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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: [email protected], [email protected]
Steven Haberman
Affiliation:
Faculty of Actuarial Science and Statistics, Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom, Email: [email protected], [email protected]
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Abstract

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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

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