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Insurance as a Giffen Good under a Bonus-Malus System and its Effect on Adverse Selection

January 10, 2006

Shinichi Kamiya

Department of Actuarial Science, Risk Management, and Insurance School of Business, University of Wisconsin-Madison

975 University Avenue 3164 Grainger Hal Madison, WI, 53706 (608)265-4189 [email protected]

©2006 Shinichi Kamiya, All Right Reserved, Preliminary Draft

Insurance as a Giffen Good under a Bonus-Malus System and its Effect on Adverse Selection

Shinichi Kamiya1

January 10, 2006

Abstract A bonus-malus system (BMS) serves not only as a fair rating system but also as a mechanism to avoid adverse selection by developing an applied premium based on the claim experience of a policy. This paper analyzes how a BMS could work for these two concepts: achieving fairness among policies and mitigating adverse selection. The purpose is to understand how the transition rules, the number of classes, and the premium scale affect the behavior of a consumer's insurance purchase. By utilizing the mean asymptotic premium and the elasticity (Loimaranta (1972), Lemaire (1995, 1998)), this article defines the asymptotic critical value characterizing automobile insurance as a good and the condition for the slope of insurance demand curve under a bonus-malus system. As an empirical analysis, current and past Japanese bonus-malus systems are introduced in chronological order, and it is shown that no negative slope could be found for any combination of systems and claim frequency. This implies that promoting fairness in the way of lower premiums for good drivers and assigning higher penalties for bad drivers might increase the occurrence of adverse selection, contrary to the expected result. Key words: bonus-malus system, elasticity, Giffen good, adverse selection

1.

Introduction

It is common in automobile insurance to see insurance companies adjust their premiums based on past

claim history. The bonus-malus system (BMS) has been widely used in European and Asian countries as a risk classification method. In BMS, a premium reduction is offered if the prior policy did not pay for any claim. The primary reason that many countries adopted BMS for their automobile policies is to assign appropriate premiums that fairly reflect the inherent risk of policy. Since it is inevitable that heterogeneous risks belong to the same risk group, adverse selection may also exist in an insurance system. Insurance companies and supervising authorities may collaborate to mitigate

1

Ph.D student in Department of Actuarial Science, Risk management, and Insurance, School of Business, University of Wisconsin-Madison ([email protected])

adverse selection. In other words, it is similar to controlling insurance demand by motivating low risk policies to purchase their coverages and high risk policies to not abuse the system. A rating method which can properly evaluate the insured's risk may be one solution. That is why BMS is anticipated to mitigate adverse selection. Another major risk classification method, an priori rating system, is also commonly used in insurance. In an a priori rating system, features of a policy (e.g. insured's age, gender, marital status, vehicle type, use of car) are evaluated for the calculation of the premium. However, this risk classification method is based on the aggregate propensity of policies, where the same characteristics tend to be shared. Such statistical methods are not well equipped to evaluate and estimate a policy's inherent risk. Even if a policyholder correctly recognizes its risk, the information could not be shared. There is no reason to reveal information to an insurer when a policyholder realizes the information could negatively affect his or her premium rating, and this leads to asymmetric information between the two parties. Low risk policies have to pay higher premiums than the actuarial fair value and they may also be motivated to reduce the coverage. On the other hand, high risk policies are willing to purchase full coverage, or more if possible, for their exposure since the premium is lower than actuarial fair value. Thus, these unfavorable shifts of demand may make adverse selection more serious. It is known that the claim experience of a policy, a posteriori variable, can be the best variable to forecast future claims. Both the BMS and no-claim bonus system used in the US and Canada utilize the data. BMS rating is different than the no-claim bonus system in that BMS determines the next policy year's premium only from the information about the number of prior claims2 and the BMS class. The premium is adjusted every policy year. In the process of reflecting past claim experiences, the premium converges to the policy's true premium level. However, since claim experience is not enough to evaluate the inherent risk of a policy in the first few policy years, the premium has to rely on risk characteristics. Once sufficient claim experience data is collected, the premium no longer needs to rely heavily on prior information. Instead, it depends more on posterior data, which can reflect the policy's inherent risk and improve the rating over time. It is therefore reasonable to say that a BMS has positive effects on mitigating adverse selection in the long run even though those effects are not expected in the short run. Numerous studies on BMS have been conducted and summarized by Lemaire (1995, 1998). Although research about the BMS evaluation measure tend to focus on the effect of claim frequency on the premium, i.e. fairness; from an actuarial perspective, the evaluation method featuring adverse selection has not been well

2

except for Korean BMS, see Lemaire (1995)

defined. While some measures regarding the steady-state premium level intuitively suggest the possibility of working against adverse selection, it does not theoretically imply the magnitude of the influence of BMS on insurance demand. In the economics literature, economists have tackled the adverse selection problem with asymmetric information3. However, economists are mainly interested in examining the equilibrium in various competitive market settings rather than the influence of the practical premium rating method on the adverse selection. This study focuses on the relationship between fairness among policyholders and adverse selection. This study takes an optimization approach rather than the market equilibrium, and analyzes the relationship between the structure of a BMS and the impact on adverse selection. It also hopes to provide useful evaluation techniques to control insurance demand. This article uses the elasticity measure of premium with respect to claim frequency. The measure was defined by Loimaranta (1972) and is called elasticity. This article is structured as follows. First, research about BMS evaluation are revisited; the meaning of elasticity in BMS is discussed in detail. Second, an optimal insurance purchase model under a bonus-malus rating is derived by solving the one-period expected utility maximization problem. Comparative static analysis of the insurance demand provides the condition and the asymptotic critical value to determine of properties of a good4, which implies how the revision of BMS may affect the purchase of policies with heterogeneous risks. Third, Japanese past and current bonus-malus systems are analyzed empirically. The study calculates the elasticity of each BMS and the asymptotic critical values to determine the properties of a good. The interpretation of research results shed light on how a change of BMS may affect the demand for insurance and how adverse selection can be effectively controlled.

