Developing a robust-fuzzy multi-objective optimization model for portfolio selection

Transcription

1 Developing a robust-fuzzy multi-objective optimization model for portfolio selection COMPUTATIONAL MANAGEMENT SCIENCE University of Bergamo, Italy May 31, 217 Mohammad Salehifar 1 PhD student in finance, Science and Research Branch of Islamic Azad University, Tehran, Iran mohammadsalehifar@gmail.com Soheila Naderi 2 PhD student in finance, Science and Research Branch of Islamic Azad University, Tehran, Iran satia_queen@yahoo.com Hashem Nikoomaram 3 Professor, Science and Research Branch of Islamic Azad University, Management and Economics faculty, Tehran, Iran h-nikoumaram@srbiau.ac.ir

2 Contents 1 Introduction 2 Portfolio Optimization 3 Methodology 4 Conclusion

3 Portfolio Optimization Markowitz (1952) Risk on the one hand, to minimize the risk measure Rate of Return on the other hand, to ensure that the rate of return will be more than a specific amount, based on the decision maker idea

4 Sharpe (1963) proposed a new model for the portfolio selection problem. He introduced systematic risk that measures the sensitivity of a stock's return to the market return. Some other models try to reduce the difficulties of the portfolio selection problem. We can point to Chiodi et al. (23), Kellerer et al. (2), Konno (1999), Mansini et al. (23), Michalowski and Ogryczak (21), Papahristodoulou and Dotzauer (24), and Rockafellar and Uryasev (2). All mentioned models are single-objective, that is, they consider only one objective function; on the other hand, multi-objective models consider more than one objective.

5 Multi-Objective Models Some techniques exist for solving multi-objective models, one of them is goal programming (GP), a multi-objective programming technique that was developed by Charnes et al. (1955). Charnes and Cooper (1961), Ignizio (1976), Ijiri (1965), and Lee (1972) are those who work on it. The first GP model in finance was developed by Lee and Lerro (1973). Then other approaches to portfolio selection using GP were presented by Alexander and Resnick (1985), Bilbao et al. (26), Booth and Dash (1977), Kumar and Philippatos (1979), Kumar et al. (1978), Lee and Chesser (198), Levary and Avery (1984), Li and Xu (27), Marasovic and Babic (211), Muhlemann et al. (1978), Pendaraki et al. (24), Sharma and Sharma (26), Stone and Reback (1975), Tamiz et al. (1996), and Wu et al. (27).

6 Uncertainty of Data To consider the uncertainty of parameters in multiobjective portfolio selection problem, we refer to Alimi et al. (212), Ghazanfar Ahari et al. (211), Gupta et al. (213), and Sadjadi et al. (211) In most papers, fuzzy approach was used to consider uncertainty. In this paper, we suggest robust-fuzzy optimization approach to consider uncertainty of parameters.

7 Soyster, Bertsimas and Sim The application of GP encounters some problems when the decision situation is marked by uncertainty. To consider the parameters of uncertainty, the recent research trend is to develop new robust optimization approaches. Soyster (1973), proposed a linear optimization model for constructing a solution that is feasible for all data that belong to a convex set. The solutions of Soyster's model are, however, too conservative; using his approach, decision maker has to sacrifice considerable amount of optimality from the nominal problem in order to ensure robustness. To overcome this problem, Ben-Tal and Nemirovski (2) developed a new robust formulation. In Ben-Tal and Nemirovski's approach, the decision maker can control the conservatism of the solution. However, their model is nonlinear. Bertsimas and Sim (24) developed a robust approach that is linear programming and controllable. In their approach as well, the decision maker can control the conservatism of the solution. The advantage of this approach is that the robust counterpart is linear programming.

8 Methodology We report on the development of a robust-fuzzy model for the portfolio selection problem by using the GP approach. We use Lee and Chesser's (198) model, which is a linear GP approach that uses systematic risk and rate of return, both of which are uncertain in the real world. We use Bertsimas and Sim's approach to address the parameter uncertainty in Lee and Chesser's model.

9 Lee and Chesser's Goal Programming Model s. t min W % d ' % + W ) (d + ) + d + ',, ) + W, ' )', _ /12 d / + W 2 /1'2 d / + + W 4 d )'2 91% x 9 + d + % d ' % = BC 91% R 9 x 9 + d + ) d ' ) = DR 91% B 9 x 9 + d +, d ', = B(BC) x 9 + d + 9', d ' 9', = V 9 + x 9 + d )')'9 ' d )')'9 = D 9 BC + + ' 91% R 9 x 9 + d )'2 d )'2 = M (1-1) (2-1 ) (3-1 ) (4-1 ) (5-1 ) ( 6-1 ) (7-1 )

10 Parameters: W1 to W5 Xj BC Rj Pt Dt DR Bj B Vj Dj M The advantage of objectives based on the decision maker idea Decision variable. Quantity of money invested in security j The quantity of the budget The expected rate of return, based on the formulation R 9 = ln P E + D E P E+% price at a specific time (t) divided profit at time t total revenue based on the decision maker idea systematic risk of security j expected systematic risk limit of investment on security j quantity of money that should be invested in security j based on Bj a large number

11 Parameters: Objective (2) considers the budget limit. Objective (3) focuses on the portfolio's rate of return that should be more than DR (total revenue based on decision maker idea). Objective (4) concentrates on portfolio systematic risk. If an investor predicts that the market will improve in the future, he/she should make his/her portfolio close to the market beta. This paper uses this assumption in this model, and therefore, in Eq. (4) we maximize the portfolio beta based on the decision maker idea. Objectives (5) and (6) consider the limit of investment for each security and objective Equation (7) focuses on maximization of the sum of the budget and the portfolio return.

