A Numerical Experiment in Insured Homogeneity Joseph D. Haley, Ph.D., CPCU * Abstract: This paper uses a numerical experiment to observe the behavior of the variance of total losses of an insured group, as the group is continually divided and subdivided. In the tradition of Rothschild and Stiglitz (1976) only loss frequency is analyzed. The results of the experiment suggest that an insurer need only divide a large group of insureds into a relatively small number of subgroups (10 15) to achieve most of the efficiency gains that are available. T his paper addresses the issue of what happens to the expected total losses, E(T), and the variance of total losses, Var(T), of a large group of N heterogeneous insureds as the group is continually divided and subdivided into multiple subgroups. The group is heterogeneous only with respect to loss frequency. 1 This issue has received little treatment in the literature. The most direct reference, Hoeffding (1956), shows that a convex function of the number of successes in a series of independent trials is maximized when the probabilities of success on each trial are equal. But Hoeffding makes no mention of any specific application that would give rise to such a convex curve. My results show that the behavior of Var(T) is convex as heterogeneity is identified in the group. Three other, more peripheral, references Feller (1957), Hogg and Klugman (1984), and Cummins (1991) present, as a curious example, an illustration of how heterogeneity in an insured group reduces the Var(T) of the group. The remainder of this paper is divided into four parts. The first part relies on the Rothschild and Stiglitz (1976) model of fair-odds insurance pricing to describe the economic incentives of insurers to subdivide a group * Department of Finance, Insurance, and Real Estate, College of Business, St. Cloud State University, St. Cloud, MN 56301. 51 Journal of Insurance Issues, 1999, 22, 1, pp. 51 60. Copyright 1999 by the Western Risk and Insurance Association. All rights reserved.
52 JOSEPH D. HALEY of heterogeneous insureds. Two notions of efficiency are presented aggregate efficiency and individual equity. The second part of the paper presents a numerical experiment displaying the behavior of E(T) and Var(T) as a hypothetical insured group is continually subdivided into smaller groups. The third portion of the paper addresses how the results of the numerical experiment can be applied to bonus-malus systems in automobile insurance, while the final portion contains some concluding remarks. COMPETITION AND PRICING EFFICIENCY The economic principle underlying an insurance market s tendency to divide and subdivide a group of heterogeneous insureds is price competition. The fair-odds model of insurance equilibrium described by Rothschild and Stiglitz (1976) demonstrates this principle. Their model, like my experiment, assumes that insureds are homogeneous in all respects except the probability of loss. 2 Rothschild and Stiglitz (1976) have, as a general result, that an insured s equilibrium (fair-odds) premium is equal to πl, where π is the individual s probability of loss and L is the expected loss. A fair-odds premium results in no change in the expected wealth of the individual. If an individual pays something other than a fair-odds premium (and purchases full coverage), she will experience either a net gain or loss in expected wealth. When an insurer evaluates a large group of potential insureds, it could equate total revenue to E(T) by charging each member of the group a premium equal to ΠL, where Π is the average loss probability for the group. But this situation, of course, would not persist as competing insurers will attempt to identify, as costs permit, those lower-risk individuals who are paying a premium greater than the fair-odds premium. Once identified, these lower-risk individuals are charged a rate lower than the group rate, causing the first insurer to suffer adverse selection. The limiting case is having N groups of one insured each, where each insured is charged the fair-odds price and experiences no change in her expected wealth. 3 It will be shown in the next section that the limiting case of N groups of one insured each reveals the true (and minimum) value of Var(T), whereas the case of underwriting the N insureds as a single group results in the maximum (theoretical) value of Var(T). A conceptual difficulty arises here and the perspective of this paper must be made clear. If it is assumed that the insurer knows the group to be heterogeneous, but only the average loss rate Π is known, the true value of Var(T) will be unknown. As the insurer investigates and discovers some of the individual probabilities of loss, the insurer is also acquiring knowl-
