Life Insurance. Bertrand Villeneuve, Handbook of Insurance chapter 27 G. Dionne (ed.), Kluwer, 2000.

Life Insurance Bertrand Villeneuve, Handbook of Insurance chapter 27 G. Dionne (ed.), Kluwer, 2000. Abstract This survey reviews the micro-economic foundations of the analysis of life insurance markets. The first part outlines a simple theory of insurance needs based on the life-cycle hypothesis. The second part builds on contract theory to expose the main issues in life insurance design within a unified framework. We investigate how much flexibility is desirable. Flexibility is needed to accommodate changing tastes and objectives, but it also gives way to opportunistic behaviors from the part of the insurers and the insured. Many typical features of actual life insurance contracts can be considered the equilibrium outcome of this trade-off. JEL Classification Numbers: G220, D910, D820. Institut D Economie Industrielle (IDEI) and Commissariat à l Energie Atomique (CEA) at the University of Toulouse. Many thanks to Helmuth Cremer, Georges Dionne, Jeff Myron and the anonymous referees. All errors and imprecisions are my responsibility. 1

Contents I Overview 3 II Possibilities and needs in life insurance 5 1 Life insurance possibilities 5 1.1 Theproductionset... 6 1.2 Typicallifeinsurancecontracts... 8 1.3 IndicesandRatesofreturn... 9 2 Life insurance needs 11 2.1 Thelife-cyclehypothesis... 11 2.2 Constraints... 14 2.3 Portfoliochoice... 18 2.4 Lifeinsuranceandsocialsecurity... 21 III A contract theory of life insurance 21 3 Incompleteness 22 3.1 Limitedcommitment... 22 3.2 Incompletecontracts... 23 4 Asymmetric information 24 4.1 Asymmetricinformationbeforesigning... 24 4.2 Asymmetricinformationappearingovertime... 30 5 Conclusion 31 A Appendix 37 2

Part I Overview Life insurance serves to guarantee a periodic revenue or a capital to dependents of the policyholder (the spouse, or the children, sometimes the parents or any other person) in case of his death, or to himself, in case he survives. Life insurance economics is undoubtedly a question of applied theory and most useful ideas originated in other fields: savings theory or contract theory flourished well before their interests for insurance were perceived. Rather than trying to be complete and fair with respect to the valuable studies in saving theory, contract theory, the economics of the family, and standard insurance theory, we cite essentially papers that have reinterpreted these ideas and applied them to life insurance particularities. We will not always follow this line, especially when certain such transfers have not yet been effected. This survey will therefore give a personal view of the state of the art and will suggest certain extensions that remain to be formalized. We start by providing in section 1 a description of insurance supply or insurance possibilities. The theory of contingent claims has improved the understanding of life insurance contracts as bundles of elementary assets whose costs for the insurer are rather easy to measure. With this actuarial view, we come up with a production set which will serve as a basis for further investigation. We want to build a consistent theory of life insurance needs. Needs are determined of course by the policyholder s tastes and the stage of the lifecycle that is considered, but also by his economic conditions, the structure of his family, etc. In section 2, we discuss the factors affecting the portfolio choice between ordinary savings, life insurance, and life annuities in an ideal financial environment. We give some indications on the so-called bequest motive, which is often a blackbox in insurance and saving models. This section being more formalized than the others, the reader may want to skip certain technicalities. In the last part (sections 3 4), this survey provides ultimately a basis for a theory of life insurance contracts. Markets do not work as perfectly as suggested by the theoretical benchmark described in the first two sections. In practice the contracts offered to the consumers are limited to a few typical structures; explaining these features and the stability of this selection is the role assigned to the economic theory of insurance. The main limitations to the implementation of first-best contracts are the parties inability to commit, asymmetric information before signing, and asymmetric infor- 3

mation emerging during the life of the contract. The existence of options (extension of coverage, renewability, surrender values, etc.) needs a particular and thorough treatment. In any case, an understanding of each party s objective is an imperative condition for characterizing incentive compatible contracts: indeed, the actuaries must be aware of self-selection effects (i.e. actuarial non-neutrality) in choices between the offered options. We have to clarify and model under which circumstances they would be exercised. Starting from the fact that life insurance contracts are incomplete, we propose in section 3 some clarification of the reasons for the existence of options in contracts. This latter fact appears to be linked to renegotiation possibilities that kill inter-temporal insurance to some extent, and to the fact that essential information (shocks in tastes to be short) may not be observable by both parties, which explains why a degree of discretion at some points is desirable. Section 4 discusses the importance of adverse selection (and moral hazard) in life insurance markets. These markets are interesting in two respects: the first is that there exist relatively close substitutes to life insurance, which is only part of a balanced saving portfolio; the second is that it is almost impossible to ensure exclusivity, policyholders being typically able to secretly hold as many contracts as they want. The modeling of markets and the power of public regulation are deeply affected by these particularities. The conclusion in section 5 gives a series of modest reflections on the value of theory for designing life insurance contracts. Two important limitations of this study must be mentioned. The first one concerns the literature on investment policy, a topic that has not been related to well-structured insurance demand models. Some intuition on the effect of financial uncertainty on saving strategies (precautionary saving, risk premia on securities, structure of the portfolio, etc.) may be found in several other contributions to this handbook. Though strictly speaking, our study cannot be orthogonal to these concerns, we think that a complete model of the effects of risk to life has to be built first in a simpler framework. The second limitation concerns the effect of taxes on insurance demand. At first sight, taxes simply distort prices of the contingent claims that insurance contracts bundle. This simple picture is rarely valid. In general, the tax system gathers non-linear benefits and penalties. Insurance supply is also affected by the efforts of actuaries to find and sell fiscal niches. Moreover, a serious analysis of taxes on life insurance would require a clear notion of the aim of the public authority. This last requirement is the most disappointing. A mere description of actual practices is definitely not within the scope of this survey. 4

