Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-31T19:53:15.341Z Has data issue: false hasContentIssue false

Economic Capital Allocations for Non-negative Portfolios of Dependent Risks

Published online by Cambridge University Press:  17 April 2015

Edward Furman
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON, Canada, M3J 1P3, E-mail: efurman@mathstat.yorku.ca
Zinoviy Landsman
Affiliation:
Department of Statistics, University of Haifa, Haifa, Mount Carmel 31905, Israel, E-mail: landsman@stat.haifa.ac.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we explore the problem of economic capital allocations in the context of non-negative multivariate (insurance) risks possessing a dependence structure. We derive a general result and illustrate it with a number of useful examples. One such example, for instance, develops explicit expressions for the discussed economic capital decomposition rule when the underlying portfolio consists of dependent compound Poisson risks.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2008

Footnotes

1

Corresponding author, tel.:+1-416-736-2100 (Ext 33768), fax:+1-416-736-5757.

References

Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking & Finance 26(7), 14871503.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9, 203228.CrossRefGoogle Scholar
Bildikar, S. and Patil, G.P. (1968) Multivariate exponential type distributions. Annals of Statistics 39(4), 13161326.CrossRefGoogle Scholar
Brazauskas, V., Jones, B.L., Puri, M.L. and Zitikis, R. (2008) Estimating Conditional Tail Expectation with Actuarial Applications in View. Journal of Statistical Planning and Inference 138(11), 35903604.CrossRefGoogle Scholar
Cai, J. and Li, H. (2005) Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability 42(3), 810825.CrossRefGoogle Scholar
Chiragiev, A. and Landsman, Z. (2007) Multivariate Pareto portfolios: TCE-based capital allocation and divided differences. Scandinavian Actuarial Journal 4, 261280.CrossRefGoogle Scholar
Denault, M. (2001) Coherent allocation of risk capital. Journal of Risk 4(1), 721.CrossRefGoogle Scholar
Denneberg, D. (1994) Non-additive measure and integral. Kluwer, Dordrecht.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial theory for dependent risks — measures, orders and models. John Wiley & Sons.CrossRefGoogle Scholar
Dhaene, J., Henrard, L., Landsman, Z., Vandendorpe, A. and Vanduffel, S. (2008) Some results on the CTE-based capital allocation rule. Insurance: Mathematics and Economics, (in press).CrossRefGoogle Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M., Kaas, R., Tang, Q. and Vyncke, D. (2006) Risk measures and comonotonicity: a review. Stochastic Models 22, 573606.CrossRefGoogle Scholar
Furman, E. (2008) On a multivariate Gamma distribution. Statistics and Probability Letters, 78(15), 23532360.CrossRefGoogle Scholar
Furman, E. and Landsman, Z. (2005) Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics 37(3), 635649.Google Scholar
Furman, E. and Landsman, Z. (2006a) On some risk-adjusted tail-based premium calculation principles. Journal of Actuarial Practice 13, 175191.Google Scholar
Furman, E. and Landsman, Z. (2006b) Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bulletin 36(2), 433462.CrossRefGoogle Scholar
Furman, E. and Zitikis, R. (2008a) Weighted premium calculation principles, Insurance: Mathematics and Economics 42(1), 459465.Google Scholar
Furman, E. and Zitikis, R. (2008b) Weighted risk capital allocations. Insurance: Mathematics and Economics 43(2), 263269.Google Scholar
Furman, E. and Zitikis, R. (2008c) General Stein-type decompositions of covariances: revisiting the Capital Asset Pricing Model. Available at SSRN: http://ssrn.com/abstract=1103333 Google Scholar
Hürlimann, W. (2003) Conditional Value-at-Risk bounds for compound Poison risks and a normal approximation. Journal of Applied Mathematics 3, 141153.CrossRefGoogle Scholar
Hürlimann, W. (2004) Multivariate Fréchet copulas and conditional value-at-risk. International Journal of Mathematics and Mathematical Sciences 7, 345364.CrossRefGoogle Scholar
Hürlimann, W. (2006) A note on generalized distortion risk measures. Finance Research Letters 3(4), 267272.CrossRefGoogle Scholar
Hürlimann, W. (2007a) Positive dependence properties of the multivariate reduction class. Far East Journal of Theoretical Statistics 21(2), 157169.Google Scholar
Joe, H. (1997) Multivariate models and dependence concepts. Chapman & Hall, London.Google Scholar
Jorgensen, B. (1997) The theory of dispersion models. Chapman & Hall, London.Google Scholar
Kaas, R., Goovaerts, M.J., Dhaene, J. and Denuit, M. (2001). Modern actuarial risk theory. Kluwer Academic Publishers.Google Scholar
Landsman, Z. and Valdez, E. (2003) Tail conditional expectation for elliptical distributions. North American Actuarial Journal 7(4), 5571.CrossRefGoogle Scholar
Landsman, Z. and Valdez, E. (2005) Tail conditional expectations for exponential dispersion models. ASTIN Bulletin 35(1), 189209.CrossRefGoogle Scholar
Leavens, D.H. (1945) Diversification of investments. Trusts and Estates 80, 469473.Google Scholar
Mardia, K.V. (1970) Families of bivariate distributions, Griffin, London.Google Scholar
Mcneil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative risk management, Princeton University Press, Princeton and Oxford.Google Scholar
Panjer, H.H. and Jia, J. (2001) Solvency and capital allocation . Institute of Insurance and Pension Research, University of Waterloo, Research Report N 01-14.Google Scholar
Patil, G.P. and Ord, J.K. (1976) On size-biased sampling and related form-invariant weighted distributions, Sankhya, Ser. B 38, 4861.Google Scholar
Patil, G.P. and Rao, C.R. (1978) Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics 34, 179189.CrossRefGoogle Scholar
Tasche, D. (2002) Expected shortfall and beyond. Journal of Banking and Finance 26(7), 15191533.CrossRefGoogle Scholar
Vernic, R. (2006) Multivariate skew-normal distributions with applications in insurance. Insurance: Mathematics & Economics 38, 413426.Google Scholar
Wang, S.S. (1995) nsurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics 17, 4354.Google Scholar
Wang, S.S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin 26, 7192.CrossRefGoogle Scholar
Wang, S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of Insurance prices. Insurance: Mathematics and Economics 21, 173183.Google Scholar
Zaik, E., Walter, J., Kelling, G. and James, C. (1996) RAROC at bank of America: from theory to practice. Journal of Applied Corporate Finance 9(2), 8393.CrossRefGoogle Scholar