A NUMERICAL EXPERIMENT IN INSURED HOMOGENEITY 53 edge of the true value of Var(T). Since knowledge of the true value of Var(T) influences risk charges, it is of importance to the insurer. I refer to the proximity of an insurer s knowledge to the true value of Var(T) as a measure of aggregate efficiency. A more obvious and more often discussed benefit of an insurer s effort to discover the individual loss probabilities of the members of the group is the achievement of individual equity. Each member is charged the appropriate fair-odds price. Lower-risk individuals pay lower premiums, while the higher-risk individuals pay higher premiums. It is also necessary, for the purpose of clarity, to point out that the reduction in Var(T) analyzed here is not simply a portfolio effect. In the numerical experiment that follows, I am not combining loss exposures to reduce the variance of total losses. I am analyzing what degree of refinement is needed, under a class-rating scheme, to discover the true value of Var(T). A NUMERICAL EXPERIMENT The numerical experiment begins with a single large group of heterogeneous insureds. The individual loss probabilities follow a Beta distribution, with parameters a and b. In other contexts, this characterization of insureds is used as a mixing distribution. 4 The values of a and b are chosen a such that.1 (the mean) and.007 (the variance). These choices conform to values reported in Lemaire (1988) and a ----------- + b = ab ---------------------------------------------- ( a + b + 1) ( a + b) 2 = Venter (1991). It needs to be noted here that by not forming a mixture of distributions, I am looking only at loss/no-loss situations. This differs from previous work, which uses mixed distributions to model the probabilities of 0,1,2,3, claims occurring for each individual. This difference is not drastic, though, because the vast majority of individuals fall into the loss/ no-loss category. A 1976 observed claim distribution from a Belgian insurer revealed that 99.29% of the loss portfolio had zero claims or only one claim. 5 Using the individual probabilities, the average loss probability for any i = 1 group is easily computed as π g = --------------, where π i is the loss probability n g for person i, and n g is the size of group g. The polar cases are having a single group of N insureds, and N groups of one insured each. π i
54 JOSEPH D. HALEY Now the following set of expressions for E(T) and Var(T) can be formed. Single Group: ET ( ) = Π N L (1.1) Var( T) = Π ( 1 Π) N L 2 (1.2) g Subgroups: ET ( ) = π i n i L (2.1) g g i = 1 Var( T) = π i ( 1 π i )n i L 2 i = 1 (2.2) As g N the subgroup expressions become ET ( ) = Π N L (3.1) N Var( T) = L 2 Π ( 1 Π) N ( Π π i ) 2 i = 1. (3.2) L is the loss amount and will henceforth be set equal to 1. All the E(T) expressions equal one another. Expressions (1.2) and (3.2) are the respective maximum and minimum variance values. Cummins (1991) presents these two equations as an example of how it can be desirable not to have perfectly homogeneous insured groups. He presents (1.2) as representing a group of homogeneous insureds, while (3.2) represents a group of heterogeneous insureds. The heterogeneous group, the expressions reveal, has a lower Var(T). Hogg and Klugman (1984) and Feller (1957) describe the comparison of equations (1.2) and (3.2) as surprising and striking. 6 The behavior of Var(T) between these minimum and maximum points is the focal point of my experiment. This behavior is observed by repeatedly slicing the large group of heterogeneous insureds into smaller and smaller subgroups. With each slice I compute the value of Var(T) using equation (2.2). The results show that aggregate efficiency, as I just defined it, is achieved relatively quickly. Table 1 gives the (partial) results of the first experiment. The number of insureds is N = 4,800. The repeated subgrouping of the insureds is done by first arranging the individual loss probabilities in ascending order and then splitting them into equal-sized subgroups with the use of all the integer divisors of N. This procedure generated 42 cases.