Part II Possibilities and needs in life insurance 1 Life insurance possibilities The purpose of this section is to present a simple description of the technical and financial constraints that are imposed on life insurance contracts. Though we acknowledge that probabilities are tightly connected to statistical observations, probabilities are seen in the sequel primarily as a measure of information, notably because this modern view will enable us to explore the evolution of information over time. The minimal requirement is that insurance contracts must be measurable, at each date, with respect to the available information. This preliminary remark makes sense for three reasons. The firstreasonisthatinfinancial markets, there is an almost continuous flow of information and the funds invested by the insurance company are managed so as to accommodate with maximal foresight the movements of the rates of return of the various possible assets. This aspect of insurance contracts is particularly worthy of mention for long term arrangements where benefits are typically somehow linked to financial performance. The second reason is that time allows for some learning of policyholders abilities and preferences. Often for the best: the contract will take into account essential changes in the policyholder s objective. But, even in the case of symmetric evolution of knowledge, if commitments on both parts are not total, the contractual relationship may be disrupted in certain contingencies, e.g. if the policyholder proves too risky. Though legal restrictions moderate this threat, there exists serious obstacles to the sustainability of most desirable long term contracts. The third reason that makes a powerful information structure indispensable is that in the standard modelling, informational asymmetries are not due to someone being wrong, but rather on different precisions in the information possessed by the parties. Modeling how parties interpret each other s actions requires a well-suited formal setting. For example, a question that can be addressed with this methodology is the effect of prohibiting the use for contract design of certain pieces of information (anti-discriminatory laws) in spite of their objective relevance. 5

1.1 The production set Actuarial approach A life insurance contract is a financial agreement between an insurer and a policyholder, signed at a date t 0, specifying monetary transfers at certain dates {t 0 + d,...,t 0 + d + m} where d is a delay and m the maximal duration (0 d<+ and 0 m). We work with discrete time throughout the paper. The flows either go from the policyholder to the insurerortheotherwayaround. Wedenotebyp i t(s) apaymentatdatet from the insurer to beneficiary i (i =1 n), conditionally upon the arrival of state of the world s I t where I t istheinformationsetatdatet (an element of I t contains all the available information, and {I t } t t0 is a filtration to capture the fact that information is more and more accurate). A negative p i t(s) is interpreted as a premium or contribution, a positive p i t(s) as an indemnity or benefit. The two substantial elements in this definitionarethatpaymentsare contingent on a potentially very rich algebra of events, and that they are assigned to named persons (the beneficiaries). In practice, payments contingent upon survival or death can be explicitly specified quite simply in contracts, nevertheless, all contingencies are not listed in details, or are used in a crude manner: for example, financial performanceisoftenutilizedundertheformofsomesimplesharingrule. Non-anonymity is the major difference with purely financial assets. For obvious moral hazard reasons, there is a legal prohibition on betting on other people s lives. This is not a neutral limitation: if, for example, your income is highly dependent on the survival of your associate, you cannot hedge against that eventuality without his agreement. In other words he has to agree to purchase life insurance with you as beneficiary, possibly in exchange of some compensation. The production set is defined by the following standard economic principle: expected profits must be positive. Formally: Ã! t=tx 0 +d+m X Xi=n a t (s) p i t(s) 0 (1) s I t i=1 t=t 0 where a t (s) represents the cost (at date t 0 ) ofoneunitatdatet in state s. In an economy with complete markets, the a t (s) represent the prices of the Arrow Debreu assets; otherwise, they represent marginal value for shareholders and may contain the shadow cost of liquidity constraints or reserve regulation. In any case, discount factors (interest rates, probabilities, risk premia, etc.) are embodied in the a t (s). To simplify, I t can be structured as F t M t, where F t represents states of financial markets and M t is a list 6