2. Fairness of a Bonus-Malus System

Among numerous contributions summarized by Lemaire (1995), this article is directly related to the elasticity concept introduced by Loimaranta (1972) and the mean asymptotic premium. Since these measures assume a stationary distribution as asymptotic property of Markov Chain and its expected premium, it can be

3

4

The theoretical studies about adverse selection primarily focus on the effects of asymmetric information on equilibrium in competitive market setting. Rothechild and Stiglitz (1976) works on one-period expected utility model, and Dionne and Lasserre (1985), Watt and Vazquez (1997), and Janeesn and Karamychev (2005) on multi-period model. Especially, those multi-period models deal a dynamic perspective of insurance contract where a posteriori rating system is introduced. However, these theoretical studies implicitly assume posteriori rating systems to be perfect elastic, and seem no interest in the case of the elasticity is less than perfect or even close to zero for some risks. It is essential matters for this study if insurance could be Giffen and if decreasing absolute risk aversion hypothesis would be still valid, because those significantly affect the result of comparative static analysis. Researches related those topics will be mentioned as those are relevant.

determined without knowing the distribution of each class at any particular time. This property allows us to compare different kinds of BMS with respect to the performance of fairness. In particular, the elasticity measure is defined as the ratio of a change in the mean asymptotic premium to the change in the claim frequency, and reflects how BMS succeeds in reflecting a policy's claim frequency in the long run. Lemaire compares thirteen European and Japanese BMS and analyzes thirty different kinds of BMS. (Lemaire, 1988; Lemaire & Hongmin,1994). Lemaire (1995, 1998) summarizes contributions of BMS and evaluates how BMS is effective, and shows five measures: 1) the relative stationary average level, 2) the coefficient of variation of the insured's premium, 3) elasticity of the mean asymptotic premium with respect to the claim frequency, 4) the average optimal retention, and 5) the rate of convergence of BMS. This article works utilizes elasticity because it can be an important variable that connects BMS and the demand for coverage. It should be noted that little attention is paid to the low observed elasticity. As shown in the next section, elasticity should be ideally the one (called perfectly elastic) that can attain the ideal level. However, only some very exceptional combinations of systems and claim frequencies can reach the ideal fairness. This implies that most good drivers with low elasticities have to subsidize bad-drivers. This may be partially explained via the existence of other risk classification methods. Richaudeau (1999) studies adverse selection and moral hazard in French automobile insurance. He concludes that risk classification with the combination of insured's characteristics and BMS works effectively, and that the observed adverse selection would be associated not with Rothschild-Stiglitz concept, but simply with the frequency of driving, which is not eliminated in French automobile insurance rating. This paper assumes that cross-subsidization would significantly influence consumer's insurance purchase behavior.

3. Elasticity

Elasticity represents the response of BMS to a change in claim frequency. It is used to measure the fairness of a system. Lemaire's studies (1995, 1998) provide a good interpretation about elasticity and other measures. Here, the justification of a perfectly elastic BMS is revisited to provide an intuitive sense of unfairness that might change the consumer's optimal purchase of coverage. One strong assumption which should be kept in mind is that the asymptotic condition of assuming BMS as a closed system and the distribution of claim frequency as constant over time. The premium used in calculating the elasticity is called the mean asymptotic premium. It is the expected premium level calculated by the stationary distribution of in each class. The premiums for each class are determined by the BMS.

Through this article, a Poisson distribution with parameter , where (0, ) is assumed for the number of claims, and b() represents the mean asymptotic premium. The elasticity5 () is defined by the claim frequency and the mean asymptotic premium b().

( )

db( ) b( ) b( ) = d b( )

(1)

An actuarially fair premium, equal to an expected claim payment, could be calculated by a claim frequency times an average claim amount. Particularly, if a unit claim payment is assumed, then it has to be equal to the claim frequency . b() (2)

When the claim frequency changes by 100h%, equivalently (1+h), the change in the mean asymptotic premium could be expressed as6:

b ( (1 + h)) b ( )(1 + ( ) h)

(3)

Hence, if the elasticity ()1, then the mean asymptotic premium has a proportional relationship to claim frequency. However, if ()(0, 1), then >0,

d 1 ln (b ( ) ) = ( ( ) -1) < 0 d

This guarantees that b()/ is a monotonic decreasing function.

lim

0

(4)

b( ) b( ) = , lim =0

7

(5)

Figure 1 illustrates the monotonic decreasing property of the Japanese BMS(83) satisfying the condition

()(0,1). This property implies that there exists a unique claim frequency 0 such that b(). If the

insured's true claim frequency is 1, where 0<1, and the applied premium is b(), then b(1)<1 (6)

5 6 7

Calculation technique of the mean asymptotic premium and the elasticity can be found in Lemaire (1995). b()=()b()/ BMS(83) means the bonus-malus system revised in 1983. See Appendix 1.

50 40 30 b/ 20 10 0 0.0 0.1 0.2 0.3 0.4

Figure 1: Monotonic Decreasing Property of b/ (BMS(83))

This means that bad drivers with claim frequency 1 don't pay premiums to cover their risk, and that good drivers with 2, where 0>2, subsidize the bad drivers by paying more than their risk. b(2 )>2 Thus, it is shown that a favorable elasticity () is one that assigns fair premiums to each policyholder. (7)

4. Insurance Demand and Elasticity

In this section, an insurance demand function is derived by solving the one-period expected utility maximization problem. Then, the insurance demand model is examined if insurance could have the property of an inferior good, and then if it could be a Giffen good conditioning on an inferior good. Finally, the condition is provided to define the property of a good.

4.1.

Optimal Insurance Consider a consumer with initial wealth W0 considering whether to purchase automobile insurance

covering the economic loss L. Assume the loss amounts are same for each accident. The number of claims N in one period has Poisson distribution pn with parameter , and the distribution is assumed to be constant over time. The applied BMS has attained a steady-state, and the mean asymptotic premium b() associated with a claim frequency is assumed. Let I denote the insurance claim per accident and let u represent the consumer's

utility function where u>0, u<0. A consumer's one-period expected utility can be expressed as:

E u (W1 ) = pn un (W0 - nL + nI - Ib) n =0

(8)

where pn =

ne- , n = 0,1, 2, n!

