On Compulsory Per-Claim Deductibles in Automobile Insurance

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1 The Geneva Papers on Risk and Insurance Theory, 28: 25 32, 2003 c 2003 The Geneva Association On Compulsory Per-Claim Deductibles in Automobile Insurance CHU-SHIU LI Department of Economics, Feng Chia University, Taiwan, ROC CHWEN-CHI LIU Department of Insurance, Feng Chia University, Taiwan, ROC Received November 14, 2001; Revised July 15, 2002 Abstract The purposes of this paper are to analyze the theoretical characteristics of the compulsory deductible system and to verify the rationality of an increasing per-claim deductible in automobile insurance. We derive the optimal variable per-claim deductible by assuming the insurers are financially balanced and the expected utility of the insured is maximized in the absence of moral hazard. Our result suggests that a variable per-claim deductible increasing with the number of claims per year is not optimal. Instead, deductibles should be charged in a decreasing rate forming a second-best solution. Key words: automobile insurance, deductible, per-claim, second-best JEL Classification No.: G22 1. Introduction Designing the optimal form of an insurance contract is a major issue in insurance economics literature. Previous studies have shown that a preferred insurance policy contract by a riskaverse individual includes a deductible [Arrow, 1963; Raviv, 1979]. While attempting to derive the optimal solution, most studies analyze the case of a single risk and describe a deductible in terms of the expense of the total claim amount. The implicit assumption behind this description is that the claim occurs only once (but with a variable accident probability). Beyond the analyses of a single loss and a one-time deductible, Raviv [1979] shows that, with multiple losses, a single deductible for the aggregate loss is optimal. Gollier and Schlesinger [1995] discuss the insurance contract of one risk in the presence of other independent risks and conclude that individual deductibles are always second-best optima. Paulsen [1995] considers the possibility of a per-year per-claim deductible that is assumed constant. While some of the previous analyses provide the possibility of separate deductibles for multiple losses, this paper considers the case that the deductible of a specific risk is decided contingent only upon the number of past losses. The insurance contract with per-claim deductibles rather than aggregate deductibles is not uncommon in insurance markets. The reimbursement system of medical insurance is a

2 26 LI AND LIU typical example. The insured receives insurance indemnity after paying a fixed deductible for each claim up to a total cap limit. Although it appears in the form of an aggregate deductible, Arrow s first-best insurance contract can be interpreted as a policy in which the overall insurance indemnity equals the sum of the multiple losses minus a straight deductible. The deductible for each loss may be reinterpreted as a stochastic amount determined by previous losses and indemnity. A recent example of a per-claim deductible contract is the new paradigm of automobile collision insurance implemented in Taiwan in Under the new system, the insured is required to pay a deductible that increases with the number of claims per year. The deductible is 3,000 NTD for the first claim, 5,000 NTD for the second, and 7,000 NTD for the third and final claim. Intuitively, the design of an increasing perclaim deductible seems to be to punish negligent or reluctant behavior. But it is interesting to explore the nature of the variable form of deductibles. To analyze the theoretical foundation of variable per-year per-claim deductibles with random sequential loss severities, we want to understand the rational design. Should perclaim deductibles be constant, decreasing, or increasing? This paper derives the optimal variable per-year per-claim deductible by assuming that the insurers are financially balanced and that the expected utility of the insured is maximized in the absence of moral hazard. Our result suggests that an increasing variable per-claim deductible cannot meet the requirements of optimal conditions. Instead, deductibles should be charged in a decreasing rate providing a second-best solution. In our examination, the decreasing marginal utility of income plays a crucial role. If the insured needs to pay the deductible twice for two claims, for example, the marginal utilities per dollar are different in each payment. In particular, the marginal utility per dollar for the first deductible is lower than that for the second deductible. Therefore, in order for the insured to maximize expected utility while the insurer remains financially balanced, the deductible for the second claim should be less than that for the first one. Section 2 of the paper outlines the analytical model. Section 3 presents a numerical illustration and Section 4 contains our conclusions. 2. The model Assume for each claim i = 0, 1,...,N, the claim size X i has a deductible of D i, and that X 0 = 0 and D 0 = 0 indicating no initial loss. For an insurer, the total claim expense is: Y (D) = N max{x i D i, 0}. (1) i=0 For the insured, out-of-pocket loss in addition to the premium paid is: Z(D) = N min{x i, D i }. (2) i=0 Most actuarial literature presumes that the claim number N is independent of the claim size and that X and N follow a Poisson distribution with expectation λ (see for instance Lemaire [1995], Ch. 15). 1