ofindicatorsofwhoisaliveandwhoisdead. Ifoneassumesthatmortality is independent of interest rates, then for all f,f 0 F t and for all m, m 0 M t :Pr{f m} =Pr{f m 0 } and Pr{m f} =Pr{m f 0 }. We can write for all s =(f,m) :a t (s) =ϕ t (f) µ t (m). The aim of this last factorization is to show that insurers technical ability, summarized by the a t (s), stems from two independent expertises: actuarial estimates ϕ t, and asset management µ t ; these two dimensions are of course complementary for assessing the global performance of a given insurer. The reader should retain for the moment that (essentially) the technical dimension of insurance boils down to a single constraint. The determination of the optimal contracts under this constraint requires of course also a good understanding of policyholders tastes. To illustrate the non-triviality of the actuarial dimension, one should keep in mind that life expectancy has increased steadily in developed countries during the last fifty years. Insurers have to extrapolate somehow the past trend when using mortality tables, since actual mortality tables are not applicable directly to younger customers. Mullin and Philipson (1997) developed methods to estimate the mortality rates implicit in competitive prices of life insurance policies, in other words, the anticipation of the market on the evolution of longevity for the current generations. They claimed that the increase of longevity is expected (by insurers) to follow at least the same pace as observed recently. Incentive compatibility In principle, in a world of symmetric verifiable information and full rationality, decisions nodes are useless in contracts since the optimal plan was completely specified ex ante and continuations are mechanically determined by the observation of the state of the world. Leaving aside for methodological reasons bounded rationality problems, there are two essential assumptions behind this view of contracts. The first is that parties are committed to the complete implementation of the contract. The second is that no information relevant to the optimal continuation can emerge asymmetrically during the life of the contract (not to speak of asymmetries at the time the contract is signed). Under a more realistic view, it may be optimal, under identified informational constraints, to leave the policyholder choose an option at certain dates, his choice being determined by his current interest. Accordingly, we have to add decisions by agents in the definition of the states. Typically, insurance policies contain renewal options, without medical examination, for a limited number of additional periods; they also specify surrender values, that is, the money the policyholder can get if he dismisses 7

the contract. Another common option, though not always seen as such, is due to the legal requirement that if the insured stops paying his contributions, the insurance company can only reduce the benefits in proportion to the missing contributions, the contract being totally kept in force. The difficulty now is that it becomes indispensable to ensure consistency of the contracts in the sense that probabilities put on the decision tree have to be compatible with actual behavior of the party taking decisions. Now we are leaving the comfortable realm of purely statistical evaluation of contracts: technical ability cannot be disentangled from the ability to understand behavior. The exact nature of the restrictions imposed by incentive compatibility will be explored in the third part of this survey. Taxes We will not deal with the important question of tax rules applied to life insurance. However, the reader must keep in mind that these rules are an important determinant of life insurance yields. We just mention the fact that the tax system in this matter is often intended to give incentives to financing old-age incomes (typically, contributions are deductible from the taxable income), while trying to ensure that they are not used for other purposes (by putting penalties on premature withdrawals). Our choice is to give an extensive pure theory of life insurance, i.e. to offer a theory of needs in life insurance, a theory of production, and a theory of the impact of asymmetric information. Though in practice, taxes do not have a marginal effects in life insurance (it is even often stated that most of life insurance demand is tax-driven), we think that taxes are of secondary importance for understanding life insurance. Once the theory is clear, the effect of taxes becomes a relatively easy problem, conceptually at least. It should be mentioned in passing that the rational foundations of the fiscal doctrine in life insurance has not been seriously studied by public economists. 1.2 Typical life insurance contracts We give indications of the principal characteristics encountered in practice. Basically, life insurance contracts serve to guarantee a revenue to dependents of the policyholder (the spouse, or the children, sometimes the parents or any other person) in case of his death, or to himself, in case he survives. The benefitsmaydependonwhoisaliveinthehouseholdinapotentially sophisticated way. In the following, we shall insist on survival/death of the policyholder and beneficiaries in the definition of a state of the world. Depending on how these states are utilized, we can outline the broad categories of insurance 8

contracts. Here we follow (approximately) the classification and definitions proposed by Huebner and Black (1976). Life insurance A term policy in life insurance is a contract that furnishes life insurance protection for a limited number of years (m is typically 5 20 years), payments to beneficiaries being effected only if death occurs during the stipulated term, and nothing being paid in case of survival. Instead of specifying a duration of coverage, whole-life insurance contracts provide payment in case of death to the beneficiary whenever it happens (m =+...). Life annuities A life annuity may be defined as a periodic payment made during the duration of a designated life. A life annuity may be either whole or temporary (the payments contingent upon survival being then terminated after a fixed period). Typically, pensions are annuities. Endowment insurance Endowment insurance provides the payment of the face value of the policy upon the death of the insured during the fixed term of years, and also the payment of the full face value at the end of the term if the insured is living. We recognize a sort of mix of term life insurance (the first part) and a term (with a single payment) annuity. Miscellany Contracts where benefits in case of death of the insured are annuities for the beneficiary are common: pension benefits for widows, minimum income until adulthood or until a child s college graduation, settlement for a handicapped child. 1 Disability insurance can be linked to life insurance for the reason that the breadwinner needs in fact coverage against permanent income losses, not against death per se. 2 In practice, contracts have a finite duration since they are conditioned upon a finite number of lives. There maybeadelay(d>0) between the signature of the contract and the first transfer. 1.3 Indices and Rates of return Market conditions and macroeconomic factors play a role in the evolution of contributions and benefits over time. Using a correction for (anticipated or random) inflation is a way of securing stable purchasing power for, e.g. a life annuity. The beneficiary may prefer a variable payment, adjusted for 1 See, e.g., Gustavson and Trieschmann (1988). 2 See Cox, Gustavson and Stam (1991) for empirical evidence on demand of these insurances. 9

