Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T05:42:04.059Z Has data issue: false hasContentIssue false
  • English
  • Français

Pricing General Insurance Using Optimal Control Theory

Published online by Cambridge University Press:  17 April 2015

Paul Emms  and
Steven Haberman
Paul Emms
Affiliation:
Faculty of Actuarial Science and Statistics, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom, Email: p.emms@city.ac.uk, s.haberman@city.ac.uk
Steven Haberman
Affiliation:
Faculty of Actuarial Science and Statistics, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom, Email: p.emms@city.ac.uk, s.haberman@city.ac.uk
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.

Insurance premiums are calculated using optimal control theory by maximising the terminal wealth of an insurer under a demand law. If the insurer sets a low premium to generate exposure then profits are reduced, whereas a high premium leads to reduced demand. A continuous stochastic model is developed, which generalises the deterministic discrete model of Taylor (1986). An attractive simplification of this model is that existing policyholders should pay the premium rate currently set by the insurer. It is shown that this assumption leads to a bang-bang optimal premium strategy, which cannot be optimal for the insurer in realistic applications.

The model is then modified by introducing an accrued premium rate representing the accumulated premium rates received from existing and new customers. Policyholders pay the premium rate in force at the start of their contract and pay this rate for the duration of the policy. It is shown that, for two demand functions, an optimal premium strategy is well-defined and smooth for certain parameter choices. It is shown for a linear demand function that these strategies yield the optimal dynamic premium if the market average premium is lognormally distributed.

Keywords

Competitive demand modelOptimal premium strategiesMaximum principleBellman equation
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 2 , November 2005 , pp. 427 - 453
Copyright
Copyright © ASTIN Bulletin 2005

References

Bender, C. and Orszag, S. (1978) Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill.Google Scholar
Brockett, P.L. and Xia, X. (1996) Operations Research in Insurance: A Review. Transactions of the Society of Actuaries, 47, 1–74.Google Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries. Chapman and Hall.Google Scholar
Emms, P., Haberman, S. and Savoulli, I. (2004) Optimal strategies for pricing general insurance. (in review) Cass Business School, City University, London.Google Scholar
Fleming, W. and Rishel, R. (1975) Deterministic and Stochastic Optimal Control. New York: Springer Verlag.CrossRefGoogle Scholar
Gelfand, I.M. and Fomin, S.V. (2000) Calculus of Variations. Dover.Google Scholar
Gerrard, R.J. and Glass, C.A. (2004) Optimal premium policy in motor insurance; discrete approximation. Working paper, Cass Business School, London.Google Scholar
Hipp, C. and Plum, M. (2003) Optimal investment for investors with state dependent income, and for insurers. Finance Stochast., 7, 299–321.CrossRefGoogle Scholar
Højgaard, B. and Taksar, M. (1997) Optimal Proportional Reinsurance Policies for Diffusion Models. Scand. Actuarial J., 2, 166–180.Google Scholar
Hürlimann, W. (1998) On Stop-Loss Order and the Distortion Pricing Principle. ASTIN Bulletin, 28(1), 119–134.CrossRefGoogle Scholar
Lilien, G.L. and Kotler, P. (1983) Marketing Decision Making. Harper & Row.Google Scholar
Paulsen, J. and Gjessing, H.K. (1997) Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics, 20(3), 215–223.Google Scholar
Rantala, J. (1988) Fluctuations in Insurance Business Results: Some Control Theoretic Aspects. In: 23rd International Congress of Actuaries.Google Scholar
Sethi, S.P. and Thompson, G.L. (2000) Optimal Control Theory. 2nd edn. Kluwer Academic Publishers.Google Scholar
Taylor, G.C. (1986) Underwriting strategy in a competitive insurance environment. Insurance: Mathematics and Economics, 5(1), 5–77.Google Scholar
Wang, S. (1996) Premium Calculation by Transforming the Layer Premium Density. ASTIN Bulletin, 26(2), 71–92.CrossRefGoogle Scholar
Yong, J. and Zhou, X.Y. (1999) Stochastic controls: Hamiltonian systems and HJB equations. Springer-Verlag.CrossRefGoogle Scholar