increases. Fig. 4 presents the efficient frontiers from the model (Q0) under three different upper bounds on investment. The short sell is still not permitted, i.e., li = 0,
Mean – Variance portfolio
–3
2.6
x 10
without transaction cost with transaction cost
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.0108
0.011
0.0112
0.0114
0.0116
0.0118
risk(variance)
Fig. 3. Efficient frontiers with and without transaction cost.
0.012
10
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12 Mean – Variance portfolio
–3
2.6
x 10
ub=0.325 ub=0.3 ub=0.275
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.011
0.0112
0.0114
0.0116
0.0118
0.012
0.0122
0.0124
0.0126
0.0128
risk(variance)
Fig. 4. Efficient frontiers under difference investment upper bound.
i = 1, 2, . . . ,n. It can be observed from the figure that higher in values of the upper bound, higher the efficient frontier. It also be found from the calculation results that with higher values in upper bound, the diversify of the resulting portfolio increases. Thus, the transaction cost increases, and the net return of the portfolio decrease under the same level of risks. It can also be observed from the figure that the differences between efficient frontiers increase as the level of risks increases. Larger the upper bound, higher the increment on the net return. This implies that the net returns can be increased by increasing the upper bound on the limitation of the investment without increasing the level of risks. Fig. 5 presents the efficient frontiers with transaction cost under different low bounds on investment. The upper bound is ui = 0.3 and the values on low bound are li = 0.01, 0, 0.01, i = 1, 2, . . . ,n. It can be observed from the figure that the lower the low bound on the investment, the higher the position of the efficient frontier. This is because when the low bound on investment decreases, the diversity of the resulting investment reduces, that is, the investment will be on few assets, and hence the transaction cost will be reduce and the net return increases. This will give the higher efficient frontier. Another observa-
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12 Mean–Variance portfolio
–3
2.6
11
x 10
lb=0.0 1 lb=0 lb=0.01
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.011
0.0115
0.012
0.0125
risk(variance)
Fig. 5. Efficient frontiers under difference investment low bound.
tion from the figure is that under the same return requirement, the portfolio with lower value of the low bound on investment has lower level of risks that the portfolio with higher values of low bound. This implies that investors can reduces their risks by reducing their low bound on investment without reducing the requirement on net returns, and hence, short sells are a valuable investment strategy in reducing risks on investment.
5. Conclusion In this paper, the classical mean–variance portfolio model is modified to take the concave transaction costs so that it can approximate the conditions in real market well. A branch and bound algorithm is given for the solution of the mean–variance portfolio optimal models. Numerical tests on the model using the branch-and-bound algorithm with the data of 9 stocks in Shan Xi province show that the efficient frontier with the transaction costs taken into account will be lower than the efficient frontier without transaction cost. Tests with different low bounds and upper bounds on investment have been made to show the effects of the transaction cost.
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References [1] J.E. Falk, R.M. Soland, An algorithm for separable nonconvex programming problems, Manage. Sci. 15 (1996) 550–569. [2] P.E. Gill, W. Murray, M.H. Wright, Practical Optimazition, Academic Press. Inc, London, 1982. [3] R. Horst, H. Tuy, Global Optimization Deterministic Approaches, third ed., Springer, Berlin, New York, 1997. [4] R. Horst, P.M. Pardalos, N.V. Thoai, Introduction to Global Optimization, Kluwer Academic Publishers, London, 1995. [5] H. Konno, A. Wijayanayake, Portfolio optimazation problem under concave transaction costs and minimal transaction unit constrains, Math. Program., Ser. B 89 (2001) 233–250. [6] H. Konno, P.T. Thach, H. Tuy, Optimization on Low Rank Nonconvex Structures, Kluwer Academic Publishers, Dordrecht, 1997. [7] P. Krokhmal, J. Palmquist, S. Uryasev, Portfolio optimization with conditional Value-at-Risk objective and constrains, Research Report 2001-9, ISE Department, University of Florida, September 2001. [8] H.M. Markowitz, Portfolio selection, J. Finan. 7 (1) (1952) 77–79. [9] M.C. Steinbach, Markowitz revisited: single-period and multi-period mean–variance models, Woking Paper, Konrad-Zentrum fu¨r Informationstechnik Berlin, Preprint SC-99-30, August, 1999. [10] Q.P. Thai, Le Thi Hoai An, Pham Dinh Tao, Decomposition branch and bound method for globally solving linear constrained indefinate quadratic programming problems, Operat. Res. Lett. 17 (1995) 215–220. [11] H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998. [12] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [13] H. Xue, C. Xu, A branch and bound algorithm for Separable Concave Programming, J. Computat. Math. 6 (2004) 895–904. [14] H. Xue, C. Xu, A branch and bound algorithm for solving a class of D-C programming, Appl. Math. Comput. 165 (2) (2005) 291–302.
