Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-03T05:38:40.516Z Has data issue: false hasContentIssue false
  • English
  • Français

Premium Calculation for Fat-tailed Risk

Published online by Cambridge University Press:  17 April 2015

Roger Gay
Roger Gay*
Affiliation:
Dept. of Accounting and Finance, Monash University, Clayton, Australia, 3168, 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.

When insurance claims are governed by fat-tailed distributions considerable uncertainty about the value of the tail-index is often inescapable. In this paper, using the theory of risk aversion, a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) is established and its properties investigated. Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims, for which the ratio of the actual claims is an unbiased as well as a consistent estimator. Whereas thin-tailed claim premiums are determined largely by the first two moments of the claims distribution, fat-tailed claim premiums are determined by the first two extremes. The context of risk-aversion leads to a natural model for incorporating tail-index uncertainty into premiums, which nevertheless leaves the basic ratio structure unaltered. To illustrate the theory, possible ‘premiums’ for US hurricane data are examined, which utilize the consistent pattern of observed extremes.

Keywords

Exponential principlepower principleconstant risk aversiontail-index uncertaintyexpected value of extremes
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 163 - 188
Copyright
Copyright © ASTIN Bulletin 2005

References

Albrecht, P. (1992) Premium calculation without arbitrage, ASTIN Bulletin, 22, 247–256.CrossRefGoogle Scholar
Arnold, B., Balakrishnan, N. and Nagaraja, H. (1993) A first course in order statistics, J. Wiley.Google Scholar
Arrow, K.J. (1971) Theory of risk aversion in Essays on the theory of risk bearing, 90–120, Markham, Chicago.Google Scholar
Balkema, A.A. and de Haan, L. (1974) Residual lifetime at great age, Annals of Probability, 2, 792–804.CrossRefGoogle Scholar
Beirlant, J., Joossens, E. and Segers, J. (2004) A new model for large claims, North American Actuarial Journal, to appear.Google Scholar
Beirlant, J., Teugels, J.L. Vynckier, P. (1996) Practical analysis of extreme values, Leuven University Press.Google Scholar
Beirlant, J., Teugels, J.L., and Vynckier, P. (1994) Extremes in non-life insurance, Galambos, J. et al. eds, Extreme value theory and applications, 489–510, Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Risk Theory Society of Actuaries, USA.Google Scholar
Coles, S. (2001) An Introduction to Statistical Modelling of Extreme Values, Springer-Verlag, London.CrossRefGoogle Scholar
Coles, S.G. and Powell, E.A. (1996) Bayesian methods in extreme value modelling; a review and new developments, International Statistical Review, 64, 119–136.CrossRefGoogle Scholar
Coles, S.G. and Tawn, J.A. (1996) A Bayesian analysis of extreme rainfall data, Applied Statistics, 45, 463–478.CrossRefGoogle Scholar
Cox, D.R. and Hinkley, D.V. (1974) Theoretical Statistics, Chapman and Hall, London.CrossRefGoogle Scholar
Danielsson, J., De Haan, L., Peng, L. and De Vriess, C.G. (2001) Using the bootstrap method to choose the sample fraction in tail index estimation, J. Multivariate Anal., 76, 226–248.CrossRefGoogle Scholar
David, H.A. and Nagaraja, H.N. (2003) Order Statistics, 3rd. Edition, Wiley-Interscience.CrossRefGoogle Scholar
De Haan, L. (1994) Extreme value statistics, Galambos, J. et al. eds, Extreme value theory and applications, 93–122, Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Dekkers, A.L.M. and De Haan, L. (1993) Optimal choice of sample fraction for extreme value estimation. J. Multivariate Anal., 47, 173–195.CrossRefGoogle Scholar
Dekkers, A.L.M., Einmahl, J.H.J., and De Haan, L. (1989) A moment estimator for the index of an extreme value distribution, Ann. Statist., 17, 1833–1855.Google Scholar
Drees, H., Ferreira, A. and De Haan, L. (2004) On maximum likelihood estimation of the extreme value index, Annals of Applied Probability, 14, 1179–1201.CrossRefGoogle Scholar
Drees, H. and Kaufmann, E. (1998) Selecting the optimal sample fraction in univariate extreme value estimation, Stochastic Proc. Appl., 75, 149–172.CrossRefGoogle Scholar
Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Springer-Verlag.CrossRefGoogle Scholar
Feller, W. (1968) An introduction to probability theory and its applications, Vol. I, 3rd edn. J. Wiley.Google Scholar
Feller, W. (1971) An introduction to probability theory and its applications, Vol. II, J. Wiley.Google Scholar
Feuerverger, A. and Hall, P. (1999) Estimating the tail exponent by moment departure from a Pareto distribution, Ann. Statist., 27, 760–781.CrossRefGoogle Scholar
