Numerical Evaluation of Continuous Time Ruin Probabilities for a Portfolio with Credibility Updated Premiums

Abstract

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.

MEASURING THE IMPACT OF A BONUS-MALUS SYSTEM IN FINITE AND CONTINUOUS TIME RUIN PROBABILITIES FOR LARGE PORTFOLIOS IN MOTOR INSURANCE

“…There, a mix of calculation and simulation procedure is used. The same authors adapted the same model for the calculation of ruin probabilities where premia are updated using credibility estimation [see Afonso et al (2010)]. In this manuscript, we adapt it for portfolios with bonus-malus updated premia, it allows to get fast results for a finite horizon and continuous time framework.…”

“…Obviously, insurers benefit from a good global, portfolio behavior, or are penalized otherwise. In Afonso et al (2010), it assumed a credibility theory approach, where premia can be different, and so individualized, according to each risk characteristic, assigned by a risk parameter, so that the portfolio is not completely homogeneous and we have a collection of different risk types. We can consider that the parameter is known or unknown.…”

“…This procedure brings a higher variation in the premia during the lifetime of the portfolio [see e.g., Lemaire (1995)], when compared to a standard procedure. The model in Afonso et al (2009), that we retrieve, is built under the assumption of an homogeneous classical compound Poisson process, with Afonso et al (2010), we bring some sort of heterogeneity to the portfolio. Under a BMS, we often work under a mixed Poisson framework and premium varying on claim counts only.…”

On the ruin probability of a generalized Cramér–Lundberg model driven by mixed Poisson processes

“…Trufin and Loisel (2013) used a similar model, and proved that the adaptive premium policy decreases the ruin probability under some assumptions. Afonso (2008) and Afonso et al (2010) studied a continuous-time model with discretely varying premiums, and presented numerical evaluations of the ruin probability. Asmussen (1999) assumed that the company continuously modifies the premium using the sample mean of the number of observed claims, even though credibility theory or the Bayes method is not incorporated.…”

On a discrete-time risk model with claim correlated premiums

“…It is further shown that there is an intrinsic relationship between such a stationary distribution and the probability of ruin in the risk theoretical model. Moreover, continuous-time risk models with varying premiums, experience-based premiums or credibility updated premiums, have been studied in recent literature (see Afonso et al ., 2010; Li et al ., 2015 and references therein).…”