Mean–variance portfolio optimal problem under concave transaction cost Hong-Gang Xue
a,*
q
, Cheng-Xian Xu b, Zong-Xian Feng
c
a
c
School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710049, China b Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China JinHe Center for Economic Research, Xi’an Jiaotong University, Xi’an 710049, China
Abstract In this paper, the classical mean–variance portfolio model is modified for calculating a globally optimal portfolio under concave transaction costs. A non-decreasing concave function is employed to approximate origin transaction cost function. The resulting model is a D-C (difference of two convex functions) programming and a branch and bound algorithm is designed to solve the problem. A series of numerical experiments on the model is presented. The history data of nine stocks in Shan Xi province is used in experiments, and efficient frontiers generated from the resulting model with different limitations on investments are presented to show the effect of the model and the efficiency of the algorithm solving the model. 2005 Elsevier Inc. All rights reserved. Keywords: Mean–variance; Concave transaction cost; Globally optimal portfolio; Branch and bound algorithm; Efficient frontier
q
This work is supported by National Natural Key Product Foundations of China 10231060. Corresponding author. Address: Faculty of Science, XiÕan Jiaotong University, XiÕan 710049, China. E-mail address: [email protected] (H.-G. Xue). *
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.05.005
2
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
1. Introduction The mean variance model for portfolio selection was first proposed by Markowitz in 1952, and has been used by many companies and investors in financial market. Assume that there exist n risky assets with expected return rate ri of ith asset, i = 1, 2, . . . ,n. Then the model can be expressed as [8] min
l T x Vx ð1 lÞRT x 2
s.t.
x 2 D \ S;
ðM-VÞ
where x = (x1, x2, . . . , xn)T denotes the investment weight vector, R ¼ T ðr1 ; r2 ; . . . ; rn Þ the expected return rate vector and V = (rij)n·n the variance– covariance matrix of the return distributions of these n assets. Parameter l 2 (0, 1] reflects the investorÕs attitude towards risk (which is measured by the variance of the portfolio). The set D is a polytope that is generally formulated by a series of linear P inequalities such as Ax 6 b. One typical case is the capital budget constraint ni¼1 xi ¼ 1. The set S = {li 6 xi 6 ui, i = 1, 2, . . . , n} is a rectangle and gives the low bound l = (l1, l2, . . . , ln)T and the upper bound u = (u1, u2, . . . , un)T on the vector x. In general, the variance–covariance matrix V is symmetric and positive definite, and problem (M-V) is a convex quadratic programming. Effective methods are available for the solution of problem (M-V) [9]. The transaction cost does not considered in problem (M-V). However, recent researches show that the transaction cost is considerable in some investment actions. Konno and Wijayake [5] studied a portfolio optimization problem under concave transaction cost. The absolute deviation (AD) of the portfolioÕs return distribution is used to measure the portfolioÕ risk, and the resulting problem is a Mean–AD (MAD) model under concave transaction cost. A branch-and-bound algorithm is proposed to solve the MAD model, and linear approximations are used to result in linear sub-programming [5]. Krokhmal, Palmquist and Uryasev [7] also took the transaction cost into account in their portfolio optimization problem in which the CVaR is used to measure the portfolioÕs risk and transaction cost is expressed as a linear function. In this paper, we consider the M–V model (M-V), but the transaction will included into the model. Since the concave transaction cost is employed, the resulting model is a D-C (difference of two convex functions) programming, and it is difficult to find itÕs global solution. However, the transaction cost funcPn tion C(x) is separable, i.e., CðxÞ ¼ i¼1 C i ðxi Þ, a branch and bound algorithm that is similar to [1,5,10,13,14] is proposed to solve the problem by using its structure. Konno [5,6] proposed a branch and bound algorithm based on linear underestimation to the concave transaction cost function to solve large scale portfolio optimal problems with MAD model and concave transaction cost.
