A Genetic Algorithm improving tariff variables reclassification for risk segmentation in Motor Third Party Liability Insurance.

Transcription

1 A Genetic Algorithm improving tariff variables reclassification for risk segmentation in Motor Third Party Liability Insurance. Alberto Busetto, Andrea Costa RAS Insurance, Italy SAS European Users Group International May 1997 Palacio de Congresos de Madrid, Spain Abstract A careful assessment of the risk covered by the insurer in Motor Third Party Liability insurance is a primary competitive factor. Actually, rating structures are heavily based on these risk estimates and, as a consequence, a bad portfolio risk segmentation may imply a bad premiums attribution. Unfortunately, there is at present a big number of variables involved in personalized premium calculation, and the number of possible combinations between class levels often exceeds some millions. For these reasons the insurers frequently reclassify the variables, summarizing them for easily manage their portfolio risks. However, the reclassification of variables affects the estimation of these heterogeneous risks and the adopted criterion should take into account all the interactions between the variables, while reducing the dimensionality of the problem (in statistical language it means a multivariate criterion). Here we suggest an unconditioned way of facing the problem, based on the principles of the so-called Genetic Algorithms. This approach follows the evolutionary idea (derived from biological sciences) that better individuals can be produced from one generation to the next by selecting the best individuals in the first one and by making them reproduce, generating a new breed. In this case the individuals are the level boundaries of reclassified variables, compared in terms of fitness of the statistical model used to estimate risk level. So, a first random generated population of bounds is ranked in terms of statistical fit and best cases are, in some way, mixed and reproduced in a new population. The process is then iterative and model free in the reproducing phase (obviously depending on the statistical measure of fit) and it permits to explore the large set of possibilities in an efficient way. Overview In Motor Insurance Line, over 50% of the total Italian market is shared by the 10 largest insurance companies. The total insured vehicle base amounts to 40 million, 30 million of which being private cars (Penco, 1995). In 1995, RAS ranked at the fourth place in this line, having a market share of 5,8% (Data source: ANIA). The respective collected premiums amounted to more than Billions Lira (about 50% of the overall nonlife lines result), with a portfolio size of about 2 millions units. In particular, compulsory Motor Third- Party Liability (TPL) insurance market was regulated up to July 1994 by the Government, prescribing by ministerial decrees rating structures and premiums. For this purpose, an appropriate commission produced annually the estimates of economical requirements, basing on market data (corresponding at present to more than Billions Lire of TPL premiums - Data source: ANIA). 1

2 Nevertheless, during 23 years of market regulation, the Italian companies get a serious level of loss ratio in this line: the imposed tariff led in fact to chronic undercharging, caused by the attention paid by the Ministry of Industry to general economic and social policies. Moreover, premiums were very similar from one policyholder to another, putting a high degree of mutual aid between them. In fact very expensive categories like young male drivers (typically bad drivers) paid like middle aged men, a much better business. Actually, the adopted rating structure was not able to discriminate policyholders, subdividing them only by four variables: vehicle taxable engine power (not its actual power), Bonus- Malus class, Geographical areas and Third Party insurance coverage. Finally, following III Loss EEC directive (no. 49/1992), the market was deregulated after July This circumstance introduced a new situation of price competition in a previously still market, forcing insurers to improve their database quality and starting an appropriate data mining. So the major companies slowly introduced new tariff variables, trying to improve their portfolio management, and get homogeneous results in each market segment. Now, looking for badly rated targets, insurers try to produce personalized tariffs, making good categories pay less then bad ones. The main goal is to attract potential customers and to retain them longer together with the customers already in the portfolio. 1. Theoretical aspects 1.1 The statistical problem At present, statistical models used in the study of the portfolio characteristics can be very complex. Unfortunately, this implies a large number of variables (typically class variables or effects) involved in personalized premium calculation, often with a wide range of class levels. We call here a priori variables the set of characteristics of the policyholder and the vehicle, useful to estimate in advance his actual claim cost. In addition to this, there is an important a posteriori variable, the Bonus-Malus class, that ranks insured clients following their own claim occurrence. All these individual effects are commonly used in Generalized Linear Models (GLMs; McCullagh and Nelder, 1989) to produce a set of multiplicative relativities with respect to the base premium, defining a personalized risk score. The space spanned by these variables is multidimensional, and the number of possible combinations between their levels often exceeds some millions. For this reasons the insurers frequently reclassify the variables, grouping original class levels (or discretizing continuous variables), reducing this way the number of combinations for easily manage their portfolio risks. There is an even practical reason of simplicity to do so, selling insurance policies by tied-agents that cannot be asked to mess about with complex tariffs or big rating guides. The considered variables are, so, classified in macro-classes, generally grouping more class levels. For example variable age (a continuous variable entered in practice as class variable ranging from level 18 to level 80) becomes a factor ranging from level 1 to level 6, and so on for variables like car engine capacity, occupation and geographical districts. However, a realistic set of typically used variables can produce more than valid combinations. Anyway, different reclassification of variables can produce very different results in terms of discrimination between insured clients. Grouping, for example, young male drivers and old female drivers produce a mean premium that is correct for this class but too low for the first ones and too high for the last ones: grouping heavily affects tariff personalization quality. It should be clear now the relevant role of a good variables reclassification in the assessment of heterogeneous risks. The term good here means that the adopted criterion should take into account all the interactions between the variables while reducing the dimensionality of the problem (in statistical language it means a multivariate selection criterion). This implication is sometimes neglected, when focusing the attention on clustering problems. Nevertheless, popular algorithms like AID and its implementation CHAID (producing a clustering tree based on Chi-square statistic; Kass, 1980) are 2