The terminal wealth considers the initial wealth, losses, insurance payments, and premium payment. Furthermore, the one-period model is also justified as long as the steady-state is assumed, in which the mean asymptotic premium is unchanged with respect to a particular risk . Implementing a second order Taylor series expansion around (W0-Ib) differentiating with respect to the insurance amount I, and dividing by -u gives the following approximation.

E u (W1 ) pn (n - b) n( I - L)R -1 I n=0

(9.1)

where R is Pratt-Arrow measure of the absolute risk aversion coefficient defined as R-u/u, and is assumed to be positive. The first order condition can be expressed as the expectation of the number of claims. And, it is easy to see that the second order condition is negative8.

E (n - b)(n ( I - L) R -1) = 0

Solving the equation in terms of I, the optimal insurance purchase I* is derived.

(9.2)

I* = L -

b - 2R

(10)

4.2.

Property of Insurance as a Good Differentiating (10) with respect to wealth yields the condition of whether insurance is an inferior good9.

I * b - 1 =- 2 W W R

(11)

Here, the risk tolerance is increasing, (1/R)/W>010, or equivalently, the decreasing absolute risk aversion hypothesis11 is assumed. Even though many empirical studies have attempted to estimate the behavior of

8 9

See Appendix 2 As pointed out by Mossin (1968) 10 It is meant that decreasing absolute risk aversion, equivalently constant or decreasing relative risk aversion is assumed. 11 Pratt-Arrow hypothesis of decreasing absolute risk aversion. As discussed by Mossin (1968) the hypothesis is just a formulation of a particular preference ordering. Some empirical researches provide evidence of a decreasing absolute

absolute and relative risk aversion, no conclusive evidence has been provided. The mean asymptotic premium b is not restricted to be smaller than the claim frequency. So, insurance is a normal good if b<. However, where b> is anticipated, the result is consistent with Mossin (1968). Consider how the optimal purchase of insurance changes as the premium changes. Differentiating equation (10) with respect to the premium b() yields the slope of the insurance demand curve.

I * 1 =- 2 b R

(12)

Equation (12) is strictly negative, since the denominator is assumed to be positive. Therefore, it is shown that insurance can always be an ordinary good and has a negative demand curve slope as long as a perfectly elastic premium is assumed. This assumption has to be relaxed in order to incorporate the elasticity into the model to examine how the elasticity of a BMS much less than unity affects the condition derived above. To show the insurance demand as a function of the elasticity measure, differentiating equation (10) with respect to the claim frequency defines the relationship between insurance demand and claim frequency as

b 2b - 2 1 + I = 4 R

*

(13)

The left hand side of equation (13) can be expressed as the following:

I * I * b = ( ) b

(14)

Substituting (14) into (13) and rearranging the equation establishes that the slope of the insurance demand curve is defined as a function of the elasticity measure.

I * (2b - ) b ( ) - 1 = b 2R

(15)

Considering the sign condition of equation (15), the numerator becomes a determinant of the sign condition. Therefore, by (11) and (15), the sign condition to be Giffen12 is obtained in (16). Normally, b> is anticipated

risk aversion implied by a constant relative risk aversion (Szpiro (1986), Friend and Blume (1975), Siegal and Hoban (1982)) and a decreasing relative risk aversion (Cohn et al (1975), Bellanti and Saba (1986) and Levy (1994)) Also, other studies (Friend and Blume (1975), Morin and Suarez (1983), Szpiro (1983), Eisenhauer and Halek (1999), and Halek and Eisenhauer (2001) show evidence of an increasing relative risk aversion implying a constant or increasing absolute risk aversion. Thus, no consistent result couldn't be found to determine the direction of an absolute risk aversion. The results seem to significantly depend on consumer characteristics, type of data, and method of estimation. Hoy and Robson (1981) first explicitly investigated the possibility of Giffen when CRRA is assumed, and concluded

12

for all claim frequencies , and then the condition for an inferior good tends not to restrict the condition to positive demand curve slopes.

I * < 0 if < b W I * > 0 if < b 2 - ( ) b

(16)

The boundary value for claim frequency to have a positive slope is defined as Cb(2-()). To interpret this result, insurance always can be observed as an ordinary good13 under the perfectly elastic condition. However, that is not necessarily true once this assumption is relaxed. By the sign condition in (16), insurance could have a positive demand curve slope if the mean asymptotic premium is larger than the critical value. This implies that there exists a particular condition such that a decrease in the premium induces policyholders to buy less coverage. This condition for the positive slope makes sense intuitively. The fair elasticity of unity allows it reduced to be b>. And, the smaller elasticity tends to lead to a positive slope. Thus, this elasticity measure could provide useful insights and the critical value can be used to evaluate the property of insurance as a good and the insurance demand.

5.

Development of Japanese Automobile Bonus-Malus Systems14

In Japanese automobile insurance, the no-fault bonus system applied to all types of vehicles was

introduced in 1963 (BMS(63)). The largest discount rate was 15%. It did not take policyholders nor insurers long to realize that this premium rating structure was not enough to ensure fairness among policies. In 1965 (BMS(65)), the maximum discount rate was increased to 50%. The system applied no discount in the first policy year, and a 10% discount was allowed if there was no claim during the year. No claim for five consecutive policy years could qualify a policyholder for the 50% maximum discount. However, once a claim occured, the cumulative discount might turn out to be zero.

13 14

that the coefficient must be greater than one, which is empirically implausible. And the following theoretical studies concerning if insurance could be Giffen and its upward-sloping (Borch (1986), Briys, Dionne, and Eeckhoudt (1989), Barzel and Suen (1992), and Weber (2001)) tend to deny the possibility, though those are not sufficient to reject it in this setting. by equation (12) Kamiya and Tojo (2005) focused on evaluating fairness of systems. See Appendix 1.

