1 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS: EVIDENCE FROM LIFE INSURANCE* IGAL HENDEL AND ALESSANDRO LIZZERI We use data on lfe nsurance contracts to study the propertes of long-term contracts n a world where buyers cannot commt to a contract. The data are especally suted to test a theory of dynamc contractng snce they nclude nformaton on the entre pro le of future premums. All the patterns n the data t the theoretcal predctons of a model wth symmetrc learnng and one-sded commtment à la Harrs and Holmstom. The lack of commtment by consumers shapes contracts n the way predcted by the theory: all contracts nvolve front-loadng (prepayment) of premums. Front-loadng generates a partal lock-n of consumers; more front-loadng s assocated wth lower lapsaton. Moreover, contracts that are more front-loaded have a lower present value of premums over the perod of coverage. Ths s consstent wth the dea that front-loadng enhances consumer commtment and that more front-loaded contracts retan better rsk pools. I. INTRODUCTION The effect of agents nablty to commt to long-term plans has been shown to have far-reachng mplcatons for economc arrangements. The goal of ths paper s to examne emprcally the role of mperfect commtment n shapng contractual relatons usng data from the lfe nsurance market. Lack of consumer commtment can generate nef cences n nsurance markets because short-term contracts do not offer nsurance aganst reclass caton rsk: bad news about the health status of a consumer resultng n ncreased premums. Lack of long-term nsurance s consdered an mportant market falure n health and lfe nsurance markets, and t has underscored much of the polcy debate concernng health reform (see Damond ). Several factors make lfe nsurance sutable for studyng commtment to long-term contracts. Learnng (about health) s a quanttatvely mportant phenomenon. Lfe nsurance contracts are smple and explct, and ssues such as asymmetrc nformaton and msreportng of accdents are not mportant n the case of lfe nsurance (see Secton V). In contrast, other contractual * We are very grateful to Glenn Daly. We also thank Andrew Funderburk for research assstance, Jonathan Levn, Derek Neal, and Steven Tadels for ther comments, and LIMRA Int. and Compulfe for generously provdng us access to ther data. Fnally, we would lke to thank Edward Glaeser and anonymous referees. The nancal support of the Natonal Scence Foundaton through grants SES and SES s gratefully acknowledged by the Presdent and Fellows of Harvard College and the Massachusetts Insttute of Technology. The Quarterly Journal of Economcs, February
2 300 QUARTERLY JOURNAL OF ECONOMICS relatons such as labor contracts are plagued by unobservables and by the presence of mplct agreements. Fnally, very rch data on contracts are avalable. 1 We rst present a theoretcal framework to understand how we should expect lack of commtment to be re ected n observed contracts. The model adapts to the lfe nsurance market the Harrs-Holmstrom  model of symmetrc learnng that was rst developed n a labor market context. The man predctons of the model are then tested usng data on the Unted States lfe nsurance ndustry. We assume that the ntal health status of the nsured s known to all, and that new nformaton about her health type s revealed over tme symmetrcally to all market partcpants. We also assume that there s one-sded commtment;.e., nsurance companes can commt to long-term contracts but buyers cannot. Because of learnng, short-term (spot) contracts nvolve reclass caton rsk: future premums wll re ect the future health status of consumers. Long-term contracts would solve the problem f consumers could commt: nsurance companes would offer future premums re ectng the average future health of the pool. Lack of commtment makes such a contract nfeasble: healthy consumers would drop out to obtan cheaper coverage from competng companes. Hence, to obtan nsurance aganst reclass caton rsk, consumers must prepay some of the premums. The prepayment, or front-loadng, locks the consumer nto the contract. To be completely successful, the front-loaded amount must guarantee that no consumers drop coverage n the future: the future premum must be no hgher than the far premum for the healthest type. Such a contract would mplement the same allocaton that would be acheved under consumer commtment: t provdes full nsurance aganst reclass caton rsk. All contracts we observe n the lfe nsurance market n the Unted States (see subsecton IV.A) are front-loaded, and some of the contracts come close to achevng full nsurance. However, most of these contracts leave consumers subject to some reclas- 1. Prevous emprcal work on dynamc contracts ncludes Chappor, Salane, and Valentn  who test a model of wage dynamcs usng data from one large French rm. They nd support for theores of learnng wth downward rgdty of wages. Murphy  nvestgates a longtudnal sample of CEOs and compares learnng wth ncentves; hs evdence s mxed. Donne and Doherty  present a dynamc model of adverse selecton and wth evdence from auto nsurance. Crocker and Moran  studes employee lock-n as a way to ncrease commtment to long-term health nsurance.
