Lithuanian Mathematical Journal, Vol. 53, No. 4, October, 2013, pp. 412–422 Optimization of expected shortfall on convex sets Christos E. Kountzakis Department of Mathematics, University of the Aegean, Karlovassi, GR-83200 Samos, Greece (e-mail: chr_koun@aegean.gr) Received March 12, 2012; revised July 11, 2013 Abstract. In this article, we prove that the minimization problem of the expected shortfall over a convex but not nec- essarily closed set of financial positions X⊆ L 1 has a solution. We provide both minimax and variational approaches on this problem. In the case where the optimization conclusions arise from the application of subgradient arguments, we need the assumption that the set of financial positions X is closed. MSC: 46B40, 91B30 Keywords: expected shortfall, coherent risk measures, expectation-bounded risk measures, saddle-points, subgradient 1 Introduction 1.1 Expected shortfall and CVaR Expected shortfall ES a , where a ∈ (0, 1) denotes a level of significance, is identical to the conditional value at risk CVaR a as [2, Cor. 4.3] indicates. CVaR a is initially defined in [2, Def. 2.5], while ES a is a coherent risk measure on L 1 (Ω, F ,μ) (see [2, Prop. 3.1]) if we suppose that the financial positions are (Ω, F )-measurable real-valued random variables with respect to the probability measure μ of Ω defined on F . As it is well known from [2] and [19], the expected shortfall ES a (X ) for a financial position X and a level of significance a ∈ (0, 1) is defined in [2, Def. 2.6] as the negative of tail-mean of X at the significance level a: ES a (X )= - 1 a ( E(X 1 {Xqa(X)} ) - q a (X ) ( a - μ ( X  q a (X ) ))) , where q a (X ) denotes the a-lower quantile of X . As it is quoted in [12], the expression ES a (X )= -(1/a) ×  a 0 q u (X )du indicates that ES a is the building block for law-invariant, coherent risk measures, according to the results containing in [12]. These properties of CVaR a may make it very attractive in applications. Also, the expected shortfall according to [9, Thm. 4.1] admits the dual representation ES a (X ) = max Q∈Za E Q (-X ), where Z a = {Q ∈ M 1 | dQ/dμ  1/a μ-a.e.} with M 1 denoting the set of μ-continuous probability measures on the measurable space (Ω, F ) and dQ/dμ ∈ L 1 (Ω, F ,μ). However, for the probability measures 412 0363-1672/13/5304-0412 c  2013 Springer Science+Business Media New York