3 affected by the structure of the contingencies tables used in the analysis, depending this way even on the effects categorization. 3

4 1.2 Genetic Algorithms In wildlife, species living in difficult environment are submitted to the so-called natural selection law. Only the best individuals survive, reproducing themselves and transmitting their genetic traits to the next generation, whereas unsuitable ones die without propagating their weak traits. The process continues from one generation to another, making the species stronger and optimized to survive. In the last years, this simple principle has been transferred even in computational problems by Genetic Algorithms (GA): given a problem and a set of possible solutions, an evolutionary iterative process select better solutions, killing the worse. Solutions are usually represented like strings of bits analogous to biological chromosomes and genes (from which the term genetic), coding some traits like in figure 1. Figure 1. An example of chromosome-like encoding District Age class Gender A The implementation of such a GA is quite simple. First of all, a random procedure generate a subset of valid solutions. The solutions are then ranked in terms of score obtained by an appropriate fit function. Finally a reproducing phase select pairs of solutions and produce new solutions mixing their characteristics (chromosomes). The core of the algorithm is then the generation of new individuals from parents selected at random, typically with probability proportional to the fitness or the ranking score. There are two fundamental genetic operators for the purpose: crossover and mutation. The crossover operator select at random a cutting point between the set of genes of two solution A and B and substitute those of A after the cutting point with those of B and vice versa, generating two new solutions with peculiarities of both parents. The mutation operator takes some newly generated solutions and changes at random some genes. In figure 2.a and 2.b are represented two examples of crossover and mutation in six-bit strings chromosomes. Figure 2.a Crossover Cutting point A A B B Figure 2.b Mutation 4

5 Mutation point A A

6 2. The case study 2.1 RAS rating structure Like other insurance companies, even RAS improved previous premium rating scheme, adding new a priori variables like gender, age, marriage status, occupation, vehicle type and supply, or modifying those existing like geographical districts. RAS discriminates in its rating structure between five classes of male and female age (normally ranging from 18 to 80), eight classes of car engine capacity (otherwise ranging from class 1 coding under 57,1cc to class 50 coding over 7.835,6cc) and seventeen geographical districts (consisting of 110 cities and districts). This way the number of possible combinations grown up to more than , but not all of them valid or represented in the portfolio. The aim of this work is to find a good way of reclassifying normally used variables, that is to say find new level bounds for each tariff variable, improving the quality of risk estimation. 2.2 The classification chromosome To accomplish the aim is necessary to find the way of recoding original variables into new macro-classes. So the classification information is coded into integer numbers strings, like in figure 3 (it can be noted that the information coded in every gene is no more binary). Figure 3. The classification chromosome Car Engine Capacity Male age Female age Geographical district st bound 3 rd bound 1 st bound 1 st bound 1 st city 3 rd city For each variable the algorithm read in the chromosome the sequence of values which separate the classes of the recoded variable. So the first car engine capacity class range in this example from 1 to 8, the second from 9 to 11, the third from 12 to 13 and so on separately for male and female age class (a way of considering the gender/age interaction). Instead, geographical districts are directly coded with a number from 1 to 17. Possible combinations become a number like 108 Billions raised to 119, a practically incommensurable number! The effectiveness of this approach is that, even exploring an infinitesimal part of this solution space, a solution can be found that can be considered a good solution, although likely suboptimal. 6