3 ON COMPULSORY PER-CLAIM DEDUCTIBLES IN AUTOMOBILE INSURANCE 27 The notation Q(D) is the premium corresponding to coverage Y (D). Here, D is the vector (D 1, D 2,...,D N ). According to the basic assumption of Paulsen [1995], we can write Q(D) = (1 + α) E[Y (D)], (3) where α is a loading factor. In this paper, we assume that the deductible D i may vary with each claim. The sizes for each claim X 1, X 2,...,X N are independent and identically distributed. In order to avoid the problem of personal bankruptcy, we assume that each claim has a support of [0, x] and that N x is less than the initial wealth. Suppose for simplicity that there are no more than two claims per year. Therefore, the total expected claim expense for the insurer is: { x } E[Y (D)] = [1 F(X)] dx P 1 D 1 { x x } + [1 F(X)] dx + [1 F(X)] dx P 2 (4) D 1 D 2 where F(X) denotes the distribution function of each claim with density f (X), P i represents the probability of an insured with claim number i in a given time period has an accident, i = 0, 1, 2, and P 0 is the probability of no claim. The signs of the first and second order derivatives with respect to D i can be obtained as follows: and Q(D) D i = (1 + α) [1 F(D i )] (1 P 0 P 1 ) 0, (5) 2 Q(D) Di 2 = (1 + α) f (D i ) (1 P 0 P 1 ) 0, (6) where P 0 + P 1 + P 2 = 1. It is assumed that each insured is risk-averse and pursues expected utility maximization. Let EU and W 0 denote the expected utility and the initial endowment respectively. In the one-period model, we consider the case for which each claim X i has a different D i, i = 1, 2. The final wealth of the insured after paying the deductible is: W (X,D) = W 0 Q(D) Z(D). (7) The insured chooses the levels of D 1 and D 2 under the conditions X i 0(i = 1, 2) so as to maximize expected utility EU(W ). It is necessary to prove the following lemma to derive the first order optimal conditions for each claim deductible. Lemma 1: Suppose V (W, D) = E[U(W )], then V D i = Q(D) D i E[U (W )] [1 F(D i )] E[U (W i )], (8)

4 28 LI AND LIU where W i = W 0 Q(D) Z i (D), Q(D) is independent of D, and Z i (D) = 2 (X k, D k ) + D i, i = 1, 2 k=0 k i (See the Appendix for a proof.) By applying Eq. (8), the first order condition associated with V (W, D) is V/ D i = 0. Thus, the insured s optimization can be derived as follows: and D 1 : D 2 : Q(D) {U [W 0 Q(D)] P 0 + E{U [W 0 Q(D) Z(D)]} D 1 [1 F(D 1 )] E{U [W 0 Q(D) Z 1 (D)]}} = 0 (9) Q(D) {U [W 0 Q(D)] P 0 + E{U [W 0 Q(D) Z(D)]} D 2 [1 F(D 2 )] E{U [W 0 Q(D) Z 2 (D)]}} = 0. (10) where Z 1 = D 1 + min(x 2, D 2 ) and Z 2 = D 2 + min(x 1, D 1 ). In order to see whether D i is a local maximum, 2 we show that the second order conditions for maximization are satisfied, 2 V/ D 2 i < 0. 3 Following the results of (9) and (10), we get (1 P 0 ) (1 P 0 P 1 ) = E{U [W 0 Q(D) Z 1 (D)]} > 1. (11) E{U [W 0 Q(D) Z 2 (D)]} If P 1 is not equal to zero, then the left hand side of (11) is greater than one. This implies that E[U (W 1 )] > E[U (W 2 )], where W 1 = W 0 Q(D) Z 1, and W 2 = W 0 Q(D) Z 2. The main purpose of this paper is to analyze the rational design of per-claim deductibles for a risk-averse individual. The following result of Proposition 1 shows that if D 1 > D 2, then E[U (W 1 )] > E[U (W 2 )]. Proposition 1: Suppose that a risk-averse insured endowed with nonrandom initial wealth is subject to two random losses in a given period. If deductibles vary with according to each claim, then it is optimal to choose an insurance policy in which the first claim deductible is higher than the second claim deductible in the absence of moral hazard. That is, D1 > D 2. Proof. Since X 1 and X 2 are independent random variables representing losses, we define Y 1 = min(t, D 2 ) + D 1, and Y 2 = min(t, D 1 ) + D 2. The value of E[U ] could be integrated along the random variable t for 0 t < x. For the interval D 1 t < x we have Y 1 = Y 2 = D 1 + D 2, and thus U (W 1 ) > U (W 2 ). However, for the interval 0 t < D 1 we see Y 1 > Y 2. This implies that U [W 0 Q(D) Y 1 ] > U [W 0 Q(D) Y 2 ]. Thus, the overall integration (0 t < x) results in E[U (W 1 )] > E[U (W 2 )].