the financial performances of his fund, notablyifheisrelativelylittlerisk averse so as to prefer to bear some residual risk in exchange for a share of possible high gains in financial markets; he may prefer a less risky (but less profitable on average) agreement. These sorts of arrangements are known as variable payments. What index is used, and how payments are index-linked, are contractual agreements. Several papers calculate the rate of return implicit in life insurance and life annuities. These studies intend to isolate loadings due to commissions and administrative costs, corrections due to adverse selection, and financial performance corrected for taxes. 3 The calculated financial return can be compared to the returns of other types of assets. Babbel (1985) proposed a simple index of life insurance costs (the consumer s viewpoint is taken; costs there are the money paid above the actuarial benefits). His estimates suggest that consumers are sensitive to costs, and tend to diminish their purchases when they increase, which is, as Babbel claimed, a point in favor of economic theory and against the popular view among salesmen that life insurance is sold, not bought. Winter (1982) discussed the theoretical possibility of an index (a single number) facilitating the comparisons between life insurance policies for heterogenous consumers. Though he proposed a reasonable solution to that problem, he also made clear why the quest for an indisputable index is hopeless. The notion of rate of return makesnoexceptiontohiscritique: an index based only on the a t (s) see our definition of contracts may be right for assessing the purely technical ability of the insurers. However, the allocation of benefits across contingencies (the payments p i t(s)), however crucial they might be for policyholders, would not be captured. 4 Despite these caveats, simplifying computations are useful. Warshawsky (1985), for example, defended the idea that the decline in life insurance savings from the mid 1950s to 1981 (life insurance has boomed since that paper was written) is largely imputable to the lower rate of return on the investment part of cash-value policies. Obviously, the complex structure of these contracts makes this assertion relatively delicate to establish, but Warshawsky subjected his calculations to a sensitivity analysis by screening a large set of plausible scenarios. Warshawsky (1988) in his study of annuity markets in the United States over 1919 1984 estimated that the loading 3 In this section, the reader must be aware that our selection of papers is extremely short. The papers retained here are chosen because they associate an economic reflection on the methodology to the calculations. Purely actuarial studies (published or unpublished) on similar issues abound. 4 Using a bounded rationality approach, Puelz (1991) proposes a practical strategy for selecting a life insurance policy. 10

factor ranged from 10 cents to 29 cents per dollar of actuarial present value. 5 The major cause of the evolution would be the tendency on the part of the insurers to use assets whose yields are significantly lower than that of the reference portfolio (namely, U.S. government bonds). The aggravation of adverse selection also seems to have a non-negligible impact on the loading factor. Mitchell et al. (1997) defend the view that costs have declined. The period covered by their study includes more recent years. 2 Life insurance needs The aim of this section is to provide a relatively simple theory of needs in life insurance, i.e. demand in an ideal world where markets would be complete and competitive. The model is compatible with most views and formal studies of life insurance demand. We adopt this terminology (needs) to grasp the multidimensional aspect of life insurance contracts that demand would not suggest. To start with, we offer an analysis of the life-cycle theory and of the so-called bequest motive. 2.1 The life-cycle hypothesis Suppose the individual knew the date of his death. The allocation of his wealth over time may be assumed to derive from the maximization of the following inter-temporal objective (the Fisherian model after Fisher (1930), to retain Yaari s (1965) terminology): 6 Xt=T U t (c t )+V T +1 (b T +1 ) (2) t=1 where U t ( ) is the period t felicity derived from current consumption c t, and V T +1 (b T +1 ) is the value of bequest b T +1 left at date T +1. 7 We retain a discrete approach mainly because it facilitates the introduction of imperfect markets and the analysis of long term contracts. The drawback is that calculations in the simplest cases become less compact than with continuous time modeling. 8 5 See also Poterba (1997). 6 See also Fischer (1973) or Karni and Zilcha (1986). 7 We could also enrich the model by giving value to inter-vivos transfers at other dates. 8 For examples of this last category, see, e.g., Yaari (1965) and Pissarides (1980) where perfect markets are assumed. 11

When the horizon is random, one can assume that the individual maximizes expected utility with respect to the distribution of T, T being the upper limit of the support Xt=T E T { U t (c t )+V T +1 (b T +1 )}. (3) t=1 The main restriction embodied in this objective function is the additive separability over time and states of the world: the marginal rate of substitution between two consumptions is independent of the other consumptions. Still, this formulation allows for time dependent utilities: it is consistent with the frequent assumption that future utility is discounted, and with an evolution of risk aversion over time. Rearranging we get Xt=T {q t U t (c t )+(q t 1 q t )V t (b t )} + q T V T +1 (b T +1 ) (4) t=1 where q t denotes the probability of living at least until period t; in particular q t >q t+1,q 0 =1and q T +1 =0, and the mortality rate at the end of period t is 1 q t+1 q t. Compared to the certainty case, future consumption is further discounted by the survival probability; moreover, in all periods where death is probable, the bequest has a value. In the objective above, the value of bequests is not built on primitives, and a rationale for particular specifications or properties is rarely even mentioned in studies interested in saving-consumption choice. Still a literature has developed an analytical description of the bequest motive, notably in view of deriving testable implications of the theory. The first point is that we know little about the specification of V t ( ) as compared to U t ( ), and about how it should evolve period after period. Life insurance being after all only a financial tool for controlling inter-personal transfers, references to the theory of transfers (bequest, gifts, inter-vivos transfers) are necessary. The reader interested in this literature could for example refer to Bernheim, Shleifer and Summers (1985), Hurd (1987, 1989) and Ando, Guiso and Terlizzese (1993), the latter providing a Probit estimation of the determinants of life insurance demand. Abel and Warshawsky (1988) presented a useful discussion and implementation of how bequest motives could be specified and calculated, starting from simple principles. Lewis (1989) extended Yaari s model by exploring explicitly how the bequest motive should be formed when it is intended to take into account the direct utilities of the dependents to be protected. In particular, he calculated theoretically and tested empirically the impact of the number of beneficiaries on life insurance demand. To this end, he modelled the way beneficiaries 12

