On admissible efficient portfolio selection problem

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Abstract

The expected return and risk of asset cannot be predicted accurately because of uncertain factors that affect the finical markets. In this paper, the admissible efficient portfolio model is proposed under the assumption that the expected return and risk of asset have admissible errors to reflect the uncertainty in real investment actions. The upper and lower admissible efficient portfolios can be defined by the spreads of the portfolio expected returns and risks from the upper and lower bounds of admissible errors. The admissible efficient portfolio frontiers are derived explicitly when short sales are not allowed. A numerical example of a portfolio selection problem is given to illustrate our proposed effective means and approaches.

Introduction

The mean–variance methodology for the portfolio selection problem, posed originally by Markowitz [1], [2], has played an important role in the development of modern portfolio selection theory. It combines probability and optimization techniques to model the behavior investment under uncertainty. The return is measured by mean, and the risk is measured by variance, of a portfolio of assets. The Markowitz's mean–variance model for portfolio selection can be formulated mathematically in two ways: minimizing risk when a level return is given, maximizing return when a level risk is given. In the mean–variance portfolio selection problem, previous research includes Sharpe [3], Markowitz [4], Merton [5], Szegö [6], Perold [7], Pang [8], Elton and Gruber [9], etc. There exist a number of studies about efficient algorithms for finding efficient portfolio from solving mean–variance model (see, e.g. [10], [11], [12], [13], [14], [15], [16]). They have pointed out that it would be very difficult to obtain the efficient portfolios in closed form under short sales are not allowed for correlated assets (see, e.g. [11], [12], [13]). On the other hand, the main input data of the mean–variance model are expected returns and variance of expected returns of assets. The basic assumption for using mean–variance models is that the situation of asset markets in future can be correctly reflected by asset data in the past, that is, the mean and covariance of assets in future is similar to the past one. It is hard to ensure this kind of assumption for real ever-changing asset markets.

The portfolio selection model based on fuzzy probabilities has proposed by Tanaka et al. [17]. The mean vector and covariance matrix in Markowitz's model are replaced by the fuzzy weighted average vector and covariance matrix, respectively. It can be regarded as a natural extension of Markowitz's model because of extending probability into fuzzy probability. Its approach permits the incorporation of expert knowledge by means of a possibility grade, to reflect the degree of similarity between the future state of asset markets and the state of previous periods.

Since the return of asset is in a fuzzy uncertain economic environment and varies from time to time, the expected return and risk cannot be predicted accurately. Based on this fact, it is reasonable to the assumption that the expected return and risk have admissible errors. This paper deals with the portfolio selection problem based on the assumption that the expected return and risk of asset have admissible errors to reflect the uncertainty in real investment problem. We present the admissible efficient portfolio model and define the upper and lower admissible efficient portfolio frontiers by the spreads of the portfolio expected returns and risks from the upper and lower bounds of admissible errors in Section 2. The admissible efficient portfolio frontiers are derived explicitly when short sales are not allowed in Section 3. A numerical example of a portfolio selection problem is given to illustrate our proposed effective means and approaches in Section 4.

Section snippets

Definition and model of admissible efficient portfolio frontier

We consider a portfolio selection problem with n risky assets. Let rj be the expected return rate of asset j and let xj be the investment rate to asset j, j=1,…,n. In order to describe conveniently, we set x=(x1,x2,…,xn), r=(r1,r2,…,rn) and l=(1,…,1). We use prime (′) to denote matrix transposition and adopt the convention that all non-primed vectors are column vectors.

Then the expected return and variance associated with the portfolio x=(x1,x2,…,xn) are, respectively, given byE(r)=rx,D(r)=

The closed form solution of the admissible efficient portfolio

In this section we formulate an explicit solution to (9) based on some essential assumptions. Our results about (9) will require the following assumptions to be satisfied.

Assumption 1

(i) a+Φkl, for any k∈R, (ii) Σ+Vε is a positive definite matrix.

Assumption 1(i) is essential. Assumption 1(ii) is easily satisfied by a proper selection of Vε.

We introduce the following notations:e=(a+Φ)(Σ+Vε)−1(a+Φ),f=l(Σ+Vε)−1l,d=(a+Φ)(Σ+Vε)−1l,δ=ef−d2.Some properties of e, f, d and δ are given by Lemma 1.

Lemma 1

Let Assumption 1

Numerical example

In order to illustrate the proposed procedure and analytic formulas of the admissible efficient portfolios in this paper, let us consider a practical example shown in Table 1 introduced by Markowitz in 1959 (see [2]). Columns 2–10 represent American Tobacco, A.T.&T., United States Steel, General Motors, Atcheson & Topeka & Santa Fa, Coca-Cola, Borden, Firestone and Sharon Steel securities data, respectively. In this example, Φh and Φl are given byΦh

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