Value‐at‐Risk Bounds With Variance Constraints

Abstract

We study bounds on the Value-at-Risk (VaR) of a portfolio when besides the marginal distributions of the components its variance is also known, a situation that is of considerable interest in risk management. We discuss when the bounds are sharp (attainable) and also point out a new connection between the study of VaR bounds and the convex ordering of aggregate risk. This connection leads to the construction of an algorithm, called Extended Rearrangement Algorithm (ERA), that makes it possible to approximate sharp VaR bounds. We test the stability and the quality of the algorithm in several numerical examples. We apply the results to the case of credit risk portfolio models and verify that adding the variance constraint gives rise to significantly tighter bounds in all situations of interest. IntroductionIn this article, we study bounds on the Value-at-Risk (VaR) of sums of risks with known marginal distributions (describing the stand-alone risks) under the additional constraint that a bound on the variance of the sum is known. The variance of the sum partially describes the dependence among the risks, as it involves their correlations. This setting is of significant interest as in many practical situations it corresponds closely to the maximum information at hand when assessing the VaR of a portfolio. For example, in the context of credit risk portfolio models, one typically has knowledge regarding the marginal risks (through the so-called PD, EAD, and LGD models), and the variance of the aggregate risk (sum of the individual losses) is also often available, as obtained from default correlation models or through statistical analysis of observed

A new approach to assessing model risk in high dimensions

“…However, note that the variables Z c i , which played crucial roles in deriving the bounds in Proposition 3.1 can also be expressed as Bernard et al (2015).…”

“…In these papers it is shown, among other results, that in several theoretical cases of interest one can construct a dependence among the risks that lead to mixability. Furthermore, in many practical cases one can construct a dependence such that the quantile function of the sum becomes approximately flat; see e.g., Embrechts et al (2013) and Bernard et al (2015) for illustrations. In this regard, we note that a large class of distributions exhibits asymptotic mixability implying that in high-dimensional problems the lower bound A that is stated in Proposition 3.1 is expected to be approximately sharp; see e.g., and Puccetti and Rüschendorf (2013).…”

“…Some previous papers have dealt explicitly with the presence of (partial) information on the dependence structure: Embrechts and Puccetti (2010) and Embrechts et al (2013) consider the situation in which some of the bivariate distributions are known; Denuit et al (1999) study VaR bounds assuming that the joint distribution of the risks is bounded by some distribution; and Bernard et al (2015) compute VaR bounds when the variance of the sum is known. However, the setup in these papers is often difficult to reconcile with the information that is available in practice; or, it does not make use of all available dependence information.…”

Centers of Probability Measures Without the Mean

“…Mixability serves as a building block for the solutions of many optimization problems under marginal-distributional constraints. Applications are found in optimal transportation [14], quantitative finance [6,2] and operations research [8,1].…”

“…. + x n = C = 1, (2) if and only if C satisfies condition (1). In general, for any probability measure µ on R, the set of C ∈ R satisfying (2) for some λ ∈ Γ(µ, .…”