5 ON COMPULSORY PER-CLAIM DEDUCTIBLES IN AUTOMOBILE INSURANCE 29 The economic rationality behind Proposition 1 is the diminishing marginal utility of income. For a risk-averse individual, the marginal utility of insurance given a higher level of wealth is less than that with a lower one. When the insured suffers a collusion, he pays either the first claim deductible or the realized loss depending on which value is lower, and the final wealth is less than the initial wealth with no loss. Thus, given that the first claim deductible (or realized loss) has been paid, the insured chooses the lower second deductible for the second claim since the marginal utility of wealth is higher at this stage. As can be seen in earlier studies [Arrow, 1963; Gollier and Schlesinger, 1995], a single aggregate deductible would be a first-best solution when there are multiple losses. This kind of contract implies that the optimal deductible is decreasing in the number of claims, but in a stochastic way. In contrast, Eeckhoudt et al. [1991] show that the optimal deductible per claim is a constant. Their model is based on the assumption that the total loss for a given period is scattered equally for each claim and the deductible for each claim is independent on the realization of other losses. This is in a sense a third-best solution. 4 In fact, our study provides a second-best solution by restricting coverage to a decreasing deductible for each realized past loss and is intermediary between the first-best stochastically decreasing deductible and the third-best constant deductible. The key idea is that, inspired by the newly designed automobile insurance contracts in Taiwan, we focus on the relationship between deductibles of sequential claims. In comparison with the case of Eeckhoudt et al. [1991] in which each claim is scattered equally, our model reflects the realistic situation of random claims. Since past losses definitely reduce the wealth of the insured, the net wealth after loss will affect the choice of deductible. Therefore it is rational to assume that the deductible for each claim can be made contingent upon the number of past accidents. We find that in an insurance policy designed with per-claim deductibles, a second-best solution of decreasing deductibles is preferable. 3. Numerical illustration This section provides numerical examples to illustrate the model above. Since the probability of the third accident is very small, we only consider the case of two claims; that is, the policy is designed using the first deductible D 1 and the second deductible D 2 only. It is assumed that claim amounts are exponentially distributed, with expected value 30 for a sample of 5000 insured. The number of claims is Possion distributed with parameter λ = (Lemaire [1995], Ch. 15). It is further assumed that the insured s utility is approximated exponentially, U(W ) = e aw, where a is the level of absolute risk aversion and W is the final wealth. The initial endowment W 0 is 200 and the loading factor α is 0.1. Table 1 shows the different solutions of (D1, D 2 ) for various values of a. As expected, the size of the first claim deductible is greater than the second claim. 4. Generalization to N claims The simple model of two claims can be easily generalized to N claims per year. The insurance contract covers losses for one year and the deductible is on each claim. The key

6 30 LI AND LIU Table 1. Optimal deductibles for various level of absolute risk aversion (λ = , X = 30, W 0 = 200,α = 0.1). a D D issue of this paper is to explore the situation in which the per-claim deductible is not a constant. We examine any two successive claims i and j, where i = j 1. In particular, we try to indicate whether there is any relationship between D i and D j. The insured can select the levels of D i and D j when X i 0 and X j 0 under the condition of maximizing the expected utility E[U(W )]. In the general case, the optimal conditions for each per-claim deductible, Dl, can be derived as follows: 5 D l : Q(D) D l E[U (W )] [1 F(D l )] E{U [W 0 Q(D) Z l (D)]} =0 (12) where Z l = D l + N k=0 k l min(x k, D k ), and l = i, j. Let W i and W j be the final end-of-period wealth for i and j, respectively. By applying the results of the basic case as shown in Eqs. (10) and (11), we derive the following result: E[U (W i )] > E[U (W j )] for i = j 1, (13) where W i = W 0 Q(D) Z i (D) and W j = W 0 Q(D) Z j (D). It is easy to see that E[U (W i )] is a monotonically increasing function with respect to the variable D i, for i = 1, 2,...,N, where W i = W 0 Q(D) Z i (D), Q(D) is a constant, and Z i (D) = N k=0 k i (X k, D k ) + D i. Thus, for a risk-averse individual with a utility function satisfying U > 0 and U < 0, the result of Eq. (13) can be applied to show that Di > D j for i = j 1. The following proposition shows that the optimal deductible per claim is a decreasing function of past claims. Proposition 2: Suppose that a risk-averse individual faces N i.i.d. risks of losses X 1,...,X N for a given period. If deductibles vary according to each claim, then the risk-averse insured chooses an insurance policy in which the higher the claim number, the lower the size of the deductible in the absence of moral hazard. That is, D1 > D 2 > D 3 > > D N.