7 2.3 The fit function and the starting population The successive step is to define what is the fit function to be tested. In this case chromosomes are ranked in terms of statistical fit of a model used to estimate a function of the average cost per policy. Then a random population of 30 individuals is created to start the process. To provide a good starting point, in the first iteration one of these chromosomes is forced to reply the used tariff classification. After the population ranking, a routine selects at random couples of parents to reproduce themselves and their cutting point, obtaining only a son at a time (a little different version than usual) until a new full population is created. The reproducing selection probability is a linear decreasing function of the rank, whereas the cut point is selected with uniform probability. Finally, some individuals undergo a mutation (the mutation factor is determined by the individual mutation probability and by single gene mutation probability) and the process is repeated. Fit function values after iterations on a records sample are displayed in figure 5. Figure 5. Fit function values 25% 24% 23% 22% 21% 20% 20,8% 22,3% 21,2% 22,7% 23,0% 24,1% 23,8% 24,4% 19% 18% 17% 16% 15,7% 15% Iteration number The function grows monotonically because the best generation solution is certainly reproduced unchanged in the next one. According to this, the average population fit can even decrease from one generation to another (following an increase in the fit scores variance), but best population fit cannot be lower than the preceding one. It is important to consider that the fit measure (still based on GLMs) reflects a real estimates improvement, being the procedure forced to maintain the degrees of freedom of the model. The specification of the whole set of parameters involved in the algorithm is a critical step for the success of the attempt, and this often imply a long trial and error process. Particularly, an important role in the final efficiency of the algorithm is played by the genetic pressure, that is to say the mutation factor level combined with the number n of poor solutions that surely will not reproduce (limiting the set of possible parents to first 30- n solutions). This circumstance can be seen in this example between the 600 th and the 900 th iteration. In this phase the mutation probability, continuously tuned with iterations, was too high to make the process improve the fit, altering too much the sons of good solutions. For this sample the fitness finally passed (after x 30 = attempts) from 15,7% to 24,5% (an improvement 56%!) first 100 iterations raising the fit by 32,5%. 7

8 Conclusions The algorithm proved to be effective and the results applicable to the whole portfolio, confirming robustness of the approach. A careful inspection of newly created classes find them logical and not a pure numerical circumstance, encouraging the authors to continue following this apparently daring way. References John H. Holland, Algoritmi Genetici, Le Scienze n. 289, September 1992, pp Internet Usenet Group: comp.ai.genetic, Frequently asked questions: Part 1 - Introduction, September the 20, Internet Usenet Group: comp.ai.genetic, Frequently asked questions: Part 1/6 - Part 6/6, September the 20, G. V. Kass, An Explanatory Technique for Investigating Large Quantities of Categorical Data, Appl. Statist. 29, No. 2, 1980, pp P. McCullagh and J.A. Nelder, Generalized Linear Models, Chapman & Hall, Salvatore R. Mangano, A Genetic Algorithm White Paper - An Introduction to Genetic Algorithm Implementation, Man Machine Interface Inc., Internet pages. Paolo Penco, Segmentation and rating in motor insurance, SCOR NOTES, June Appendix: Algorithm flow chart Used tariff classification 1 st Random chromosome 29 th Random chromosome STARTING POPULATION (1) RANKING... th Random chromosome New random chromosome New random chromosome NEW POPULATION AFTER MUTATION 10% 8% 6% (2) PARENTS SELECTION Reproducing selection probability 4% (4) MUTATION 2% 0% th Random chromosome New random chromosome New random chromosome NEW POPULATION (3) REPRODUCTION 8