3.5 3.0 2.5 premium 2.0 1.5 1.0 0.5 0.0

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

minimum premium maximum premium

year

Figure 2: Maximum and Minimum Premiums of Japanese BMS

Compared to recent systems, this penalty for incurring one claim was tough. At the same time, a new penalty system was also introduced. Under the new penalty system, conditioning on a number of claims greater than or equal to three, 150% of the premium was assigned if the loss ratio in the previous policy year was less than or equal to 200%. And 200% of the premium was applied if the loss ratio was more than 200%15. The problem of this penalty system was that the information about the loss ratio depended on the policyholder's voluntary announcement. It was reasonable to assume that the penalty system did not work efficiently. The separately established discount and penalty systems had experienced several modifications without substantial change until July, 1984 In July, 1984, a newly established system that integrated the previous discount and penalty systems was introduced. The system was completely different from the previous systems in that past claim history reflected the class belonging to the policy year. It has been the prototype of BMS that is currently used by Japanese insurers. Some basic rules such as all policyholders starting from class 6, one class going up in case of no claims in the previous year, and 60% as the maximum discount rate, are equivalent to the minimum premium 0.4. On the other hand, there are some primary differences between the BMS(84) and BMS(04). The current system has twenty classes as opposed to sixteen classes in BMS(84). The maximum premium was 1.3 which is smaller than the current maximum premium of 1.6. For good drivers, the system requires only eight consecutive no-claim years to reach class 14 with the minimum premium level of 0.4 from the initial class 6. The current system requires twelve consecutive no-claim years to attain class 18 with the maximum. These revisions seem to provide more penalties for bad drivers and lower discounts for good drivers.

15

Some rules are neglected or simplified for a computational purpose.

Table 1: Revise of BMS and the Mean Asymptotic Premium

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.85 0.86 0.86 0.87 0.87 0.87 0.88 0.88 0.88 0.88 0.90 0.91 0.92 0.93 0.94 0.94 0.95 0.96 0.53 0.56 0.58 0.60 0.63 0.65 0.67 0.68 0.70 0.72 0.78 0.83 0.87 0.91 0.94 0.97 1.00 1.03 0.53 0.56 0.58 0.61 0.63 0.65 0.67 0.69 0.71 0.73 0.81 0.87 0.93 0.97 1.02 1.06 1.11 1.15 0.51 0.52 0.53 0.54 0.56 0.57 0.59 0.61 0.62 0.64 0.74 0.84 0.92 0.99 1.06 1.11 1.16 1.21

The Mean Asymptotic Premium BMS(63) BMS(65) BMS(70) BMS(74) BMS(83) BMS(84) BMS(86) BMS(93) BMS(98) BMS(99) BMS(04)

0.41 0.42 0.43 0.45 0.46 0.48 0.50 0.52 0.54 0.57 0.69 0.81 0.91 0.99 1.05 1.11 1.16 1.21

0.40 0.40 0.40 0.40 0.41 0.41 0.42 0.43 0.45 0.47 0.70 0.99 1.15 1.22 1.25 1.27 1.28 1.28

0.40 0.40 0.40 0.40 0.41 0.41 0.42 0.43 0.45 0.47 0.72 1.04 1.23 1.33 1.37 1.40 1.42 1.44

0.40 0.40 0.41 0.41 0.42 0.44 0.46 0.49 0.54 0.59 0.94 1.18 1.30 1.36 1.39 1.41 1.43 1.44

0.40 0.40 0.41 0.41 0.42 0.44 0.46 0.49 0.54 0.59 0.94 1.18 1.30 1.36 1.39 1.41 1.43 1.44

0.40 0.41 0.42 0.43 0.44 0.45 0.48 0.51 0.55 0.60 0.94 1.17 1.29 1.35 1.39 1.41 1.43 1.44

0.40 0.40 0.40 0.41 0.41 0.41 0.43 0.44 0.47 0.51 0.85 1.11 1.25 1.32 1.38 1.41 1.44 1.46

* Underlines indicate premiums closest to one

Furthermore, the system drops two classes in one claim though three classes have to be reduced in the current system. And there is a rule to distinguish the body injury claims from the claims of others in that the claim for body injury liability is counted as two other claims. The revision of 1986 (BMS(86)) increased penalties for class 1 and 2 from 130% up to 150% and 140% of the premium, respectively. It also created a new class of accident: no-count accident, where the claims are literally uncounted as claims to reduce classes. The purpose of this rule was to differentiate between accidents with the insured at or not at fault. To focus on the insured's inherent risk, some claims with only the passengers' medical payment or payment for comprehensive coverage were considered as no-count accident. The transition rule with two classes down for one claim and two-claim-count rule in the claim of bodily injury liability were transformed into a simple three class down rule for all claims in 1993. Meanwhile, no-count accident, in which the increase of classes was possible regardless of the claim payment, was changed to no-upward shift accident, in which the next year's class had to remain the same if there was a claim payment for the class of accident. Since the automobile compulsory rating system was repealed in 1998, each insurer had started to revise its own BMS. And some differences between the systems could be observed among insurers at the time of complete deregulation. However, in 2004, all 16 insurers, affiliated in the Japanese non-life insurance association and dealing automobile insurance in Japanese market, have used the same system (BMS(04)).

Table 2: Revise of BMS and the Elasticity

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.04 0.05 0.05 0.05 0.06 0.06 0.05 0.05 0.05 0.09 0.12 0.15 0.17 0.18 0.19 0.20 0.21 0.21 0.22 0.22 0.22 0.22 0.22 0.24 0.25 0.27 0.05 0.10 0.13 0.16 0.18 0.20 0.21 0.22 0.23 0.24 0.26 0.27 0.28 0.29 0.30 0.32 0.35 0.37 0.02 0.04 0.07 0.10 0.13 0.17 0.20 0.24 0.27 0.30 0.41 0.43 0.42 0.40 0.39 0.38 0.39 0.40 0.07 0.14 0.23 0.31 0.40 0.49 0.57 0.64 0.71 0.76 0.85 0.75 0.61 0.50 0.44 0.41 0.41 0.41

Elasticity BMS(63) BMS(65) BMS(70) BMS(74) BMS(83) BMS(84) BMS(86) BMS(93) BMS(98) BMS(99) BMS(04)

