# Aspects of optimal multiperiod life insurance.

Aspects of Optimal Multiperiod Life Insurance

ABSTRACT

A multiperiod model is used to analyze aspects of whole life and term insurance contracts within the context of the lifetime consumption-investment problem. Not only the life status, but also the state of health of the consumer is considered in examining optimal insurance purchases. Particular attention is given to the policy loan and guaranteed reinsurability options of the whole life insurance contract. When viewed in this context, whole life insurance, term insurance, and savings are shown likely to coexist in an optimal consumption-investment plan rather than act as substitutes for each other. In this paper it is shown how the bounds of the insurance pricing parameters can be derived to ensure that both kinds of insurance will continue to be sought by rational consumers, while avoiding adverse selection and lapsation.

Introduction

Over the past three decades there has been a conundrum in the economic and business literature. Why do so many people purchase whole life insurance when it has been shown that a combination of term insurance and savings dominates a whole life policy? Equally puzzling is that many people hold both (individual or group) term life and whole life insurance. Are consumers irrational or simply poorly informed?

This article demonstrates that rational, well-informed consumers will choose to hold both term and whole life policies. This result is achieved by considering aspects of whole life insurance which have heretofore escaped formal modeling--features which render the whole life contract distinguishable from linear combinations of term life insurance and a savings program. The simultaneous demand for both kinds of insurance is shown to occur under standard actuarial pricing practices, both with actuarially fair pricing and with typical premium loadings.

This article comes at a juncture characterized by a pause in the demand for whole life insurance. During the past variations of life insurance (e.g., universal life) and new investment vehicles (e.g., interest rate options), which effectively unbundle features of the whole life policy that had rendered it unique, were actively marketed. Consequently, consumers have been afforded the opportunity of seeking only those policy options they deem useful while avoiding payment for the others.

The past several years have raised doubts, however, that these newer insurance policies will ever return a profit to insurers, notwiethstanding a remarkable consumer demand for them. Actuaries claim the problem lies in pricing that is so competitive as to preclude their ability to properly charge for the options granted policyholders. While this situation cannot persist in the long run, it has created a strong desire among some insurers to be more circumspect in the kinds and amounts of policies they write. Many insurers are returning to more traditional products, where the bundling together of options leaves them less susceptible to simultaneous utilization. Most have attempted to offer revised policy provisions in their old book of business that strip the policies of some of their more troublesome options. For example, consumers can opt for a revised contract with a variable-rate policy loan feature and in return receive higher dividends or cash value crediting rates reflecting their utilization of policy loans. Some newly offered whole life policies feature dual premium schedules--a low-premium schedule for insureds who periodically provide evidence of insurability and another rate for those who do not provide such evidence. Other policies link the cash value accumulation patterns to actual market values of the assets supporting these policies.

Most of the innovations in life insurance policies sold today represent some combination of elements in the traditional whole life policy with term insurance and/or a variable savings program. Thus, traditional products remain the starting point in a comprehensive analysis.

Review of the Literature

Among the important advances in financial economics in recent years is the development of multiperiod investment theory. [See, for example, Hakansson (1970, 1971a, 1971b, 1974, 1979), Leland (1971), Mossin (1968), Ross (1974), and Samuelson 1969).] In the wake of this development have come numerous studies focusing on the structure of optimal multiperiod contracts in the presence of informational asymmetries [e.g., Townsend (1982)]. Various types of multiperiod contracts have been investigated. To give a few examples, Arnott (1982) studied the structure of multiperiod employment contracts with incomplete insurance markets, Merton (1981) viewed the role of social security as a means for efficient inter-generational risk bearing in an economy where human capital is not tradable, and Venezia and Levy (1982) derived optimal multiperiod automobile insurance contract.

