Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T14:46:11.982Z Has data issue: false hasContentIssue false

Tail Conditional Expectations for Exponential Dispersion Models

Published online by Cambridge University Press:  17 April 2015

Zinoviy Landsman
Affiliation:
Actuarial Research Center, Department of Statistics – University of Haifa, Mount Carmel, Haifa 31905, Israel, E-mail: [email protected]
Emiliano A. Valdez
Affiliation:
School of Actuarial Studies, Faculty of Commerce & Economics – University of New South Wales, Sydney, Australia 2052, E-mail: [email protected]
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.

There is a growing interest in the use of the tail conditional expectation as a measure of risk. For an institution faced with a random loss, the tail conditional expectation represents the conditional average amount of loss that can be incurred in a fixed period, given that the loss exceeds a specified value. This value is typically based on the quantile of the loss distribution, the so-called value-at-risk. The tail conditional expectation can therefore provide a measure of the amount of capital needed due to exposure to loss. This paper examines this risk measure for “exponential dispersion models”, a wide and popular class of distributions to actuaries which, on one hand, generalizes the Normal and shares some of its many important properties, but on the other hand, contains many distributions of nonnegative random variables like the Gamma and the Inverse Gaussian.

Type
Workshop
Copyright
Copyright © ASTIN Bulletin 2005

References

Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999) Coherent Measures of Risk. Mathematical Finance 9, 203228.CrossRefGoogle Scholar
Barndoff-Nielsen, O.E. (1978) Hyperbolic Distributions and Distributions on Hyperbolae. Scandinavian Journal of Statistics 5, 151157.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., and Nesbitt, C.J. (1997) Actuarial Mathematics 2nd edition. Society of Actuaries, Schaumburg, Illinois.Google Scholar
Brown, L.D. (1987) Fundamentals of Statistical Exponential Families. IMS Lecture Notes – Monograph Series: Vol. 9, Hayward, California.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Jorgensen, B. (1986) Some Properties of Exponential Dispersion Models. Scandinavian Journal of Statistics 13, 187198.Google Scholar
Jorgensen, B. (1987) Exponential Dispersion Models (with discussion). Journal of the Royal Statistical Society, Series B, 49, 127162.Google Scholar
Jorgensen, B. (1997) The Theory of Dispersion Models. Chapman & Hall, London.Google Scholar
Kaas, R., Dannenburg, D., Goovaerts, M. (1997) Exact Credibility for Weighted Observations. Astin Bulletin, 27, 2, 287295.CrossRefGoogle Scholar
Kagan, A.M., Linnik, Y.V. and Rao, C.R. (1973) Characterization Problems in Mathematical Statistics. John Wiley, New York.Google Scholar
Klugman, S., Panjer, H. and Willmot, G. (1998) Loss Models: From Data to Decisions. John Wiley, New York.Google Scholar
Landsman, Z. (2002) Credibility Theory: A New View from the Theory of Second Order Optimal Statistics. Insurance: Mathematics & Economics, 30, 351362.Google Scholar
Landsman, Z. and Makov, U. (1998). Exponential Dispersion Models and Credibility. Scandinavian Actuarial Journal, 1, 8996 CrossRefGoogle Scholar
Landsman, Z. and Valdez, E.A. (2003) Tail Conditional Expectations for Elliptical Distributions. North American Actuarial Journal, 7(4), 5571.CrossRefGoogle Scholar
Nelder, J.A. and Verrall, R.J. (1997) Credibility Theory and Generalized Linear Models. Astin Bulletin, 27, 7182.CrossRefGoogle Scholar
Nelder, J.A. and Wedderburn, R.W.M. (1972) Generalized Linear Models. Journal of the Royal Statistical Society, Series A, 135, 370384.CrossRefGoogle Scholar
Panjer, H.H. (2002) Measurement of Risk, Solvency Requirements, and Allocation of Capital within Financial Conglomerates. Institute of Insurance and Pension Research, University of Waterloo Research Report 01-15.Google Scholar
Tweedie, M.C.K. (1947) Functions of a Statistical Variate with Given Means, with Special Reference to Laplacian Distributions. Proceedings of the Cambridge Philosophical Society, 49, 4149.CrossRefGoogle Scholar