3 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 301 s caton rsk. The model provdes an explanaton for the varety of contracts, lnkng them to consumers heterogenety n wllngness to front-load. More mportantly, gven the varety of observed contracts, the model wll generate testable predctons about the cross secton of contracts. Front-loadng can be consdered as the ex ante prce that consumers pay for future caps on premums, a result of ther nablty to commt. Good rsks can stll reenter the market n the future whle bad rsks are protected aganst steeper premum ncreases f they reman n the front-loaded polcy pool. The key comparatve statcs generated by the model are that contracts that are more front-loaded lock n consumers to a greater extent. Thus, more front-loaded contracts retan a better rsk pool and have a lower present value of premums. As predcted by the model, lfe nsurance contracts are offered n several varetes, nvolvng dfferent tme pro les of premums whch dffer n how steeply premums ncrease over tme. Dfferent premum pro les represent dfferent degrees of front-loadng (commtment). We use the contract dversty to test the mplcatons of the model. In accordance wth the contract theoretc predctons, we nd that vrtually every contract s front-loaded and that there s a negatve correlaton between front-loadng and the present value premums. Remarkably, most of the (large) dsperson n the present value of premums can be accounted for by the extent of front-loadng. We con rm the role of front-loadng as a commtment devce by lookng at the relaton between lapsaton (a term commonly used n the actuaral lterature that means voluntary termnaton of coverage) and frontloadng. As predcted, more front-loaded contracts have lower lapsaton. Thus, all the patterns n the data suggest that contracts are desgned n a way that ts a model of symmetrc learnng wth one-sded commtment. Alternatve commtment assumptons are rejected by the data. Standard asymmetrc nformaton models delver predctons that are nconsstent wth the data. Fnally, understandng and quantfyng the nef cency from the lack of blateral commtment n lfe nsurance s of nterest for the polcy debate on health nsurance. Lack of renewablty and the absence of nsurance of reclass caton rsk are some of the concerns rased about health nsurance (e.g., n Clnton s reform proposal, new state regulaton durng the 1990s). However, the health care market suffers from a varety of other problems (see Cutler ). Isolatng the problems due to the lack of blateral
4 302 QUARTERLY JOURNAL OF ECONOMICS commtment should be useful for understandng the source of nef cency n health nsurance (see Concludng Remarks). II. A. The Contracts II. THE CONTRACTS AND THE DATA The lfe nsurance market s segmented n Term and Cash Value contracts. 2 We focus on the market for Term nsurance for two reasons. (1) term s smpler snce t nvolves pure nsurance: a xed sum s pad upon death of the nsured f death should occur wthn the perod of coverage. In contrast, Cash Value offers a combnaton of nsurance and savngs. (2) we do not have data on cash values. Term contracts represented over 37 percent of all ordnary lfe nsurance n 1997 accordng to LIMRA . All contracts are unlateral (see McGll ): the nsurance companes must respect the terms of the contract for the duraton, but the buyer can look for better deals at any tme. The contract becomes vod once the buyer stops payng premums, and there are no cash ows upon termnaton. These features t a model of one-sded commtment. 3 The man dstncton among term contracts s the premum pro le. Annual Renewable Term (ART) contracts charge premums that ncrease yearly. Level Term contracts (LTn) offer premums that ncrease only every n years, rangng from 2 to 30 years. Term contracts are renewable, wthout medcal examnaton, up to a prespec ed age (typcally 70 or older). Future terms for renewal are spec ed at the moment of underwrtng. 4 Column 2 of Table I llustrates an aggregate ART. Ths contract has premums that vary exclusvely by age. Column 3 llustrates a ten-year Level Term contract (LT10). We dscuss the 2. We focus on the ndvdual market. In employment-related nsurance commtment s enhanced by job lock-n. 3. Severance payment would enhance buyers ablty to commt to stay n the contract. However, for legal reasons they are not used n any lfe nsurance contract. Severance payments would be consdered a stpulated penalty, and would not be enforced by a Court. A Common Law tradton prevents Courts from enforcng terms stpulatng damages that exceed the actual harm caused by breach [Cooter and Ulen 1999]. Insurance companes do not suffer any ex post damage from the loss of a consumer. 4. Premums are guaranteed for a prespec ed perod of tme, rangng from a year to twenty years. Snce no medcal examnaton s needed to renew, prce changes are not nsured spec c. Furthermore, nsurers rarely change the announced premums: Consumer Reports (July 1993) found that only one ncreased term premums between 1989 and Hence, preannounced premums are the future premums.
