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
Contents
I
Overview
3
II
Possibilities and needs in life insurance
5
1 Life insurance possibilities 5
1.1 The production set . . . 6
1.2 Typical life insurance contracts . . . 8
1.3 Indices and Rates of return . . . 9
2 Life insurance needs 11 2.1 The life-cycle hypothesis . . . 11
2.2 Constraints . . . 14
2.3 Portfolio choice . . . 18
2.4 Life insurance and social security . . . 21
III
A contract theory of life insurance
21
3 Incompleteness 22 3.1 Limited commitment . . . 223.2 Incomplete contracts . . . 23
4 Asymmetric information 24 4.1 Asymmetric information before signing . . . 24
4.2 Asymmetric information appearing over time . . . 30
5 Conclusion 31
Part I
Overview
Life insurance serves to guarantee a periodic revenue or a capital to depen-dents 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 the-ory 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 in-surance 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 life-cycle 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 typ-ical structures; explaining these features and the stability of this selection is the role assigned to the economic theory of insurance. The main limita-tions to the implementation of first-best contracts are the parties’ inability to commit, asymmetric information before signing, and asymmetric
infor-mation emerging during the life of the contract. The existence of options (extension of coverage, renewability, surrender values, etc.) needs a particu-lar 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 insur-ance 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.
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 tech-nical 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 insur-ance contracts must be measurable, at each date, with respect to the available information. This preliminary remark makes sense for three reasons.
The first reason is that in financial 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 ac-count 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 moder-ate this threat, there exists serious obstacles to the sustainability of most desirable long term contracts.
The third reason that makes a powerful information structure indispens-able is that in the standard modelling, informational asymmetries are not due to someone being wrong, but rather on different precisions in the infor-mation 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.
1.1
The production set
Actuarial approach A life insurance contract is a financial agreement be-tween an insurer and a policyholder, signed at a date t0,specifying monetary
transfers at certain dates {t0 + d,. . . , t0+ 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 insurer or the other way around. We denote by pi
t(s) a payment at date t
from the insurer to beneficiary i (i = 1 · · · n), conditionally upon the arrival of state of the world s ∈ It where It is the information set at date t (an
ele-ment of It contains all the available information, and {It}t≥t0 is a filtration
to capture the fact that information is more and more accurate). A negative pi
t(s) is interpreted as a “premium” or contribution, a positive pit(s) as an
“indemnity” or benefit.
The two substantial elements in this definition are that payments are 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 explic-itly specified quite simply in contracts, nevertheless, all contingencies are not listed in details, or are used in a crude manner: for example, financial performance is often utilized under the form of some simple sharing rule.
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 princi-ple: expected profits must be positive. Formally:
t=tX0+d+m t=t0 Ã X s∈It at(s)· i=n X i=1 pit(s) ! ≤ 0 (1)
where at(s) represents the cost (at date t0) of one unit at date t in state
s. In an economy with complete markets, the at(s) represent the prices of
the Arrow—Debreu assets; otherwise, they represent marginal value for share-holders 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 at(s). To simplify, It can be structured
of indicators of who is alive and who is dead. If one assumes that mor-tality is independent of interest rates, then for all f, f0 ∈ F
t and for all
m, m0 ∈ Mt: Pr{f|m} = Pr{f|m0} and Pr{m|f} = Pr{m|f0}. We can write
for all s = (f, m) : at(s) = ϕt(f )· µt(m).
The aim of this last factorization is to show that insurers’ technical ability, summarized by the at(s), stems from two independent expertises: actuarial
estimates ϕt, and asset management µt; 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 dimen-sion 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 coun-tries 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) devel-oped 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 infor-mational 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
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 behav-ior. 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 fis-cal doctrine in life insurance has not been seriously studied by public econo-mists.
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 benefits may depend on who is alive in the household in a potentially 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
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 pay-ment 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, mini-mum 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 may be a delay (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
1See, e.g., Gustavson and Trieschmann (1988).
