Matrix-variate risk measures under Wishart and gamma distributions

Matrix-variate distribution theory has been instrumental across various disciplines for the past seven decades. However, a comprehensive examination of financial literature reveals a notable gap concerning the application of matrix-variate extensions to Value-at-Risk (VaR). However, from a mathematical perspective, the core requirement for VaR lies in determining meaningful percentiles within the context of finance, necessitating the consideration of matrix c.d.f. This paper introduces the concept of “matrix-variate VaR” for both Wishart and Gamma distributions. To achieve this, we leverage the theory of hypergeometric functions of matrix argument and integrate over positive definite matrices. Our proposed approach adeptly characterizes a company's exposure by into a comprehensive risk measure. This facilitates a readily computable estimation of the total incurred risk. Notably, this approach enables efficient computation of risk measures under Wishart, exponential, Erlang, gamma, and chi-square distributions. The resulting risk measures are expressed in closed analytic forms, enhancing their practical utility for day-to-day risk management.