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

• Published date: 01 June 2020
• Author: Gracinda R. Guerreiro,Lourdes B. Afonso,Rui M. R. Cardoso,Alfredo D. Egídio dos Reis
• DOI: http://doi.org/10.1111/jori.12300
• Date: 01 June 2020

© 2019 American Risk and Insurance Association. Vol. 87, No. 2, 501–522 (2020).

DOI: 10.1111/jori.12300

RUIN PROBABILITIES AND CAPITAL REQUIREMENT FOR OPEN

AUTOMOBILE PORTFOLIOS WITH A BONUS‐MALUS SYSTEM

BASED ON CLAIM COUNTS

Lourdes B. Afonso

Rui M. R. Cardoso

Alfredo D. Egídio dos Reis

Gracinda R. Guerreiro

ABSTRACT

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 surplusneeded to maintain

afixed level of the ruin probability.

INTRODUCTION AND MOTIVATION

The main goal of this work is to calculate finite time ruin probabilities for large motor

insurance portfolios where a Markovian Bonus‐Malus System (brieflyBMS)based on

claimcountsisputinplaceasexperiencerating.ThepaperbyAfonsoetal.(2017)shows

LourdesB.Afonso,RuiM.R.Cardoso,andGracindaR.GuerreiroareatFCTNOVAandCMA,

Universidade Nova de Lisboa. Afonso can be contacted via e‐mail: lbafonso@fct.unl.pt. Cardoso

can be contacted via e‐mail: rrc@fct.unl.pt. Guerreiro can be contacted via e‐mail: grg@fct.unl.pt.

Alfredo D. Egídio dos Reis is at the ISEG and CEMAPRE, Universidade de Lisboa. Egídio dos

Reis can be contacted via e‐mail: alfredo@iseg.ulisboa.pt. The authors gratefully acknowledge to

MagentaKoncept—Consultores, Lda and CMA‐FCT‐UNL for the computational support. Also,

we gratefully acknowledge financial support from Fundação para a Ciência e a Tecnologia/

Portuguese Foundation for Science and Technology (FCT/MEC)through national funds and

when applicable co‐financed by FEDER, under the Partnership Agreement PT2020, through

programmes UID/Multi/00491/2019 (Centre for Applied Mathematics and Economics

[CEMAPRE])and UID/MAT/00297/2019 (Centro de Matemática e Aplicações [CMA]).

501

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. The classical BMS model has implicitly expressed the idea that the policies that

may exit the portfolio in some period of time will be compensated by incoming ones. In

the real world we do not necessarily have this behavior. Indeed, in the very competitive

motor insurance market, we assist great market movements among insurers, where

insureds try often to get better deals, lower premia, and insurers try to increase their

sales. Besides, every insurer can build their own bonus scale. Often insureds are quite

conservative and try not to deal with or have too many different insurers when they buy

several coverages, that is, if an insured move a policy to another insurer they are likely to

move the whole portfolio.

Furthermore, classical BMS model assumes the existence of fixed entry bonus class

for all the portfolio newcomers. Nowadays, this is not appropriate since insurance

regulators provide insurers with the past record of a policyholder irrespective of

previous insurers. This leads us to consider that a portfolio newcomer (at least in

theory)can enter at any bonus class in a new portfolio if he changes insurer at some

time. Better information results in better risk classification, then more appropriate

premia are to be charged and also a better evaluation or estimation of capital re-

quirements for the insurers’business can be made.

Afirst question is: Does this new idea have an impact on ruin probabilities? Also: Would

it lead to a substancial change in the ruin probability figures shown by. Afonso et al.

(2017), for instance? We believe they may. In fact, we know already that there is an effect

on optimal scales, see Guerreiro, Mexia, and Miguens (2014). Our aim is wider as we

intend to show that modeling an open portfolio may lead to a significant change in ruin

probabilities, when compared with the classical BMS models. Also, they may contribute

to a re‐evaluation of capital requirements of an insurance company, once a level for the

ruin probability has been fixed, whether in finite or infinite horizon. Furthermore, we are

interested in evaluating the impact on existing optimal scales, in premia and ruin

probabilities, when applying an open model. Bonus classes may be allowed to be less

dispersed (bonuses not as high or maluses as low as in the classic formulation).Longrun

behavior is also important as, in general, most of existing BMS tend to concentrate most

of the insureds in higher bonus classes.

There aren’t many authors calculating ruin probabilities in the presence of a BMS, cer-

tainly even less when we consider an open BMS formulation. Lemaire (1995)is clearly a

classical reference for BMS, an important and more recent reference is Denuit et al. (2007).

These only deal with the classical model and do not calculate ruin probabilities. Reference

Afonso et al. (2017)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. Chosen examples are Andrade e Silva and Centeno (2001), Guerreiro,

Mexia, and Miguens (2014)and Mahmoudvand and Aziznasiri (2014).

The manuscript is organized as follows. Next section is devoted to the presentation of the

base model framework, including definitions, risk model and assumptions, BMS in open

portfolios, scenarios and ruin probability formulae, procedures and estimation. Third

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