0.00 0.00 0.01 0.03 0.05 0.09 0.16 0.25 0.38 0.55 1.37 0.98 0.48 0.25 0.14 0.09 0.06 0.04

0.00 0.00 0.01 0.03 0.05 0.09 0.16 0.25 0.39 0.57 1.42 1.03 0.52 0.29 0.19 0.13 0.10 0.08

0.00 0.01 0.03 0.07 0.15 0.26 0.42 0.61 0.82 1.01 1.04 0.55 0.30 0.19 0.14 0.10 0.08 0.07

0.00 0.01 0.03 0.07 0.15 0.26 0.42 0.61 0.82 1.01 1.04 0.55 0.30 0.19 0.14 0.10 0.08 0.07

0.01 0.03 0.05 0.10 0.16 0.26 0.40 0.57 0.76 0.94 1.02 0.56 0.31 0.20 0.14 0.11 0.09 0.07

0.00 0.01 0.01 0.03 0.06 0.12 0.22 0.38 0.62 0.90 1.26 0.66 0.39 0.27 0.21 0.18 0.15 0.13

* Underlines indicate elasticity more than one

6. Elasticity and Bountary Claim Frequency of a Good

6.1. The Elasticity of the Japanese BMS Table 1 shows the mean asymptotic premiums with respect to claim frequencies for some representative BMS. We can see similar premiums for the systems revised after 1984 regardless of claim frequency, since they have similar transition rules and the number of classes as mentioned earlier. For example, the minimum mean asymptotic premiums after BMS(84) are approximately 0.4 because the minimum premium for those systems are consistently 0.4. Similar mean asymptotic premiums can be seen even for the average claim frequency =0.116. This result indicates that those recent BMS do not adequately reflect the impact of low claim frequencies on the premium level. Also, the premiums of recent systems have attained unity for claim frequencies around 0.3 or 0.4. This may be interpreted as the base premium rate17 for those risks. Furthermore, in chronological order, the claim frequency with unity of the mean asymptotic premium has decreased over time for all BMS, except the current BMS. By focusing on the claim frequency =1, current BMS increases the premium level to 1.46, which is associated with the increase of maximum penalty of class 1 to 160%. Obviously, the BMS revised after

16 17

Refer to Jidoushahoken no Gaikyo If other risk classification methods are taken into account, or if a steady-state is not assumed, then the interpretation would not be valid.

deregulation decreases the premium level in the long-run for good drivers and even more for the average drivers. Yet it increases for extremely bad drivers in a competitive market. Table 2 shows the corresponding elasticities. It should be noted that the most values of the elasticities for all BMS are much lower than the ideal value of one, especially for small claim frequencies. The systems before BMS(84) had lower maximum elasticities for every claim frequency. And those provided relatively high elasticity for extremely high claim frequencies. The BMS(83) with elasticity of 0.4 for the average claim frequency, around =0.1, can be considered as the most efficient average policy. One year later, the BMS is transitioned to a new system. And systems after BMS(84) have provided extremely low elasticities to good drivers and bad drivers with claim frequencies higher than 0.4, but high elasticities for drivers with claim frequencies between 0.2 and 0.4. Thus, Japanese BMS are typically not efficient for both high and low risk drivers. One of the reasons for the low elasticities of good and bad drivers is that the current BMS has only twenty classes. In a steady-state, policies with low and high claim frequencies reach to the end of classes and stay there. Most of the distribution belongs to the highest and lowest classes. The limited number of classes causes lower elasticities for those drivers. Another reason is the small variability of premium scales. The same premium is applied to some classes close to the highest. For the current system, classes from 18 to 20 have the same premium of 0.4. Only within 10% of difference of premium is provided for upper seven classes. On the other hand, 160% of the premium is the toughest penalty, which is attainable with only two claims from the initial class.

6.2. Critical Value for a Good and the Boundary Claim Frequency This section discusses the situation where the policyholder regards insurance as a Giffen good. For example, consider a Japanese policy with an average claim frequency of =0.1 under BMS(04). since its mean asymptotic premium is 0.41 and the elasticity is 0.06, then the critical value is 0.80. And since the critical value is much larger than the claim frequency 0.1, then automobile insurance should be a Giffen good. Contrary to our intuition, the average risk policies tend to reduce coverage if the premium becomes less expensive and tend to purchase more coverage if the premium becomes more expensive.

Table 3: Revise of BMS and the Asymptotic Critical Values

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.70 1.71 1.71 1.72 1.72 1.72 1.73 1.73 1.74 1.74 1.76 1.78 1.79 1.81 1.82 1.84 1.85 1.86 1.03 1.06 1.09 1.12 1.15 1.17 1.20 1.23 1.25 1.28 1.39 1.48 1.56 1.62 1.67 1.71 1.75 1.78 1.03 1.06 1.09 1.12 1.15 1.18 1.20 1.23 1.26 1.28 1.40 1.51 1.59 1.67 1.73 1.78 1.83 1.87 1.01 1.02 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.19 1.32 1.46 1.59 1.70 1.79 1.87 1.94

Asymptotic Critical Value BMS(63) BMS(65) BMS(70) BMS(74) BMS(83) BMS(84) BMS(86) BMS(93) BMS(98) BMS(99) BMS(04)

0.79 0.78 0.77 0.76 0.74 0.73 0.72 0.71 0.70 0.70 0.79 1.01 1.26 1.48 1.64 1.76 1.85 1.93

0.80 0.80 0.80 0.80 0.79 0.79 0.78 0.76 0.72 0.68 0.44 1.01 1.74 2.14 2.32 2.42 2.48 2.51

0.80 0.80 0.80 0.80 0.79 0.79 0.78 0.75 0.72 0.68 0.42 1.01 1.82 2.27 2.49 2.62 2.70 2.75

0.80 0.80 0.80 0.79 0.78 0.76 0.73 0.68 0.63 0.58 0.90 1.71 2.20 2.45 2.59 2.68 2.74 2.78

