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Distributionally Robust Inference for Extreme Value-at-Risk
- Publication Year :
- 2019
- Publisher :
- arXiv, 2019.
-
Abstract
- 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.
- Subjects :
- Statistics and Probability
Economics and Econometrics
021103 operations research
Optimization problem
Linear programming
Euclidean space
0211 other engineering and technologies
Mathematics - Statistics Theory
02 engineering and technology
Statistics Theory (math.ST)
01 natural sciences
Infimum and supremum
010104 statistics & probability
FOS: Mathematics
49K30, 62G32, 62G35
Applied mathematics
Uncountable set
0101 mathematics
Statistics, Probability and Uncertainty
Special case
Extreme value theory
Finite set
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....9118b7efc18e978905adcb4afdd7952f
- Full Text :
- https://doi.org/10.48550/arxiv.1902.05853