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12
3
Numerical tests on 200 assets and 60 simulation sceneries showed that the algorithm can solve large scale problems effectively. Subdivision processes play an important role in proving the convergence of a branch and bound method. The concept of ‘‘normal rectangular subdivision’’ introduced in [3,4,11] will be used in branch and bound algorithms for the solution of the problem with set S. The class of normal subdivision includes wellknow exhaustive bisection, x-subdivision, adaptive bisection. Xue and Xu [13,14] proposed a branch and bound algorithm based on the largest distance bisection, and it is proved that the largest distance bisection is also a normal rectangular subdivision. Solve these problems from 20 to 160 variables effectively. Some evidence showed that the x-subdivision is better than the exhaustive bisection and the adaptive bisection [10], while numerical tests in [13,14] showed that the largest distance bisection is more efficient than the x-subdivision. Hence, we will use the largest distance bisection in the proposed branchand-bound algorithm for the solution of the resulting D-C programming problems. The rest of the paper is organized as follows. In Section 2, we will construct the mean–variance model under concave transaction cost and describe the new branch and bound algorithm for solving it. In Section 3, we will disrupt the detail process of the largest distance bisection method. In Section 4, we will compare the case that the transaction cost exist with the other case that the transaction cost exist, and we will analysis the effective of the investment upper bound and low bound for efficient frontier by numerical tests. Conclusions are given in Section 5.
2. The M–V portfolio problem under concave transaction cost In this section, we will give the mean–variance portfolio problem under concave transaction costs. The transaction cost associated with a portfolio x = (x1, x2, . . . , xn)T is usually defined as the sum of individual transaction cost on each asset n X C i ðxi Þ; CðxÞ ¼ i¼1
where Ci(xi) is the individual cost on the ith asset. Ci(xi) generally is a nondecreasing concave function up to certain point ^xi . It is assumed that Ci(xi), i = 1, 2, . . . ,n are smooth enough. Then the first-order derivative of the transaction cost function has the property C 0i ðxi Þ P 0 and the second-order derivative of Ci(x) satisfies C 00i ðxi Þ < 0 for all i = 1, 2, ,n. Such properties of the cost function coincide with the rule of margin cost decreasing, and can be described by the curve in Fig. 1.
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Fig. 1. concave transaction cost.
However, the unit transaction cost increases beyond the point ^xj , due to the ‘‘ill-liquidity’’ effects. Konno and Wijayanayake explained this economic phenomena in [5], and made an assumption that the amount of investment on the jth asset is below the critical point ^xj . The value of ^xj can be obtained by finding the inflexion point of original transaction cost function. Under such an assumption, the transaction cost is a well specified concave function and can be calculated by the transaction cost table of the agent. Under the concave transaction cost, the portfolioÕs net return is RTx C(x), and the portfolio optimal problem can be expressed as follows: min
f ðxÞ
s.t.
x 2 D \ S;
ðQ0 Þ
where the objective function f ðxÞ ¼ pðxÞ þ uðxÞ; l pðxÞ ¼ xT Vx 2 is the convex part which denotes the portfolioÕs risk, and n X ui ðxi Þ; ui ðxi Þ ¼ ð1 lÞðC i ðxÞ ri xi Þ; i ¼ 1; 2; . . . ; n uðxÞ ¼
ð1Þ
ð2Þ
i¼1
is the portfolioÕs net return. Problem (Q0) is a D-C programming. The constraints in (Q0) is the same as in problem (M-V). In general, 1 6 ui 6 ^xi . If short sells are not permitted, then li P 0, else li 6 0. If we set n 1 xP weight vector x 2 S, and i ¼ n for all i = 1, 2, . . . , n, then the 1 investment n 1 T x ¼ 1, so x 2 D. That is, x ¼ ; . . . ; is a feasible point, and hence, i i¼1 n n the feasible set is nonempty. Theorem 2.1. Suppose the concave transaction cost functions Ci(xi) i = 1, 2, . . . , n are continues, then the global solution of the problem (Q0) exists.