respond to the protection they receive. In some cases, their incomes may be sufficiently high relatively to that of the potential policyholder for life insurance to be unnecessary. 9 Fitzgerald (1987) and Bernheim (1991) used information on the levels of pensions to explore the effects of social security (treated as exogenous) on life insurance demand. Given that insurance is crowded out on the one hand, and that increased pensions increase actual wealth on the other hand, the net effect is ambiguous a priori. Fitzgerald s data confirm that the impact of marginal pension differs according to who is the principal beneficiary (husband or wife) of the supplement. Bernheim s project is more focussed on the estimation of bequest motives. His estimates support the view that the differences in insurance purchases (whether people buy life insurance or annuities, or neither) are significantly determined by the differences in the generosity of the pension benefits: the better the pensions, the larger the bequests; the lower the pensions, the larger the propensity to cover oneself with private annuities. Auerbach and Kotlikoff (1986, 1991) questioned whether women are wellcovered by the life-insurance plans of their husbands. The normative standpoint is that a sound protection should allocate savings and insurance in view of maximizing a weighted sum of the spouses utilities, taking into account their survival probabilities. The 1986 paper examined the case of the elderly and the 1991 one explored and found confirmation of the inadequacy of insurance coverage of younger households, who, given that a large part of their lifetime resources is tied up in human wealth, were supposedly more in need of protection. Fitzgerald (1989) proposed, with the help of a structural econometric model, to study the dependency on age of the relative (expected or actual) economic well-being of widows according to whether the husband lives or not. He suggested that, because economies of scales in households are not very large, the standard-of-living falls after death of the husband are not as dramatic as previously reported, and is even contradicted in certain groups. Most authors find convenient to let the market be the only institution where insurance is available, and to assume that there is a single decisionmaker (or at least a leader) involved in purchase decisions. Nevertheless a few papers have scrutinized the insurance demand of households in imperfect contexts. Despite their interest, they have unfortunately not yet given rise to econometric applications. Kotlikoff and Spivak (1981) viewed the family as an institution able to replace inefficient annuity markets. They assumed that markets are incomplete: 9 With our notations, it amounts to say that the marginal utility V 0 t (0) may be low. 13

there exist no life insurance or annuity markets; members of the household can only save. In a single agent household, savings would be lost in case of death in the sense that they provide no utility to the decision-maker; here, savings are mutually bequeathed, and they are lost only in case of simultaneous deaths, an event of relatively low probability. Kotlikoff and Spivak proved that this risk sharing arrangement over few household members is sufficient to approximate first-best allocations very closely. Browning (1994) insisted on the strategic issue. The difference here with standard individualistic models is that each household member has access to life insurance markets (in fact life annuities) and savings, but they act noncooperatively. The source of inefficiency is that consumption being in the model a purely public good, each household member free-rides the other s savings. The consequence is that one member only (typically the husband) will subscribe life annuities. Though indisputably a caricature, this alternative approach (as compared to the single decision-maker tradition) is original and deserves attention and development. Empirical studies like Arrondel and Masson (1994), Sachko Gandolfi and Miners (1996) or Goldsmith (1983) document the determinants of household demand. Typically, wealth, income, the number of children have significant positive effects. Arrondel and Masson also show that professions where human wealth (expected future income from labor) is relatively large are more likely to demand life insurance. 2.2 Constraints An image of financial markets variables: Let us define additional control and state A t and L t : the face value (i.e. the benefit) of, respectively, short term annuities and short term life insurance, to be paid to the beneficiary at date t in the corresponding contingency; Rt A and Rt L : the gross rate of return of, respectively, annuities and life insurance received at date t; contributions A t and L Rt A t arepaidatdatet 1; Rt L taxes are ignored throughout, but may be included in the rates of return; R t : the gross rate of return on saving Wt A and Wt L : the disposable wealth of the agent at the beginning of period t if he is alive (respectively if he is just dead); Y t : the exogenous income (wage or pension) which is conditional on the consumer being alive; Wt A + Y t c t A t+1 L t+1 : shorttermsavingattheendofperiodt. Rt A Rt L 14

