Upper bounds on value-at-risk for the maximum portfolio loss

Abstract

Extremal dependence of the losses in a portfolio is one of the most important features that should be accounted for when estimating Value-at-Risk (VaR) at high levels. Multivariate extreme value theory provides a principled framework for the modeling and estimation of extremal dependence. In practice, however, this involves dealing with a challenging infinite dimensional parameter such as the spectral measure. Here, following recent developments in Schlather and Tawn (Extremes, 5(1) 87–102; 2002), Molchanov (Extremes, 11, 235–259; 2008), and Strokorb and Schlather (2013), we propose to represent extremal dependence of a multivariate portfolio via the so–called Tawn–Molchanov (TM) model, which is finite dimensional. Every max–stable random vector X can be associated with a TM max-stable vector Y = TM(X) so that the extremal coefficients of X and Ymatch and at the same time Y stochastically dominates X in the lower orthant order. This result readily yields an optimal upper bound on the value-at-risk \(\text {VaR}_{\alpha }(\mathbf {X}^{\vee })\) of the maximum portfolio loss \(\mathbf {X}^{\vee }:=\max _{j=1,\ldots ,d}X_{j}\). We develop a statistical methodology for estimating TM models from data and illustrate the resulting upper bounds on \(\text {VaR}_{\alpha }(\mathbf {X}^{\vee })\) with simulations and real data. Fitting TM models to portfolio data may be of independent practical interest, since their coefficients provide a qualitative picture of the degree and nature of diversification to extreme shocks.