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Some remarks on the value-at-risk and the conditional value-at-risk. (English) Zbl 0994.91031
Uryasev, Stanislav P. (ed.), Probabilistic constrained optimization. Methodology and applications. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 49, 272-281 (2000).
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.
For the entire collection see [Zbl 0959.00019].

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91B28 Finance etc. (MSC2000) 60E05 Probability distributions: general theory 60E15 Inequalities; stochastic orderings