Value-at-Risk and Expected Shortfall for Quadratic portfolio of securities with mixture of elliptic Distributed Risk Factors

TL;DR

This paper develops a method to estimate the Value-at-Risk and Expected Shortfall for quadratic portfolios of equities with elliptic distributed risk factors, avoiding derivatives Greeks and simplifying to a one-dimensional integral.

Contribution

It introduces a novel approach to compute quadratic VaR and Expected Shortfall for portfolios with elliptic distributions, bypassing Delta-Gamma approximations.

Findings

Method applies to mixture of normal and t-student distributions.

Quadratic VaR estimation reduces to a one-dimensional integral.

Provides a way to estimate Expected Shortfall for elliptic distributed risks.

Abstract

Generally, in the financial literature, the notion of quadratic VaR is implicitly confused with the Delta-Gamma VaR, because more authors dealt with portfolios that contains derivatives instruments. In this paper, we postpone to estimate the Value-at-Risk of a quadratic portfolio of securities (i.e equities) without the Delta and Gamma greeks, when the joint log-returns changes with multivariate elliptic distribution. We have reduced the estimation of the quadratic VaR of such portfolio to a resolution of one dimensional integral equation. To illustrate our method, we give special attention to the mixture of normal and mixture of t-student distribution. For given VaR, when joint Risk Factors changes with elliptic distribution, we show how to estimate an Expected Shortfall .