A NUMERICAL EXPERIMENT IN INSURED HOMOGENEITY 55 Table 1. Behavior of the Standard Deviation of Total Losses for Different Numbers of Equal-Sized Groups (Beta Distribution of Loss Probabilities) No. of groups Standard deviation Percentage reduction No. of groups Standard deviation Percentage reduction 1 20.82 15 20.02 3.82 2 20.36 2.22 16 20.02 3.84 3 20.22 2.88 20 20.01 3.87 4 20.15 3.20 24 20.01 3.89 5 20.11 3.39 25 20.01 3.90 6 20.09 3.51 30 20.01 3.92 8 20.06 3.64 32 20.00 3.92 10 20.05 3.72 12 20.03 3.78 4,800 19.99 3.98 E(T) for each case: $481.865 Note: The information in this table corresponds to Figure 1, but the horizontal axis in Figure 1 is not No. of Groups ; rather, it is Obs. Number. The first observation is one group, while the last observation is 4,800 groups. The vertical axis in Figure 1 is the percentage change in the standard deviation of total losses. It is clear in Table 1 that most of the potential aggregate efficiency gains are achieved with only 10 15 subgroups. Over half of the gains are reached by just splitting the group into two equal parts. Virtually all of the gains are achieved with 32 groups. Figure 1 graphically displays the percentage change in the standard deviation of total losses as the group is split into equal-sized subgroups. The curve is convex and, as such, fits the general description provided by Hoeffding (1956). I performed two additional numerical experiments one using the Beta distribution of individual probabilities, but subgroups of unequal size, and another using equal-sized subgroups and an Inverse Gaussian distribution of individual probabilities. The results of these experiments are largely similar to the results reported in Table 1 and are detailed in the appendix. AN APPLICATION: BONUS-MALUS SYSTEMS FOR AUTOMOBILE LIABILITY INSURANCE Automobile liability insurance underwriting classifications can be made with a priori variables or can be done on a posterior basis. Some commonly used a priori variables are age of operator, type of automobile,
56 JOSEPH D. HALEY Fig. 1. Percentage change in standard deviation of total losses for different numbers of equal-sized groups (Beta distribution of loss probabilities). gender, and geographic region. 7 Insurers use these to assess the risk of individual applicants. Underwriting automobile coverage on a posterior basis looks at the individual s claim record and seeks to adjust the premium accordingly. A poor driving record leads to higher premiums as the individual is classified as high risk. Bonus-malus systems rely exclusively on posterior-based (or merit-based) classification of insureds. 8 All individuals who are new to a bonus-malus system are put into the same starting class (with possible differences between business and personal use). From the starting class, each individual will move to a discount premium class or a malus class depending upon claims activity. Important features of bonus-malus systems are (1) the premium of the starting class, (2) the number of classes, (3) the transition rules dictating how individuals are moved from one class to another, and (4) what the discounts and maluses are that accompany interclass moves. The success of a bonus-malus system is determined by the interaction of these features. Increasing the number of classes will not improve a system if the transition rules do not result in the insureds being properly classified. A system with a well-coordinated set of classes and transition rules will not function optimally if the starting premium, or the discounts and maluses, are insufficient. The focus of this research has been restricted to the number of classes to be used for a group of insureds. Table 2 presents the number of classes used in the various bonus-malus systems around the world. The average
A NUMERICAL EXPERIMENT IN INSURED HOMOGENEITY 57 Table 2. The Number of Classes in Bonus-Malus Systems Around the World Belgium* 23 Singapore 6 Brazil 7 The Netherlands 14 Denmark 10 Norway* 22 Finland* 17 Portugal 6 Germany* 22 Spain 5 Hong Kong 6 Sweden 7 Italy* 18 Switzerland 22 Japan* 16 Taiwan 9 Kenya 7 Thailand 7 Luxembourg* 24 U.K. 7 The average number of classes is 12.65. Source: Bonus-Malus Systems in Automobile Insurance, by Jean Lemaire (1995). *Lemaire reported on new and old systems in these countries. Only the new system is considered here. In all instances, movement to a new system either held the number of classes the same or increased them. number of classes in these systems is 12.65, which falls in line with my results indicating that only 10 15 classes are needed to obtain a measure of aggregate efficiency. Forming a system with more than 15 classes would (potentially) improve individual equity, while making only minor improvements in aggregate efficiency. Since it is the regulatory authorities of each country who determine to what extent competitive forces will be allowed to operate, it is the same authorities who will determine how strongly individual equity will be pursued. The more competitive the environment, the more vigorous the effort will be to achieve individual equity. CONCLUDING REMARKS This paper has analyzed, by way of a numerical experiment, the question of how refined a class-rating insurance scheme should be. The results of the experiment indicate that only 10 15 underwriting classes are necessary to achieve a good measure of aggregate efficiency. Such a conclusion gives weight to the notion of using community-based rating instead of a highly refined class underwriting system. The issue of whether to use community rating or class rating could depend upon the regulatory status of the insurance coverage. For example, automobile liability and workers compensation, which are required in most states, may be good choices for
58 JOSEPH D. HALEY a community-based approach. Such a determination is, of course, beyond the scope of this paper. APPENDIX I performed a second and a third numerical experiment. The second experiment uses the same set of individual probabilities as the first, but divides the large group into subgroups of unequal size. I investigated only one case with this procedure since the results are quite similar to the first experiment. The standard deviation of total losses with the five subgroups is 20.07. The results of this experiment are displayed in Appendix Table 1. The third experiment uses an Inverse Gaussian distribution to generate the individual loss probabilities. The parameter values of µ = 1 and λ =(1/7) are used to form the distribution. The size of the large group is the same as in the other experiments, N = 4,800. The results of this experiment are similar to the others. Aggregate efficiency is achieved quickly, though somewhat more slowly than in the first (Beta distribution) experiment. Appendix Table 2 presents the results of the third experiment. Appendix Table 1. The Standard Deviation of Total Losses for One Set of Unequal-Sized Groups Group 1: Range: 0 < π i.05 E(Losses) = 40.57 Group 2: Range:.05 < π i.12 E(Losses) = 134.37 Group 3: Range:.12 < π i.24 E(Losses) = 197.49 Group 4: Range:.24 < π i.45 E(Losses) = 103.16 Group 5: Range:.45 < π i 1 E(Losses) = 6.27 No. of persons in the group: 1,624 Std. Deviation = 6.290 No. of persons in the group: 1,638 Std. Deviation = 11.106 No. of persons in the group: 1,181 Std. Deviation = 12.824 No. of person in the group: 344 Std. Deviation = 8.500 No. of person in the group: 13 Std. Deviation = 1.802 π 1 =.0250 π 2 =.0820 π 3 =.1672 π 4 =.3000 π 5 =.4825 E(T) for all five subgroups: $481.865. StdDev(T) for all five subgroups: 20.07.