The constraints to respect are the following: A t 0 (5) L t 0 (6) (5) and (6) are short-sale constraints. They need an explanation. Saving and insurance leave three assets for two states of the world. In terms of benefits, one unit of ordinary saving is replicable by simultaneously purchasing one unit of life insurance and one unit of life annuities for the same period. If the costs were the same, (5) and (6) would be purely arbitrary and would only serve to fix a terminology: for example, if A t >L t, we could apply the convention that the agent saves and purchases annuities. But in practice, insurance prices are loaded and the equivalence is not true. Intuitively, high mortality individuals tend to buy life insurance whereas low mortality individuals tend to buy annuities; in consequence, actual purchases provide some information to the insurers on the consumer s riskiness that is taken into account in the prices charged. Short-sale constraints represent how markets deal with adverse selection: by setting (simplified) non-linear prices. Technically, short-sale constraints prevent the arbitrage argument to work, as a consequence, all three assets are needed. Most papers assume either that insurance is available at an actuarial price(theinterestratecorrectedformortality)orthatitisnotavailable at all, which leads to comparisons of the profiles of consumption in the two contexts. 10 When insurance is actuarial, we find: Rt A = q t q t+1 R t and Rt L = q t q t q t+1 R t. For example in Fischer (1973), assuming perfect markets amounts to imposing 1 + 1 = 1 Rt A Rt L R t, which explains why he findsnegativepurchasesof life insurance (i.e. implicit positive purchases of annuities) in his examples. Loading factors decrease insurance yields; they are principally due to administrative costs, to adverse selection, and to fiscal regimes. Another factor that may enlarge the gap between average market returns and life insurance returns is that life insurance funds are backed by a larger proportion of low return (presumably less risky) assets in general (Friedman and Warshawsky (1990)). Papers where both imperfect markets (life insurance and annuities) are modelled at the same time are scarce. For a simple theoretical example and an empirical application, see Bernheim (1991). It is proved in Moffet 1 (1979a,b) or Villeneuve (1996) that if + 1 > 1 Rt A Rt L R t, then the agent never purchases life insurance and annuities at the same time for the same term: else it would be cheaper to cut life insurance and annuity benefits by an 10 Yaari (1965), Fischer (1973), Levhari and Mirman (1977), Pissarides (1980). 15

arbitrary quantity and to compensate it exactly by an increase of savings. 11 We impose that the agent s disposable wealth be always non-negative Wt A 0 (7) Wt L 0 (8) Such borrowing constraints are discussed at length by Yaari (1965). (8) simply says that negative net wealth cannot be inherited. It should be possible in principle to overcome (7) if the agent were able to publicly commit to a contingent borrowing plan for the future. In the absence of such a commitment, the policyholder would be able to play a sort of Ponzi game, i.e. a strategy of unbounded rolling debt. 12 In response to this threat, the liquidity constraint, though very conservative, is in practice easily implemented. Another practical useful interpretation of the choice between insurance and savings is the following: if the individual borrows money, he must provide a guarantee under the form of life insurance. This is the standard practice for mortgage loans. W A t+1 = R t (W A t + Y t c t A t+1 R A t W L t+1 = R t (W A t + Y t c t A t+1 R A t L t+1 )+A Rt L t+1 (9) L t+1 )+L Rt L t+1 (10) b t+1 = W L t+1 (11) (9) and (10) give the laws of motion of conditional wealth: they express the dependency of incomes in case of survival and in case of death at date t+1 on the portfolio choice at date t. (11) reminds that in case of death, the entire wealth, composed of savings and life insurance, is left to the beneficiaries. 11 If 1 R A t < 1 R t, the individual would never have interest to detain ordinary assets: his portfolio would be entirely composed of insurance. But, except if insurance if heavily subsidized, insurance companies would not be able to sustain such yields: the inequality says that no financial assets could match these liabilities. 12 Assume that, each period, the individual is able to borrow on the promise he would reimburse, with the payment of a certain interest adjusted for mortality risk, only if he lives. Without control, the individual would borrow, period after period, unbounded quantities of money, first to pay back the previous loan, and second to finance consumption. With an infinite support of life duration (and even if the probability of staying alive is extremely low), we typically enter into the Ponzi game problem, well-known in public finance. When the support of life duration is bounded, a consistency problem appears at the upper bound date, when the individual becomes certain to die: no insurance company will lend money and the individual is bankrupt. It is not clear that this is sufficient to impose discipline to the agent: punishment being necessarily limited, the Ponzi game problem remains an issue. + 1 R L t 16

Remark that it is assumed that no information is revealed over time except the date of death at the exact moment when it occurs. If the policyholder were to learn progressively his survival law, then the initial objective should be an expectation over the possible upcoming information too. If this information is observed by both parties (insurers and the policyholder) and if insurance prices depend on it, this would increase the number of necessary state variables and insurance markets. Fortunately, the dynamics of consumption would not be affected seriously since complete markets would provide insurance against these shocks. In general however, the equivalence between the optimal long term contract (agreed upon at period zero) and the optimal choice of short term contract would disappear. See Babbel and Ohtsuka (1989), and our Part III. Regimes Because of the uncertainty, we introduced additional state variables (conditional wealth) to the standard life-cycle model. Classically, under complete markets, they can be eliminated so as to give a single inter-temporal budget constraint where the present value of consumption equals the present valueofincome. Here, wehavetodetermine, asafirst step, non-binding constraints in order to reduce the complexity of the program. The sub-optimality of simultaneous purchases of life insurance and annuities opens the possibility of different regimes at different periods of the life-cycle: the individual may want to purchase life insurance at certain dates and life annuities at others. At each period t, the individual saves or borrows, but concerning insurance, he purchases either annuities or life insurance, or neither. For example, a man below fifty will be covered by life insurance; above seventy, by life annuities. It is possible in practice that he seems to have both (e.g. a pension plus life insurance). However, his net position will presumably be as we say. The consequence of the existence of these three regimes is that it is not possible a priori to write a unique constraint, since part of the individual decision is the qualitative choice of the contingent assets he needs. Confronted with this difficulty, several papers have supposed, or set conditions ensuring, that a certain regime is systematically in force (Abel (1986)), or have limited the conclusions regarding the temporal evolution of the structure and quantity of saving to a certain regime. 13 To simplify the rest of the exposition, we write the relevant budget constraint within a regime as if it were prevailing throughout the individual s life-cycle: either the individual is a permanent annuitant (regime A), or he is always covered by life-insurance (regime L), or he holds neither (regime 13 Yaari (1965), Fischer (1973), Levhari and Mirman (1977). 17

