A Multiple Objective Stochastic Portfolio Selection Program with Partial ...

IEEE.org | IEEE Xplore Digital Library | IEEE Standards | IEEE Spectrum | More Sites Access provided by: UNIVERSITY OF BAHRAIN Sign Out

Brow se Conference Publications > Com puter and Netw ork Technolo ...

A Multiple Objective Stochastic Portfolio Selection Program with Partial Information on Probability Distribution 3

Full Text Sign-In or Purchase

Masri, H. ; Fac. of Econ. Manage. & Inf. Syst., Univ. of Nizwa, Nizwa, Oman ; Ben Abdelaziz, F. ; Meftahi, I.

Author(s)

Abstract

Authors

References

Cited By

Keywords

Metrics

Similar

In this paper, we propose a multi objective stochastic model with linear partial information on probability distribution (MSPLI) for portfolio selection problem. We apply an extended chance constrained compromise programming approach to obtain the deterministic equivalent of the MSPLI model.

Published in: Computer and Network Technology (ICCNT), 2010 Second International Conference on Date of Conference: 23-25 April 2010 Page(s): 536 - 539

Conference Location : Bangkok

E-ISBN : 978-1-4244-6962-8

Digital Object Identifier : 10.1109/ICCNT.2010.97

Print ISBN: 978-0-7695-4042-9 INSPEC Accession Number: 11358108

Sign In | C reate Account

IEEE Account

Purchase Details

Profile Information

Need Help?

C hange Username/Password

Payment Options

C ommunications Preferences

US & Canada: +1 800 678 4333

Update Address

Order History

Profession and Education

Worldwide: +1 732 981 0060

Access Purchased Documents

Technical Interests

C ontact & Support

About IEEE Xplore | Contact | Help | Terms of Use | Nondiscrimination Policy | Site Map | Privacy & Opting Out of Cookies A not-for-profit organization, IEEE is the world's largest professional association for the advancement of technology. © C opyright 2013 IEEE - All rights reserved. Use of this web site signifies your agreement to the terms and conditions.

Second International Conference on Computer and Network Technology

A Multiple Objective Stochastic Portfolio Selection Program with Partial Information on Probability Distribution Hatem Masri Faculty of Economics Management and Information Systems , Un iversity of Nizwa Nizwa, Sultanate of Oman [email protected]

Fouad Ben Abdelaziz

Ines Meftahi

Engineering System Management Graduate Program College of Engineering, A merican University of Sharjah, UAE [email protected]

LA RODEC laboratory, Institut Supérieur de Gestion, University of Tunis, 41 Rue de la liberté, Le Bardo, Tunisia ines.meftehi@g mail.co m

Abstract— In this paper, we propose a multiobjective stochastic model with linear partial information on probability distribution (MS PLI) for portfolio selection problem. We apply an extended chance constrained compromise programming approach to obtain the deterministic equivalent of the MS PLI model.

[0,1] .All these cases may be modeled using the concept of linear partial information (LPI) on probability distribution. The LPI concept was introduced by Kofler [7] and it refers to the case where the probabilities pi of a g iven event [ i ,

KeywordsMultiobjective optimization, Portfolio selection

S

I.

programming,

i 1, ...,N , belong to a known polyhedral set as follows:

Stochastic

 p1 , , pN

t

N

: A P d b, ¦ pi i 1

½ 1; pi t 0, i 1,..., N ¾ (1) ¿

A (ai , j )i , j and b (bi )i are respectively an s u N and s u1 fixed matrices.

where

INT RODUCTION (HEADING 1)

Portfolio selection models address the problem of finding optimal proportions to be invested in a set of a fin ite number of securities. The basic model for portfolio selection was introduced by Markowitz [8]. The model is a biobjective mathematical program that maximizes expected returns and minimizes variances of securities. It leads to a frontier, called efficient frontier, composed by portfolios with different combinations of return and risk. The main difference between almost all investors in the stock exchange is related to the informat ion that they possess about the probability distribution of returns. Based on historical data and expertise, many models were developed to forecast the expectation and variance of these distributions [6,10]. For new and individual investors historical data and expert ise are not free of charge and may be expensive [9]. Therefore, they need for a decision model to select portfolios under partial information on the probability of returns. Let us denote by 4 {[1 ,..., [ N } the set of mutually exclusive states of the stock exchange and

