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A modified goal programming approach for the mean-absolute deviation portfolio optimization model
Applied Mathematics and Computation 171 (2005) 567–572 www.elsevier.com/locate/amc
A modified goal programming approach for the mean-absolute deviation portfolio optimization model Ching-Ter Chang Information management, National Changhua University of Education, Paisa Village, Changhua 50058, Taiwan, ROC
Abstract The purpose of this paper is to present a reformulation of the model presented by Feinstein and Thapa [C.D. Feinstein, M.N. Thapa, Notes: a reformulation of a meanabsolute deviation portfolio optimization model, Management Science 39 (12) (1993) 1552–1553]. The approach of Feinstein and Thapa has been accepted as the most efficient technique published, requiring the least number of auxiliary constraints and additional continuous variables. To solve a portfolio optimization problem with T periods, in their method would introduce T + 2 auxiliary constraints, 2T auxiliary sign constraints, and 2T additional continuous variables. This note indicates that it is still possible to reduce the number of auxiliary constraints and additional continuous variables in the model of Feinstein and Thapa. The equivalent concise model is proposed in this note, which has T + 2 auxiliary constraints, T auxiliary sign constraints, and T additional continuous variables. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Portfolio; Goal programming
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.072
568
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
1. Introduction The mean-variance portfolio model stems from the work of Markowitz [3]. In fact, it has not been used in solving the large-scale portfolio optimization problem, contrary to its theoretical reputation. Konno and Yamazaki [1] indicate that some of the reasons why it is so: (i) computational burden problem, (ii) investors perception against risk and distribution of stock prices problem and (iii) transaction/management cost and cut-off effect problem. In order to alleviate these defects in the original model of Markowitz, Konno and Yamazaki proposed a L1 risk (absolute deviation) model below. " " ## n n X X minimize wðxÞ ¼ E Rj x j E Rj xj j¼1 j¼1 subject to
n X
E½Rj xj P qM 0 ; ð1Þ
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n;
where Rj is a random variable representing the rate of return (per period) from the asset Sj; xj is the amount of money to be invested in asset Sj out of the total fund M0; E[Æ] represents the expected value of Rj in the bracket; q is scalar parameter representing the minimal rate of return required by an investor; uj is the maximum amount invested in asset Sj. Konno and Yamazaki assume that rjt is the realization of random variable Rj during period t (t = 1, . . . , T) which is assumed to be available PT through the 1 historical data or from future projection, and r ¼ E½R ¼ j j t¼1 rjt . w(x) is T P P approximated by T1 Tt¼1 j nj¼1 ðrjt rj Þxj j. Denoting ajt = rjt rj (j ¼ 1, . . . , n and t = 1, . . . , T). Program (1) can be expressed as follows. T X n 1 X minimize ajt xj T t¼1 j¼1 subject to
n X
rj xj P qM 0 ;
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
ð2Þ
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
569
Program (2) can be converted as the following linear program. minimize subject to
T 1 X y T t¼1 t n X yt þ ajt xj P 0;
yt
j¼1 n X
ajt xj P 0;
t ¼ 1; . . . ; T ; t ¼ 1; . . . ; T ;
j¼1 n X j¼1 n X
ð3Þ
rj xj P qM 0 ; xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
Clearly, there are 2T + 2 auxiliary constraints and T additional continuous variable (i.e., yt) in the above linear program (3). Later, Feinstein and Thapa [2] derived a model to solving program (2), which is substantially improved the technique of Konno and Yamazaki. In Feinstein and Thapas method, subtract nonnegative surplus variables 2vt and 2wt from the constraints in program (3) to replace the inequalities by the following equalities (4) and (5). n X ðrjt rj Þxj 2vt ¼ 0; ð4Þ yt þ j¼1
yt
n X ðrjt rj Þxj 2vt ¼ 0:
ð5Þ
j¼1
This leads to the following program. P1 minimize
T X ðvt þ wt Þ t¼1
subject to vt wt
n X
ajt xj ¼ 0;
t ¼ 1; . . . ; T ;
j¼1
vi P 0; wi P 0; n X rj xj P qM 0 ;
t ¼ 1; . . . ; T ;
j¼1
n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
ð6Þ
570
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
Clearly, there are T + 2 auxiliary constraints, 2T auxiliary sign constraints (i.e., vi P 0, wi P 0), 2T additional continuous variables (i.e., vi, wi) in the above linear program (6).
