Wishart Conditional Tail Risk Measures: An Analytic Approach

Abstract

This study introduces a new analytical framework for quantifying multivariate risk measures. Using the Wishart process, which is a stochastic process with values in the space of positive definite matrices, we derive several conditional tail risk measures which, thanks to the remarkable analytical properties of the Wishart process, can be explicitly computed up to a one- or two-dimensional integration. These quantities can also be used to solve analytically a capital allocation problem based on conditional moments. Exploiting the stochastic differential equation property of the Wishart process, we show how an intertemporal (i.e., time-lagged) view of these risk measures can be embedded in the proposed framework. Several numerical examples show that the framework is versatile and operational, thus providing a useful tool for risk management.