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Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The estimation of the spectral measure is challenging in practice and virtually impossible in high dimensions. This motivates the problem studied in this work, which is to find universal lower and upper bounds of the extreme Value-at-Risk under practically estimable constraints. That is, we study the infimum and supremum of the extreme Value-at-Risk functional, over the infinite dimensional space of all possible spectral measures that meet a finite set of constraints. We focus on extremal coefficient constraints, which are popular and easy to interpret in practice. Our contributions are twofold. First, we show that optimization problems over an infinite dimensional space of spectral measures are in fact dual problems to linear semi-infinite programs (LSIPs) – linear optimization problems in Euclidean space with an uncountable set of linear constraints. This allows us to prove that the optimal solutions are in fact attained by discrete spectral measures supported on finitely many atoms. Second, in the case of balanced portfolia, we establish further structural results for the lower bounds as well as closed form solutions for both the lower- and upper-bounds of extreme Value-at-Risk in the special case of a single extremal coefficient constraint. The solutions unveil important connections to the Tawn–Molchanov max-stable models. The results are illustrated with two applications: a real data example and closed-form formulae in a market plus sectors framework.

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.

This article considers the problem of examining time-varying causal effect moderation using observational, longitudinal data in which treatment, candidate moderators, and possible confounders are time varying. The structural nested mean model (SNMM) is used to specify the moderated time-varying causal effects of interest in a conditional mean model for a continuous response given time-varying treatments and moderators. We present an easy-to-use estimator of the SNMM that combines an existing regression-with-residuals (RR) approach with an inverse-probability-of-treatment weighting (IPTW) strategy. The RR approach has been shown to identify the moderated time-varying causal effects if the time-varying moderators are also the sole time-varying confounders. The proposed IPTW+RR approach provides estimators of the moderated time-varying causal effects in the SNMM in the presence of an additional, auxiliary set of known and measured time-varying confounders. We use a small simulation experiment to compare IPTW+RR versus the traditional regression approach and to compare small and large sample properties of asymptotic versus bootstrap estimators of the standard errors for the IPTW+RR approach. This article clarifies the distinction between time-varying moderators and time-varying confounders. We illustrate the methodology in a case study to assess if time-varying substance use moderates treatment effects on future substance use.