1. Upper bounds on value-at-risk for the maximum portfolio loss
- Author
-
Stilian Stoev and Robert Yuen
- Subjects
Statistics and Probability ,Mathematical optimization ,Multivariate random variable ,Economics, Econometrics and Finance (miscellaneous) ,Extremal dependence ,Upper and lower bounds ,Spectral measure ,Orthant ,Combinatorics ,Portfolio ,Extreme value theory ,Engineering (miscellaneous) ,Value at risk ,Mathematics - 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.
- Published
- 2014