2See Cox, Gustavson and Stam (1991) for empirical evidence on demand of these
the financial performances of his fund, notably if he is relatively little risk 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 con-sumer’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 het-erogenous consumers. Though he proposed a reasonable solution to that problem, he also made clear why the quest for an indisputable index is hope-less. The notion of rate of return makes no exception to his critique: an index based only on the at(s)–see our definition of contracts–may be right
for assessing the purely technical ability of the insurers. However, the allo-cation of benefits across contingencies (the payments pi
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
3In 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.
4Using a bounded rationality approach, Puelz (1991) proposes a practical strategy for
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
t=T
X
t=1
Ut(ct) + VT +1(bT +1) (2)
where Ut(·) is the period t felicity derived from current consumption ct, and
VT +1(bT +1) is the value of bequest bT +1 left at date T + 1.7
We retain a discrete approach mainly because it facilitates the intro-duction 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
5See also Poterba (1997).
6See also Fischer (1973) or Karni and Zilcha (1986).
7We could also enrich the model by giving value to inter-vivos transfers at other dates. 8For examples of this last category, see, e.g., Yaari (1965) and Pissarides (1980) where
When the horizon is random, one can assume that the individual max-imizes expected utility with respect to the distribution of T, T being the upper limit of the support
ET{ t=T
X
t=1
Ut(ct) + VT +1(bT +1)}. (3)
The main restriction embodied in this objective function is the additive sep-arability 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
t=T
X
t=1
{qtUt(ct) + (qt−1− qt)Vt(bt)} + qTVT +1(bT +1) (4)
where qtdenotes the probability of living at least until period t; in particular
qt > qt+1, q0 = 1 and qT +1 = 0, and the mortality rate at the end of period
t is 1 −qt+1
qt . 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 Vt(·) as
compared to Ut(·), 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 esti-mation of the determinants of life insurance demand. Abel and Warshawsky (1988) presented a useful discussion and implementation of how bequest mo-tives could be specified and calculated, starting from simple principles.
Lewis (1989) extended Yaari’s model by exploring explicitly how the be-quest 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 beneficia-ries on life insurance demand. To this end, he modelled the way beneficiabeneficia-ries
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 im-pact 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 well-covered by the life-insurance plans of their husbands. The normative stand-point is that a sound protection should allocate savings and insurance in view of maximizing a weighted sum of the spouses’ utilities, taking into ac-count 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 decision-maker (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 re-place inefficient annuity markets. They assumed that markets are incomplete:
9With our notations, it amounts to say that the marginal utility V0
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 simul-taneous 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 non-cooperatively. 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 alterna-tive 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 hu-man wealth (expected future income from labor) is relatively large are more likely to demand life insurance.
2.2
Constraints
An image of financial markets Let us define additional control and state variables:
At and Lt : 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;
RA
t and RLt : the gross rate of return of, respectively, annuities and life
insurance received at date t; contributions At
RA t and
Lt
RL
t are paid at date t − 1;
taxes are ignored throughout, but may be included in the rates of return; Rt: the gross rate of return on saving
WA
t and WtL: the disposable wealth of the agent at the beginning of period
t if he is alive (respectively if he is just dead);
Yt : the exogenous income (wage or pension) which is conditional on the
consumer being alive; WA
t + Yt− ct−ARt+1A t −
Lt+1
RL
The constraints to respect are the following:
At ≥ 0 (5)
Lt ≥ 0 (6)
(5) and (6) are short-sale constraints. They need an explanation. Sav-ing and insurance leave three assets for two states of the world. In terms of benefits, one unit of ordinary saving is replicable by simultaneously pur-chasing 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 At> Lt, 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. In-tuitively, 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 (the interest rate corrected for mortality) or that it is not available at all, which leads to comparisons of the profiles of consumption in the two contexts.10 When insurance is actuarial, we find: RA
t = qt qt+1Rt and R L t = qt
qt−qt+1Rt.For example in Fischer (1973), assuming perfect markets amounts
to imposing R1A t + 1 RL t = 1
Rt,which explains why he finds negative purchases of
life insurance (i.e. implicit positive purchases of annuities) in his examples. Loading factors decrease insurance yields; they are principally due to ad-ministrative 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 (1979a,b) or Villeneuve (1996) that if 1
RA t + 1 RL t > 1
Rt, 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
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
WtA ≥ 0 (7)
WtL ≥ 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 possi-ble in principle to overcome (7) if the agent were apossi-ble to publicly commit to a contingent borrowing plan for the future. In the absence of such a commit-ment, 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 liquid-ity 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. Wt+1A = Rt(WtA+ Yt− ct− At+1 RA t − Lt+1 RL t ) + At+1 (9) Wt+1L = Rt(WtA+ Yt− ct− At+1 RA t − Lt+1 RL t ) + Lt+1 (10) bt+1 = Wt+1L (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.