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Proof. It is clear that the feasible set D \ S is nonempty, closed and bounded. While the assumption implies that the objective function is continues on the feasible region. Therefore, the conclusion comes from the fact that the minimum exists for a continuous function on a closed and bounded nonempty set. h
3. The branch and bound algorithm In this section, we will present the proposed branch and bound method in which the largest distance rectangular subdivision process will be used to divide problems into sub-problems. 3.1. The description of the algorithm Let S0 = {li 6 xi 6 ui, i = 1, 2, . . . ,n} be the initial rectangle. We replace the individual concave functions ui(xi) in u(x) by underestimated linear functions w0i ðxi Þ over S0 (see Fig. 2) w0i ðxi Þ ¼ di xi þ gi ;
i ¼ 1; 2; . . . ; n;
ð3Þ
where di ¼
ui ðui Þ ui ðli Þ ; ui li
gi ¼ ui ðli Þ di li ;
i ¼ 1; 2; . . . ; n.
ð4Þ
Let g0 ðxÞ ¼ pðxÞ þ w0 ðxÞ; then g0(x) Pnis the convex envelope of the function f(x) over the set S0, where w0 ðxÞ ¼ i¼1 w0i ðxi Þ. We solve the quadratic underestimated approximation to (Q0),
Fig. 2. The underestimated linear function.
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minfg0 ðxÞ ¼ pðxÞ þ w0 ðxÞjx 2 D; l 6 x 6 ug.
ðQ0 Þ
(Q0 ) is a convex quadratic programming problem. The active set method in [2,12] can be used to effectively solve ðQ0 Þ. Let x0 be an optimal solution ðQ0 Þ, then we obtain an lower bound g0 ðx0 Þ and a upper bound f ðx0 Þ of the optimal value f(x*) of the problem (Q0) according to the following theorem. Theorem 3.1. Let x0 be an optimal solution of problem ðQ0 Þ and x* be a global optimal solution of problem (Q0). Then the following relations hold g0 ðx0 Þ 6 f 6 f ðx0 Þ;
ð5Þ
where f* = f(x*). Proof. It follows from g0(x) 6 f(x) "x 2 [l, u], that: g0 ðx0 Þ ¼ minfg0 ðxÞjx 2 D; l 6 x 6 ug 6 minff ðxÞjx 2 D; l 6 x 6 ug ¼ f 6 f ðx0 Þ. This gives the conclusion.
h
Theorem 3.1 indicates that if uðx0 Þ w0 ðx0 Þ 6
ð6Þ
is satisfied with a given tolerance , then x0 is an approximate solution of (Q0) with error less than . If (6) does not hold, we will use an NRS process to divide the problem (Q0) into two subproblems: minff ðxÞjx 2 D; x 2 S 1 g
ðQ1 Þ
minff ðxÞjx 2 D; x 2 S 2 g;
ðQ2 Þ
and
where the sub-rectangles S1 and S2 are generated from S0 S 1 ¼ fx j ls 6 xs 6 hs ; lj 6 xj 6 uj ; j ¼ 1; 2; . . . ; n; j 6¼ sg;
ð7Þ
S 2 ¼ fxjhs 6 xs 6 us ; lj 6 xj 6 uj ; j ¼ 1; 2; . . . ; n; j 6¼ sg:
ð8Þ
Using a similar way, we can get two underestimated quadratic programming programs to the branched subproblems (Q1) and (Q2) by replacing the function u(x) with new underestimated linear functions w1(x) and w2(x), where X 0 wi ðxi Þ þ w1s ðxs Þ ð9Þ w1 ðxÞ ¼ i6¼s
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and w2 ðxÞ ¼
X
w0i ðxi Þ þ w2s ðxs Þ.