The ubiquitous multiperiod life insurance policy has also been a subject of theoretical inquiry. Typically the approach taken has been to posit the life insurance policy as a series of single-period or instantaneous term contracts [e.g., Yaari (1965, Hakansson (1969), Richard (1974)]. This approach has meant that some of the complexities associated with the ever popular whole life policy (1) have escaped formal analysis within the context of multiperiod consumption-investment models. The degree of abstraction customarily employed in these models has rendered the whole life policy indistinguishable from a constructible linear combination of single-period term insurance and a savings program of some sort [Richard (1974)], with the latter usually adjudged superior to the former [e.g., Fischer (1973, Moffitt (1979, and Winter (1981)]. Fischer explicitly considered a multiperiod term insurance contract and concluded that it either presents the individual with arbitrage possibilities, or makes no difference to his welfare. Moffitt has extended the analysis to the whole life policy and reached a similar conclusion under well specified conditions. Winter (1981), who took into account the option of withdrawal, also concluded that the fair cash value whole life policy will not be purchased in equilibrium. When fortune (1973) introduced (with a two-period, three time-point model) a policy featuring some aspects usually associated with a whole life policy [e.g., maturity or surrender cash value, level premiums, and (implicitly) guaranteed reinsurability], he was criticized by Klein (1975) and others on the grounds that his whole life policy was little more than a combination of pure insurance and savings, and therefore was dominated by a strategy of purchasing single-period term insurance and investing the difference.

More recently the whole life policy has been perceived as a package of options and not simply a linear combination of single-period term insurance and a savings program [e.g., Smith (1982), Walden (1985), Outreville (1981), and Berkovitch and Venezia (1987)]. (2) Hence, the notion that term and whole life forms of insurance are substitute goods has been called into question. It should be emphasized here that the earlier analyses are correct within the frameworks they propose. Only by adding new dimensions to the framework can the multiperiod life insurance policy be more fully analyzed.

The following model extends the work of Peter Fortune and Stanley Fischer by allowing for an analysis that includes the two most prominent options of a whole life contract--the policy loan option and guaranteed reinsurability option--within the context of a multiperiod consumption-investment framework. (3) Basic features of the whole life policy, such as level premiums and surrender cash values, are also incorporated into the analysis. Other options that are often part of the contract [see Smith (1982) for a list] are not modeled here.

Model Development

The consumer is hypothesized to possess state-dependent utility for consumption, U(C), and bequest V(B). The underlying functional form of consumer utility across all states is assumed to be of the generalized logarithmic class. this class was chosen due to its many attractive theoretical properties [see Rubinstein (1977, 1978) and Hakansson (1979)] as well as its consistency with observed empirical facts [see, for example, Brown and Gibbons (1984) and Grauer and Hakansson (1982)]. (4)

The model proposed here is comprised of three periods (four points in time): two productive earning periods, and a retirement period covered by retirement plan benefits and savings, after which survivors are assumed to reach the end of their lifetimes. Three full periods are the minimum number necessary for adequately modeling the salient features of whole life vs. term insurance.

At the beginning of the first period, the consumer is endowed with wealth and earns a certain income, which sum to [Y.sub.0]. This total wealth is then divided among comsumption, [C.sub.0], and whole life insurance W at a fixed premium rate of [pi] per unit of W, term life insurance [T.sub.0] at a first period rate of [[pi].sub.0], and saving, [S.sub.0] (which may be positive or negative--borrowing), which are placed in a one-period bond (or taken in a single-period loan) that will return (or cost) [R.sub.0.S.sub.0] at the conclusion of the first period. Whenever a new insurance policy is desired, the state of health of the consumer is assessed by a medical expert and reported to both the insurer and the consumer. The premium rates stated above are based upon a clean bill of health.

At the end of the first period, the consumer could be in one of three states of health: good (insurable), bad (uninsurable via new policies), or dead, with probabilities [p.sub.1g.,p.sub.1b.,p.sub.1d.], where the number on the subscript denotes end of the first period and the letter denotes health status. If the insured is dead, no additional income is earned. However, the estate receives a bequest [B.sub.1], comprised of proceeds from any life insurance policies in force, W and [T.sub.0], and savings with interest, [R.sub.0.S.sub.0]. If the consumer survives the first period but is in poor health, he is ineligible for new term life insurance, but is assumed to maintain in force whatever amount of whole life insurance had earlier been purchased by making an additional premium payment of [pi]. The consumer has available for consumption reduced wages [Y.sub.1b], if any, as well as [R.sub.0.S.sub.0] plus the proceeds from any policy loan undertaken, [Q.sub.1.W]. (5) The latter is assumed to be incurred whenever the second-period (after-tax) market rate of interest exceeds that charged on policy loans (after tax). (6) This loan can be used as a low cost source of funds for current consumption or for additional investment at the favorable, higher market interest rate. Second-period consumption and saving in the event of poor health will depend, in part, upon investment opportunities, captured in the model by (one plus) the interest rate, [R.sub.1l] (l for low rate) or [R.sub.1h] (h for high rate). Subscripts are placed on the interest, savings, consumption and bequest variables throughout to indicate point in time (0, 1, 2, 3).