N). The difference between regime N and the other two is that then it is not possible to eliminate wealth variables (Wt A,Wt L ) to build a single budgetconstraint:thereisonlyonecontrolvariable(saving)perperiodfortwo arguments in the utility. In contrast, in regime A, where annuities demand is always non-negative, a single constraint can be used: Xt=T µ s=t Y t c t RA t 1 R t 1 Y b t Rs A =0. (12) R t 1 t=0 In regime L, where life insurance demand is always non-negative: ( Xt= T µ s= ) R t 1 Y T R s Rs L Y t c t b Rt 1 L t =0, (13) R t 1 Rs L R s t=0 When prices are fair, the notion of regime, as said before, is only semantic, and we find in all cases: Xt=T s=t (q Y ty t q t c t (q t 1 q t )b t ) R s =0 (14) t=0 In all cases, we see that what matters is the present value of income discounted (implicitly in (12) and (13) or explicitly in (14)) by the survival probability (future incomes are conditional on living); the correction also applies to consumption and bequests. 2.3 Portfolio choice Within each regime, the Euler condition indicates the forces driving the short-run evolution of saving and insurance purchases. (Calculations are not detailed; hints are given in the appendix.) In case A (annuities every period) Ut+1(c 0 t+1 ) Ut(c 0 t ) Vt 0 (b t ) Ut(c 0 t ) = 1 R A t s=t s=t s=t q t q t+1, (15) = RA t 1 R t 1 R t 1 In case L (life insurance every period) µ Ut+1(c 0 t+1 ) 1 = 1 Ut(c 0 t ) R t Rt L Vt 0 (b t ) R t 1 = Ut(c 0 t ) Rt 1 L R t 1 18 q t q t 1 q t. (16) qt q t+1, (17) q t q t 1 q t. (18)

In any case we see that the variations of consumption depend positively upon the interest rate and negatively upon mortality. The sensitivity to these two effects is proportional to the elasticity of substitution between periods. For example, when U t+1 ( ) =βu t ( ), (15) can be approximated by c t+1 c t c t ' ³ 1 q t βr A t q t+1 ctu 00 t (ct) U 0 t (ct), (19) We recover the usual result that consumption profiles depend directly on the comparison of interest rates with the discount factor. 14 Consider the case where the individual purchases annuities throughout his life. When insurance markets are perfect, the first order conditions above are simplified and the marginal rate of substitution of consumption between two periods becomes equal to the interest rate, exactly as without uncertainty. Expected utility over states and additivity over time have the advantage, already noticed by Yaari, that weights attached to felicities (partial utilities) on the one hand, and prices on the other hand, are proportional. Except for the effects of risk-free interest rates and taxes, programmed consumption is extremely smooth. Intheperiodswhentheindividualismoreriskaverse(becauseofold age, or because dependents are more in need of protection), consumption and bequests tend to be less sensitive to price incentives and are therefore more likely to be protected by an insurance contract. On the role of risk aversion (or resistance to inter-temporal substitutability), see for example Karni and Zilcha (1986) or Hu (1986). When the rate of return on annuities increases in a given period, contingent consumption that period becomes cheaper; the utility being additively separable, the income effect works in the same direction as the substitution effect to increase current consumption. Still, the increase in the rate of return of insurance decreases the value of incomes earned after that period (this is a well-known paradoxical property of life-time earnings); the consequence of that particular effect is a decrease of insurance demand. The net effect is ambiguous. The same indeterminacy occurs for life insurance: budget constraint (13) shows the depreciation of the present value of earnings Y s for s t +1provoked by an increase in life insurance returns Rt L. 15 Fischer (1973) noticed this inferior good nature of insurance in his particular specification without market imperfections. Predictions are complicated by 14 Again, Yaari (1965), Hakanson (1969), Fischer (1973), Levhari and Mirman (1977), Pissarides (1980). 15 The reader must be careful that the present value in that reasoning is taken in terms of first period disposable income. 19