0.80 0.80 0.80 0.79 0.78 0.76 0.73 0.68 0.63 0.58 0.90 1.71 2.20 2.45 2.59 2.68 2.74 2.78

0.80 0.81 0.81 0.81 0.80 0.79 0.76 0.73 0.68 0.64 0.92 1.69 2.18 2.44 2.58 2.67 2.73 2.78

0.80 0.80 0.80 0.80 0.79 0.78 0.76 0.71 0.65 0.56 0.63 1.49 2.01 2.29 2.46 2.57 2.66 2.73

2.0 1.8 1.6 excess of critical values 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 claim frequency BMS(65) BMS(83) BMS(86) BMS(04)

Figure 3: Excess of Critical Value for an Ordinary Good

A similar calculation shows that policies with risk parameter =0.2 have b=0.51, =0.90, and C= 0.56, and concludes that they consider insurance as a Giffen good as well. Table 3 shows critical values with respect to claim frequencies for each system. If a policy's claim frequency is larger than the critical value, then the insurance is an ordinary good. It turns to be a Giffen good if a policy's is smaller than the critical value. It is interesting to note that automobile insurance is Giffen for all observed claim frequencies under all

systems adopted in Japan. Figure 3 exhibits the excess of critical values to be an ordinary good. The least excess for all combination of systems and claim frequencies can be obtained for BMS(86) and =0.3. It can be seen that recent systems tend to have a V-shaped curve as shown in Figure 3. The bottoms of the curves vary between systems, but are approximately between =0.2 and 0.3. Thus, a very limited number of combinations tend to be close to the condition of an ordinary good. This result might seem odd, compared with past research of Giffen goods 18 . However, unfairness represented by an unfavorably priced mean asymptotic premium and small elasticities serves as an expensive premium loading, and then it is found the reasonable condition that insurance can be Giffen without considering premium loading. Thus, it is clear that the structure of BMS greatly impacts insurance demand. The mismanagement of premium rating may reduce low-risk policies' demand and increase high-risk policies' purchase of coverage.

7.

Optimal Control of BMS to Mitigate Adverse Selection

It is reasonable that deregulation of the Japanese insurance market may lead to market pressure and may

motivate insurers to reduce the premium level. As a matter of fact, the current system lowers its mean asymptotic premiums for all claim frequencies less than 0.8, compared to the previous versions. Theoretical analysis might show that a competitive market setting induces premium reduction toward the equilibrium, maximizing welfare of both parties. However, it should be noted that there could exist policies which recognize insurance as a Giffen good under a risk classification method with a low elasticity of premium with respect to the risk. Reviewing the revisions to BMS(04), the revision that lowers the premiums for most claim frequencies is supposed to reduce the coverage demand for all policies with <0.8. This should not affect the condition of adverse selection. Similar analysis can be applied to future revisions of BMS. If insurers intend to reduce the premium level in a similar manner, market volume continues to shrink. The most appropriate BMS control for the specific purpose of mitigating adverse selection in this circumstance is to increase the premium level for policies with the claim frequency less than the average, <0.1119, and to decrease the premium level for those with >0.11. This control implies that diminishing fairness among policyholders leads to an increase in insurance demand for low risk policies and a decrease of the purchase for worse-than-average-risk policies,

18

19

Hoy and Robson (1981) and Briys, Dionne, and Eeckhoudt (1989) concludes Giffen is not feasible as CRRA is assumed. Borch (1986) shows the similar conclusion under more general approach. Japanese average claim frequency in 2004, referred to Jidoushahoken no Gaikyo (2004)

providing favorable results for insurers. That might result in mitigating adverse selection under BMS(04). Therefore, improving individual fairness under BMS does not necessarily imply that it works for adverse selection. One way to achieve both individual fairness and fairness of insurance system as a whole could be obtained by letting insurance be an ordinary good. Once insurance has down-slope demand curve, low risk policies purchase more coverage if it becomes less expensive, and high risk policies demand less if the premium goes up. Those consumer behaviors could improve fairness for both individual and insurance system.

8.

Conclusion

The primary purpose of this article is to examine how bonus-malus systems widely used in European and

Asian countries work for the adverse selection of automobile insurance. BMS reflects the insured's heterogeneous inherent risk on the premium level. It plays an important role in reducing the unfairness with respect to premium sharing. However, the elasticity calculated in many countries' systems have indicated low values not equal to the ideal value of unity. It is assumed that this observation significantly influences insurance demand and adverse selection. In order to examine the assumption and to show the relationship between the structure of BMS and the coverage demand, the elasticity concept is utilized. As long as the premium is perfectly elastic, automobile insurance seems to be a ordinary good, and the result is consistent with past research about Giffen goods. However, once the elasticity measure is incorporated into the model, automobile insurance is not necessarily an ordinary good. Actually, under a small elasticity and unfavorable premium, insurance is likely to be a Giffen good, in which reduction of the premium results in decreasing the demand of coverage. As an empirical study, Japanese past and current BMS are introduced and analyzed. In the analysis, the mean asymptotic premiums, the elasticity, and the critical value determining the property of a good are calculated. The common feature of elasticity for all BMS is that the calculated values are lower than one except for a particular range of claim frequencies. Particularly, the low values are found for good drivers. Recent systems tend to be more severe on policies with claim frequencies between 0.2 and 0.3. The main reason that the system can be severe only for those with such claim frequencies is caused by the limited number of classes, such as twenty classes in the current system, moderate discount, or the penalty for those with low and high claim frequencies.

The observed critical values for all systems and claim frequencies are larger than the corresponding claim frequencies. The result implies that all policies with (0,1) regard automobile insurance as a Giffen good. It is interesting to note that current BMS which are used by all insurers without any regulations provides a lower premium level than the previous system for almost all claim frequencies. The reason may be that insurers who are under market pressure have the incentive to reduce the premium to attract policies and to increase those demands. Otherwise, competitive market setting might lead the equilibrium. However, the revision may not work expectedly. This is because policyholders that get lower premium rate should be motivated to purchase less if compared with what they purchased in the past due to their perception of insurance as a Giffen. On the other hand, only extremely bad drivers that get higher premiums should purchase more insurance due to their recognition of insurance as an ordinary good. In conclusion, the theoretical analysis shows that the last transition of the system does not only decrease the good drivers' demand for coverage but also worsens adverse selection.

