Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window

Abstract

Abstract:We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the bonus-malus (BM) feature of pricing car insurance. We discuss first the asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process, and via an appropriate change of measure, ruin probabilities could be computed to a closed-form formulae. Additionally, we employ the importance sampling simulations to derive ruin probabilities, which further permit an in-depth analysis of a few concrete cases.

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

“…Recently, Li et al (2015) considered computing ruin probabilities where the Poisson parameter is a continuous random variable and use credibility theory arguments to adjust the premium rate a posteriori. Even more recently, Constantinescu et al (2016) discuss ruin probabilities in a model with dependence on the number of claims that can be viewed as an application to a "no claims discount" system (where only bonuses are allowed, not malus). Again, in both cases, only ruin probabilities in an infinite time horizon are considered.…”

The Exponential Estimate of the Ultimate Ruin Probability for the Non-Homogeneous Renewal Risk Model

“…Other authors worked with the claim sizes, which have a common distribution function and not necessarily identically distributed inter-arrival times (see Šiaulys 2015, 2017;Burnecki and Giuricich 2017;Mao et al 2017). Other authors dealt with identically distributed claims and inter-arrival times, but there may be some kind of dependence between them (see Chen and Ng 2007;Huang et al 2017;Constantinescu et al 2016;Liu and Gao 2016;Li and Sendova 2015;Shen et al 2016;Yang and Yuen 2016;Yang and Konstantinides 2015;Yang et al 2014;Wang et al 2013). Some authors consider models in which claim amounts are divided in several lines by supposing some dependence relations between these lines (see Fu and Ng 2017;Guo et al 2017;Yang and Yuen 2016).…”

Ruin probabilities in a Sparre Andersen model with dependency structure based on a threshold window

“…Another popular dependence structure is often established between inter-claim times and successive inter-claim times or claim sizes (e.g. Albrecher & Boxma, 2004; Constantinescu et al ., 2016). These were normally studied within a Sparre Andersen model.…”

“…In fact, a specific dependence structure between successive inter-claim times has been constructed in Constantinescu et al . (2016) in order to reflect the Bonus–Malus feature at a collective risk level. In that paper, the distribution of the current inter-claim time depends on whether the previous inter-claim time exceeds a fixed threshold which we refer here as the threshold window.…”

“…AbstractWe analyse ruin probabilities for an insurance risk process with a more generalised dependence structure compared to the one introduced in Constantinescu et al . (2016). In this paper, we assume that a random threshold window is generated every time after a claim occurs.…”