Pure risk premiums under deductibles

Abstract

It is a common practice in most insurance lines for the coverage to be restricted by a deductible. For example, it is often incorporated in motor, health, disability, life, and business insurance. The main idea of a deductible is, firstly, to reduce claim handling costs by excluding coverage for the often numerous small claims and, secondly, to provide some motivation to the insured to prevent claims through a limited degree of participation in claim costs (Daykin, Pentikainen, and Pesonen, 1994; Sundt, 1994; Klugman, Panjer, and Willmot, 1998). We mention the following properties of a deductible: (i) loss prevention - as the compensation is reduced by a deductible the retention of the insured is positive; This makes out a good case for avoiding the loss; (ii) loss reduction - the fact a deductible puts the policyholder at risk of obtaining only partial compensation provides an economic incentive to reduce the extend of the damage; (iii) avoidance of small claims where administration costs are dominant - for small losses, the administration costs will often exceed the loss itself, and hence the insurance company would want the policyholder to pay it himself; (iv) premium reduction - premium reduction can be an important aspect for the policyholders, they may prefer to take a higher deductible to get a lower premium. 428 19 Pure Risk Premiums under Deductibles There are two types of deductibles: an annual deductible and a per occurrence deductible, the latter being more common. We quote now an example from the American market. Blue Shield of California, an independent member of the Blue Shield Association, is California's second largest not-for-profit health care company, with 2 million members and USD 3 billion annual revenue. Blue Shield of California offers 4 Preferred Provider Organization (PPO) Plans, each offering similar levels of benefits with a different deductible option: USD 500, 750, 1500, and 2000, respectively. For example, the Blue Shield USD 500 Deductible PPO Plan has a USD 500 annual deductible for most covered expenses. This is just the case of the fixed amount deductible, which is exploited in Section 19.2.2. The annual deductible does not apply to office visits or prescription medications. Office visits and most lab and X-ray services are provided at a USD 30 copayment. This is also the case of the fixed amount deductible. For other covered services, after the annual deductible has been met, you pay 25% up to an annual maximum of USD 3500. This is a case of the limited proportional deductible, which is examined in Section 19.2.4. In Section 19.2 we present formulae for pure risk premiums under franchise, fixed amount, proportional, limited proportional, and disappearing deductibles in terms of the limited expected value function (levf), which was introduced and exploited in Chapter 13. Using the specific form of levf for different loss distributions, we present in Section 19.3 formulae for pure risk premiums under the deductibles for the log-normal, Pareto, Burr, Weibull, gamma, and mixture of two exponential distributions. The formulae can be used to obtain annual pure risk premiums under the deductibles in the individual and collective risk model framework analysed in Chapter 18. We illustrate graphically the influence of the parameters of the discussed deductibles on the premiums considering the Danish fire loss example, which was studied in Chapter 13. It gives an insight into an important issue of choosing an optimal deductible and its level for a potential insured and a proper pricing of the accepted risk for an insurer. © Springer-Verlag Berlin Heidelberg 2005.