12 Robust Portfolio Optimization according to Bertsimas and Sim Approach and Lee and Chesser's GP Model min W % d ' % + W ) (d ' ) + d ' '4 ' )'5 _, ) + W, /12 d / + W 2 d / /1'5 ' + W 4 d )'6 (1-2) 91% x 9 + d + % d ' % = BC 91% R 9 x 9 + d + ) d ' ) + Z % Γ % + 91% P %9 = DR 91% B 9 x 9 + d +, d ', +Z ) Γ ) + 91% P )9 = B(BC) x 9 + d 2 + d 2 ' = V 9 x 9 + d 4 + d 4 ' = D 9 BC 91% R 9 x 9 + d + K d ' K + Z % Γ % + 91% P %9 = M Z % + P %9 RM 9 y 9 Z ) + P )9 BM 9 y 9 y 9 x 9 y 9 P 9, y /, Z %, Z ) (2-2) (3-2) (4-2) (5-2) (6-2) (7-2) (8-2) (9-2) (1-2) (11-2)

13 Robust Portfolio Optimization according to Bertsimas and Sim Approach and Lee and Chesser's GP Model This section develops a robust optimization for the portfolio selection problem. We use Lee and Chesser GP model. Kouchta (24) developed robust GP. The constraints should be changed with uncertainty parameters such as ax b. The objective function of Eqs. (1) to (7) changes as follows: )'4 /1'4 min W % d ' % + W ) d ' ) + d ' '2 ', + W, /12 d / + W 2 + ' d / + W 4 d )'K we introduce a parameter, Γi, which is not necessarily an integer, that takes values in the interval [, Ji ] which i={1,2}

14 Robust-Fuzzy Portfolio Optimization based on Bertsimas and Sim Approach and Lee and Chesser's GP Model Max Z = λ (1-3) s.t W % d ' % + W ) d ' ) + d ' T'2 ' )'4 _, + W, /12 d / + W 2 d / /1'4 ' + W 4 d )T'K + λz U Z V (2-3) 91% x 9 + d + % d ' % = BC 91% R 9 x 9 + d + ) d ' ) + Z % Γ % + 91% P %9 = DR 91% B 9 x 9 + d +, d ', +Z ) Γ ) + 91% P )9 = B(BC) (3-3) (4-3) (5-3) (6-3) x 9 + d + 2 d ' 2 = V 9 (7-3) x 9 + d + 4 d ' 4 = D 9 (8-3) BC 91% R 9 x 9 + d + K d ' K + Z % Γ % + 91% P %9 = M ) Z ) + P )9 + λ BW XYZ [ BW X[T [ y 9 BW \/ y 9 (1-3 Z % + P %9 + λ RW XYZ [ RW X[T [ y 9 RW \/ y 9 y 9 x 9 y 9 (11-3) P 9, y /, Z %, Z ), λ 1 (12-3) (9-3)

15 Robust-Fuzzy Portfolio Optimization based on Bertsimas and Sim Approach and Lee and Chesser's GP Model Azar and Rabiah (211) developed Bertsimas and Sim robust model applying fuzzy logics in uncertain environment. In robust optimization models like Bertsimas and Sim, the number at the middle of intervals is named as a nominal value. But in some cases, determining the interval is not easy and determination of interval s length is ambiguous. Then the conservatism would be vulnerable to the volatilities. To overcome this problem, fuzzy logic can help researchers by presenting the right hand side coefficients in fuzzy numbers.

16 Data: The Data of 2 big companies in Tehran Stock Exchange (TSE) for 7 years from Jan. 1, 21 to Jan. 1, 217 are tested in the model. Tehran stock exchange. The budget (BC) in the problem is 1,, monetary units, the revenue (DR) based on the decision maker idea is 2, monetary units, the systematic risk (B) based on the decision maker idea is.9, and the limit for investment in each stock is 15, monetary units. The minimum investment in securities based on systematic risk is equal to zero. This model was coded by Lingo. The data of the real market are presented in Table 1. Wi : W1=1; W2=1.5; W3=W4=:5; W5 =1 Γi is the price of robustness that is selected by the decision maker to fix his or her degree of pessimism with respect to each goal. For the i-th goal, Γi expresses how many of the left-hand side coefficients of this goal can reach their least favorable value.

17 Table 1: Data stocks Rate of return Systematic risk stocks Rate of return Systematic risk X X X X X X X X X X X X X X X X X X X X

18 (2,2) (15,15) (1,1) (9,9) (8,8) (7,7) (6,6) (5,5) (4,4) (3,3) (2,2) (1,1) (,) i(γ 1, Γ 2 ) X X X X X X X7 X X X X X X X X15

19 X X X X X Rate of return (2,2) (15,15) (1,1) (9,9) (8,8) (7,7) (6,6) (5,5) (4,4) (3,3) (2,2) (1,1) (,) Γ / (Γ %, Γ ) )

20 Conclusion: conservatism of the solution increases when the price of robustness increases. The model considers both uncertainty and the decision maker idea. Because it considers multi-objectives, we expect to be more practical in the real world, however we can see some unexpectedly and economically volatilities. The trend of rate of return is descending, however in some periods it experiences ascending order because of some special economic situations.

21 Limitations and Suggestions: q There are different constant limitations which should be controlled during study. q Multi-objective techniques require more data with more eyes. q Decision maker plays an important role. ü Evaluate more data in different periods with less constant features. ü Test with other multi-objective techniques.

22 Thank You