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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

References

Artzner, P. (1999) Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal 3(2), 11–25.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1997a) Definition of coherent measures of risk. Risk Management Symposium of the European Finance Association Meeting, Vienna, 08 27-30, 1997.Google Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1997b) Thinking coherently. RISK 10, no. 11 (11 1997), 68–70.Google Scholar
Ballmann, M., and Hürlimann, W. (2000) Zur Modellierung des ökonomischen Risikokapitals in der Lebensversicherung. Manuscript.Google Scholar
Bäuerle, N., and Müller, A. (1998) Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bulletin 28(1), 59–76.CrossRefGoogle Scholar
Blackwell, D. and Dubins, L.E. (1963) A converse to the dominated convergence theorem. Illinois Journal of Mathematics 7, 508–514.CrossRefGoogle Scholar
Block, H.W. and Sampson, A.R. (1988) Conditionally ordered distributions. Journal of Multivariate Analysis 27, 91–104.CrossRefGoogle Scholar
Bondesson, L. (1992) Generalized gamma convolutions and related classes of distributions. Lecture Notes in Statistics 76. Springer-Verlag.CrossRefGoogle Scholar
Dellacherie, C. (1970) Quelques commentaires sur les prolongements de capacités. Séminaire de Probabilités V, Strasbourg. Lecture Notes in Mathematics 191. Springer-Verlag, Berlin and New York.Google Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral. Theory and Decision Library, Series B, vol. 27. Kluwer Academic Publishers.CrossRefGoogle Scholar
Denuit, M., Dhaene, J. and Van Wouve, M. (1999) The Economics of Insurance: a review and some recent developments. Bulletin of the Swiss Association of Actuaries, 137–175.Google Scholar
Dhaene, J., Wang, S., Young, V. and Goovaerts, M.J. (2000) Comonotonicity and maximal stop-loss premiums. Bulletin of the Swiss Association of Actuaries, 99–113.Google Scholar
Dickson, D.C.M., and Waters, H.R. (1993) Gamma processes and finite time survival probabilities. ASTIN Bulletin 23(1), 259–272.CrossRefGoogle Scholar
Dubins, L.E. and Gilat, D. (1978) On the distribution of the maxima of martingales. Transactions of the American Mathematical Society 68, 337–38.Google Scholar
Dufresne, F., Gerber, H.U. and Shiu, E.S.W. (1991) Risk theory with the Gamma process. ASTIN Bulletin 21(2), 177–192.CrossRefGoogle Scholar
Embrechts, P. (1995) A survival kit to quantile estimation. UBS Quant Workshop, Zürich.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, Th. (1997) Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Embrechts, P., McNeil, A. and Straumann, D. (1998) Correlation and dependency in Risk Management: properties and pitfalls. Preprint, ETHZ.Google Scholar
Fisz, M. (1973) Wahrscheinlichkeitrechnung und mathematische Statistik. VEB Deutscher Verlag der Wissenschaften. Berlin.Google Scholar
Goovaerts, M.J. and Dhaene, J. (1999) Supermodular ordering and stochastic annuities. Insurance: Mathematics and Economics 24, 281–290.Google Scholar
Goovaerts, M.J., Dhaene, J. and De Schepper, A. (2000) Stochastic upper bounds for present value functions. Journal of Risk and Insurance Theory 67, 1–14.CrossRefGoogle Scholar
Hardy, G.H. and Littlewood, J.E. (1930) A maximal theorem with function-theoretic applications. Acta Mathematica 54, 81–116.CrossRefGoogle Scholar
Hürlimann, W. (1998a) Risk measurement and financial gain: a paradox and a reconciliation. Proceedings 2nd International Congress on Insurance: Mathematics and Economics, Lausanne, 07 20-22, 1998. Unpublished improved manuscript.Google Scholar
Hürlimann, W. (1998b) On distribution-free safe layer-additive pricing. Insurance: Mathematics and Economics 22, 277–85.Google Scholar
Hürlimann, W. (1998c) On stop-loss order and the distortion pricing principle. ASTIN Bulletin 28(1), 119–34.CrossRefGoogle Scholar
Hürlimann, W. (1998d) Inequalities for lookback option strategies and exchange risk modelling. Proceedings of the 1st Euro-Japanese Workshop on Stochastic Modelling for Finance, Insurance, Production and Reliability, Brussels, 1998.Google Scholar
