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

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.

Ruin Probabilities And Capital Requirement for Open Automobile Portfolios With a Bonus‐Malus System Based on Claim Counts

“…In this section, we summarize briefly our basic framework for the calculation of the probability of ruin in finite and continuous time. The base model was taken from Afonso, Egídio dos Reis, and Waters () and it is already summarized in Afonso et al (). As for the modeling of BMS in open portfolios we will follow the work by Guerreiro, Mexia, and Miguens () with some developments obtained in Esquível, Fernandes, and Guerreiro ().…”

“…The main goal of this work is to calculate finite time ruin probabilities for large motor insurance portfolios where a Markovian Bonus‐Malus System (briefly BMS) based on claim counts is put in place as experience rating. The paper by Afonso et al () shows a way to do this calculation/estimation in the presence of a classical BMS model. Our aim is to update their model to provide the implementation of an open BMS as we believe that the resulting ruin probabilities have a better or realistic representation for the business.…”

“…Also: Would it lead to a substancial change in the ruin probability figures shown by. Afonso et al (), for instance? We believe they may.…”

“…These only deal with the classical model and do not calculate ruin probabilities. Reference Afonso et al () particularly is concerned with finite time ruin probabilities for BMS in automobile insurance. There are several references about open BMS; however, they are not devoted to ruin probability calculation, they mostly work with bonus scales and model efficiency.…”

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

“…For simplicity, we assume everything to be exponential distributed with τ ∼ Exp(λ 1 ),τ ∼ Exp(λ 2 ) and Y ∼ Exp(β), respectively. Recall our process described by (2), note that ruin happens only at claim arrivals σ k = ∑ k i=1 τ i and σ 0 = 0. From time σ k to σ k+1 , the distribution of the increment S(σ k+1 ) − S(σ k ) is only dependents on the relation between τ k and ξ.…”