11If 1 RA t + 1 RL t < 1
Rt, 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.
12Assume 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.
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 policy-holder 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 nec-essary 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 vari-ables (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 value of income. Here, we have to determine, as a first step, non-binding constraints in order to reduce the complexity of the program.
The sub-optimality of simultaneous purchases of life insurance and an-nuities 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 de-cision 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 lim-ited 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 con-straint 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
N). The difference between regime N and the other two is that then it is not possible to eliminate wealth variables (WA
t , WtL) to build a single
bud-get constraint: there is only one control variable (saving) per period for two arguments in the utility.
In contrast, in regime A, where annuities demand is always non-negative, a single constraint can be used:
t=T X t=0 µ Yt− ct− Rt−1A − Rt−1 Rt−1 bt ¶s=TY s=t RAs = 0. (12) In regime L, where life insurance demand is always non-negative:
t= ¯T X t=0 (µ Yt− ct− Rt−1 RL t−1− Rt−1 bt ¶s= ¯YT s=t RsRLs RL s − Rs ) = 0, (13)
When prices are fair, the notion of regime, as said before, is only semantic, and we find in all cases:
t=T X t=0 (qtYt− qtct− (qt−1− qt)bt) s=TY s=t Rs = 0 (14) 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) U0 t+1(ct+1) U0 t(ct) = 1 RA t qt qt+1 , (15) Vt0(bt) U0 t(ct) = R A t−1− Rt−1 Rt−1 qt qt−1− qt . (16)
In case L (life insurance every period) U0 t+1(ct+1) U0 t(ct) = µ 1 Rt − 1 RL t ¶ qt qt+1 , (17) V0 t(bt) U0 t(ct) = Rt−1 RL t−1− Rt−1 qt qt−1− qt . (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 Ut+1(·) = βUt(·), (15) can be approximated by
ct+1− ct ct ' 1− qt βRA tqt+1 ³ −ctUt00(ct) U0 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.
In the periods when the individual is more risk averse (because of old 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, contin-gent 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 re-turn of insurance decreases the value of incomes earned after that period (this is a well-known paradoxical property of life-time earnings); the conse-quence of that particular effect is a decrease of insurance demand. The net effect is ambiguous. The same indeterminacy occurs for life insurance: bud-get constraint (13) shows the depreciation of the present value of earnings Ys for s ≥ t + 1 provoked by an increase in life insurance returns RLt.15
Fis-cher (1973) noticed this “inferior good” nature of insurance in his particular specification without market imperfections. Predictions are complicated by
14Again, Yaari (1965), Hakanson (1969), Fischer (1973), Levhari and Mirman (1977),
Pissarides (1980).
15The reader must be careful that the present value in that reasoning is taken in terms
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 probabili-ties a given period should decrease the demand for annuiprobabili-ties 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, Lev-hari 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 de-creased initial consumption. They conclude that the first effect dominates with Cobb-Douglas utility functions provided that, for a given discount fac-tor, 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
16Kessler 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.
