A continuous time solution for optimal claim limits in vehicle insurance
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A continuous time solution for optimal claim limits in vehicle insurance JS Dagpunar* University of Edinburgh, UK The traditional method of obtaining optimal claim limits for vehicle insurance is to discretise the state space and use successive approximations. In this paper we show how the stochastic dynamic programming equations reduce to a set of differential equations, in which these are easily solved to provide exact continuous time solutions. The resulting model can be used for evaluating alternative levels of excess. Keywords: dynamic programming; stochastic processes; insurance
Introduction In this paper we describe a new and ef®cient method for deciding whether or not an insured person should claim on a vehicle insurance policy following an accident. The incentive for not claiming is well known by every driver. It is that depending upon the claims history, there may be a reduction in the premium at the next renewal. This is known as the `no claims bonus or discount'. Operational Research has played a part in the development of mathematical models within insurance. This is not surprising since decision making and risk assessment ®gure prominently in such problems. A useful review is given in Haehling von Lanzenauer and Wright.1 For vehicle insurance, several authors have explored the problem to be dealt with here. Hastings2 constructed a basic discrete time model for determining `claim limits' for accident costs, above which it was sensible to claim, rather than for the insured to pay out of his own pocket. In Hastings' model the claim limits do not depend upon the time remaining until the next renewal. Haehling von Lanzenauer3 and Norman and Shearn4 recognised this as unrealistic. They split the interval between renewals into sub-intervals and solved an associated discrete time, discrete space stochastic dynamic program. Chappell and Norman5 considered simpli®ed decision rules for `protected bonus' policies, where a no claims bonus will only be reduced if there are more than a speci®ed number of claims within a speci®ed period. A negative exponential distribution for accident cost is assumed in several of these papers, while Tijms6 presented a comparison of lognormal and gamma distributed accident costs. *Correspondence: Dr JS Dagpunar, Department of Mathematics and Statistics, University of Edinburgh, JCMB, King's Buildings, May®eld Road, Edinburgh EH9 3JZ, UK. E-mail: [email protected]
A number of questions arise in the scenario we explored. Some relate to the insured, others to the insurer. These typical questions are: (1) if an accident occurs should the insured claim?; (2) when there is a choice of voluntary excesses, which is the `best buy' for the insured?; and (3) for the insurer, what is the long-run average pro®t and premium collected per policy per annum? This present paper provides a method for answering these questions and improves previous approaches in two respects. Firstly
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