2. Modification of Feinstein and Thapa’s model Introducing a continuous variable dt, program (2) can be expressed as a concise program below. P2 ! T n X X minimize 2d t ajt xj t¼1
j¼1
subject to d t
n X
ajt xj P 0;
t ¼ 1; . . . ; T ;
ð7Þ
j¼1
d t P 0; n X
t ¼ 1; . . . ; T
ð8Þ
rj xj P qM 0 ;
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
Proposition 1. P1 and P2 are equivalent in the sense that they have the same optimal solutions. Proof P P (i) If nj¼1 ajt xj > 0 then d t P nj¼1 ajt xj (+) (from (7)), di P 0 (from (8)), will PT Pn PT Pn force t¼1 j¼1 ajt xj (i.e., t¼1 ð2 j¼1 ajt xj Pn the object function to be a x ÞÞ in the minimization program. j¼1 jt j P Pn n (ii) If j¼1 ajt xj < 0 then d t P j¼1 ajt xj () (from (7)), di P 0 (from (8)), will Pn PT PT force t¼1 ð j¼1 ajt xj Þ (i.e., t¼1 ð0 Pn the objective function to be j¼1 ajt xj ÞÞ in the minimization program. Pn (iii) If j¼1 ajt xj ¼ 0 then dt P 0 (from (7)), di P 0 (from (8)), will force the PT objective function to be zero (i.e., t¼1 ð0 0Þ). Which is obviously the same as P1 = P2. This completes the proof of Proposition 1. h
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
571
Table 1 Size of constraints and variables
No. of additional continuous variables No. of auxiliary constraints No. of auxiliary sign constraints
Konno and Yamazakis model [1]
Feinstein and Thapas model [2]
Proposed model (P2)
T 2T + 2 –
2T T+2 2T
T T+2 T
Clearly, there are T + 2 auxiliary constraints, T auxiliary sign constraints (i.e., dt P 0), and T additional continuous variables (i.e., dt) in P2. The number of additional continuous variables and auxiliary constraints used in Konno and Yamazaki, Feinstein and Thapa, and P2 are listed in Table 1. From Table 1 we realize that the proposed model requires fewer additional variables and fewer auxiliary constraints to formulate the L1 risk (absolute deviation) model.
3. Computational experience The superiority of the proposed approach can also observed through some test examples. These test examples have the same pattern as in the model of Feinstein and Thapa and the proposed model. Five groups of the test examples are characterized by the number of T periods (n = 5,10,15). For each of the five groups, 8 programs are randomly generated. Thus, a set of 40 test problems is formed. Each of test example is formulated as Feinstein and Thapas model (FTM) and the proposed model (PM) and then solved by LINDO (1994) on
Table 2 Relative performance of PM and FTM (CPU time) Number of T periods
5
10
15
FTM/PM FTM/PM FTM/PM
1.21 1.24 1.51
2.36 2.15 2.72
4.16 3.61 2.91
Table 3 Relative performance of PM and FTM (no. of iterations) Number of T periods
5
10
15
FTM/PM FTM/PM FTM/PM
3.82 4.13 4.41
5.62 4.94 5.43
8.90 9.21 9.43
572
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
a PC/586. The average relative performance of FTM and PM, measured by CPU time and number of iterations, is compared in Tables 2 and 3. From Tables 2 and 3, we can see that, compared with PM, FTM takes 243% of the average CPU time and 621% of the number of iterations. The performance of PM becomes much better when the number of T periods is increased. This is an expected outcome because PM reduced the number of constraints and variables.
References [1] H. Konno, H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science 37 (5) (1991) 519–531. [2] C.D. Feinstein, M.N. Thapa, Notes: a reformulation of a mean-absolute deviation portfolio optimization model, Management Science 39 (12) (1993) 1552–1553. [3] P. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91. [4] L. Schrage, LINDO Release 5.3, LINDO System Inc., 1994.
