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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.

Max-stable processes provide canonical models for the dependence of multivariate extremes. Inference with such models has been challenging due to the lack of tractable likelihoods which has motivated use of composite likelihood methods Padoan et al. (J. Amer. Stat. Assoc. 105(489):263–277, 2010). In contrast, the finite dimensional cumulative distribution functions (CDFs) appear natural to work with, and are readily available or can be approximated well. Motivated by this fact, in this work we develop an M-estimation framework for max-stable models based on the continuous ranked probability score (CRPS) of multivariate CDFs. We start by establishing conditions for the consistency and asymptotic normality of the CRPS-based estimators in a general context. We then implement them in the max-stable setting and provide readily computable expressions for their asymptotic covariance matrices. The resulting point and asymptotic confidence interval estimates are illustrated over popular simulated models. They enjoy accurate coverages and offer an alternative to composite likelihood based methods.