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Dynamic Pricing of General Insurance in a Competitive Market

Published online by Cambridge University Press:  17 April 2015

Paul Emms*
Affiliation:
of Actuarial Science and Insurance, Cass Business School, City University, London
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Abstract

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A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer’s objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation.

In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer’s risk aversion which leads to reduced market exposure and lower overall wealth generation.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2007

References

Asmussen, S. and Taksar, M. (1997) Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 20, 115.Google Scholar
Benth, F.E., Karlsen, K.H. and Reikvam, K. (2003) A semilinear Black and Scholes partial differential equation for valuing American options. Finance and Stochastics, 7, 277298.CrossRefGoogle Scholar
Britton, N.F. (1986) Reaction-Diffusion Equations and Their Applications to Biology. Academic Press.Google Scholar
Browne, S. (1995) Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Oper. Res., 20, 937958.CrossRefGoogle Scholar
Campbell, J.Y. and Viceira, L.M. (2001) Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. New York: Oxford University Press.Google Scholar
Daykin, C.D. and Hey, G.B. (1990) Managing Uncertainty in a General Insurance Company. Journal of the Insitute of Actuaries, 117(2), 173277.CrossRefGoogle Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries. Chapman and Hall.Google Scholar
Emms, P. and Haberman, S. (2005) Pricing general insurance using optimal control theory. Astin Bulletin, 35(2), 427453.CrossRefGoogle Scholar
Emms, P., Haberman, S. and Savoulli, I. (2007) Optimal strategies for pricing general insurance. Insurance: Mathematics & Economics, 40(1), 1534.Google Scholar
Fleming, W. and Rishel, R. (1975) Deterministic and Stochastic Optimal Control. New York: Springer Verlag.CrossRefGoogle Scholar
Fleming, W.H. and Soner, H.M. (1993) Controlled Markov Processes and Viscosity Solutions. Springer-Verlag.Google Scholar
Friedman, A. (1964) Partial Differential Equations of Parabolic type. Prentice-Hall.Google Scholar
Gerber, H.U. and Pafumi, G. (1998) Utility Functions: From Risk Theory to Finance. North American Actuarial Journal, 2(3), 74100.CrossRefGoogle Scholar
Hipp, C. (2004) Stochastic Control with Application in Insurance. Working Paper, Universität Karlsruhe, Germany.CrossRefGoogle Scholar
Hipp, C. and Plum, M. (2000) Optimal Investment for Insurers. Insurance: Mathematics and Economics, 27, 215228.Google Scholar
Højgaard, B. and Taksar, M. (1997) Optimal Proportional Reinsurance Policies for Diffusion Models. Scand. Actuarial J., 2, 166180.Google Scholar
Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance. Springer.Google Scholar
Kolmanovskii, V.B. and Shaikhet, L.E. (1996) Control of Systems with Aftereffect. Translations of Mathematical Monographs, vol. 157. Americal Mathematical Society.Google Scholar
Lilien, G.L. and Kotler, P. (1983) Marketing Decision Making. Harper & Row.Google Scholar
Merton, R.C. (1971) Optimum Consumption and Portfolio Rules in a Continuous Time Model. Journal of Economic Theory, 3, 373413.CrossRefGoogle Scholar
Merton, R.C. (1990) Continuous-time Finance. Blackwell.Google Scholar
Murray, J.D. (2002) Mathematical Biology. Springer.CrossRefGoogle Scholar
Pentikäinen, T. (1986) Discussion of “Underwriting strategy in a competitive insurance environment”. Insurance: Mathematics and Economics, 5, 8183.Google Scholar
Pratt, J.W. (1964) Risk Aversion in the Small and in the Large. Econometrica, 32, 122136.CrossRefGoogle Scholar
Samuelson, P.A. (1969) Lifetime Portfolio Selection by Dynamic Stochastic Programming. Review of Economics and Statistics, 51, 239246.CrossRefGoogle Scholar
Smith, G.D. (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.Google Scholar
Taylor, G.C. (1986) Underwriting strategy in a competitive insurance environment. Insurance: Mathematics and Economics, 5(1), 5977.Google Scholar