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Analytical Evaluation of Economic Risk Capital for Portfolios of Gamma Risks

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
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Abstract

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Based on the notions of value-at-risk and expected shortfall, we consider two functionals, abbreviated VaR and RaC, which represent the economic risk capital of a risky business over some time period required to cover losses with a high probability. 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). Quantitatively, RaC is equal to VaR plus an additional stop-loss dependent term, which takes into account the average amount at loss. Furthermore, RaC is additive for comonotonic risks, which is an important extremal situation encountered in the modeling of dependencies in multivariate risk portfolios. Numerical illustrations for portfolios of gamma distributed risks follow. As a result of independent interest, new analytical expressions for the exact probability density of sums of independent gamma random variables are included, which are similar but different to previous expressions by Provost (1989) and Sim (1992).

Keywords

Value-at-riskexpected shortfallrisk-adjusted capitalcomonotonicityadditivitysupermodular orderstop-loss ordergamma convolutions
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 31 , Issue 1 , May 2001 , pp. 107 - 122
Copyright
Copyright © International Actuarial Association 2001

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