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. Micro-econometric 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
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 possibili-ties. 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 pe-riod. 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. Incomplete-ness, namely (a) limited commitment, and (b) incomplete contracts (certain continuations are too complicated to write explicitly); 2. Asymmetric infor-mation (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 policy-holders should continue to hold whole life insurance, though, if prices are considered naively, this strategy seems dominated by a combination of
suc-cessive term contracts and saving. The argument runs as follows: future health conditions being random, the average term contracts might turn inac-cessible to certain individuals; consequently, the “dominant” strategy would be not implementable and the arbitrage argument does not work.
Cochrane (1995), in an article dedicated to health insurance organization, offered an interesting reflection on the various channels through which com-mitment could be escaped, stressing the importance of regulation. There, information over health is improved symmetrically over time. The main problem is that, if competition were unregulated, lucky persons would like to switch to an other insurer proposing better prices; this would break the cross-subsidies implicit in the first-best arrangement. He proposes a solu-tion to overcome the market failure: insurance contracts last one period only (long-term commitment is not forced); insurers are free to charge the price they want to their clients, and markets are competitive; people have per-sonal accounts to which their insurers pay severance payments when they are denied average price insurance; this money is given to the unlucky to fi-nance their higher premia. This payment is intended to compensate exactly for the discrimination these people suffer. Basically, Cochrane shows that the desirable cross-subsidies are exactly implementable by this minimal new institution preserving competition and incentives.
In practice, in life insurance as well as in health insurance, asymmetries of information are multi-dimensional and the motivations of the agents are so diverse that the implementability of the first-best is not warranted (insuring taste shocks is considerably more difficult than probability shocks). The proposal however goes in the right direction and is capable of correcting a non-negligible part of the undesirable effects of discrimination.
The diffusion of group insurance seems to be a second-best response of certain communities in a world where the absence of commitment affects welfare significantly. These insurance programs are efficient in terms of cov-erage of long term risk since they are not discriminatory in general, and tax incentives plus economies of scale are substantial. However, group insurance may be efficient within the group, but different groups may be treated very differently, not to speak of people who are not eligible to these programs. The appropriate public policy at the highest level is not totally clear in this context.
3.2
Incomplete contracts
Optimal contracts should take into account an enormous quantity of detailed information. In practice, we do not observe a high degree of complexity in the way, e.g., incomes are utilized. Contracts are incomplete in the sense that
a contract may become sub-optimal in terms of its dependency on income given some new information coming up: if his future incomes are from a certain date on expected to follow a certain path that was unlikely ex ante (inheritance, wage rise), the insured may want to reallocate his contributions and benefits in a way that significantly differs from that the contract would pursue. This creates situations where renegotiation becomes desirable, i.e. where mutually advantageous arrangements could be found. Other examples could be found in the way the composition of the family (birth, divorce) can affect the desirable continuation of the contracts.
Incomplete commitment interacts in an important way with incomplete contracts. The problem of mortgages renegotiation is well-known in the finance literature: a decrease in the interest rate may induce premature re-payments financed by cheaper new mortgages. In life insurance, a cause of premature termination of a contract would be an increase in interest rates, new contracts becoming more attractive.
In order to limit the effects of renegotiation, and given that more complete contracts cannot reasonably be conceived, it may be optimal to circumvent, within certain limits, this problem by setting penalties for premature termi-nation. We are not aware of any formal literature drawing conclusions on the constrained best contracts.
4
Asymmetric information
4.1
Asymmetric information before signing
Empirical evidence Adverse selection in life insurance markets is a well-known phenomenon, one well-known by actuaries before it became popular in economic theory after Akerlof (1970) or Pauly (1974). This asymmetry of information may be due to the inobservability of certain mortality factors (health, life-style) or a consequence of legal restrictions on the use of certain observable information (sex). It should be clear that the asymmetric infor-mation we are talking about is the residual (small or large) after classification has taken place.
