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Analytical Bounds for two Value-at-Risk Functionals

Published online by Cambridge University Press:  29 August 2014

Werner Hürlimann
Werner Hürlimann*
Affiliation:
Value and Risk Management, Winterthur Life and Pensions, Postfach 300 – CH-8401 Winterthur – Switzerland, Tel. +41-52-2615861, E-mail: werner.huerlimann@winterthur.ch
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Abstract

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Based on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interest.

Keywords

Value-at-riskConditional value-at-riskEconomic risk capitalOrdering of risksStochastic boundsComonotonic risks
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 2 , November 2002 , pp. 235 - 265
Copyright
Copyright © International Actuarial Association 2002

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