Gay, R. (2004a) Pricing risk when the distributions are fat-tailed. J. Appl. Prob. 41A, Special Volume: Festschrift for Chris Heyde; Stochastic Methods and Their Applications, 157–175.CrossRefGoogle Scholar
Gay, R. (2004b) The power principle and tail-fatness uncertainty, Monash University, Dept of Econometrics, Working Paper Series, (WP01/04). Available online at: http//:www.buseco.monash.edu.au/depts./ebs/pubs/wpapers/ Google Scholar
Grimshaw, D. (1990) A unification of tail-estimators, Comm. Statist. Theory Methods, 19, 4841–4857.CrossRefGoogle Scholar
Hall, P. and Tajvidi, N. (2000) Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data, Statistical Science, 15, 153–167.CrossRefGoogle Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution, Ann. Statist., 3, 1163–1174.CrossRefGoogle Scholar
Hill, B.M. (1994) Bayesian forecasting of extreme values in an exchangeable sequence, Journal of Research of the National Institute of Standards and Technology, 99, 521–538.CrossRefGoogle Scholar
Hsieh, P.-H. (2001a) Robustness of conditional moments: an application to premium calculation of reinsurance treaties, Risk Analysis, 21, 225–234.CrossRefGoogle ScholarPubMed
Hsieh, P.-H. (2001b) On Bayesian predictive moments of the next record value using three-parameter gamma priors, Communications in Statistics; Theory and Methods, 30, 729–728.CrossRefGoogle Scholar
Hsieh, P.-H. (2004) A data-analytic method for forecasting the next record catastrophic loss, Journal of risk and insurance, 71, 309–322.CrossRefGoogle Scholar
Huisman, R., Koedijk, K., Kool, C. and Palm, F. Tail index estimation in small samples, Journal of business and economic statistics, 19, 208–16.CrossRefGoogle Scholar
Kendall, M.G. and Stuart, A. (1969) The advanced theory of statistics, Vol. 1. Griffin.Google Scholar
Leadbetter, M. (1991) On a basis for ‘peaks over threshold’ modeling, Statistics and Probability Letters, 12, 357–362.CrossRefGoogle Scholar
Mikosch, T. (1997) Heavy-tailed modelling in insurance, Commun. Statist. – Stochastic models, 13, 799–815.CrossRefGoogle Scholar
Nagaraja, H.N. (2004) An introduction to extreme order statistics and actuarial applications, Paper presented at 2004 ERM Symposium April 26, 2004.Google Scholar
Oberhettinger, F.L. (1974) Tables of Mellin Transforms, Springer-Verlag, N.Y. CrossRefGoogle Scholar
Pandey, M.D., Van, Gelder P.H.A.J.M. and Vrijling, J.K. (2001) The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach, Structural Safety, 23, 179–192.CrossRefGoogle Scholar
Pickands, J. (1975) Statistical inference using extreme order statistics, Annals of Statistics, 3, 119–131.Google Scholar
Pickands, J. III (1994) Bayes quantile estimation and threshold selection in tail index estimation; in Galambos, J. et al. eds, Extreme value theory and applications (Dordrecht: Kluwer Academic).Google Scholar
Pratt, J.W. (1964) Risk aversion in the small and in the large, Econometrica, 32, 122–136.CrossRefGoogle Scholar
Puri, M.L. and Sen, P.K. (1971) Non-parametric methods in multivariate analysis, Wiley J. Google Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation and Point Processes, Springer, New York.CrossRefGoogle Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic processes for insurance and finance, Wiley.CrossRefGoogle Scholar
Smith, R.L. (2000) Measuring risk with extreme value theory. In Risk Management: Theory and Practice, edited by Dempster, M., Cambridge University Press, also as Chapter Two of Extremes and Integrated Risk Management, edited by Embrechts, P., Risk Books London, 19–35.Google Scholar
Smith, R.L. (1987) Estimating the tails of probability distributions, Annals of Statistics, 15, 1174–1207.CrossRefGoogle Scholar
Smith, R.L. (1985) Maximum likelihood estimation in a class of non-regular cases, Biometrika, 72, 67–90.CrossRefGoogle Scholar
Smith, R.L. and Goodman, D. (2000) Bayesian Risk Analysis, Chapter 17 of Extremes and Integrated Risk Management, Embrechts, P., Ed. Risk Books London, 235–251.Google Scholar
Smith, R.L. and Naylor, J.C. (1987) A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Applied Statistics, 36, 358–369.CrossRefGoogle Scholar
Teugels, J.L. and Vanroelen, G. (2003) Box-Cox transformations and heavy-tailed distributions, J. App. Prob. (Special Volume) 43a ‘Festschrift for C.C. Heyde, 2004’, Applied Probability Trust.Google Scholar
Wrench, J.W. Jr. (1968) Concerning two series for the gamma function, Math. Comput., 22, 616–626.CrossRefGoogle Scholar
Yuan, A. and Clarke, B.S. (1999) A minimally informative likelihood for decision analysis, Canad. J. Statist., 27, 649–665.CrossRefGoogle Scholar