In life insurance econometrics, testing the adverse selection hypothesis presents methodological problems similar to, and perhaps worse than, those encountered in automobile insurance (on this, see e.g. Chiappori (this vol-ume)). Policyholders are submitted to medical examinations leading to the exclusion of certain categories, and to complex pricing. These two factors give rise to selection and self-selection biases that are extremely difficult to correct. Applicants themselves are self-selected, due to the fear (or simply
the certitude) of denial by those who would show tangible signs of higher mortality.17 Among applicants, new policyholders tend to be healthier on
average than the population (lower mortality). The econometrician has to be careful that he might have less data available than the insurers. If this is the case, spurious regressions could bias the interpretations. See also Dionne, Gouriéroux and Vanasse (1999).
Even if observed, the classification techniques used by the actuaries should not be taken for granted. The insurance industry has been perhaps exces-sively conservative in risk selection. We see in Cummins et alii (1982) that the degree of sophistication of scoring methods can be pushed extremely far, and the statistician may wonder whether the corrective factors that are ap-plied to premia are reliable. The series of statistical models from which their were estimated are not likely to be consistent which each other. Overall, major identification problems are obvious.
There is one modest way to escape this difficulty: working on insurance contracts offered without medical examination, like annuities where age is the only parameter that is used. Indeed testing for adverse selection is rela-tively easy in this market: it suffices to calculate the mortality experienced, age by age, by policyholders and to compare these with mortality tables, and to relate the differences to the quantities purchased.18 In this case basically,
adverse selection can be proved because classical antidotes are hardly used (selection, non-linear pricing, etc.). However, one cannot draw valuable con-clusions from this on the degree of sophistication that insurers are able to reach in general.
The classical paper of Rothschild and Stiglitz (1976), adapted for life in-surance or annuities, has given rise to several direct applications.19 Beliveau
(1984) illustrated this view with a reduced form analysis of adverse selection. Her estimates support the hypothesis that people demanding more life insur-ance are charged a higher price, a result in accordinsur-ance with Rothschild’s and Stiglitz’s prediction. However, given that she observes very few variables (presumably fewer than the insurers), it is not clear that there was really asymmetric information.
Cawley and Philipson (1996) also started from the assumption that life
17Attanasio and Hoynes (1995) showed on micro data that there is a positive correlation
between survival prospects and saving. This phenomenon is presumably not negligible for life insurance applicants.
18See Philippe (1987) for a study of this type.
19Carlson and Lord (1986) stressed an unsurprising consequence of prohibiting
discrim-ination between men and women: it creates a problem of adverse selection. Rea (1987) explained how that problem could be circumvented (not solved) by offering menus of contracts à la Rothschild and Stiglitz.
insurance contracts were designed according to Rothschild’s and Stiglitz’ premises and predictions. They tested whether larger coverages were charged larger prices per dollar, and the answer is no. To us, the main interest of that study is that it shows the shortcomings of standard theory as a description of life insurance markets. Indeed, they have extrapolated the conclusions of the classical static model to a situation where several of its assumptions are not met (heterogeneity on probabilities only, one-shot game, exclusiv-ity, no saving, no loadings, etc.). All these hypotheses being tested at the same time, one cannot conclude that adverse selection, rather than any other assumption, is rejected.
When efficiency and regulatory problems arise, models suitable for finer descriptions are required. We now see papers addressing the most remarkable particularities of life insurance markets.
Quantities The number of periods involved changes completely the no-tion of quantity (is it the face value? the durano-tion of the contract?) and the notion of exclusivity (do we have exclusivity–one insurer at once–each period only, or also over the lifetime of the insured?). These factors inflat-ing the dimension of the space of contracts over which the insurers design their strategies, the elaboration of simple robust empirical methods becomes messy.