Decision modeling with linear partial information on probability distribution was first introduced by Abel [ 1]. Ben Abdelaziz and Masri in [3] provide a general framework to solve stochastic programs with linear partial information on probability distribution (SPLI). They extended the main stochastic programming approaches, namely chance constrained approach, recourse approach and multistage approach, to solve the SPLI problem. They also proposed in [3] a solution strategy for a multiobjective version of the SPLI problem based on a chance constrained approach and a compro mise approach. In this paper, we propose a multiobjective stochastic program with linear partial information on probability distribution (MSPLI) to the portfolio selection problem. We consider two conflicting objectives to be maximized wh ich are the return and the liquidity. The risk is treated via the system of constraints which includes in addition the diversification constraints. In section 2, we introduce to the MSPLI model for the portfolio selection problem. In section 3, we propose under predefined hypotheses of the compromise chance constrained approach [2] to get a certainty equivalent program to the MSPLI model. In section 4, we illustrate all results with real data fro m the Tunisian stock exchange.

P : ( p1 ,..., pN )t the discrete probability d istribution that characterizes the possible state of the stock exchange. Part ial information on the probability distribution P may be encountered for example when experts are more co mfortable giving an interval of probability rather than a precise probability: D d pi d E , where D and E belong to 978-0-7695-4042-9/10 $26.00 © 2010 IEEE DOI 10.1109/ICCNT.2010.97

­ ®P : ¯

536

II.

THE MSPLI MODEL FOR PORTFOLIO SELECTION

For the system of constraints we consider the following four constraints: 1. The budget constraint: the initial sum of money

We define the portfolio selection problem for one period. We assume that the markets are perfect and that securities’ prices reflect their real values. Transaction costs are not considered and short sales are not permitted. We assume as usual a beginning of a holding period, an end of a holding period and an initial sum of money to be invested in n securities. We denote by xi , i=1,…,n, the proportion to be invested in the ith security. For the objective functions of the MSPLI problem, we maximize t wo conflicting objectives which are the return and the liquidity. An investor usually invests his wealth in securities with high prospects of return and which can be convert into cash rapidly. The two objectives are presented as follows:

n

should be totally invested:

Max

Maximize the rate of return:

¦ R x

i i

2.

The diversification constraint: i=1,…,n.

3.

The high risk constraint::

4.

The low risk constraint:

2.

Maximize the liquidity:

Max

¦ L x , where i i

Max ¦ Li xi i

Li

s.t.

Th

¦ xi t lr n

¦ xi

J i d xi d N i

(2) In the next section, we present a solution strategy for the MSPLI problem (2), by providing a certainty equivalent mathematical program to it. III.

is

S h and if the absolute value of E i is les than a pre-set value

T l than the security belongs to Sl . Finally, we can state the two sets Sh and S l as follows: Sl

{i, Ei  Tl }

THE COMPROMISE CHANCE CONST RAINED A PPROACH

Based on the compromise approach and the chance constrained approach, Ben Abdelaziz and Masri [2] solution strategy to build a certainty equivalent to the MSPLI problem (2) is as follows: x Random constraints are transformed to deterministic ones using the chance constrained approach, x The obtained chance constraints are linearized using the concept of p-Level Efficient point (pLEP) proposed by Prékopa et al. [11], x Multiple objectives are aggregated into a single objective using the compromise programming approach The chance constrained approach aims to find the optimal solution that remains feasible as much as possible and allows the decision maker to introduce exp licitly the reliability of constraints and to retain the most accepted solutions [4]. To illustrate the use of the chance constrained approach with the uncertain constraints in program (2), we first replace the high risk constraint by an equivalent one defined as

than the security belongs to

{i, Ei ! T h }

1

i

hr for high risk securities

Sh

¦ xi d hr

iSh

iSl

and lr for low risk securities, are random and that they follow the distribution P of the future market prices. This may be exp lained by the fact that the DM will be a risk taker when the market prices will increase and then he will invest more in high risk securities. Otherwise, he will be risk averse and try to invest more in low risk securities. To define the set of high risk securities Sh and low risk securities S l , we use the correlation coefficient beta E i for higher than a pre-set value

t lr

i

for the proportion xi to be invested in the security i, i=1,…,n. To deal with risk, we follow Zopounidis et al. [14] and we suppose that the decision maker (DM) is able to fix the proportions of investment that he or she would place in high risk securities and in low risk securities. Zopounidis et al. [ 14] suppose that these proportions are fixed. Tero l et al. [12] considered that these proportions are fuzzy. In this paper, we

Ei

i

n

and an upper bound

each security i, i=1,…,n. If the absolute value of

¦x

n

N i  [0,1]

suppose that these proportions,

d hr .