A NUMERICAL EXPERIMENT IN INSURED HOMOGENEITY 59 Appendix Table 2. Behavior of the Standard Deviation of Total Losses For Different Numbers of Equal-Sized Groups (Inverse Gaussian Distribution of Loss Probabilities) No. of groups Standard deviation Percentage reduction No. of groups Standard deviation Percentage reduction 1 20.82 15 20.08 3.54 2 20.47 1.69 16 20.08 3.56 3 20.34 2.32 20 20.07 3.63 4 20.27 2.65 24 20.06 3.68 5 20.22 2.87 25 20.05 3.69 6 20.19 3.02 30 20.04 3.73 8 20.15 3.22 32 20.04 3.74 10 20.12 3.35 12 20.10 3.45 4,800 20.00 3.94 E(T) for each case: $481.866 NOTES 1 The insureds are homogeneous with regard to loss severity. 2 For a discussion on the validity of this assumption see Lemaire (1995), page 71. 3 Lemaire (1998) contains an example showing that one insurer will eventually take over a (two-insurer) market by creating an additional discounted underwriting class, if the other insurer doesn t make a like response. 4 See Lemaire (1995). Lemaire uses a similarly configured Gamma distribution as the mixing distribution. 5 See Lemaire (1995), page 25. 6 See Hogg and Klugman (1984), page 39, and Feller (1957), pages 216 217. 7 See Webb et al. (1984), page 273. 8 For a thorough description of bonus-malus systems, see Lemaire (1995). REFERENCES Cummins, J. David (1991) Statistical and Financial Models of Insurance Pricing and the Insurance Firm, The Journal of Risk and Insurance, 58, pp. 261 302. Feller, William (1957) An Introduction to Modern Probability Theory and Its Applications, Vol. 1, 2nd edition. New York: John Wiley & Sons.
60 JOSEPH D. HALEY Hoeffding, Wassily (1956) On the Distribution of the Number of Successes in Independent Trials, Annals of Mathematical Statistics, 27, pp. 713 721. Hogg, Robert V., and Stuart A. Klugman (1984) Loss Distributions. New York: John Wiley & Sons. Lemaire, Jean (1988) A Comparative Analysis of Most European and Japanese Bonus-Malus Systems, The Journal of Risk and Insurance, 55, pp. 660 681. Lemaire, Jean (1995) Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer. Lemaire, Jean (1998) Bonus-Malus Systems: The European and Asian Approach to Merit-Rating, North American Actuarial Journal, vol. 2, no. 1. Rothschild, Micheal, and Joseph Stiglitz (1976) Equilibrium in Competitive Insurance Markets: An Essay on Economics of Imperfect Information, Quarterly Journal of Economics, 90, pp. 629 649. Venter, Gary G. (1991) A Comparative Analysis of Most European and Japanese Bonus-Malus Systems, The Journal of Risk and Insurance, 58, pp. 542 547. Webb, Bernard L., Launie, J. J., Rokes, Willis Park and Baglini, Norman A. (1984) Insurance Company Operations, Vol. 1. Malvern, PA: American Institute for Property and Liability Underwriters.