Townley and Boadway (1988) made an original contribution to the de-scription of life annuity markets. Offers consist of constant term annuities at a given price per unit of income. People can purchase as many annuities as desired but they have to choose the term (choice takes place at retirement time and no supplementary annuity can be purchased afterwards). The term will be used for self-selection: indeed, the marginal rate of substitution be-tween the duration and the quantity of annuities is dependent on the type of the individual. A standard single-crossing argument à la Mirrlees sug-gests that sorting agents by type is possible: as expected, short-lived people tend to give relatively less value to long terms than long-lived people. How-ever, the shapes of the indifference curves are difficult to characterize, which makes the problem slightly different from Rothschild’s and Stiglitz’. Now, pooling contracts are not excluded. The normative part of the analysis is less convincing for game theoretic reasons. Forcing linear prices, they assumed implicitly that instantaneous exclusivity was not possible (see the discussion in the following). But this interpretation is at odds with the fact that one term only is possible, since the individual could after all request different terms of different insurers. Their bounds on the insurers strategy space is hard to justify as an equilibrium outcome of a well-specified game.
In a related paper (where the contracts permitted are the same), Town-ley (1990) shows that, in equilibrium, in a world of adverse selection, whole life insurance with non-linear prices is inefficient (rationing terms is more efficient). Moreover, most simple public interventions (uniform compulsory plan, forcing linear prices) cannot guarantee efficiency gains. This is the origin of the “policy dilemma”: public policy in that field can only be redis-tributive (rather than Pareto improving). Theoretically, the case is clear: the fact that contracts and policy tools are incomplete is a systematic obstacle to the elimination of ambiguity.
Non-exclusivity and unobservable savings As we have seen in the life-cycle model of demand, life insurance is only part of a portfolio. Ordinary savings being a substitute to life insurance, insurers are forced to keep their prices relatively low even in a situation where competition is weak. Moreover, the threat of arbitrage opportunities limits their ability to propose effective non-linear prices. The papers we cite now are based on the particularities of asymmetric information in life insurance.
Exclusive insurance contracts and unobservable savings. In Eichenbaum and Peled (1987), agents have no bequest motives (they don’t give value to their assets in case of death), therefore a replication argument shows that an-nuities are the dominant asset (strictly dominant if yields are calculated on the basis of a strictly positive mortality) compared to ordinary saving. How-ever, mortality being private information, the equilibrium menu of annuity contracts must take into account the fact that the low mortality type (=high cost risk) can combine the contract assigned to the other type plus savings to bypass incentives taken naively. The consequence of this additional incentive constraint (compared to the exclusivity case in Rothschild and Stiglitz) is that the high mortality type (=low cost risk) is more drastically rationed as far as annuities are concerned.
Moreover, the only people that save (in addition to purchasing annuities) in equilibrium are the high mortality people, i.e. those who are least in need of second period income! The authors show that in an overlapping generation model, this characterizes an inefficient level of saving. They also show that a mandatory annuity program which is actuarially fair on average results in an equilibrium without involuntary bequests (proceeds of savings in case the agent dies at the end of the first period) that Pareto-dominates the laissez-faire equilibrium. This conclusion is related to the known result that a small amount of mandatory insurance is welfare-enhancing when the Rothschild-Stiglitz equilibrium is inefficient.
con-tracts, and this is unobservable by insurers, finding non-linear pricing equi-libria in pure strategies is possible when endogenous communication between firms is allowed (Jaynes, 1978). However, without exclusivity and commu-nication, linear pricing is not theoretically well-founded: in any linear equi-librium, there exist non-linear deviations that kill the equilibrium. At best, linear pricing may be seen as a self-restriction of the strategy space by the insurance industry. This restriction is convenient for applied theory, and is taken for simplicity in the following studies.