Max ¦ Ri xi

,

is the random rate of liquidity of security i. To guarantee diversification in the optimal portfolio, we

J i  >0,1@

i

The final portfolio selection model is the following MSPLI problem:

i

fix a lower bound

¦x

iSl

Ri is the random rate of return of security i. n

J i d xi d N i ,

iS h

i

where

1.

i

i

n

1.

¦x

537

follows: the sum of proportions to be invested in securities outside the set Sh should be greater than

¦x

i

§ ¦ xi · ¨ iSh ¸ t u z i i ¨ ¦ xi ¸ ¦ ¨ iS ¸ i d N © l ¹ ¦ ui 1

1  hr :

t 1  hr

iSh

Under a chance constrained approach, the uncertain constraints as follows:

id N

¦ xi t 1  hr and ¦ xi t lr are transformed

iSh

ui  {0,1}

iSl

§ ¦ xi t 1  hr · ¨ iSh ¸ P¨ ¸ t D , P  S  t x l ¦ ¨ iS i r ¸ l © ¹

(6) The compromise programming (CP) approach, first introduced by Zeleny [13], transforms a mu ltiobjective program to a uniobjective program by minimizing the deviations between the achieved objectives and the ideal values of each objective.

(3)

Let us denote by R and L the ideal values of the return and liability objective function, in the program (2), respectively. We propose to define these two ideal values as follows:

where P is the probability distribution defined to belong to the polyhedral set S and D is the satisfaction level specified by the decision maker. Based on the concept of lower probability [5], we replace the constraint (3) by the following equivalent constraint:

§ ¦ xi t 1  hr · ¨ iSh ¸ F¨ ¸ tD  xi t lr ¸ ¨ i¦ © Sl ¹

X

i

A Ž 4 with N the nu mber of realization of random parameters ( 4 {[1 ,..., [ N } ).

for all the sets

To enumerate all p LEP’s, we use propose to use the algorithm proposed by Prékopa [11]. This algorith m ends

^z

(1)

`

, z (2) ,...., z ( M ) of all p LEP’s and

therefore the constraint (4) can be rewritten as follows:

§ ¦ xi · ¨ iSh ¸ t z (i ) , for at least one i {1,..., M } ¨¨ ¦ xi ¸¸ © iSl ¹

i

i

,

Lˆi

`

max Li [i 4

and

1, J i d xi d N i .

Min O1G1 ([ )  O2G 2 ([ ) s.t. G1 ([ ) R  R t ([ ) x G 2 ([ ) L  Lt ([ ) x G1 t 0, G 2 t 0

(7) where O1 and O2 are weights that the DM assigns to characterize the importance of the two objectives, the return objective ( O1 ) and the liability objective ( O2 ). The function described in (7) is called the compro mise function [2]. The compro mise function is random and considering its expected value as an objective function needs to specify which probability distribution among those in S , we must consider. In this paper, we suppose that the decision maker is pessimist and is aware of the worst case and bad scenarios. This is a common behavior for many investors and the hypothesis of pessimism, in our point of v iew, is no t restrictive. Based on this additional hypothesis, the objective function is Max E p [C ( x, [ )]

y  4 such that y d z , y z z and F ( w d y ) t D .

Z

^

xX

[i 4

n

C ( x, [ )

Prékopa et al. [11] proposed the notion of p-Level Efficient point (pLEP) to linearize chance constraints. Ben Abdelaziz and Masri [2] extended this notion to deal with the constraint (4). An event z 4 is a pLEP regarding the lower probability F, if F ([ d z ) t D and there is no

with the set

Max ¦ Lˆi xi

max Ri

x ƒn / ¦ xi

i

n

Under a CP approach, we transform the two objectives in program (2) to a single objective where we minimize a weighted sum of the difference between the ideal values of the return and liability, and the obtained values of the return and liability objective functions, respectively:

