No abstract

Motor insurance is a very competitive business where insurers operate with quite large portfolios, often decisions must be taken under short horizons and therefore ruin probabilities should be calculated in finite time. The probability of ruin, in continuous and finite time, is numerically evaluated under the classical Cramér–Lundberg risk process framework for a large motor insurance portfolio, where we allow for a posteriori premium adjustments, according to the claim record of each individual policyholder. Focusing on the classical model for bonus-malus systems, we propose that the probability of ruin can be interpreted as a measure to decide between different bonus-malus scales or even between different bonus-malus rules. In our work, the required initial surplus can also be evaluated. We consider an application of a bonus-malus system for motor insurance to study the impact of experience rating in ruin probabilities. For that, we used a real commercial scale of an insurer operating in the Portuguese market, and we also work on various well-known optimal bonus-malus scales estimated with real data from that insurer. Results involving these scales are discussed.

In this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

The probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating.We consider a portfolio of risks which satisfy the assumptions of the Bühlmann (1967, 1969) or Bühlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.

For a large motor insurance portfolio, on an open environment, we study the impact of experience rating in finite and continuous time ruin probabilities. We consider a model for calculating ruin probabilities applicable to large portfolios with a Markovian Bonus-Malus System (BMS), based on claim counts, for an automobile portfolio using the classical risk framework model. New challenges are brought when an open portfolio scenario is introduced. When compared with a classical BMS approach ruin probabilities may change significantly. By using a BMS of a Portuguese insurer, we illustrate and discuss the impact of the proposed formulation on the initial surplus required to target a given ruin probability. Under an open portfolio setup, we show that we may have a significant impact on capital requirements when compared with the classical BMS, by having a significant reduction on the initial surplus needed to maintain a fixed level of the ruin probability.