7 ON COMPULSORY PER-CLAIM DEDUCTIBLES IN AUTOMOBILE INSURANCE Concluding remarks This paper indicates the importance of extending previous models to incorporate the empirical features of the deductible system with random sequential loss severities. It adopts the usual assumption that the probabilities of claims are independent of the size of the deductibles. Our results show that a risk-averse insured prefers a per-claim deductible policy that varies with the number of claims. We also show that the size of the optimal per-claim deductible is decreasing when the number of claims is greater than one in a given time period. It may be asked, in the absence of moral hazard, what is the rationale for the existence of increasing deductibles? This may be explained by the presence of non-negligible administrative costs for handling each claim, making it sub-optimal to cover claims too small to justify the administrative costs. For cases where the moral hazard issue is important (i.e. where the size and probability of claims do vary with the size of the deductibles), an increasing variable per-claim deductible may serve to reduce the inefficiency of excessive moral hazard. This moral hazard consideration may be the reason for the new pattern of increasing per-claim deductibles introduced in Taiwan in Whether the importance of the effect in reducing moral hazard (that suggests increasing per-claim deductibles) offsets the effect of risk aversion (that suggests decreasing per-claim deductibles) is beyond the scope of this paper. Appendix Proof of Lemma 1. Notice V = Q(D) E[U (W )] + E{φ[Z(D)]}, (A1) D i D i D i where φ(z(d)) = U(W 0 Q(D) Z(D)), and Q(D) is assumed to be a constant. Our analysis will focus on the last term in (A1), D i E{φ[Z(D)]}. For simplicity, we first derive the derivative form for D 1, that is, { [ ]} E{U[W 0 Q(D) Z(D)]} = 2 E U W 0 Q(D) min(x k, D k ) D 1 D 1 k=1 = { x D1 U[W 0 Q(D) X 1 min(x 2, D 2 )] df(x 1 ) df(x 2 ) D 1 X 2 =0 X 1 =0 x x } + U[W 0 Q(D) D 1 min(x 2, D 2 )] df(x 1 ) df(x 2 ) X 2 =0 X 1 =D 1 x x = U [W 0 Q(D) D 1 min(x 2, D 2 )] df(x 1 ) df(x 2 ) X 2 =0 X 1 =D 1 = [1 F(D 1 )] E{U [W 0 Q(D) D 1 min(x 2, D 2 )]}. (A2)

8 32 LI AND LIU By applying the same procedure for D 2 as shown above, we could obtain the following result: D 2 E{U[W 0 Q(D) Z(D)]} = [1 F(D 2 )] E{U [W 0 Q(D) D 2 min(x 1, D 1 )]}. (A3) Thus, following the results of (A1), (A2), and (A3), we have proved Eq. (8). Acknowledgments We are grateful to the editor and anonymous referee for many helpful comments and instructions. We are also grateful to Yew-Kwang Ng and Li-Chien Lin for valuable suggestions. Notes 1. In fact, the results of this paper would be hold without this Poisson assumption. 2. With multiple losses, it is reasonable to obtain that the deductibles equal the losses for early losses. 3. According to Eq. (8), V/ D i depends on the values of E[U (W )] and E[U (W i )]. The first term in Eq. (8) is derived when we assume Z(D) is a constant while the second term is derived by assuming Q(D) is a constant. By definition W = W 0 Z(D) Q(D), and when D i increases, Q(D) decreases but W increases if we treat W 0 Z(D) as a constant in the first term of Eq. (8). Thus, an increase in D i lowers the value of E[U (W )]. In contrast, when D i increases, W i decreases since W i = W 0 Q(D) Z i (D), as we treat D i as the only variable in W i. Thus, when D i increases, E[U (W i )] increases. This implies that an increase in D i decreases V/ D i ; that is, the second order conditions of maximizing the expected utility are satisfied. 4. We are indebted to a referee for pointing out the relationship between the second-best and third-best solutions and that the deductible in the first-best solution can be expressed as D i = max(0, D t 1 i=1 X i ). 5. All features in this model can be generalized to N claims. The proof for the general model is not given here due to the complex notation; interested readers may request it from the corresponding author. References ARROW, K. [1963]: Uncertainty and the Welfare Economics of Medical Care, American Economic Review, 53, EECKHOUDT, L., BAUWENS, L., BRIYS, E., and SCARMURE, P. [1991]: The Law of Large (Small?) Numbers and the Demand for Insurance, Journal of Risk and Insurance, GOLLIER, C. and SCHLESINGER, H. [1995]: Second-Best Insurance Contract Design in an Incomplete Market, Scandinavian Journal of Economics, 97(1), LEMAIRE, J. [1995]: Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer Nijhoff. PAULSEN, J. [1995]: Optimal Per Claim Deductibility in Insurance with the Possibility of Risky Investments, Insurance: Mathematics and Economics, 17, RAVIV, A. [1979]: The Design of an Optimal Insurance Policy, American Economic Review, 69, SCHLESINGER, H. [1981]: The Optimal Level of Deductibility in Insurance Contracts, Journal of Risk and Insurance, 48,

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