5 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 303 TABLE I TYPES OF TERM CONTRACTS S&U ART Polcy year Age ART LT These are contracts offered n 7/1997 to a preferred nonsmoker, male, by Northwestern Mutual (ART and LT10) and Jackson Natonal (S&U ART) for $500,000 of coverage. ART 5 annual renewable term polcy. LT10 5 term polcy wth level premums for ten years. S&U ART 5 annual contract that allows for reclass caton, by showng good health. dfference between these contracts n detal later. For now, note that contracts dffer substantally n the tme pro le of premums; the premums for the ART start below those for the LT10 but ncrease much more rapdly. The rght sde of Table I llustrates a Select and Ultmate (S&U) ART. These contracts offer state contngent prces: low premums are contngent on the nsured showng that he s stll n good health. Thus, premums vary by ssue age and duraton snce underwrtng (last date that good health was proved). Rows show the age at latest underwrtng; columns the years snce underwrtng (captured by Polcy Year). For nstance, at age 41 the nsured would pay $475 ( rst row, second column of matrx) for renewng hs one-year-old polcy f he does not requalfy (pass the medcal). If he had requal ed, he would have pad $385 (second row, rst column). The ncentve to requalfy ncreases wth age; at 59 the nsured who does not requalfy pays $1750 nstead of $1340. The bg dfference n premums n the alternatve scenaros of a S&U ART shows that learnng about the health status of an ndvdual s an mportant phenomenon, and that consumers may be left to suffer sgn cant reclass caton rsk. A 59 year old could be charged between $1340 and $6375. The market shares of these contracts are ARTs (aggregate
6 304 QUARTERLY JOURNAL OF ECONOMICS and S&U) 22.4 percent, LT percent, LT percent, and LT percent (LIMRA ). Ths ntal look at contracts hghlghts some patterns that we wll nvestgate more systematcally n what follows. Frst, there are both contngent (e.g., S&U ART) and noncontngent (e.g., aggregate ART) contracts. Second, the tme pro le of premums dffers across contracts: some have a atter pro le (are more front-loaded). Furthermore, contracts that have a low ntal premum, have hgh premums later on. II. B. The Data The source of nformaton on contracts s the Compulfe Quotaton System (July 1997), an nformaton system compled for the use of nsurance agents whch contans quotes by the man Unted States lfe nsurers (240 rms). Premums are reported by consumer characterstcs such as age, smokng status, gender, etc. It s an accurate source of prcng data snce agents are forbdden to offer dscounts by the antrebatng laws. For the purpose of testng the predctons of the model, the unque feature of Compulfe, s that t reports the whole pro le of future premums faced by a buyer at the moment of underwrtng. Thus, we observe the entre contract tme pro le, not just the startng premum. The vrtue of these data s that they are generated by usng the prcng rule of dozens of nsurance companes. We punch n the demographcs and face value nto the Compulfe prcng software, and we obtan the premums for all future dates and contngences. Ths s deal for testng the model snce, when we compare contracts, we lterally x all demographc characterstcs and only vary the contract pro le. In our sample we ncluded nsurers that satsfy a dual crteron of sze and solvency. They must be n categores IX and above of A.M. Best for sze (n terms of sales of term polces) and ratng A or better n terms of solvency (accordng to A.M. Best). Ths guarantees product homogenety and avods nonrepresentatve polces. Ffty- ve nsurers satsfy the crtera. 5 Table II presents descrptve statstcs of the contracts n our sample. The table presents the mean premum and standard devaton by type of contract. The thrd and fourth columns con- 5. Twenty-one of the rms n our sample held 46.8 percent of the 2937 bllon dollars of n-force face amount of term nsurance n 1996 [Best s Revew July 1996]. We do not have sales nformaton for the other companes n our sample, but clearly a large share of the market s covered.
7 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 305 TABLE II CONTRACT DESCRIPTIVES Contracts Premum at age 40 PV 20 years of coverage Type Observatons Mean Std dev Mean Std dev All types , ,244.7 LT , ,785.1 LT , ,090.7 LT , ,839.9 Aggregate ART , ,984.7 S&U ART , ,251.8 Data Source: Compulfe Quotaton System (July 1997). Contracts offered by the top 55 nsurers n the Unted States accordng to sze and solvency crtera. LTX 5 term polcy wth level premums for X years. Aggregate ART 5 annual renewable term polcy wth premums that depend only on age. S&U ART 5 annual renewable term wth premums that depend on age and tme elapsed snce last medcal examnaton. PV of the twenty years of coverage startng at age 40, and an 8 percent nterest rate. cern the rst-year premum for a 40 year old male nonsmoker who just passed the medcal screenng. The fth and sxth columns report the present value of premums for twenty years of coverage. When the contract s state contngent, because premums depend on whether the consumer does or does not requalfy, we need to specfy what contngency we look at. The present value s computed under the assumpton that the buyer never requal- es. As wll be shown later, ths s a useful benchmark because t allows a theoretcally meanngful comparson of contracts. We use an 8 percent nterest rate n computng present values. 6 Table II shows substantal varablty n the present value of premums, n partcular, across contract types. Ths varablty wll be used to test the model. 7 III. THEORY Before presentng the formal analyss, we nformally dscuss the man deas of the model. The key ngredents of the model are 6. We chose a hgh nterest rate for two reasons: to account for both tme preference and for the probablty of survvng to a spec c date, and because results are renforced f the nterest rate s lower. We expermented wth other nterest rates and our qualtatve results reman unchanged. 7. Prce dsperson may seem nconsstent wth competton. However, Wnter  found that, after controllng for contract characterstcs, only a small fracton of the dsperson remans unexplaned. He concluded that observed premums are consstent wth the hypothess of competton. We wll control for these polcy characterstcs.
8 306 QUARTERLY JOURNAL OF ECONOMICS symmetrc learnng (nformaton about the health type of the nsured s revealed over tme), one-sded commtment (nsurers commt, buyers do not), and buyer heterogenety n the cost of front-loadng. Consder a two-perod model where nformaton about rsk types (health states) s revealed at the begnnng of the second perod. A contract that guarantees full nsurance aganst reclass caton rsk has second-perod premums that are ndependent of the health state. If buyers cannot commt to stay wth a contract, ths constant premum should be smaller than the actuarally far premum for the best health state. Otherwse, the healther buyers would nd better rates on the spot market and drop coverage. However, ths contract (that retans even the healthest buyers) suffers second-perod losses amountng to the dfference between the second-perod premum (calculated to retan the healthest buyer) and the cost of coverng the entre pool. Therefore, ths contract must frontload premums: a surcharge n the rst perod to cover future expected losses. It s clear then that front-loadng s bene cal: payng premums up front reduces reclass caton rsk. One of our emprcal ndngs s that almost all contracts dsplay front-loadng. However, many contracts leave buyers to suffer consderable reclass caton rsk, suggestng that many buyers are not wllng to front-load enough to fully nsure. There are several reasons why consumers may nd front-loadng costly. For nstance, wth captal market mperfectons, the tghter the credt constrants the more costly t s to front-load premums. If t s costly for consumers to front-load, equlbrum contracts offer mperfect nsurance aganst reclass caton rsk: n return for the front-loadng, polces offer a cap on the premum requred n the second perod. To capture the varety of degrees of front-loadng offered n the lfe nsurance market, we assume that consumers are heterogeneous n the cost of frontloadng. Consumers wth more resources n the rst perod (those that face looser credt constrants) wll front-load more and face a lower cap on the premum n the second perod, whch translates nto lower reclass caton rsk. Front-loadng and the comparatve statc wth respect to the premum pro les wll provde the man testable mplcatons of the model.
