# The Role of Commitment in Dynamic Contracts: Evidence from Life Insurance

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9 III.2 Solving the Model The approach we follow to obtain the equilibrium contracts is the following. In a competitive equilibrium, allocations must maximize consumers expected utility subject to a zero profit constraint and a set of no lapsation constraints that capture consumers inability to commit. Solving this constrained maximization problem delivers the set of premiums and face amounts that must be available to a consumer in a competitive market. However, there are several ways in which this set of premiums and face amounts can be made available to consumers. The constrained maximization problem below is constructed via fully contingent contracts that involve no lapsation. We later describe how to obtain the same allocation with non contingent contracts that are quite common in the life insurance market. In a competitive equilibrium, premiums and face amounts that are fully contingent on the health state Q 1, F 1, (Q 1 2, F2 1 ),..., (Q N 2, F2 N ) must maximize consumers expected utility. (1) N pv(f 1 ) + (1 p)u(y 1 g Q 1 ) + (1 p) π i [p i v(f2) i + (1 p i )u(y 2 + g Q i 2)]. i=1 Subject to the zero profit condition: N (2) (1 p)q 1 pf 1 + (1 p) π i [(1 p i )Q i 2 p i F2] i = 0 i=1 and the no lapsation constraints: for i = 1,..., N, and for all Qi 2, F i 2 such that (1 p i ) Q i 2 p i F i 2 > 0, (3) p i v(f i 2) + (1 p i )u(y 2 + g Q i 2) p i v( F i 2) + (1 p i )u(y 2 + g Q i 2). The constraint in equation (2) requires that insurance companies break even on average. This must hold under competition. The no-lapsation constraints defined by equation (3) are the additional constraints imposed by lack of consumer commitment. They require that, at every state in the second period, the buyer prefers staying in the long term contract than switching to a competing insurance company. In other words, an equilibrium contract must be such that there does not exist another contract that is profitable and that offers the buyer higher utility in any state of period two. We say that a consumer gets full event insurance in state i of date 2 (the definition for date 1 is analogous) if (4) v (F2) i = u (y 2 + g Q i 2). We call Q i 2(F I) the fair premium for full insurance in state i, namely, the premium that guarantees zero profits. 9

11 model predicts that the most front-loaded contracts have flat premiums, which is exactly what we observe in the data. III.3 Contract Equivalence and Empirical Implementation of the Model Proposition 1 obtains equilibrium allocations via fully contingent contracts that involve no lapsation. However, as we saw in Section, both contingent and non contingent contracts are common in the life insurance market. We now show that non contingent contracts can also be optimal. We then show how we account for lapsation, which is quite common in life insurance, and we discuss how we compute the present value of premiums for contingent contracts. To gain an idea of how equivalence is possible, note that a buyer who chooses an LT20 or an aggregate ART (which are non contingent) is free to drop out of the contract at any date to (for instance) purchase an S&U ART. More formally, fix an equilibrium contingent contract C. A non contingent contract N C is also an equilibrium contract if, in each state, the consumer for whom the C contract is optimal can obtain the same utility with the N C contract and the insurance companies offering the contracts obtain the same profits with either contract. Clearly, this requires that the consumer drops out of NC in the appropriate states. From part (ii) of Proposition 1, the terms of a C contract are constant: Q i 2 = Q s 2 and F i 2 = F s 2 for i = s,..., N. In states i = 1,... s 1, the premiums and face amounts equal those offered in the spot market: the buyer pays the actuarially fair premium. Fix the terms of the non contingent contract NC to be the same as for contract C in the first period, and (Q s 2,F s 2 ) in the second period. Thus, the two contracts are equivalent by construction in the first period and in the bad states of the second period. However, in states i = 1,... s 1 contract C has better terms. But in those states the consumer can obtain the same terms by dropping out of contract NC and purchasing spot contracts. Because by Proposition 1, in each state i = 1,... s 1, contract C has the same terms as a spot contract, the consumer obtains the same utility via NC and C. Insurers are indifferent between selling the two kinds of contracts as well: in state s through N consumers stay in the contract under both contracts and the terms are exactly the same. In states 1 to s 1, the two alternative contracts induce different behavior by the consumer, but equal (zero) profits for the firms: in those states premiums in contract C are actuarially fair. Thus, in those states, firms are indifferent between retaining and not retaining the consumers. This shows that the two ways of achieving the equilibrium allocation are equivalent. This equivalence result will help us in two ways: First, by explaining within our model the existence of non contingent contracts in the life insurance market. Second, the result helps us compare contingent and non contingent contracts. All we have to do is transform each contingent contract into its equivalent non contingent contract, along the lines of the previous discussion. The importance of having comparable non contingent contracts comes from the next proposition, 11

