Covariance of uncertain random variables and its application to portfolio optimization

Abstract

Covariance is a device to measure the joint variability of two uncertain random variables. If the greater values of one uncertain random variable associated with the greater values of the other uncertain random variable, and the same state holds for the lesser values, i.e., the uncertain random variables have similar behavior, the covariance is positive. On the other hand, when the greater values of one uncertain random variable associate with the lesser values of the other, i.e., the variables tend to show opposite behavior, the covariance is negative. Since, interpretation of covariance is not easy, we consider the concept of correlation coefficient for two uncertain random variables as a normalized version of covariance. For calculating the covariance of uncertain random variables, some formulas are provided through the inverse uncertainty distribution. As an application of variance-covariance, portfolio selection problem is optimized by mean-variance model. The main results are explained by using several examples.