Sharp Bounds on the Value-at-Risk for the Sum of Dependent Losses

• Activity Number: 176
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Business and Economic Statistics Section
• Author(s): Robert Alohimakalani Yuen* and Stilian Stoev and Dan Cooley
• Companies: University of Michigan and University of Michigan and Colorado State University
• Keywords: value-at-risk ; extremes ; portfolio losses ; regular variation

Abstract

When estimating extreme value-at-risk for the sum of dependent losses, it is imperative to determine the nature of dependencies in the tails of said losses. Characterizing the tail dependence of regularly varying losses involves working with the spectral measure, an infinite dimensional parameter that is difficult to infer and in many cases intractable. Conversely, various summary statistics of tail dependence such as extremal coefficients are manageable in the sense that they are finite dimensional and efficient estimates are obtainable. While extremal coefficients alone are not sufficient to characterize tail dependence, it was not previously known how they constrain the theoretical range of value-at-risk. The answer involves optimization over an infinite dimensional space of measures. In this work, we establish the solution and determine exact bounds on the asymptotic value-at-risk for the sum of regularly varying dependent losses when given full or partial knowledge of the extremal coefficients. We show that in-practice, the theoretical range of value-at-risk can be reduced significantly when relatively few, low dimensional extremal coefficients are given.