Jean Lemaire with Sojung Park and Kili Wang

Transcription

1 Jean Lemaire with Sojung Park and Kili Wang

2 ASTIN (Actuarial Studies in Non-Life Insurance) was founded in New York on October 16, 1957 First ASTIN Colloquium: La Baule, June 11 / 12, 1959 Only topic: Bonus-Malus systems () According to ASTIN legend, Général De Gaulle wanted insurers to introduce. French actuaries needed all the help they could get, and convened the first ASTIN Colloquium Since then, a steady stream of papers has been published, in the ASTIN Bulletin and other journals Most countries have adopted some form of

3 Main goals of : Induce policyholders to drive carefully Response to adverse selection Sequentially estimate driver s claim frequency However: Only national systems will impose strong penalties; with rating freedom, insurers may devise their own, and commercial pressures often make them weak Telematics may provide the driver information we need (mileage, night driving, excessive speeds, type of road travelled ) With low claim frequencies, the estimation process is abysmally slow Prosperity has made policyholders more risk-averse; were very popular in the 1960s; much less so nowadays

4 A huge literature deals with bonus-malus design independently of other variables which creates a potential double-counting problem Authors seems to be aware of the problem The use of more a priori classification variables is expected in free market countries, which decreases the need for a sophisticated Adverse selection and insurer s lack of knowledge of driving behavior is often cited as main reasons to introduce, with annual mileage frequently as main example But did not model it, until Taylor (1997) Setting a bonusmalus scale in the presence of other factors. ASTIN Bull, 27

5 Complicated Bayesian model, but simple simulation Classical negative binomial model: Poisson accidents for every single driver, with Poisson parameter distributed according to a Gamma The claims history of each policyholder is simulated for 60 years. So we know the claim history of every single driver The portfolio is subdivided into ten rating classes according to the a priori classification variables We then compare claim frequencies in the classes, with and without taking into account the classification variables Unfortunately, Taylor has no access to real data, so he created an example, approximating the Australian situation Policyholders are subdivided into ten classes due to unspecified classification variables Two nine-class BM, one hard (-1,+4), one soft (-1,+2)

6 Risk Group Mean cell claim frequency % Within-cell coefficient of variation Average 15.7

7 Class Stationary % Raw claim frequency % Expected claim frequency % Ratio raw / expected %

8 Class Premium ignoring covariates Premium recognizing covariates

9 Recognizing the impact of covariate variables, the ratio between highest and lowest premium levels decreases by 46% (soft ) and 38% (hard ) Pitrebois, Denuit and Walhin (ASTIN Bull, 2003) applied Taylor s model to a Belgian portfolio with claim frequency of 11.25%, using the two Taylor, and found significant but lower interactions The impact of numerous rating variables (age, gender, territory, engine power, etc) reduces the ratio between highest and lowest premiums levels by 34% (soft ) and 32% (hard )

10 Lemaire, Park and Wang (ASTIN Bull, 2016) discovered unique data in Taiwan Database with car information from leading manufacturer in country, claim information from leading insurer, and annual mileage information from network of oil change facilities (250,000 policy-years, ) Negative binomial regression models showed that mileage, by far, the best predictor of the number of claims-at-fault (ASTIN Bull, 2016) However remains extremely significant Other variables like driver age, car age, territory, engine power are also significant but less so These data provide the opportunity for the first real-life application of Taylor s model (EAJ, 2015)

11 Mileage class (km/day) Mean cell claim frequency % Within-cell coefficient of variation Average:

12 Miles All

13 Class Stationary % Raw claim frequency % Expected claim frequency % Ratio raw / expected %

14 Class Current level Premium ignoring covariates Premium recognizing covariates

15 Factoring in the interaction with annual mileage, by far the most significant rating variable, the ratio between highest and lowest BM premium levels only decreases by 8%, a far cry from Taylor s 42% and Pitrebois 34% Our interpretation of this negative result: Low range of class claim frequencies, from 3.5% to 8.4%, averaging 5.45% High within-cell variability, with coefficients of variation all in excess of 4.15, resulting in much overlap between classes Taylor (EAJ 2016) provided insightful comments on our result His model is of little significance if the portfolio is close to homogeneity The overlap in our risk group indicates that our prior (mileage) is close to being uninformative Taylor conjectures that we will get better results with more covariates The spread of claim frequencies will increase Within-class variability of claim frequency will decrease

16 In EAJ (2016) we re-ran Taylor s model with more variables Mileage (3 classes) Driver age (3 classes) Engine power (2 classes) Territory (2 classes) Car age (2 classes) Best results were obtained with 36 classes The spread of claim frequencies greatly increased, from 1.6% to 11.9% But variability of claim frequencies only marginally decreased, with coefficients of variation between 3.5 and 8.9 As a result, the ratio between highest and lowest BM premium decreases by 18%, much better than our initial 8%, but nowhere near Taylor s 46% So we have to conclude, in agreement with Taylor, that the Taiwanese portfolio is fairly homogeneous and that classification variables and BM achieve little differentiation

17 0,25 0,2 0, Taylor 0, Pitrebois 0, Taiwan Belgium Other countries Belgium trend

18 Very low claim frequencies inevitably create a nearhomogeneous portfolio, with 95.7% of policyholders not reporting a claim in any given year. In the vast majority of years, no information comes in to better evaluate a policyholder They create a huge clustering of policies in the bottom BM classes, negating the main goal of BM Convergence of the BM to stationarity is very slow (60 years) It is therefore not impossible that bonus-malus systems will fade away, and be replaced by driver characteristics obtained by on-board computers