9 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 307 III. A. Model Consder a market wth nsurance buyers and sellers. Buyers wsh to nsure a stream of ncome for ther dependents. When consderng a future perod, a buyer has two dstnct sources of utlty. If he wll be alve, and he consumes c $ 0, hs future utlty s gven by u(c). However, f he wll be dead, hs future utlty s gven by the consumpton of hs dependents. In ths case, f hs dependents consume c $ 0, hs future utlty s gven by v(c). The functons u and v are both assumed to be strctly concave and twce dfferentable. There are two perods. In perod 1 all agents have dentcal death probabltes p. We can thnk of p as representng a spec c health status n perod 1. The model can be readly extended to allow several rsk categores. In perod 1 there are 3 stages: n stage 1 nsurance companes offer contracts. At stage 2 buyers choose a contract. At stage 3 uncertanty about death s resolved, and consumpton takes place. In perod 2 there are four stages. In stage 0 uncertanty about the health status of each buyer s realzed. A consumer wth health status des wth probablty p n perod 2. We order the health status so that p 1, p 2..., p N. We assume that p 1 $ p (health worsens over tme). The probablty of beng n state s denoted by p. Uncertanty (learnng) about the second-perod rsk category creates reclass caton rsk, snce t potentally leaves the buyer facng uncertan premums n the second perod. We assume that nformaton s symmetrc: at the end of stage 0 all nsurance companes observe the health status of all buyers. 8 Stages 1 through 3 are the same as n the rst perod. We assume that there s perfect competton between nsurance companes. 9 We also assume one-sded commtment: nsurance companes can commt to future premums, whereas consumers freely choose between stayng wth ther perod 1 contract and swtchng to one of the spot contracts offered n perod 2. Therefore, the set of feasble rst-perod contracts s the set of unlateral contracts,.e., those contracts that termnate the mo- 8. We dscuss ths assumpton n detal n Secton V. 9. The market for term lfe nsurance s qute compettve: there are 400 competng companes [Lfe Insurance Factbook], and term nsurance s ncreasngly vewed as a commodty (Record 1997). There s a regulatory presence whch manly pertans to nancal solvency, not prcng. Lfe nsurance rates for ndvdual nsurance are regulated only n a most ndrect sense. It s felt that competton s an adequate regulator over any tendences toward rate excessveness [Black and Skpper 1994].
10 308 QUARTERLY JOURNAL OF ECONOMICS ment the buyer stops payng the premums; there are no cash ows upon termnaton. 10 A rst-perod contract conssts of rst-perod premum Q 1 and face amount F 1, and a vector of premums and face amounts (Q 2 1,F 2 1 ),..., (Q 2 N,F 2 N ) ndexed by the second-perod health status of an ndvdual. Contract terms can depend on the nformaton revealed n the second perod. A second-perod contract conssts of a premum and face amount (Q 2,F 2 ) ndexed by the second-perod health status: n the second perod, rms can offer dfferent contracts to buyers n dfferent rsk categores. A rst-perod contract s a long-term contract to whch the company unlaterally commts, whereas a second-perod contract s a short-term (spot) contract. Lack of consumer commtment per se need not preclude the possblty of achevng full nsurance. In a world wth no other frcton, all consumers would front-load suf cently to guarantee that they have no ncentve to drop out of the contract n the future. An mplcaton of such a soluton s that all consumers would be fully nsured aganst reclass caton rsk. Furthermore, there would be no contract varety: all consumers would choose the same contract. Explanng the exstng varety of contracts, and obtanng predctons about them, requres understandng why most consumers are reluctant to fully front-load. For smplcty, and to be closer to the formulaton of Harrs and Holmstrom, we assume that captal markets are completely absent. 11 The results would be unchanged f we assumed a borrowng rate that s hgher than the lendng rate. Ths knd of captal market mperfecton seems to be qute reasonable gven the nterest rates charged on unsecured debt (e.g., medan nterest rate charged on credt card balances s 18 percent). Also, note that to explan the varety of behavor, we do not need all buyers to face such mperfectons. We capture consumer heterogenety by assumng dfferences n the ncome process. 12 If alve, the consumer receves an ncome of y 2 g n the rst perod and y 1 g n the second perod, wth g $ These are exactly the knd of contracts offered n the U. S. lfe nsurance market; see McGll  or Daly . 11. Note that ths assumpton does not mean that consumers are constraned n ther front-loadng. Consumers have enough resources to suf cently front-load ther premums, but, absent captal markets, front-loadng requres gvng up current consumpton, whch ntroduces the trade-off between consumpton smoothng and nsurance aganst reclass caton rsk. 12. Others sources of heterogenety n front-loadng are dscussed n the workng paper [Hendel and Lzzer 2002]. These nclude heterogeneous and uncertan needs for lfe nsurance and heterogeneous belefs probablty of requal caton.