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ð10Þ
i6¼s
The following is the underestimated quadratic programming to problem (Q1): minfg1 ðxÞ ¼ pðxÞ þ w1 ðxÞjx 2 D; x 2 S 1 g: ðQ1 Þ If problem (Q1 ) is infeasible, then problem (Q1) is also infeasible, and we will delete problem (Q1). Otherwise, let x1 be an optimal solution of (Q1 ), then we obtain an low bound g1 ðx1 Þ and a upper bound f ðx1 Þ for the optimal value f(x1) of problem (Q1), where x1 is an optimal solution of (Q1). If g1 ðx1 Þ > f ðx0 Þ, then f ðx1 Þ P g1 ðx1 Þ > f ðx0 Þ (according to Theorem 3.1), and problem (Q1) will be deleted from further consideration. Otherwise, if f ðx1 Þ g1 ðx1 Þ < , then problem (Q1) is solved with x1 being an approximate solution, if f ðx1 Þ < f ðx0 Þ, x1 will replace x0 as an approximation to the optimal solution. If f ðx1 Þ g1 ðx1 Þ P , the set S1 will be further divided into two subsets to generate two new subproblems. Repeat this process until no subproblems exists. Now we give the detailed description of the proposed branch-and-bound algorithm. Algorithm 1 (The branch-and-bound algorithm). Step 0. Let l0 = l, u0 = u, and give tolerance > 0. Solve ðQ0 Þ to obtain an optimal solution x0 , and set x0 ¼ x0 ; a0 ¼ f ðx0 Þ; b0 ¼ bðQ0 Þ ¼ g0 ðx0 Þ, Q = {Q0} and k = 0. Step 1. Delete all Qi from Q with b(Qi) > ak . Let Q be the set of remaining subproblems. If Q = /, terminate and xk is an global optimal solution of (Q0). Step 2. Select a problem (Qk) from Q minff ðxÞjx 2 V ; x 2 S k g
ðQk Þ
such that bk ¼ bðQk Þ ¼ minfbðQj Þ; Qj 2 Qg and divide Sk into Sk,1 and Sk,2 using a NRS process, generate two subproblems Qk,1 and Qk,2, Set Q = Qn(Qk). Step 3. For i = 1, 2, solve quadratic underestimate problem Qk;i to obtain optimal solutions xk;i . If Qk;i is infeasible, then Q = Q, else Q = Q [ {Qk,i, and set bk;i ¼ gk;i ðxk;i Þ; ak;i ¼ f ðxk;i Þ. Step 4. Set ak+1 = min{ak, ak,1, ak,2}, xk+1 = argmin{ak+1}, k = k + 1 and then go to Step 1. The convergence of the algorithm is given by the following theorem. Since the proof process is similar to one of Theorem 2.2 in [14] and hence is omitted here.
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Theorem 3.2. The sequence xk generated by the algorithm above converges to an optimal solution of (Q0) as k ! 1. 3.2. Normal rectangle subdivisions Various NRS processes for rectangle subdivision are available [10,13]. Numerical tests in [13] show that the the largest distance bisection (LDB) strategy is efficient for solving D-C programming [10,13,14], and hence the LDB strategy will be employed in Algorithm 1. Suppose that a rectangle S k ¼ fxjlki 6 xi 6 uki ; i ¼ 1; 2; ; ng is selected for further division in Algorithm 1. For simplicity, we will denote the two sub-rectangles obtained from a bisection method as S+1, S+2, that is S þ1 ¼ fxjlks 6 xks 6 hs ; lkj 6 xkj 6 ukj ; j ¼ 1; 2; . . . ; n; j 6¼ sg;
ð11Þ
S þ2 ¼ fxjhs 6 xks 6 uks ; lkj 6 xkj 6 ukj ; j ¼ 1; 2; . . . ; n; j 6¼ sg.
ð12Þ
Different choices in the index s and the point hks generate different rectangle subdivisions. The largest distance bisection choose the index s and the point hks in the following way. The largest distance bisection Algorithm 2 (Largest distance bisection (LDB)). Step 1. Calculate the slopes of lines of the underestimation functions wki ðxi Þ di ¼
ui ðuki Þ ui ðlki Þ ; uki lki
i ¼ 1; 2; . . . ; n:
Express the distance between ui(xi) and wki ðxi Þ for xi 2 ½lki ; uki d i ðxi Þ ¼ ui ðxi Þ wki ðxi Þ ¼ ui ðxi Þ di xi gi ;
i ¼ 1; 2; . . . ; n:
Step 2. Maximize the distance function di(xi) to get the solution hki . Let d i ðhki Þ be the maximum. Step 3. Determine the index s from d s ðhks Þ ¼ maxfd i ðhki Þ; i ¼ 1; 2; . . . ; ng:
ð13Þ
Step 4. Determine the bisection point hks . One method is to set hks ¼ xks , and another is to set hks as the point calculated at Step 2. It is proved in [13] that the LDB method is a NRS process that ensures the convergence of Algorithm 1.
