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An interval portfolio selection problem based on regret function

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Abstract

Different approaches, besides the traditional Markowitz’s model, have been proposed in the literature to analyze portfolio selection problems. Among them we can cite the possibilistic portfolio models, which treat the expected return rates of the securities as fuzzy or possibilistic variables, instead of random variables. Such models, which are based on possibilistic mathematical programming, describe the uncertainty of the real world as ambiguity and vagueness, rather than stochasticity. Actually, another way to treat the uncertainty in decision making problems consists of assuming that the data are not well defined, but are able to vary in given intervals. Interval analysis is thus appropriate to handle the imprecise input data. In this paper we consider a portfolio selection problem in which the prices of the securities are treated as interval variables. In order to deal with such an interval portfolio problem, we propose the adoption of a minimax regret approach based on a regret function.

Introduction

In many real decision problems, data are usually imprecise and ambiguous. For this reason, in the specialized literature, there has been a growing interest in handling optimization problems by means of stochastic, fuzzy or possibilistic programming (see, for example, Esogbue and Hearnes, 1998, Inuiguchi et al., 1990, Inuiguchi and Tanino, 2002).

In particular, fuzzy and possibilistic programming have recently been employed in order to deal with portfolio selection problems, in addition to the traditional Markowitz’s model (see Li Calzi, 1990, Inuiguchi and Ramik, 2000, Inuiguchi and Tanino, 2000, Tanaka et al., 2000).

Whereas in Markowitz’s model the expected return rates of the portfolios are random variables, in possibilistic portfolio models they are fuzzy or possibilistic variables and the uncertainty of the investment decisions is thus described as ambiguity and vagueness, rather than stochasticity.

The meaning of various kinds of uncertainties (ambiguity and vagueness) is clarified in Inuiguchi and Ramik (2000), which highlights also the advantages and disadvantages of the use of a fuzzy approach with respect to a stochastic programming in a portfolio selection context. One of the advantages of fuzzy and possibilistic programming approaches is that they are generally more tractable than those based on stochastic programming and allow the inclusion of the knowledge of the experts in the model. Indeed, possibility portfolio models are based on a possibility distribution that is constructed by using experts’ judgments. With regard to this, see Tanaka et al. (2000).

In the decision-making literature another approach, based on interval analysis, allows handling imprecise input data. This approach consists in assuming that the data of a decision-making problem are not well defined but may vary in given intervals.

On this subject, in Alefeld and Mayer (2000) both theory and some applications of interval analysis are presented. Moreover, many papers deal with interval linear programming problems (see, for example, Chinneck and Ramadan, 2000, Inuiguchi and Sakawa, 1995).

In particular, Inuiguchi and Sakawa (1995) analyze a linear programming problem with interval objective function coefficients and give a new solution concept, based on the minimax regret criterion.

It is interesting to note that the minimax regret criterion has been also adopted to formulate possibilistic portfolio selection problems (see Inuiguchi and Ramik, 2000, Inuiguchi and Tanino, 2000). The idea is that an investor, who is supposed to know the value of the return rates after making its investment decision, wants to minimize the worst (maximum) regret. The worst regret represents the maximum deviation between the return that the investor could receive if he/she invests in the optimal portfolio and the portfolio return that he/she actually realizes.

In this paper we formulate and solve a portfolio selection problem using a regret approach, as in Inuiguchi and Tanino (2000). However, the imprecision and uncertainty characterizing the investment decision problems are not addressed within a possibilistic programming framework, but rather through the use of interval analysis.

Recently many applications of interval programming to portfolio selection can be found in Wang and Zhu (2002); among them the contribution of Lai et al. (2002) extends the Markowitz’s model to an interval programming model by quantifying the expected return and the covariance as intervals. Moreover, Ida (2003) solved a multiobjective portfolio selection problem with interval coefficients, in a Markowitz framework.

Differently from this approach, we formulate a minimax regret portfolio selection problem in which the prices of the securities are considered as interval variables. We discuss the properties of the regret function in detail and present some results. In particular, under given conditions, the initial interval problem can be transformed into a set of optimization problems.

The paper is organized as follows. In Section 2 some approaches adopted in the portfolio selection are summarized. In Section 3 the definition of the regret function is given and some theoretical results about its properties are presented. The interval portfolio selection problem is formulated in Section 4, and in Section 5 some special cases are discussed. The last section reports some concluding remarks. The proofs of some propositions and theorems are presented in the Appendix A.

Section snippets

Some approaches to portfolio selection

Let us consider n securities with return rate ci (i = 1, …, n) and denote by xi (with xi  0 and i=1nxi=1) the proportion of total amount of funds invested in the i-th security. A portfolio selection problem consists in finding the investment rate vector x = (x1, x2, …, xn) which maximizes the total portfolio return i=1ncixi.

Since investors usually make their decisions within an uncertain environment, the traditional approaches to portfolio selection treat the return rates as a random variables vector

An interval regret function

Let us consider a classical portfolio model. Let n be the total number of securities, xRn the investment rate vector and cRn the scaled price vector. The function cTx represents the total portfolio’s return, the equation eTx = 1 is the traditional balance constraint and the non negative constraint x  0 does not allow short selling. From a mathematical point of view, we consider prices instead of returns because prices can take only positive values. Moreover, we suppose to have scaled all the

The interval portfolio selection model

In this paper we analyze the following portfolio selection problem in which investor is assumed to minimize the upper bound of interval variable R(x):minxXRsup(x)=minxXmaxcCr(x;c).

From (3.7) the minimax portfolio problem can be written asminxXmaximaxciCiFci,cixi+jicjinfxj.

We can observe that our portfolio model is similar to that presented in Inuiguchi and Tanino (2000) for the formulation based on minimax regret criterion; on the other hand, it differs since the prices are interval

Some special cases

It can be interesting to analyze some special cases of functional forms of regret. The simplest case which satisfies the hypothesis of Proposition 5, is such that F is a linear function. Moreover a second case in which F is a fractional linear function is presented, fulfilling the assumption of Proposition 5. On the one hand, it represents a more realistic situation, on the other hand, it satisfies neither the hypothesis of Theorem 2 nor the one of Theorem 3.

Conclusions

The regret method is a well known approach in decision theory and in operational research. It has been considered in this work to formulate and solve a portfolio minimax regret problem, with interval variables. A part from rationality properties, the regret function can be quite general. One of the main results obtained shows that the interval portfolio selection problem can be tackled, under suitable conditions on F, by solving as many optimization problems as the securities number. These

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