11 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 309 Thus, all consumers have the same permanent ncome, but dfferent consumers have dfferent ncome growth, as represented by the parameter g. If the buyer s dead, then hs dependents receve no ncome except for the face amount of the nsurance. Because of the absence of credt markets, ndvduals wth hgher g are more tghtly constraned. Heterogenety n g s what drves the dfferent choce of contracts n our model. III.B. Solvng the Model The approach we follow to obtan the equlbrum contracts s the followng. In a compettve equlbrum, allocatons must maxmze consumers expected utlty subject to a zero pro t constrant and a set of no-lapsaton constrants that capture consumers nablty to commt. Solvng ths constraned maxmzaton problem delvers the set of premums and face amounts that must be avalable to a consumer n a compettve market. However, there are several ways n whch ths set of premums and face amounts can be made avalable to consumers. The constraned maxmzaton problem below s constructed va fully contngent contracts that nvolve no lapsaton. We later descrbe how to obtan the same allocaton wth noncontngent contracts that are qute common n the lfe nsurance market. In a compettve equlbrum, premums and face amounts that are fully contngent on the health state Q 1, F 1, (Q 2 1,F 2 1 ),..., (Q 2 N,F 2 N ) must maxmze consumers expected utlty: (1) pv~f 1! 1 ~1 2 p!u~ y 1 2 g 2 Q 1! N 1 ~1 2 p! O p v~f 2! 1 ~1 2 p!u~ y 2 1 g 2 Q 2!#, 51 subject to the zero pro t condton, N (2) ~1 2 p!q 1 2 pf 1 1 ~1 2 p! O 2 p!q 2 2 p F 2 # and the no lapsaton constrants: for 5 1,..., N, and for all Q 2, F 2 such that (1 2 p )Q 2 2 p F 2. 0, (3) p v~f 2! 1 ~1 2 p!u~ y 2 1 g 2 Q 2! $ p v~f 2! 1 ~1 2 p!u~ y 2 1 g 2 Q 2!.
12 310 QUARTERLY JOURNAL OF ECONOMICS The constrant n equaton (2) requres that nsurance companes break even on average. Ths must hold under competton. The no-lapsaton constrants de ned by equaton (3) are the addtonal constrants mposed by lack of consumer commtment. They requre that, at every state n the second perod, the buyer prefers stayng n the long-term contract rather than swtchng to a competng nsurance company. In other words, an equlbrum contract must be such that there does not exst another contract that s pro table and that offers the buyer hgher utlty n any state of perod 2. We say that a consumer gets full event nsurance n state of date 2 (the de nton for date 1 s analogous) f (4) v9~f 2! 5 u9~ y 2 1 g 2 Q 2!. We call Q 2 (FI) the far premum for full nsurance n state, namely, the premum that guarantees zero pro ts. The followng proposton, whch s proved n the Appendx, provdes a characterzaton of equlbrum contracts. PROPOSITION 1. In the equlbrum set of contracts: () All consumers obtan full event nsurance n perod 1 and n all states of perod 2. () For every g there s an s such that Q 2 5 Q 2 (FI) for 5 1,..., s 2 1, and Q 2 5 Q s 2 for 5 s,..., N, where Q 2, Q 2 (FI) for 5 s 1 1,..., N. () There s a ĝ such that, f g, ĝ, then there s front-loadng. (v) More front-loaded contracts provde more nsurance aganst reclass caton rsk: contracts wth hgher Q 1 nvolve a lower s. Buyers wth hgher g choose contracts wth less front-loadng. Part () follows from the fact that, under competton, event nsurance s offered at a far rate hence, buyers wll choose to fully nsure event rsk. 13 Part () says that there s a premum cap: at health state s or 13. Parts () and () of Proposton 1 are adaptatons of Harrs and Holmstrom  to our envronment. Part () was also shown by De Gardel . Cochrane  studes a smlar problem. He shows that lack of commtment does not preclude full reclass caton rsk nsurance. Hs result s seemngly at odds wth ours, as well as wth Harrs and Holmstrom . However, there s no such nconsstency: under hs assumptons consumers are eager to front-load suf cently to obtan full nsurance. In our context ths means that s 5 1, whch obtans when p 1 s much larger than p and g s small. He only looks at a subcase
13 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 311 worse, the premums are capped at a common prce whch s cheaper than the spot rate for each of those states. In the optmal contract, consumers transfer ncome from the rst perod to the bad states n the second perod. The reason why premums n the bad states are all the same s the followng: gven that the nolapsaton constrants are nonbndng n those states, transferrng resources across these states s costless, leadng to an elmnaton of all varablty n premums at the optmum. If, n contrast, the consumer s healther than n state s, the premums are actuarally far. These are the states where the no-lapsaton constrants are bndng. In those states, buyers must be offered spot market rates. Part () says that contracts are front-loaded so long as ncome does not ncrease too rapdly. Part (v) says that nsurance companes should offer a menu of contracts dfferng n the trade-off between front-loadng and reclass caton rsk. Less front-loaded contracts appeal to buyers wth lower rst-perod ncome,.e., consumers who nd frontloadng more costly; g captures the margnal rate of substtuton between a dollar today and a dollar n the future. Dfferent contracts are offered to cater to consumers wth dfferent wllngnesses to front-load n exchange for dfferent degrees of reclass- caton rsk nsurance. It can be shown that for g # 0, the model predcts that premums are upwardly rgd. Ths s a drect counterpart of the downwardly rgd wages n Harrs and Holmstrom. Thus, the model predcts that the most front-loaded contracts have at premums, whch s exactly what we observe n the data. III. C. Contract Equvalence and Emprcal Implementaton of the Model Proposton 1 obtans equlbrum allocatons va fully contngent contracts that nvolve no lapsaton. However, as we saw n Secton II, both contngent and noncontngent contracts are common n the lfe nsurance market. We now show that noncontngent contracts can also be optmal. We then show how we account for lapsaton, whch s qute common n lfe nsurance, where no tenson between nsurance and consumpton smoothng arses. See also Pauly, Kunreuther, and Hrth .