Reference

[1] Barzel, Y. and Suen, W. (1992), "The Demand Curves for Giffen Goods are Downward Sloping", The Economic Journal, 102, 896-905. [2] Bellanti and Saba (1986), "Human Capital and Life-Cycle Effects on Risk Aversion", Journal of Financial Research, 9, 41-51. [3] Borch, K. (1986), "Insurance and Giffen's Paradox", Economics Letters, 20, 303-306. [4] Briys, E., Dionne, G., and Eeckhoudt, L. (1989), "More on Insurance as a Giffen Good", Journal of Risk and Uncertainty, 2, 415-420. [5] Cohn, R. A., Lewellen, W. G., Lease, R. C., and Schlarbaum, G. G. (1975), "Individual Investor Risk Aversion and Investment Portfolio Risk Composition", Journal of Finance, 30, 605-620. [6] Eisenhauer, J.G. and Halek, Martin (1999), "Prudence, Risk Aversion, and the Demand for Life Insurance", Applied Economics Letters, 6, 239-242. [7] Friend, I. and Blume, M. E. (1975), "The Demand for Risky Assets", American Economic Review, 65, 900-922. [8] Halek, M and Eisenhauer, J. G. (2001), "Demography of Risk Aversion", Journal of Risk and Insurance,

68, 1-24. [9] Hoy, M. and Robson, A.J. (1981), "Insurance as a Giggen Good", Economics Letters, 8, 47-51. [10] Institute of Japanese Actuaries (2000), Sonpo-Suuri [11] Janssen, M.C.W. and Karamychev, V.A. (2005), "Dynamic Insurance Contracts and Adverse Selection", Journal of Risk and Insurance, 72, 45-59. [12] Jidoushahokenn-Ryouritusanteikai (1994), Jidoushahoken RyouritsuSeido no Hensen, 4th edition. [13] Kamiya, S. and Tojo, K. (2005), "Does Japanese Bonus-Malus System Work Efficiently?", Proceedings of the Japanese Association of Risk, Insurance and Pensions, 2005, 165-179. [14] Lasserre, P. and Dionne, G. (1985), "Adverse Selection, Repeated Insurance Contracts and Announcement Strategy", Review of Economic Studies, 52, 719-723. [15] Lemaire, J. and Hongmin, Z. (1994), "A Comparative Analysis of 30 Bonus-Malus Systems", ASTIN Bulletin, 24, 287-309. [16] Lemaire, J. (1985), Automobile Insurance: Actuarial Models. Boston: Kluwer. [17] Lemaire, J. (1988), "A Comparative Analysis of Most European and Japanese Bonus-Malus Systems", Journal of Risk and Insurance, 55, 660-681. [18] Lemaire, J. (1995), Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer. [19] Lemaire, J. (1998), "Bonus-Malus Systems: The European and Asian Approach to Merit-Rating", North American Actuarial Journal, 2, 26-47. [20] Levy, H. (1994), "Absolute and Relative Risk Aversion: An Experimental Study", Journal of Risk and Uncertainty, 8, 289-307. [21] Loimaranta, K. (1972), "Some Asymptotic Properties of Bonus Systems", ASTIN Bulletin, 6, 233-245. [22] Morin, R.A. and Suarez, A.F. (1983), "Risk Aversion Revisited", Journal of Finance, 38, 1201-1216 [23] Mossin, J. (1968), "Aspects of Rational Insurance Purchasing", Journal of Political Economy, 76, 553-568. [24] Richaudeau, D. (1999), "Automobile Insurance Contracts and Risk of Accident: An Empirical Test Using French Individual Data", Geneva Papers on Risk and Insurance Theory, 24, 97-114. [25] Siegel, F. W. and Hoban, J. P. (1982), "Relative Risk Aversion Revisited", Review of Economics and Statistics, 64, 481-487. [26] Stiglitz, J. and Rothschild, M. (1976), "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information", Quarterly Journal of Economics, 90, 629-649.

[27] Szpiro, G. G. (1983), "The hypotheses of Absolute and Relative Risk Aversion: An Empirical Study Using Cross-Section Data", Geneva Papers on Risk and Insurance, 8, 336-349. [28] Szpiro, G. G. (1986), "Measuring Risk Aversion: An Alternative Approach", Review of Economics and Statistics, 68, 156-159. [29] Watt, R. and Vazquez, F, J. (1997), "Full Insurance, Bayesian Updated Premiums, and Adverse Selection", The Geneva Papers on Risk and Insurance Theory, 22, 135-150 [30] Weber, C. E. (2001), "Actuarially Unfair Insurance and Downward-Sloping Demand Curves for Giffen Goods", Manchester School, 69, 377-386

Appendix 1: Development of Japanese Bonus-Malus System

BMS(63)

Class 1 2 3 Premium 0 1.00 0.90 0.85 2 3 3 Class After Number of Claims 1 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

BMS(65)

Class Premium 0 1 2 3 4 5 6 7 2.00 1.00 0.90 0.80 0.70 0.60 0.50 2 3 4 5 6 7 7 1 2 2 2 2 2 2 2 Class After Number of Claims 2 3 4 5 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1

6 1 1 1

7+ 1 1 1

6 1 1 1 1 1 1 1

* Starting class: 1 * Minor modification is applied

7+ 1 1 1 1 1 1 1

* Starting class: 2 * Minor modification is applied

BMS(70)

Class 1 2 3 4 5 6 7 8 9 Premium 0 3.00 2.00 1.50 1.00 0.90 0.80 0.70 0.60 0.50 4 4 4 5 6 7 8 9 9 1 4 4 4 4 4 4 4 4 4 Class After Number of Claims 2 3 4 5 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1

BMS(74)