The individual who retains goods health until the start of the second period may choose whether to purchase a new term policy at the revised rates for newly issued policies, [[pi].sub.1h] of [[pi].sub.1l]. These rates will reflect the current interest rate levels, [R.sub.1h] or [R.sub.1l], as well as any updated information about the mortality experience obtained from period 1. The individual can either surrender or continue the whole life policy. If he or she elects to maintain in force the whole life insurance, he or she may choose to obtain a policy loan either for consumption or investment provided that market rates of interest are high. (If markt rates are low, the consumer would do better to access funds at the lower market rates.) The individual will again need to allocate available funds from earned income ([Y.sub.1g], physical capital ([R.sub.0.S.sub.0]}, and any policy loans ([Q.sub.1.W]) among current consumption, insurance and savings.

Individuals who were uninsurable at the start of the second period are assumed not to survive beyond the end of the period. At this time, their estates are settled, with any remaining funds going toward their bequests. If a policy loan had been taken out earlier, the loan plus interest ([GQ.sub.1.W]) is subtracted from the proceeds of the whole life policy.

Individuals who started the second period in good health may either be alive or dead (a or d) at the start of the third period. (7) If they are not alive, their estates are settled along similar lines to that described earlier, where the amount available for request will depend, in part, upon prior decisions about what kind and how much insurance to purchase. If alive, they once again are faced with allocating retirement income and wealth among consumption, savings, and whole life insurance (provided that the policy has continued in force). The option of policy surrender at this late date was not modeled explicitly, because during the last period, the amount available for a policy loan should equal the present value of a sure payment of W one period hence, or arbitrage opportunities present themselves. If higher, the loan could be taken and later repaid at the bargain rates. If lower, it would be better to obtain desired funds through an outside loan loan, because the cash values are accumulating at more favorable rates than those available outside. In neither case would the policy be surrendered.

AT the end of the retirement period, all available funds net of any loan repayments become the bequest available for heirs. The amount of these funds will depend upon prior decisions, and is influenced by the level of interest rates during the final period. In Figure 1 on the following page, the cash flows and decision variables associated with each of the 30 states described above are presented. To assist in deciphering the diagram, the notation is reviewed below:

[Y.sub.t]: Wages (after-tax) received by individual at time t. When t = 0, Y includes accumulated physical wealth.

[S.sub.t]: Savings undertaken or (if negative) loans incurred at time t.

[C.sub.t]: Consumption at time t.

W: Face amount of whole life insurance policy.

[T.sub.t]: Face amount of term life insurance policy at time t.

[B.sub.t]: Amount of bequest at time t.

[R.sub.t]: One plus the after-tax market rate of interest for saving or borrowing (assumed to be equal) at time t.

[Q.sub.t]: The proportion of the face amount of the whole life policy available for policy loan at time t.

G: One plus the after-tax rate of interest charged on policy loans (assumed to be fixed throughout life of policy). (8)

[P.sub.t]: Probability of occurrence of stated event at time t.

[pi]: The whole life insurance premium rate, per dollar of insurance in force. This periodc premium rate is fixed throughout the life of the policy.

[[pi].sub.0]: The term insurance premium rate, per dollar of insurance in force during the first period.

[[pi].sub.1]: The term insurance premium rate, per dollar in insurance in force during the second period, available only to individuals who have retained their good health.

Additional subscripts are attached to certain variables to clarify the state paths over time of these variables, or to indicate present state. Where additional subscripts are present, they refer to states of consumer health, interest rate levels, and consumer action as follows:

g,b,d,a: g and b refer to state of consumer health at time 1, where g indicates good, and b indicates bad (hence, uninsurable); d indicates dead and a indicates alive.

h,l: Market interest rate is high or low.

s,p,c: Refers to consumer choice at time 1 to surrender his policy, take out a policy loan, or continue keeping his or her insurance in force without taking out a policy loan, respectively.

