Why is CVaR superior to VaR? (c2009)

Abstract

Until recently, value-at-risk (VaR) has been a widely used risk measure in portfolio optimization. The large number of recent bank failures shows that VaR failed to account for the expected losses which resulted from the outburst of a rare event such as the global financial crisis, thereby questioning its reliability and credibility as a measure of risk. Alternatively, previous work concurs that conditional value-at-risk (CVaR) is a coherent tail risk measure, and has established the superiority of CVaR over traditional measures of risk (variance and VaR) from a theoretical standpoint. This study aims at investigating the reasons that render CVaR superior to other traditional risk measures from an empirical perspective. We develop a theoretical model that solves the mean-risk portfolio optimization problem within a unified framework for all three different measures of risk (variance, VaR, and CVaR). We test our model empirically using financial data on return indices over a period covering the financial crisis. Our results support the theoretical predictions regarding the superiority of CVaR. We find that the mean-CVAR framework can be applied to multi-model returns, unlike mean-variance (where variance is a dispersion measure) and mean-VaR (where VaR is anon-coherent risk measure) which are only valid when returns are normal. The mean-CVaR framework respects diversification, and we find that CVaR is the most conservative measure of risk.