Usage of Cholesky Decomposition in order to Decrease the Nonlinear Complexities of Some Nonlinear and Diversification Models and Present a Model in Framework of Mean-Semivariance for Portfolio Performance Evaluation

Abstract

In order to get efficiency frontier and performance evaluation of portfolio, nonlinear models and DEA nonlinear (diversification) models are mostly used. One of the most fundamental problems of usage of nonlinear and diversification models is their computational complexity. Therefore, in this paper, a method is presented in order to decrease nonlinear complexities and simplify calculations of nonlinear and diversification models used from variance and covariance matrix. For this purpose, we use a linear transformation which is obtained from the Cholesky decomposition of covariance matrix and eliminate linear correlation among financial assets. In the following, variance is an appropriate criterion for the risk when distribution of stock returns is to be normal and symmetric as such a thing does not occur in reality. On the other hand, investors of the financial markets do not have an equal reaction to positive and negative exchanges of the stocks and show more desirability towards the positive exchanges and higher sensitivity to the negative exchanges. Therefore, we present a diversification model in the mean-semivariance framework which is based on the desirability or sensitivity of investor to positive and negative exchanges, and rate of this desirability or sensitivity can be controlled by use of a coefficient.