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4. Numerical tests Numerical experiments on the proposed algorithm have been conducted on portfolio optimal problem (Q0) with nine stocks from Shan Xi province in China. Weekly prices for these stocks are used in numerical calculations. The program was coded using MatLab and tested on Pentium Pro 1794 MHZ with 256 Mbyte memory. The parameter value e = 105 is used in the algorithm to terminate the iteration for tests on both the subdivisions. Quadratic functions are used to fit transaction cost functions. In problem (Q0), we set polytop D = {xjx1 + x2 + + x9 = 1}. Fig. 3 presents the efficient frontiers generated from the the M–V portfolio models without transaction cost (the model (M–V)) and with transaction cost (the model (Q0). The low bound and upper bound on investment are li = 0 (i.e., short sell is forbidden), and ubi = 0.35, i = 1, 2, . . . , n. It can be observed that the efficient frontier without transaction is higher than the efficient frontier with transaction cost, and the difference between the returns with and without transaction cost under the same risk level increases as the risk increases. Fig. 4 presents the efficient frontiers from the model (Q0) under three different upper bounds on investment. The short sell is still not permitted, i.e., li = 0,
Mean – Variance portfolio
–3
2.6
x 10
without transaction cost with transaction cost
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.0108
0.011
0.0112
0.0114
0.0116
0.0118
risk(variance)
Fig. 3. Efficient frontiers with and without transaction cost.
0.012
10
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12 Mean – Variance portfolio
–3
2.6
x 10
ub=0.325 ub=0.3 ub=0.275
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.011
0.0112
0.0114
0.0116
0.0118
0.012
0.0122
0.0124
0.0126
0.0128
risk(variance)
Fig. 4. Efficient frontiers under difference investment upper bound.
i = 1, 2, . . . ,n. It can be observed from the figure that higher in values of the upper bound, higher the efficient frontier. It also be found from the calculation results that with higher values in upper bound, the diversify of the resulting portfolio increases. Thus, the transaction cost increases, and the net return of the portfolio decrease under the same level of risks. It can also be observed from the figure that the differences between efficient frontiers increase as the level of risks increases. Larger the upper bound, higher the increment on the net return. This implies that the net returns can be increased by increasing the upper bound on the limitation of the investment without increasing the level of risks. Fig. 5 presents the efficient frontiers with transaction cost under different low bounds on investment. The upper bound is ui = 0.3 and the values on low bound are li = 0.01, 0, 0.01, i = 1, 2, . . . ,n. It can be observed from the figure that the lower the low bound on the investment, the higher the position of the efficient frontier. This is because when the low bound on investment decreases, the diversity of the resulting investment reduces, that is, the investment will be on few assets, and hence the transaction cost will be reduce and the net return increases. This will give the higher efficient frontier. Another observa-
H.-G. Xue et al. / Appl. Math. Comput. 174 (2006) 1–12 Mean–Variance portfolio
–3
2.6
11
x 10
lb=0.0 1 lb=0 lb=0.01
2.5
return rate
2.4
2.3
2.2
2.1
2
1.9 0.011
0.0115
0.012
0.0125
risk(variance)
Fig. 5. Efficient frontiers under difference investment low bound.
tion from the figure is that under the same return requirement, the portfolio with lower value of the low bound on investment has lower level of risks that the portfolio with higher values of low bound. This implies that investors can reduces their risks by reducing their low bound on investment without reducing the requirement on net returns, and hence, short sells are a valuable investment strategy in reducing risks on investment.
5. Conclusion In this paper, the classical mean–variance portfolio model is modified to take the concave transaction costs so that it can approximate the conditions in real market well. A branch and bound algorithm is given for the solution of the mean–variance portfolio optimal models. Numerical tests on the model using the branch-and-bound algorithm with the data of 9 stocks in Shan Xi province show that the efficient frontier with the transaction costs taken into account will be lower than the efficient frontier without transaction cost. Tests with different low bounds and upper bounds on investment have been made to show the effects of the transaction cost.
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