Joe, H. (1990) Multivariate concordance. Journal of Multivariate Analysis 35, 12–30.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions (2nd ed.), vol. 1. J. Wiley, New York.Google Scholar
Kaas, R., Dhaene, J. and Goovaerts, M.J. (2000) Upper and lower bounds for sums of random variables. Paper presented at the 4th International Congress Insurance: Mathematics and Economics. Barcelona, 07 24-26, 2000.Google Scholar
Kertz, R.P. and Rösler, U. (1990) Martingales with given maxima and terminal distributions. Israel Journal of Mathematics 69, 173–192.CrossRefGoogle Scholar
Kertz, R.P. and Rösler, U. (1992) Stochastic and convex orders and lattices of probability measures, with a martingale interpretation. Israel Journal of Mathematics 11, 129–164.CrossRefGoogle Scholar
Kertz, R.P. and Rösler, U. (1993) Hyperbolic-concave functions and Hardy-Littlewood maximal functions. In: Stochastic Inequalities. IMS Lecture Notes – Monograph Series, vol. 22.Google Scholar
Landsberger, M. and Meilijson, I. (1994) Co-monotone allocations, Bickel-Lehman dispersion and the Arrow-Pratt measure of risk aversion. Annals Operations Research 52, 97–106.CrossRefGoogle Scholar
Lorentz, G.G. (1953) An inequality for rearrangements. American Mathematical Monthly 60, 178–79.CrossRefGoogle Scholar
Marshall, A.W. and Olkin, I. (1979) Inequalities, Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Mathai, A.M. (1982) Storage capacity of a dam with gamma type inputs. Annals of the Institute of Statistical Mathematics 34, 591–97.CrossRefGoogle Scholar
Meester, L.E. and Shanthikumar, J.G. (1993) Regularity of stochastic processes: a theory based on directional convexity. Probability in the Engineering and Informational Sciences 7, 343–360.CrossRefGoogle Scholar
Meilijson, I. and Nádas, A. (1979) Convex majorization with an application to the length of critical paths. Journal of Applied Probability 16, 671–77.CrossRefGoogle Scholar
Moschopoulos, P.G. (1985) The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics 37, 541–44.CrossRefGoogle Scholar
Müller, A. (1997) Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics 21, 219–223.Google Scholar
Panjer, H.H. and Willmot, G.E. (1992) Insurance Risk Models. Society of Actuaries, Schaumburg, Illinois.Google Scholar
Provost, S.B. (1989) On sums of independent gamma random variables. Statistics 20, 583–91.CrossRefGoogle Scholar
Rüschendorf, L. (1991) On conditional stochastic ordering of distributions. Advances in Applied Probability 23, 46–63.CrossRefGoogle Scholar
Schmeidler, D. (1986) Integral representation without additivity. Proceedings American Mathematical Society 91, 255–261.CrossRefGoogle Scholar
Schröder, M. (1996) The Value at Risk Approach – Proposals on a Generalization. Aktuarielle Ansätze für Finanz-Risiken, AFIR 1996, vol. 1, 151–170. Verlag Versicherungswirtschaft, Karlsruhe.Google Scholar
Seal, H.L. (1977) Approximations to risk theory's F(x,t) by means of gamma distributions. ASTIN Bulletin 9, 213–18.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J.G. (1997) Supermodular stochastic orders and positive dependence of random vectors. Journal of Multivariate Analysis 61, 86–101.CrossRefGoogle Scholar
Sim, C.H. (1992) Point processes with correlated gamma interarrival times. Statistics and Probability Letters 15, 135–141.CrossRefGoogle Scholar
Szekli, R., Disney, R.L. and Hur, S. (1994) MR/G1/1 queues with positively correlated arrival stream. Journal of Applied Probability 31, 497–514.CrossRefGoogle Scholar
Tchen, A.H. (1980) Inequalities for distributions with given marginals. The Annals of Probability 8(4), 814–27.CrossRefGoogle Scholar
Thorin, O. (1977) On the infinite divisibility of the lognormal distribution. Scandinavian Actuarial Journal, 121–148.CrossRefGoogle Scholar
Topkis, D.M. (1998) Supermodularity and Complementarity. Frontiers of Economic Research. Princeton University Press. Princeton, New Jersey.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin 26, 71–92.CrossRefGoogle Scholar
Wirch, J.L. (1999) Raising value at risk. North American Actuarial Journal 3(2), 106–115.CrossRefGoogle Scholar
Yaari, M.E. (1987) Dual theory of choice under uncertainty. Econometrica 55, 95–115.CrossRefGoogle Scholar