A modified goal programming approach for the mean-absolute deviation portfolio optimization model Ching-Ter Chang Information management, National Changhua University of Education, Paisa Village, Changhua 50058, Taiwan, ROC
Abstract The purpose of this paper is to present a reformulation of the model presented by Feinstein and Thapa [C.D. Feinstein, M.N. Thapa, Notes: a reformulation of a meanabsolute deviation portfolio optimization model, Management Science 39 (12) (1993) 1552–1553]. The approach of Feinstein and Thapa has been accepted as the most efficient technique published, requiring the least number of auxiliary constraints and additional continuous variables. To solve a portfolio optimization problem with T periods, in their method would introduce T + 2 auxiliary constraints, 2T auxiliary sign constraints, and 2T additional continuous variables. This note indicates that it is still possible to reduce the number of auxiliary constraints and additional continuous variables in the model of Feinstein and Thapa. The equivalent concise model is proposed in this note, which has T + 2 auxiliary constraints, T auxiliary sign constraints, and T additional continuous variables. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Portfolio; Goal programming
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.072
568
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
1. Introduction The mean-variance portfolio model stems from the work of Markowitz [3]. In fact, it has not been used in solving the large-scale portfolio optimization problem, contrary to its theoretical reputation. Konno and Yamazaki [1] indicate that some of the reasons why it is so: (i) computational burden problem, (ii) investors perception against risk and distribution of stock prices problem and (iii) transaction/management cost and cut-off effect problem. In order to alleviate these defects in the original model of Markowitz, Konno and Yamazaki proposed a L1 risk (absolute deviation) model below. " " ## n n X X minimize wðxÞ ¼ E Rj x j E Rj xj j¼1 j¼1 subject to
n X
E½Rj xj P qM 0 ; ð1Þ
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n;
where Rj is a random variable representing the rate of return (per period) from the asset Sj; xj is the amount of money to be invested in asset Sj out of the total fund M0; E[Æ] represents the expected value of Rj in the bracket; q is scalar parameter representing the minimal rate of return required by an investor; uj is the maximum amount invested in asset Sj. Konno and Yamazaki assume that rjt is the realization of random variable Rj during period t (t = 1, . . . , T) which is assumed to be available PT through the 1 historical data or from future projection, and r ¼ E½R ¼ j j t¼1 rjt . w(x) is T P P approximated by T1 Tt¼1 j nj¼1 ðrjt rj Þxj j. Denoting ajt = rjt rj (j ¼ 1, . . . , n and t = 1, . . . , T). Program (1) can be expressed as follows. T X n 1 X minimize ajt xj T t¼1 j¼1 subject to
n X
rj xj P qM 0 ;
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
ð2Þ
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
569
Program (2) can be converted as the following linear program. minimize subject to
T 1 X y T t¼1 t n X yt þ ajt xj P 0;
yt
j¼1 n X
ajt xj P 0;
t ¼ 1; . . . ; T ; t ¼ 1; . . . ; T ;
j¼1 n X j¼1 n X
ð3Þ
rj xj P qM 0 ; xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
Clearly, there are 2T + 2 auxiliary constraints and T additional continuous variable (i.e., yt) in the above linear program (3). Later, Feinstein and Thapa [2] derived a model to solving program (2), which is substantially improved the technique of Konno and Yamazaki. In Feinstein and Thapas method, subtract nonnegative surplus variables 2vt and 2wt from the constraints in program (3) to replace the inequalities by the following equalities (4) and (5). n X ðrjt rj Þxj 2vt ¼ 0; ð4Þ yt þ j¼1
yt
n X ðrjt rj Þxj 2vt ¼ 0:
ð5Þ
j¼1
This leads to the following program. P1 minimize
T X ðvt þ wt Þ t¼1
subject to vt wt
n X
ajt xj ¼ 0;
t ¼ 1; . . . ; T ;
j¼1
vi P 0; wi P 0; n X rj xj P qM 0 ;
t ¼ 1; . . . ; T ;
j¼1
n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
ð6Þ
570
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
Clearly, there are T + 2 auxiliary constraints, 2T auxiliary sign constraints (i.e., vi P 0, wi P 0), 2T additional continuous variables (i.e., vi, wi) in the above linear program (6).