Class Premium 0 1 2 3 4 5 6 7 8 9 3.00 2.00 1.50 1.00 0.90 0.80 0.70 0.60 0.50 4 4 4 5 6 7 8 9 9 1 4 4 4 4 4 4 5 6 7 Class After Number of Claims 2 3 4 5 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 5 2 1 1

6 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1

6 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1

* Starting class: 4 * Minor modification is applied

* Starting class: 4 * Minor modification is applied

BMS(83)

Class 1 2 3 4 5 6 7 8 9 10 11 12 Premium 0 3.00 2.00 1.50 1.00 0.90 0.80 0.70 0.60 0.50 0.45 0.42 0.40 4 4 4 5 6 7 8 9 10 11 12 12 1 4 4 4 4 4 4 5 6 7 8 8 8 Class After Number of Claims 2 3 4 5 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 5 2 1 1 6 2 1 1 6 2 1 1 6 2 1 1

BMS(84)

Class Premium 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1.30 1.30 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.45 0.42 0.40 0.40 0.40 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 1 1 1 4 2 1 1 5 3 1 1 6 4 2 1 7 5 3 1 8 6 4 2 9 7 5 3 10 8 6 4 11 9 7 5 12 10 8 6

6 1 1 1 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1 1 1 1

6 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4

* Starting class: 4 * Minor modification is applied

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

* Starting class: 6 * Minor modification is applied

BMS(86)

Class 1 2 3 4 5 6 6s 7 8 9 10 11 12 13 14 15 16 Premium 0 1.50 1.40 1.30 1.20 1.10 1.00 1.20 0.90 0.80 0.70 0.60 0.50 0.45 0.42 0.40 0.40 0.40 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 16 1 1 1 1 2 3 4 4 5 6 7 8 9 10 11 12 13 14 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 3 1 1 1 4 2 1 1 5 3 1 1 6 4 2 1 7 5 3 1 8 6 4 2 9 7 5 3 10 8 6 4 11 9 7 5 12 10 8 6

BMS(93)

Class Premium 0 1 2 3 4 5 6 6s 7 8 9 10 11 12 13 14 15 16 1.50 1.40 1.30 1.20 1.10 1.00 1.20 0.90 0.80 0.70 0.60 0.50 0.45 0.42 0.40 0.40 0.40 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 16 1 1 1 1 1 2 3 3 4 5 6 7 8 9 10 11 12 13 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 2 1 1 6 3 1 1 7 4 1 1 8 5 2 1 9 6 3 1 10 7 4 1

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

* Starting class: 6s * Minor modification is applied

* Starting class: 6s

BMS(98)

Class 1 2 3 4 5 6 6s 7 8 9 10 11 12 13 14 15 16 Premium 0 1.50 1.40 1.30 1.20 1.10 1.00 1.30 0.90 0.80 0.70 0.60 0.50 0.45 0.42 0.40 0.40 0.40 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 16 1 1 1 1 1 2 3 3 4 5 6 7 8 9 10 11 12 13 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 2 1 1 6 3 1 1 7 4 1 1 8 5 2 1 9 6 3 1 10 7 4 1

BMS(99)

Class Premium 0 1 2 3 4 5 6 6s 7 8 9 10 11 12 13 14 15 16 1.50 1.40 1.30 1.20 1.10 1.00 1.30 0.80 0.70 0.65 0.60 0.55 0.50 0.50 0.45 0.45 0.40 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 16 1 1 1 1 1 2 3 3 4 5 6 7 8 9 10 11 12 13 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 2 1 1 6 3 1 1 7 4 1 1 8 5 2 1 9 6 3 1 10 7 4 1

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

* Starting class: 6s

* Starting class: 6s

BMS(04)

Class 1 2 3 4 5 6 6s 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Premium 0 1.60 1.30 1.20 1.00 0.90 0.90 1.30 0.80 0.70 0.60 0.60 0.55 0.50 0.50 0.45 0.45 0.42 0.42 0.40 0.40 0.40 2 3 4 5 6 7 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 1 1 1 1 1 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Class After Number of Claims 2 3 4 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 3 1 1 1 4 1 1 1 5 2 1 1 6 3 1 1 7 4 1 1 8 5 2 1 9 6 3 1 10 7 4 1 11 8 5 2 12 9 6 3 13 10 7 4 14 11 8 5

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

7+ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

* Starting class: 6s

Appendix 2: Derivation of Optimal Insurance By the Taylor expansion, equation (9) is expanded to

1 2 E u (W1 ) = pn u (i) + n ( I - L)u (i) + n 2 ( I - L) u (i) + n =0 2

(A1)

Differentiating (A1) with respect to insurance amount I, the first order condition is obtained

E u (W1 ) 1 2 = pn -bu (i) + n(u (i) - b( I - L)u (i))+ n2 ( I - L)u (i) - 2 b( I - L) u (i) + I n=0

(A2)

In order to utilize the coefficient of absolute risk aversion, (A2) is divided by -u and substituted by R.

E u (W1 ) pn b - n - nb( I - L) R + n2 ( I - L)R I n=0

(A3)

Rearranging (A3) and setting equal to zero,

E u (W1 ) = pn (n - b) n( I - L)R -1 = 0 I n=0

(A4)

(A4) is also expressed as the expectation of random variable of the number of claims denoted by N.

E ( N - b) N ( I - L) R - 1 = 0

(A5)

Since assuming Poisson distribution with parameter , (A5) can be reduced to (A6)

2 ( I - L) R - + b = 0

(A6)

Solving (A6) in terms of I, optimal insurance model (11) is obtained. About the second order condition, by differentiating (A2) with respect to I again,

b 2u (i) + n (-bu (i) - bu (i) + b 2 ( I - L)u (i)) 2 E u (W1 ) = pn +n2 u i - b I - L u i - b I - L u i + I 2 n=0 ) () ( ) ( )) ( () (

(A7)

Considering up to the second order, the condition guarantees a unique maximum.

2 E u (W1 ) = pn b2u (i) - 2nbu (i)+ n2u (i) 2 I n=0 2 E u (W1 ) 2 = pn (b - n) u < 0 2 I n=0

(A8)

(A9)

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