Range Value-at-Risk bounds for unimodal distributions under partial information

In this paper, we derive upper and lower bounds on the Range Value-at-Risk of the portfolio loss when we only know its mean, variance, and feature of unimodality. In a first step, we use some classic results on stochastic ordering to reduce this optimization problem to a parametric one, which in a second step can be solved using standard methods. The novel approach we propose makes it possible to obtain analytical results for all probability levels and is moreover amendable to other situations of interest. Specifically, we apply our method to obtain risk bounds in the case of a portfolio loss that is non-negative (as is often the case in practice) and whose variance is possibly infinite. Numerical illustrations show that in various cases of interest we obtain bounds that are of practical importance.