A Markov Chain Based No-Claim-Discount System

Abstract

In automobile insurance, among other general insurance policies, it is quite common to reduce the premium by a factor in case the insured does not make any claim in a given period. This is popularly known as NCD or no-claim-discount. Equally popular is the practice of increasing (known as ‘loading’) the premium, in case a claim is made. In effect, either system amounts to a multi-layer premium policy, where any particular policyholder is required to pick the level depending on the history of claims (s)he made in the immediate past few years. In this study, we inspect the desirability of this multi-layer premium system (also referred to as NCD system); to start with we work with a given number of levels with fixed gaps, and eventually look to determine an appropriate number of levels, as well the optimum gaps between levels.

The basic framework that we consider is that of a discrete time parameter Markov chain, where the state space consists of the different levels of the premium, and the state of a particular insured shifts randomly from an year to the next. The randomness of the transition is governed by the transition probability of causing an accident in a given year. We model this probability to be varying depending on quality of the driver. For the most part, we would be considering a finitely many groups of policyholders (drivers) characterized by respective probabilities of getting involved in an accident. The claim (damage) amount is assumed to have a known distribution, a log-normal one, for the sake of illustration. The parameters of this distribution need to be consistent with the basic premium level(s). We work under two possible behavior of all the policyholders. In the simpler version, we consider a driver would claim whenever (s)he incurs a damage from accident. In the second framework, (s)he would make a claim only if the (claim) damage amount exceeds the potential loss in the form of losing NCD. In making this judgment (s)he would need to take into account the effect of any more accident in the near future, and it is assumed that (s)he would rule out the such eventuality. This assumption is reasonable in the sense that, in reality, most drivers think very positively about their own driving ability; also, one is likely to be extra careful after getting involved in an accident.

We obtain the stationary distribution for each group of policyholders. This reflects the distribution of a particular group over the various levels of premium in the long run. For example, one can obtain the percentage of ‘good’ drivers expected to receive the fully discounted rate in the long run. A comparative study of these stationary distributions over the various groups considered, form the basis of appropriateness of the assumed NCD system. We show the existence of a stationary distribution, irrespective of the possible differences in the various NCD systems.

Finally, we look at the feasibility of evolving a NCD system which would be fair in the sense that the premium collected from any specific group would cover the expected claim amount from that group asymptotically. Use of simulation, properties of Markov chain, along with the basic actuarial and statistical principles form the core of the methodologies adopted for this study.