16 IV.3 Front-loading and Lapsation The role of front-loading as a device to provide more insurance depends on locking-in consumers. We now attempt to provide further corroboration for the theoretical prediction that less frontloaded contracts suffer from more health related lapsation. Table VI presents lapsation data, from LIMRA [1996] for the period The sample includes all the contracts in force at the beginning of the policy year, for 33 top United States insurers (namely, all LIMRA members). In total, the sample contains over 7 million policies that were in force in 1993 (approximately one third all policies in the United States in 1993). Lapse rates are percentages of face amount (or number of policies) in force at their 1993 policy anniversaries, that lapse on or before their 1994 anniversary. This data is not ideal since it does not provide a complete break-down of lapsation by type of contract. It only allows us to contrast ARTs with term contracts of longer length (2 to 30 year level contracts). The model predicts that longer term contracts, which involve more frontloading, should suffer lower lapsation. Moreover, we expect the difference in lapsation to be more pronounced when there is more health related learning (namely, for older age groups). The first two columns represent lapsation measured as the percent of the face amounts. Contract Year measures the age of the contract. For example, Contract Year =1 means contracts issued during 1993, while Contract Year =2 means contracts issued in The numbers are in line with Proposition 2. Aside from contracts just issued, on which not much lock-in has occurred yet, Term (longer) contracts have lower lapsation, and a steeper decline in lapsation over time. A similar picture appears in the next two columns that measure lapsation as a percent of the number of policies. [Table 6] The second part of the table separates lapsation by age groups. We expect more health related information to be revealed in the age-group than in the Thus, we expect the effect of front-loading in reducing lapsation to be more pronounced for the older group. As expected, we find that the difference in lapsation between ARTs and Term is bigger for the older group. IV.4 Accidental Death Insurance: a Test Case Accidental death insurance, which is a special type of life insurance contract, provides an ideal way to further test whether the main forces behind the model are responsible for the shape of the available contracts in the industry. An accidental death insurance policy is essentially the same product as a term policy, with the exception that it only pays if death is accidental; it does not pay if death is due to illness. Accidental death rates are quite flat between the ages of 25 to 60, and firms are unlikely to learn much about the characteristics of the consumers. Since learning is a key force behind our results, 16

20 To see this, note that if (Q 1, F 1 ), (Q 1 2, F2 1 ),..., (Q N 2, F2 N ) maximize (1) subject to (2) and (5), then there is no state i, and no (Q i 2, F 2) i that makes positive profits and gives buyers a higher utility in that state. Thus, (3) is satisfied. Conversely, suppose that (Q i 2, F i 2) are such that (5) is violated. Then it is clear that (3) is violated as well since a competing insurance company can offer terms that are slightly better for buyers than (Q i 2, F i 2) and that still make positive profits. Let µ be the Lagrange multiplier for the constraint in (2) and λ i the Lagrange multiplier for the ith constraint in (5). The first order conditions for an optimum are: (6) u (y g Q 1 ) = µ, (7) v (F 1 ) = µ, (8) (1 p)π i u (y + g Q i 2) + (1 p)π i µ + λ i = 0 i, (9) (1 p)π i v (F i 2) (1 p)π i µ λ i = 0 i, (10) ((1 p i )Q i 2 p i F i 2)λ i = 0 i. Part (i) follows from first combining equations (6) and (7) and then combining equations (8) and (9). To prove part (ii), note first that equations (8) and (9) imply (11) F i 2 = (v ) 1 (u (y + g Q i 2)). If constraint i in equation (5) is binding, (1 p i )Q i 2 = p i F i 2. Substituting from equation (11) we obtain (12) (1 p i )Q i 2 = p i (v ) 1 (u (y + g Q i 2)). From equation (12), we see that, since u and v are decreasing functions, if i and j are two binding constraints with i > j (so that p i > p j ), then Q i 2 > Q j 2. Suppose now that constraint k in equation (5) is non binding. Then equation (8) simplifies to: (13) u (y + g Q k 2) = µ. Thus, if constraints k and l are non-binding, Q k 2 = Q l 2. If in contrast, constraint i in equation (5) is binding, 20

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