Analytical Bounds for two Value-at-Risk Functionals

Abstract

Based on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interest.

Assessing financial model risk

“…From Lemma 4.2, as both F L 0,1 and F L 0,1 are invertible, the following identities prevail for the extremum quantiles (see for instance Hürlimann 2002…”

“…Since the Expected Shortfall is monotone with respect to the stop-loss order (see for instance Bäuerle and Müller (2006)), we look at the extremal distributions for the stop-loss order on the set L 0,1 . Following Hürlimann (2002), we use the fact that the stop-loss transform for a distribution F is defined as: F (y))dy.…”

A scenario generation-based lower bounding approach for stochastic scheduling problems

“…), since Q(x, h) is non-decreasing and convex, the composite function u(Q(x, h)) is also non-decreasing and convex wrt h (Bazaraa et al, 2006 Hu¨rlimann (2002) has shown that CVaR preserves the increasing convex order. Hence, we have the following corollary:…”

Percentile residual life orders

“…On the other hand, verification of the order −rl may be a simpler matter-and it yields the same decision! Moreover, if the above engineer (or individual) has a choice between two markets that have different mixtures of aged machines, and if the original machine lifetimes in these markets satisfy (14) [here X and Y , ∈ , are the original machine lifetimes that are mixed in the two markets], then Theorem 4.6 determines which market is preferable.…”