2. Modification of Feinstein and Thapa’s model Introducing a continuous variable dt, program (2) can be expressed as a concise program below. P2 ! T n X X minimize 2d t ajt xj t¼1
j¼1
subject to d t
n X
ajt xj P 0;
t ¼ 1; . . . ; T ;
ð7Þ
j¼1
d t P 0; n X
t ¼ 1; . . . ; T
ð8Þ
rj xj P qM 0 ;
j¼1 n X
xj ¼ M 0 ;
j¼1
0 6 x j 6 uj ;
j ¼ 1; . . . ; n:
Proposition 1. P1 and P2 are equivalent in the sense that they have the same optimal solutions. Proof P P (i) If nj¼1 ajt xj > 0 then d t P nj¼1 ajt xj (+) (from (7)), di P 0 (from (8)), will PT Pn PT Pn force t¼1 j¼1 ajt xj (i.e., t¼1 ð2 j¼1 ajt xj Pn the object function to be a x ÞÞ in the minimization program. j¼1 jt j P Pn n (ii) If j¼1 ajt xj < 0 then d t P j¼1 ajt xj () (from (7)), di P 0 (from (8)), will Pn PT PT force t¼1 ð j¼1 ajt xj Þ (i.e., t¼1 ð0 Pn the objective function to be j¼1 ajt xj ÞÞ in the minimization program. Pn (iii) If j¼1 ajt xj ¼ 0 then dt P 0 (from (7)), di P 0 (from (8)), will force the PT objective function to be zero (i.e., t¼1 ð0 0Þ). Which is obviously the same as P1 = P2. This completes the proof of Proposition 1. h
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
571
Table 1 Size of constraints and variables
No. of additional continuous variables No. of auxiliary constraints No. of auxiliary sign constraints
Konno and Yamazakis model [1]
Feinstein and Thapas model [2]
Proposed model (P2)
T 2T + 2 –
2T T+2 2T
T T+2 T
Clearly, there are T + 2 auxiliary constraints, T auxiliary sign constraints (i.e., dt P 0), and T additional continuous variables (i.e., dt) in P2. The number of additional continuous variables and auxiliary constraints used in Konno and Yamazaki, Feinstein and Thapa, and P2 are listed in Table 1. From Table 1 we realize that the proposed model requires fewer additional variables and fewer auxiliary constraints to formulate the L1 risk (absolute deviation) model.
3. Computational experience The superiority of the proposed approach can also observed through some test examples. These test examples have the same pattern as in the model of Feinstein and Thapa and the proposed model. Five groups of the test examples are characterized by the number of T periods (n = 5,10,15). For each of the five groups, 8 programs are randomly generated. Thus, a set of 40 test problems is formed. Each of test example is formulated as Feinstein and Thapas model (FTM) and the proposed model (PM) and then solved by LINDO (1994) on
Table 2 Relative performance of PM and FTM (CPU time) Number of T periods
5
10
15
FTM/PM FTM/PM FTM/PM
1.21 1.24 1.51
2.36 2.15 2.72
4.16 3.61 2.91
Table 3 Relative performance of PM and FTM (no. of iterations) Number of T periods
5
10
15
FTM/PM FTM/PM FTM/PM
3.82 4.13 4.41
5.62 4.94 5.43
8.90 9.21 9.43
572
C.-T. Chang / Appl. Math. Comput. 171 (2005) 567–572
a PC/586. The average relative performance of FTM and PM, measured by CPU time and number of iterations, is compared in Tables 2 and 3. From Tables 2 and 3, we can see that, compared with PM, FTM takes 243% of the average CPU time and 621% of the number of iterations. The performance of PM becomes much better when the number of T periods is increased. This is an expected outcome because PM reduced the number of constraints and variables.
References [1] H. Konno, H. Yamazaki, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Management Science 37 (5) (1991) 519–531. [2] C.D. Feinstein, M.N. Thapa, Notes: a reformulation of a mean-absolute deviation portfolio optimization model, Management Science 39 (12) (1993) 1552–1553. [3] P. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91. [4] L. Schrage, LINDO Release 5.3, LINDO System Inc., 1994.