A MEAN-VARIANCE-SKEWNESS PORTFOLIO OPTIMIZATION MODEL

Abstract

We will propose a mean-variance-skewness(MVS) portfolio optimization model, a direct exten-sion of the classical mean-variance model to the situation where the skewness of the rate of return of assets and the third order derivative of a utility function play significant roles in choosing an optimal portfolio. The MVS model enables one to calculate an approximate mean-variance-skewness efficient surface, by which one can compute a portfolio with maximal expected utility for any decreasingly risk averse utility functions. Also, we propose three computational schemes for solving an associated nonconcave maximization problem, and some preliminary computational results will be reported. 1. Introduction In [12], the authors proposed a mean-absolute deviation-skewness (MADS) portfolio opti-mization model, in which the lower semi-third moment of the rate of return of a portfolio is maximized subject to constraints on the mean and the absolute deviation of the rate of return. This model is an extension of the standard Mean-Variance (MV) model developed by H. Markowitz [18J, and is motivated by observations on the distribution of stock data in the market and on the practitioners' perception against risk. The standard MV model is based upon the assumptions that an investor is risk averse and that either (i) the distribution of the rate of return is multi-variate normal, or (ii) the utility of the investor is a quadratic function of the rate of return. Unfortunately however, neither (i) nor (ii) holds in practice. It is now widely recognized that the real stock data do not follow a multi-variate normal distribution. To the contrary, detailed analysis of historical data observed in the Tokyo Stock Exchange (TSE) shows that the majority of them follow positively skewed distributions (See [1, 12, 17J for details). Also, many investors prefer a positively skewed distributions to a negative one, if the expected value and variance are the same. Furthermore, some investors prefer a distribution with larger skewness at the expense of larger variance. This means that utility functions of investors are not quadratic.