14 312 QUARTERLY JOURNAL OF ECONOMICS and we dscuss how we compute the present value of premums for contngent contracts. To gan an dea of how equvalence s possble, note that a buyer who chooses an LT20 or an aggregate ART (whch are noncontngent) s free to drop out of the contract at any date to (for nstance) purchase an S&U ART. More formally, x an equlbrum contngent contract C. A noncontngent contract NC s also an equlbrum contract f, n each state, the consumer for whom the C contract s optmal can obtan the same utlty wth the NC contract and the nsurance companes offerng the contracts obtan the same pro ts wth ether contract. Clearly, ths requres that the consumer drops out of NC n the approprate states. From part () of Proposton 1, the terms of a C contract are constant: Q 2 5 Q s 2 and F 2 5 F s 2 for 5 s,..., N. In states 5 1,..., s 2 1, the premums and face amounts equal those offered n the spot market: the buyer pays the actuarally far premum. Fx the terms of the noncontngent contract NC to be the same as for contract C n the rst perod, and (Q s 2,F s 2 ) n the second perod. Thus, the two contracts are equvalent by constructon n the rst perod and n the bad states of the second perod. However, n states 5 1,..., s 2 1 contract C has better terms. But n those states the consumer can obtan the same terms by droppng out of contract NC and purchasng spot contracts. Because, by Proposton 1, n each state 5 1,..., s 2 1, contract C has the same terms as a spot contract, the consumer obtans the same utlty va NC and C. Insurers are ndfferent between sellng the two knds of contracts as well: n state s through N consumers stay n the contract under both contracts, and the terms are exactly the same. In states 1 to s 2 1, the two alternatve contracts nduce dfferent behavor by the consumer, but equal (zero) pro ts for the rms: n those states premums n contract C are actuarally far. Thus, n those states, rms are ndfferent between retanng and not retanng the consumers. Ths shows that the two ways of achevng the equlbrum allocaton are equvalent. Ths equvalence result wll help us n two ways: rst, by explanng wthn our model the exstence of noncontngent contracts n the lfe nsurance market. Second, the result helps us compare contngent and noncontngent contracts. All we have to do s transform each contngent contract nto ts equvalent noncontngent contract, along the lnes of the prevous dscusson.
15 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 313 The mportance of havng comparable noncontngent contracts comes from the next proposton, whch s at the core of our emprcal analyss. It presents the man comparatve statcs across noncontngent contracts. Due to the equvalence result we wll be able to apply t to all contracts. PROPOSITION 2. Consder two contracts offered n the rst perod that are not contngent on the health state n perod 2. In equlbrum, the contract wth the hgher rst-perod premum has a lower present value of premums, t s chosen by consumers wth lower ncome growth, and has lower lapsaton (retans a healther pool of consumers). Proof. The contract wth the hgher rst-perod premum must have a lower second-perod premum snce otherwse no consumer would choose t. Thus, n the second perod ths contract wll retan a healther pool of consumers. Ths mples that the average cost of ths contract (payments to dependents of deceased polcy holders) s lower. Under competton ths mples that the present value of the premums must also be lower. The fact that the contract wth the hgher rst-perod premum s chosen by consumers wth lower ncome growth follows from Proposton 1. The mportance of Proposton 2 s that t provdes a way to test the theory by comparng noncontngent contracts that dsplay dfferent degrees of front-loadng. The equvalence result dscussed above provdes a way to extend the comparson to the entre set of contracts by tellng us how to evaluate contngent contracts. Whenever we compute the present value of premums for a contngent contract, we do so by lookng only at the nonrequalfyng states: n the termnology of the model we consder the porton of the contract (Q 1,F 1 ), (Q 2 s,f 2 s ), where s s the state obtaned n Proposton 1. The resultng contract s exactly the noncontngent contract that mplements the same allocaton (as dscussed before n Proposton 2). An example: for the S&U ART contract descrbed n Table I, ths means computng the present value of the rst row of the matrx. Note that, absent the equvalence result, we would not be able to compute the present value of premums snce we do not have data on requalfyng probabltes; these probabltes depend on the underwrtng strngency (the state s n terms of our model) for ndvdual contracts. The
16 314 QUARTERLY JOURNAL OF ECONOMICS model allows us to compare contracts even n the absence of such nformaton. To be consstent wth the theory, n the emprcal analyss of contngent contracts, we should consder requal caton to be a lapsaton,.e., when a consumer requal es, we consder hm to have started a new contract. Ths s consstent wth ndustry practce, whch treats requal caton as a new contract, and t s the theoretcally correct way to compare contracts. Of course, n our emprcal analyss we make sure that our ndngs are not an artfact of our de ntons: we also test the predctons by lookng only across contracts that are drectly comparable. Summary of predctons. In the next secton we look at the tme pro le of premums of the varous polces to test the mplcatons of the model. The model predcts the followng. 1. Optmal contracts nvolve front-loadng. 2. Contracts wth hgher frontloadng retan hgher proportons of nsureds n the long run,.e., have lower lapsaton. 