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Pricing risk when distributions are fat tailed

Part of: Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Roger Gay
Roger Gay*
Affiliation:
Monash University, Clayton, VIC 3800, Australia. Email address: [email protected]
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Abstract

In this paper, insurance claims X on [0, ∞) with tail distributions which are O(x−δ) for some δ > 1 are considered. Markets are assumed arbitrageable, the insurer setting a premium P > E[X]. Setting a premium as a fixed quantile of the loss distribution presents difficulties; for Pareto distributions with F(x) = 1 – (x + l)–δ ‘ultimately' (as δ ↓ 1) E[X] is larger than any quantile. When δ is near 1, premiums determined by weighting outcomes and a rule analogous to the expected utility principle are highly sensitive to change in δ, which is generally unknown or known only approximately. Under these circumstances, to protect insurers' interests, strategies are needed which provide some ‘premium stability' across a range of δ-values. We introduce a class of pricing functions which are functionally dependent on the governing loss distribution, and which are themselves distribution functions. We demonstrate that they provide a coherent framework for pricing insurance premiums when the loss distribution is fat tailed, and enable some degree of premium stability to be established.

Keywords

Insurancerisk theory

MSC classification

Primary: 91B30: Risk theory, insurance
Type
Part 3. Financial mathematics
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 157 - 175
Copyright
Copyright © Applied Probability Trust 2004 

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References

Albrecht, P. (1992). Premium calculation without arbitrage. ASTIN Bull. 22, 247–256.CrossRefGoogle Scholar
Arrow, K. J. (1971). The theory of risk aversion. In Essays on the Theory of Risk Bearing , Markham, Chicago, pp. 90–120.Google Scholar
Bladt, M. and Rydberg, T. V. (1998). An actuarial approach to option pricing under the physical measure without market assumptions. Insurance Math. Econom. 22, 65–73.CrossRefGoogle Scholar
Bowers, N. et al. (1986). Actuarial Mathematics. Society of Actuaries, Itasca, IL.Google Scholar
Camerer, C. F. (1989). An experimental test on several generalized utility theories. J. Risk Uncertainty 2, 61–104.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications , Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Fishburn, P. C. (1988). Nonlinear Preference and Utility Theory. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Friedman, M. and Savage, L. J. (1948). The utility analysis of choices involving risks. J. Political Economy 56, 279–304.CrossRefGoogle Scholar
Gopikrishnan, P. et al. (1999). Scaling of the distribution of price fluctuations of financial market indices. Phys. Rev. E 60, 5305–5316.CrossRefGoogle Scholar
Heyde, C. C. and Gay, R. (2003). Fractals in finance and insurance. Preprint.Google Scholar
Heyde, C. C., Liu, S. and Gay, R. (2001). Fractal scaling and Black-Scholes: the full story. J. Securities Inst. Austral. 1, 29–32.Google Scholar
Hogarth, R. and Einhorn, H. (1990). Venture theory: a model of decision weights. Manag. Sci. 36, 780–803.CrossRefGoogle Scholar
Hogg, R. V. and Klugman, S. A. (1984). Loss Distributions. John Wiley, New York.Google Scholar
Loomes, G. and Sugden, R. (1987). Regret theory: an alternative theory of rational choice under uncertainty. J. Econom. Theory 92, 805–824.Google Scholar
Loretan, M. and Phillips, P. C. B. (1994). Testing covariance stationarity of heavy tailed time series. J. Empirical Finance 1, 211–248.CrossRefGoogle Scholar
Machina, M. J. (1987). Choice under uncertainty: problems solved and unsolved. Econom. Perspectives 1, 121–154.CrossRefGoogle Scholar
Muller, U. A., Dacorogna, M. M., Olsen, R. B. and Pictet, ?. V. (1998). Heavy tails in high frequency financial data. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications , eds Adler, R. J., Feldman, R. E. and Taqqu, M. S., Birkhäuser, Boston, MA.Google Scholar
Plerou, V. et al. (1999). Scaling of the distribution of price fluctuations of individual companies. Phys. Rev. E 60, 6519–6529.Google Scholar
Quiggin, J. (1982). A theory of anticipated utility. J. Econom. Behaviour 3, 323–343.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
Schmiedler, D. (1989). Subjective probability and expected utility without additivity. Econometrica 57, 571–587.Google Scholar
Tversky, A. and Kahneman, D. (1979). Prospect theory: an analysis of decision under uncertainty. Econometrica 47, 263–291.Google Scholar
Tversky, A. and Kahneman, D. (1992). Advances in prospect theory; cumulative representation of uncertainty. J. Risk Uncertainty 5, 297–323.CrossRefGoogle Scholar
Von Neumann, J. and Morgenstern, O. (1947). Theory of Games and Economic Behaviour. Princeton University Press.Google Scholar