Risk Measures Based on Benchmark Loss Distributions

34 Pages Posted: 19 Dec 2017 Last revised: 27 Nov 2018

See all articles by Valeria Bignozzi

Valeria Bignozzi

Università di Milano Bicocca - Dipartimento di Statistica e Metodi Quantitativi

Matteo Burzoni

University of Oxford

Cosimo Munari

University of Zurich - Department of Banking and Finance; Swiss Finance Institute

Date Written: March 8, 2018

Abstract

We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), i.e.~a function that associates to each potential loss a maximal acceptable probability of occurrence. The corresponding risk measure, called Loss VaR (LVaR), determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first-order stochastic dominance. We provide a comprehensive study of the main finance theoretical and statistical properties of LVaR with a focus on their comparison with VaR and ES. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented.

Keywords: risk measures, loss distributions, tail risk, capital adequacy, portfolio management, catastrophic risk, robustness, backtestability

JEL Classification: D81, G32

Suggested Citation

Bignozzi, Valeria and Burzoni, Matteo and Munari, Cosimo, Risk Measures Based on Benchmark Loss Distributions (March 8, 2018). Swiss Finance Institute Research Paper No. 18-48. Available at SSRN: https://ssrn.com/abstract=3088423 or http://dx.doi.org/10.2139/ssrn.3088423

Valeria Bignozzi

Università di Milano Bicocca - Dipartimento di Statistica e Metodi Quantitativi ( email )

Via Bicocca degli Arcimboldi, 8
Milano, 20126
Italy

Matteo Burzoni (Contact Author)

University of Oxford ( email )

Andrew Wiles Building
Radcliffe Observatory Quarter (550)
Oxford, OX2 6GG
United Kingdom

Cosimo Munari

University of Zurich - Department of Banking and Finance ( email )

Schönberggasse 1
Zürich, 8001
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

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