3. Snce better rsk types have hgher ncentve to lapse any gven contract, hgher lapsaton mples worse pools. 4. A worse pool (lower front-loadng) translates nto hgher present value of premums for a xed perod of coverage. IV. TESTING THE IMPLICATIONS OF THE MODEL To test the model, we use the varety of offered contracts. The dfferent types of contracts are characterzed by dfferent premum pro les, or slopes (namely, steepness of premum ncreases over tme). The slope of the premums determnes the extent of front-loadng. The steeper the premums, the less front-loaded the contract. We want to establsh rst the extent of front-loadng of contracts offered, and then the relaton between front-loadng and lapsaton, as well as wth the qualty of the nsured pool. IV. A. Front-loadng Table III shows the yearly premums pad, over twenty years, by a 40 year old purchasng half a mllon dollars of coverage under three dfferent contracts. The numbers are averages over all contracts of each type n our sample. Column 1 s for aggregate ARTs. Column 3 s for twenty-year level term contracts (LT20). Columns 5 and 7 descrbe two extreme scenaros for S&U ARTs (recall from Table I that the complete descrpton of ths type of contract s a matrx): n column 5 the buyer never sats ed the
17 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 315 TABLE III PREMIUM PROFILES AND PREMIUM TO MORTALITY RATIOS S&U ART Aggregate ART LT20 NoRequal Requal Age AvgPrm Prm/Death AvgPrm Prm/Death AvgPrm Prm/Death AvgPrm , , , , , , , , , , , , , , , , ,583 AvgPrm 5 average premum of all polces of a spec c type n our sample pad at dfferent ages by a consumer who buys (qual es for) nsurance at age 40. Prm/Death 5 average premums dvded by probablty of death at each age. We use the 1985/90 select and ultmate actuaral table by the Socety of Actuares. Select and Ultmate tables re ect mortalty experence for nsured populaton who passed a medcal. We are usng death probabltes of a person passng the medcal at age 40. Prm/Death s rescaled to 100 at age 40. NoRequal 5 premum pro le of an S&U ART buyer who faled to requalfy (show he s n good health) as good rsk n later years. Requal 5 premum pro le of a buyer who requal ed (passng a medcal) year after year. crteron for requalfyng; n column 7 he was lucky and requal- ed year after year. Observe the substantal dfference n payment pro les across contracts. By de nton, premums n LT20s are constant n nomnal terms. Due to dscountng and agng, the level term contract presents a hgh degree of front-loadng. The rst payment n the twenty-year polces s 83 percent hgher than n the yearly select contracts; ths gap represents a lower bound on the magntude of front-loadng. The ntal overpayment creates consumer lock-n or commtment to the contract. When the nsured reaches age 50, he cannot be lured by a S&U ART, snce the remanng premums are lower n LT20 contracts than the premums he would pay n S&U ARTs (even f he were to keep requalfyng as a select customer). We now argue that even ARTs are front-loaded. To check ths, n Table III we report the rato of the yearly premum over the probablty of death; the rato was normalzed to equal 100 at
18 316 QUARTERLY JOURNAL OF ECONOMICS age Ths rato should be constant absent front-loadng. In contrast, the rato declnes over tme. For example, for a 40 year old and an aggregate ART polcy the rato halves wthn four years, and t keeps declnng to below a thrd of ts ntal level. Hence, contracts do not break even perod by perod. More surprsngly, even S&U ARTs, obvous canddates for nvolvng no long-term nsurance, nvolve some degree of frontloadng. Column 6 con rms ths. Absent front-loadng, the rato (column 6) should ncrease over tme, snce the denomnator re ects the entre populaton whle the numerator re ects the worsenng pool that fals to requalfy year after year. In contrast, the rato mldly declnes. Note that on average the rst year premum of aggregate ARTs s 36 percent hgher than that of select contracts despte the fact that both are yearly contracts. Ths re ects the fact that n an aggregate contract the nsurance company s offerng a lower cap on future premums n the event that the consumer s n poor health. The aggregate contract offers more nsurance of reclass- caton rsk, at the cost of hgher ntal payments. To summarze, all the avalable contracts nvolve front-loadng, as suggested by the theory. Let us contrast ths ndng wth the alternatve assumptons on commtment. If consumers could commt, front-loadng would not be necessary to acheve the ef cent allocaton. If front-loadng s costly, we would not observe t. At the other extreme, suppose that nsurance companes cannot commt to future terms of the contract. Then front-loadng would not provde any nsurance aganst reclass caton rsk. Thus, n nether of these scenaros would we observe front-loadng. IV. B. The Negatve Relaton between Front-Loadng and the Present Value of Premums We saw n the prevous subsecton that contracts dffer n the extent of front-loadng. We now show that there s a systematc 14. The death probabltes used for Table III are taken from select and ultmate tables compled by the Indvdual Lfe Insurance Experence Commttee . Select and ultmate tables re ect the mortalty experence of nsured ndvduals, namely, those who passed a medcal examnaton, as a functon of both age and tme elapsed snce they passed the medcal. Reported numbers understate the extent of front-loadng snce most contracts do not retan the best draws. One caveat concerns the trend of decreased mortalty. The mortalty tables refer to past populatons, the contracts may thus re ect an expected declne n mortalty. However, the declnes n the ratos that we observe n Table III are too bg to be explaned by ths.
