Some remarks on the value-at-risk and the conditional value-at-risk (Q2724706)

For a real-valued random variable \(Y\) with distribution function \(F\) the Value-at-Risk (VaR) and the conditional VaR are defined by the quantile \(\text{VaR}_\alpha(Y) =F^{-1}(\alpha)\) and by \(\text{CVaR}_\alpha(Y)= E(Y|Y\geq\text{VaR}_\alpha(Y))\), respectively (where \(0<\alpha <1)\). Several basic properties of these risk measures, such as translation invariance, convexity, homogeneity and monotonicity, are shown. Furthermore, portfolio optimization problems of the type ``minimize the risk under the constraint that the expected return exceeds some prespecified level'' are considered. For CVaR, but not for VaR, every local optimum is global. A fixed-point property relating the two optimization problems is proved.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00019].