Comparing possibilistic portfolios to probabilistic ones

In this contribution, we compare mean-variance portfolios based on the standard probabilistic representation of the stock returns to mean-variance portfolios built by using stock returns represented as possibilistic numbers. There exists only a definition for the probabilistic mean and only one for the probabilistic variance, whereas there exist several definitions of possibilistic mean and of possibilistic variance. With reference to the latter, in this note we focus our attention on the definitions recently proposed in literature for modeling portfolio selection problems: the lower possibilistic mean and variance ([3]), the upper possibilistic mean and variance ([3]), and the possibilistic mean and variance `a la Zhang-Zhang-Xiao ([4]), hereinafter referred to as ZZX possibilistic mean and variance.

In this paper, we compare mean-variance portfolios based on the standard probabilistic representation of the stock returns to mean-variance portfolios built by using stock returns represented as possibilistic numbers.With reference to the latter, in this note we focus our attention on the definitions recently proposed in literature for modeling portfolio selection problems. In particular, first we investigate some theoretical properties of the possibilistic portfolios and compare them to the equivalent ones of the probabilistic portfolios, then, given the assets composing the Italian stock index FTSE MIB, we empirically compare the performances of the possibilistic portfolios to those of the probabilistic one. The results show that, generally, the probabilistic approach is more flexible than the possibilistic one in solving portfolio selection problems.