Prior to the approach offered by Rothschild and Stiglitz (1976), Pauly (1974) studied linear price equilibria in insurance under adverse selection. His conclusions fit particularly well life insurance, one of the rare instances where linear prices is a reasonable assumption. The equilibrium price is higher that the average fair price since higher risk people purchase more insurance, and are thereby statistically over-represented in clienteles.
With similar assumptions on the functioning of markets, Abel (1986) ex-plored the effect of Social security on capital accumulation in an overlapping generations model. Consumers save by purchasing life annuities in a private market: differing by their survival probabilities, they differ by their insur-ance demands. Given that private markets are subject to adverse selection, whereas the public sector can propose a uniform level of pensions (i.e. com-pulsory annuities), the issue is to find the effect of increasing pensions on total wealth. Ambiguity comes from the fact that mandatory pensions force saving on high mortality people (a positive effect on total saving), while it aggravates the adverse selection problem by raising the equilibrium price in the private market (a negative effect on total saving). Overall, the effect of public pensions remains ambiguous, except when they pass from zero to a small positive level (saving decreases). Abel supposed for simplicity that all individuals participated in the annuity markets, and accordingly, he did not model life insurance. Villeneuve (1996) developed Abel’s model in view of ex-amining systematically the effect of public pensions on the functioning of life insurance and life annuity markets at the same time. It is proved that typ-ically, if public pensions were to decrease, one should observe an alleviation of adverse selection in annuity market as well as an increased participation. However, the social welfare generated by the reform would be generically negative. Factors not modelled in the paper (incentives to work, endogenous growth, etc.) could attenuate this strong conclusion.
Brugiavini (1993) started with a set-up similar to Abel’s, abandoning the overlapping generations in order to study annuities markets more in details. She assumed that the consumers live three periods and have no bequest mo-tive. In the first period (when people are young), they only know the average mortality rate. In the second period they learn privately their probability of
dying before the third period (old age). Markets are complete in all periods and prices are endogenous so as to adjust to the actual mortality of policy-holders. Brugiavini proves that no transactions will be observed in equilib-rium in the second period, though intuitively low mortality people should demand supplementary insurance. We recognize the idea that if markets were complete before the arrival of asymmetric information, then a first-best allocation is attained and no additional transaction can take place (Laffont (1985)). A practical interpretation of this result can be seen as a variety of the classical argument in favor of redistributive taxation: public intervention mimics what ex ante markets that did not exist would have achieved.
Power and Townley (1993) studied similar questions with numerical sim-ulations. The main limitation is that annuity markets are constrained to offer only whole life annuities at a certain age, there is no bequest motive (therefore annuities are given exaggerated importance in the model), and the distribution of information that individuals are supposed to have on their own riskiness is rather arbitrary. The simulations are interesting in that they show the complexity of the redistribution due to the pension system and insurance markets.
Fraud and moral hazard Moral hazard is also an interesting hypothe-sis in life insurance. Intuitively, if people care for their dependents’ welfare, i.e. if they value the money they give them, then life insurance can dimin-ish (even if very little) their incentives to live since transfers are no longer conditional upon the donator being alive. Symmetrically, an improvement of the standard-of-living (through annuities) gives an incentive to invest in goods that favor good health (see Ehrlich and Chuma (1990)). Starting from this idea, Davies and Kuhn (1992) explored the effect of Social security on longevity. Whether the impact of insurance is considerable compared to the other determinants of the “willingness to live” is not evident.
The case of suicide is particular. People determined to commit suicide may want to protect their dependents by purchasing generous life insurance. However, this should be seen not as moral hazard but rather as adverse se-lection: moral hazard would be the case where somebody would renounce to suicide if insurance were not available. It seems to us that adverse selec-tion hypothesis is more likely than moral hazard. In any case, tradiselec-tionally, suicides are eliminated as causes of death that are covered by life insurance during the first years of the policy. This rationing of benefits finds an easy explanation in both theories.
A case of fraud was quite famous in France in the 80’s. A man subscribed life insurance policies on his head at several companies; he simulated a mortal
