Stochastic Dominance and Risk Measure: A Decision-Theoretic Foundation for VAR and C-Var

Abstract

This paper is on decision theoretical foundations for various types of VaR models, including VaR and conditional-VaR, as objective measures of downside risk for financial prospects. We establish the connections of the VaRs with the first- and the second-order stochastic dominance investment criterions, and the logical argument for the equivalence between the first-order stochastic dominance and the VaR, and the equivalence between the second-order stochastic dominance and the c*-VaR as a modification of the conditional-VaR. We also discuss the usefulness and limitation of the VaRs and propose several alternative risk measures that are associated with the weaker behaviour assumptions underlying the VaRs.

Keywords: downside risk, value-at-risk, conditional-VaR, stochastic dominance, utility

JEL Classification: C0, D81, G10