19 THE ROLE OF COMMITMENT IN DYNAMIC CONTRACTS 317 TABLE IV CONTRACT SLOPES AND COST OF COVERAGE Mean Std dev Mn Max Q(1st)/Q(11th) PV 16,055 5,245 6,871 28,754 ln(pv) Q(1st)/Q(11th) 5 s the rato of the rst to the eleventh premum; we use t to capture the slope of premums. PV 5 present value of the cost of twenty years of coverage at r relaton between present value of premums and the extent of front-loadng along the lnes of Proposton 2. We look at the relaton between the slope of the premums and the present value of premums over twenty years of coverage. 15 To do ths, we proxy the slope of the premum pro le by the rato of the rst over the eleventh premum, Q(1st)/Q(11th). Table IV shows basc statstcs for contracts offered to 40 year old male nonsmokers who qualfy as preferred rsks. Note the wde range of premum slopes and present values. A hgher Q(1st)/Q(11th) rato means that more s pad upfront (there s more front-loadng). The predcton of the model s that contracts wth a hgher Q(1st)/Q(11th) should have lower present value of premums. 16 We now nvestgate whether the relaton between slopes and present values s ndeed negatve n the data. We also ask how much of the varablty n present values can be explaned by contract slopes. The rst column of Table V presents the regresson of the log of the present value of the premums on the Q(1st)/Q(11th) rato and other contract characterstcs. The front-loadng varable s hghly sgn cant; and t explans most of the varablty n the present value of premums. Excludng ths varable from the regresson drops the R 2 from 74.4 percent to 16.6 percent (column 15. Recall that the present value s calculated condtonal on the consumer not requalfyng. As argued n subsecton III.C, ths s the theoretcally meanngful measure. 16. Recall from subsecton II.B that a contract wth a lower present value of premums does not domnate a contract wth a hgher one. Consumers who choose a contract wth a low Q(1st)/Q(11th) rato (hence wth a lower present value of premums) wll drop coverage n the future f they reman healthy, snce n ths case they obtan lower rates by ntatng a new contract. The theory says that the pool of consumers who stays wth a contract n the long run s not the same for contracts wth dfferent slopes. The evdence presented n subsecton IV.C s consstent wth ths.
20 318 QUARTERLY JOURNAL OF ECONOMICS TABLE V REGRESSION: PRESENT VALUES ON SLOPE OF PREMIUMS log(pv) (1) (2) (3) (4) (5) Q(1st)/Q(11th) (216.79) (28.77) (24.84) (22.84) Guarant (2.95) (23.96) (1.74) (1.03) (1.33) Renew (21.22) (21.18) (21.00) (20.01) (0.39) Convert (3.19) (2.85) (3.41) (2.73) (1.56) Spec Cond (3.22) (0.97) (3.14) (1.83) (3.28) Constant (62.1) (33.3) (53.5) (36.7) (29.7) R N Dependent Varable log(pv) s the log of the present value (r 5 8%) of the cost to the consumer of twenty years of coverage startng at age 40. Convert 5 age untl polcy can be converted to another. Guarant 5 years premums are guaranteed. Renew 5 last age the polcy can be renewed. Spec Cond 5 specal underwrtng condtons, lke nonpreferred rsk. Column (1) ncludes all the contracts n the sample. Column (2) excludes contracts slope as an explanatory varable. Column (3) ncludes all contracts but LT20s n the sample. Column (4) ncludes only LT5s and LT10s. Column (5) ncludes only noncontngent contracts. 2). The effect s economcally sgn cant as well, one standard devaton ncrease n Q(1st)/Q(11th) translates nto a 28 percent declne n the cost of coverage. The thrd column presents a smlar regresson after omttng the LT20s from the sample. We do so to check whether these contracts, whch have no varablty n Q(1st)/Q(11th), were responsble for the negatve relaton between premums and slope. The negatve relaton s stll strong among the rest of the contracts as well. Columns 4 repeats the exercse for a sample of veand ten-year contracts. Column 5 ncludes only the noncontngent contracts n the sample. The purpose of ths column s to test the robustness of the results wthn noncontngent contracts. Recall that to compare contngent contracts we have to appeal to the equvalence result n subsecton III.C. One mght thus wonder about the robustness of these results, once we only compare contracts that do not requre such a theoretcal nterpretaton. By comparng noncontngent contracts, we test Proposton 2 drectly, wthout relyng on the equvalence result. The negatve relaton s con rmed.