regime switches, though the effects above give some clues on when switches are likely to occur. Other things equal, it is clear that an increase in mortality probabilities a given period should decrease the demand for annuities that period. Predictions however depend on whether the survival law is shifted so as to increase the weight of future, or past, consumptions. For example, Levhari and Mirman (1977) questioned whether changes in lifetime uncertainty should increase or decrease the rate of consumption. Using a stochastic ordering measuring dispersion lifetime, they point out two opposite (and paradoxical) effects: on the one hand, more uncertainty shortens the horizon (death at young ages becomes more probable), which increases the rate of consumption; on the other hand, longer lives also become relatively more probable, preserving wealth for those eventualities plays in favor of a decreased initial consumption. They conclude that the first effect dominates with Cobb-Douglas utility functions provided that, for a given discount factor, the interest rates are not too large. The approach, though interesting, remains difficult to generalize. 16 We should note in passing that there exists a limited literature examining the consistency of beliefs on mortality and economic behavior, in the framework of life-cycle theory. See for example Hamermesh (1985), or Hurd and McGarry (1995) for more details on the distribution of beliefs and their relationship to portfolio choice. Friedman and Warshawsky (1990) explained that the average American pensioner should stop purchasing short term annuities between age 60 and 70, his constant pension (or publicly provided annuities) becoming larger than his free demand for annuities. Yagi and Nishigaki (1993) insisted on the fact that within the Fisherian model itself, optimal long term annuities should not be constant but rather declining over time in real value. Leung (1994) gave a clue for these results. He showed, keeping the same assumptions as in Yaari, that there always exists an age, strictly before the upper limit, from which the individual consumes all his current income. Moreover, he proved, on the basis of simulations, that this constrained period is not negligible. According to Leung, this prediction contradicts empirical evidence since the elderly are conservative in the use of their assets. This remark is close in spirit to the disconnection between the prediction that wealth should exhibit a hump-shape over the life-cycle and the empirical finding that the elderly do not dissave in reality. To reconcile theory and evidence, Attanasio and Hoynes (1995) attempt to correct for the selectivity 16 Kessler and Lin (1989) examined the comparative statics of a choice between cash and annuities in individual retirement accounts by varying the survival law. They show that the third derivative of the cumulative survival probability distribution plays a role. 20

biases due to differential mortality across wealth groups (richer people living longer, dissaving is apparently slow since richer individuals become relatively more numerous). It seems that dissaving at old age calculated that way is more important than previously claimed, which gives new evidence in favor of the life-cycle hypothesis. 2.4 Life insurance and social security We leave aside the debate on the potential macroeconomic inefficiency of pay-as-you go systems to concentrate our attention on the specific effects on insurance markets. See also Mitchell (this volume). Pensions are publicly provided annuities, mandated either directly by the State or by the employer. The development of social security is recognized by Abel (1986, 1988) or Eckstein, Eichenbaum and Peled (1985) as a likely cause of the decline of annuity demand in the developed countries since World War II. There is no doubt that social security crowds out life annuities. Microeconometric studies like Rejda and Schmidt (1984), Rejda, Schmidt and Mc- Namara (1987), Fitzgerald (1987) or Bernheim (1991) confirm the negative effect of public pensions (or similar programs) on annuity demand and/or private pensions. Concerning life insurance, the intuition is not clear: on the one hand pensions should decrease noninsurance saving and enhance purchase of life insurance; on the other hand, most pension programs cover spouses after the death of the beneficiary. This life insurance element of pensions, frequently ignored by analysts, renders the net effect ambiguous. A study on aggregate data by Browne and Kim (1993) suggests that income and life expectancy being taken into account, the quality of social security still has a positive impact on life insurance premium volume per capita (life insurance and annuities are not separated). However, the state of the art is far from testing the microeconomic life-cycle model presented in this survey. Part III A contract theory of life insurance The first part of this survey set up the technical constraints that contracts should meet; the second gave the effect of tastes on the structures of ideal 21

contracts, thereby offering a theory of needs of life insurance. This part will show that informational constraints play a major role in the functioning of life insurance. In principle, long term contracts have the major merit that they are the only and sufficient means of taking advantage of all insurance possibilities. If information pertaining to the cost for the insurer and the value for the individual of life insurance contracts were always kept symmetric, then maximizing expected inter-temporal utility of the individual under a unique purely technical constraint would give smooth consumption paths along the life-cycle and across states, and straightforward continuations period after period. There should remain no ambiguity: a long-term contract must not be a stationary contract paying a fixed amount whatever the conditions. On the contrary, well-conceived contracts take into account the passage of time: tastes change, notably those dictated by the composition of the household, new information (the arrival of which being anticipated as a possibility, and probabilized) may arise, for example on health conditions or future income. A long term contract under symmetric information must not leave any choice to either the individual or the insurer: discretion is undesirable. Yet, we have to find explanations for the features observed in real contracts, all somehow linked to the fact that the informational situation is frequently not idyllic. Options in contracts may be desirable trade-offs between insurance needs and incentive compatibility. We can classify the main types of imperfections as follows: 1. Incompleteness, namely (a) limited commitment, and (b) incomplete contracts (certain continuations are too complicated to write explicitly); 2. Asymmetric information (a) before signing, and (b) appearing over time. We will take up each in turn. 3 Incompleteness 3.1 Limited commitment It is well recognized in insurance theory that long term contracts provide insurance against the risk of becoming a high risk, specifically against the risk of seeing future insurance applications rejected, or accepted only at high price. Therefore, the best insurance contract should be taken very early, even before any party becomes advantaged in terms of information. Babbel and Ohtsuka (1989) applied these ideas to explain why policyholders should continue to hold whole life insurance, though, if prices are considered naively, this strategy seems dominated by a combination of suc- 22