Abstract
Utility-based shortfall risk measures (SR) have received increasing attention over the past few years for their potential to quantify the risk of large tail losses more effectively than conditional value at risk. In this paper, we consider a distributionally robust version of the shortfall risk measure (DRSR) where the true probability distribution is unknown and the worst distribution from an ambiguity set of distributions is used to calculate the SR. We start by showing that the DRSR is a convex risk measure and under some special circumstance a coherent risk measure. We then move on to study an optimization problem with the objective of minimizing the DRSR of a random function and investigate numerical tractability of the optimization problem with the ambiguity set being constructed through phi -divergence ball and Kantorovich ball. In the case when the nominal distribution in the balls is an empirical distribution constructed through iid samples, we quantify convergence of the ambiguity sets to the true probability distribution as the sample size increases under the Kantorovich metric and consequently the optimal values of the corresponding DRSR problems. Specifically, we show that the error of the optimal value is linearly bounded by the error of each of the approximate ambiguity sets and subsequently derive a confidence interval of the optimal value under each of the approximation schemes. Some preliminary numerical test results are reported for the proposed modeling and computational schemes.
Highlights
Quantitative measure of risk is a key element for financial institutions and regulatory authorities
To quantify how the errors arising from the ambiguity set propagate to the optimal value of (DRSRP), we show under some moderate conditions that the error of the optimal value is linearly bounded by the error of the ambiguity set and subsequently derive finite sample guarantee (Theorem 1) and confidence intervals for the optimal value of (DRSRP) associated with the ambiguity sets (Theorem 2 and Corollary 1)
We investigate finite sample guarantees on the quality of the optimal solutions obtained from solving (DRSRP’-N), a concept proposed by Esfahani and Kuhn [8], as well as convergence of the optimal values as the sample size increases
Summary
Quantitative measure of risk is a key element for financial institutions and regulatory authorities. To see invariance under randomization, we note that SR defined as in (2) is a function on the space of random variables, it can be represented as a function on the space of probability measures, see [26, Remark 2.1] In the latter case, the acceptance set can be characterized by N := {μ ∈ P(C) : C l(−x)μ(d x) ≤ λ}, where P(C) denotes the space of probability measures with support being contained in a compact set C ⊂ IR. It might be possible to use some partial information such as empirical data, computer simulation, prior moments or subjective judgements to construct a set of distributions which contains or approximates the true probability distribution in good faith Under these circumstances, it might be reasonable to consider a distributionally robust version of (2) in order to hedge the risk arising from ambiguity of the true probability distribution,. For a sequence of subsets {SN } in a metric space, denote by lim supN→∞ SN its outer limit, that is, lim sup SN := {x : ∃ xNk ∈ SNk such that xNk → x as k → ∞}
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