¦p

[i  A

Rˆi

where

all set of possible events A Ž 4 . The constraint (4) is not a linear and to define F, we need to solve 2 N linear programs PS

xX

L

(4)

where F is the lower probability function associated with the polyhedral set S and F ( A) Inf {P( A) / P  S } , for

Min

n

Max ¦ Rˆi xi

R

(5)

In order to avoid the condition “for at least one i”, we can use the following relaxation of (5) [11]:

PS

538

Under a chance constrained approach, a compro mise approach and the hypothesis that the DM is pessimist, we end with the following certainty equivalent program

determin istic equivalent of the proposed model and we have solved it using the modified L-shaped method proposed by Ben Abdelaziz and Masri [7].

Min max E p [C ( x, [ )]

REFERENCES

PS

[1]

s.t. § ¦ xi · ¨ iSh ¸ t ¦ u z i ¨ ¦ xi ¸ i d N i ¨ iS ¸ © l ¹ ¦ ui 1

[2]

(8)

[3]

id N

ui  {0,1} n

¦ xi

[4] [5]

1

i

J i d xi d N i

[6] [7]

Ben Abdelaziz and Masri [2] noted that the compromise function (7) is piecewise linear convex on x and then it is possible to use the modified L-shaped algorithm [3] to solve the program (8). We note it was proved in [3] that this algorithm fin itely converges to an optimal solution when it exists. In this paper, we ovoid the presentation of the algorithm and we refer the reader to the paper [3] for more details. IV.

[8] [9] [10] [11]

CONCLUSION

In this work, we developed a multi objective model for portfolio selection problem in an uncertain environment. We focused on two conflicting objectives which are the return and the liquidity. The system of constraints includes, in addition to usual constraints (Budget and diversification constraint), random constraints which are h igh risk and low risk constraints. The uncertainty of the environment imp lies that the probability distribution cannot be provided precisely based on historical data. For that we assume partial information on probability distribution. We have applied the compro mise chance constrained approach to obtain the

[12] [13] [14]

539

Abel, P., Stochastic linear programming with recourse under partial information, in: Probability and Bayesian Statistics, Innsbruck 1986, pp. 1-6. F. Ben Abdelaziz and H. Masri, “A compromise solution for the multiobjective stochastic linear programming under partial uncertainty”, European Journal of Operations Research, vol.. 202 (1), pp. 55-59, 2010. F. Ben Abdelaziz and H. Masri, “ Stochastic programming with fuzzy probability distribution”, European Journal Operational Research, vol. 162 (3), pp. 619-629, 2005. A. Charnes and W. W. Cooper, “ Chance Constrained Programming”, Management Science, vol. 6 (1), pp 73-79, 1959. A. P. Dempster, “Upper and lower probabilities induced by multivalued mapping”, The annals of mathematical statistics, vol. 38, pp. 325-339, 1967. G. N. Gregoriou, “Stock Market Volatility”, Chapman & Hall, 2009. E. Kofler, “ Linear partial information with applications”, Fuzzy sets and systems, vol. 118, pp. 167-177, 2001. H. Markowitz, “Portfolio selection”, The Journal of Finance, vol. 7, pp. 77-91, 1952. J. Muntermann, “Towards ubiquitous information supply for individual investors: A decision support system design”, Decision Support Systems, vol. 47 (2), pp. 82-92, 2009. J. E. Murphy, “Stock Market Probability: Using Statistics to Predict and Optimize Investment Outcomes”, McGraw-Hill, 1994. A. Prékopa, B. Vizvari and T. Badics, “Programming under probabilistic constraint with discrete random variable”, in: L.Grandinetti, et al., (Eds.), New Trends in Mathematical Programming, Kluwer, Dordrecht, Boston, pp. 235-255, 1998. A. B. Terol, B. P. Gladish, M. A. Parra and M. V. R. Uria, “ Fuzzy compromise programming for portfolio selection”, Applied Mathematics and Computation, vol. 173, pp. 251-264, 2006. M. Zeleny, “A concept of compromise solutions and the method of the displaced ideal”. Computers and Operations Research, vol. 1, pp. 479-496, 1974. C. Zopounidis, D. K. Despotis and I. kamaratou, “Portfolio Selection Using the ADELAIS Multiobjective Linear Programming System”, Computational Economics, vol. 11, pp. 189-204, 1998.