Utility is assumed to be of the generalized logarithmic class, such that U([C.sub.t..]) = ln([C.sub.t.] - [[gamma].sub.t.]) and U([B.sub.t.]) = ln([B.sub.t.] - [[beta].sub.t.]), where [[gamma].sub.t.] and [[beta].sub.t.] can be interpreted as the minimum tolerable standard of living and minimum acceptable bequest provision, respectively, for period t and state ".". As noted by Rubinstein, the assumption of generalized logarithmic utility is not very restrictive. While all consumers are required by this functional form to have decreasing absolute risk aversion (theoretically preferred), the heterogeneity of the model permits consumers to have separate taste parameters [[gamma].sub.t.] and [[beta].sub.t.]for their consumption and bequest preference functions for each state and at each point in time, and tolerates quite diverse attitudes of proportional risk aversion. [gamma] (and/or [beta]) may be positive, zero, or negative, implying decreasing, constant or increasing proportional risk aversion, respectively. Presumably, in the context of this model, [gamma] must be positive to provide for at least some consumption, but [beta] could be either zero or positive. The higher the [gamma] (and/or [beta]), the more risk averse the consumer. Furthermore, utility is influenced by a separate state-dependent risk preference parameter [rho.] as well as a time- preference factor [[rho].sub.t]. Thus, the model accommodates a wide variety of individual time and state preference forms. (To simplify notation and computation, without any loss of generality, the state probabilities [rho][.sub.t.] can be combined with the state-department risk preference parameter [rho.] into a single parameter [[rho].sub.t.].)

The problem for the consumer, then, is to maximize exptected utility [Omega] of consumption and bequest across all periods and states by selecting appropriate levels of consumption, insurance, and investment, while making the optimal choice at time 1: surrender policy, persist with policy and incur a policy loan, or persist without incurring a policy loan. The formal problem is presented in the following equations:

[Omega] = U([C.sub.0])

[Mathematical Expression Omitted]

Observe that there are fewer arguments in the multiperiod utility function than there are states shown in Figure 1. This is because beginning with the second period, two branches of the decision tree will be dominated by their alternatives; in particular, the consumer in good health will choose either to surrender or take out a policy loan if interest rates are high in the second period, or either continue with his policy or surrender it if interest rates are low in the second period. The decision depends upon which is associated with higher utility for the consumer. Thus, the first task was to compute the utilities associated with each branch and eliminate those branches with lower utility over which the consumer could exercise control. As one might expect, the two branches entailing policy surrender are only dominated by those entailing persistence if the insurer has priced his policies within certain bounds. Thus, this stage becomes important to the insurer that wishes to avoid the problem of adverse selection. Otherwise, it may end up retaining only the high risks beyond the first period, as the low-risk individuals will be able to get insurance cheaper elsewhere. The consumer then maximizes expected utility subject to the solvency constraints given in the boxes of Figure 1.

Consumer Demand Equilibria and Comparative Statics

To solve a maximization problem of this complexity, dynamic programming would seem appropriate. Unfortunately, an algebraic solution was implausibe. (9) Hence, the authors were compelled to rely on numerical analysis to gain insights into the dynamics of the decision variables over time. Numerical methods were employed to solve the equations for widely ranging values. In selecting ranges of values, several plausible values were substituted for the variables, where the values chosen adhered to certain known relationships. For example, it was assumed that the probability of death increased over time for a healthy individual, that retirement income was lower than income during working years, that term insurance rates increase with age ([[pi].sub.0] [is less than] [[pi].sub.1]) and begin at levels lower than whole life insurance ([[pi].sub.0] [is less than] [pi]), that surrender cash values increase over time ([Q.sub.1] [is less than] [Q.sub.2] [is less than] 1), and that policy loan rates are as high as current market rates but lower than an arbitrarily high level of potential future market interest rates (G [iss less than] [R.sub..h]). All interest rates are assumed to be positive. A further technical requirement arising from the use of logarithmic utility was that C [is greater than] [gamma] and B [is greater than] [beta], to ensure that utility is always defined. Another constraint imposed was that both term and whole life insurance were priced at or above their actuarially fair cost. This article reports the results for a representative range of the many values used in experiments. Details of the numerical analyses are provided in the appendix.

Whole life demand was positive for a wide range of values, as was the demand for term insurance. Many levels of feasible parameter values were found for which there was a simultaneous, positive demand for both types of insurance. This contrasts with other studies, in which only one form of insurance would appear in an optimal portfolio of a rational individual. When loading factors became excessive on the premiums of one type of insurance relative to the other, however, it was not uncommon for an optimal portfolio to include only one kind of insurance.

In the figures below, equilibrium values of whole life and term insurance purchases are shown for two cases: actuarially fair insurance, and standard loaded premiums. Loadings of 27.5 percent and 110 percent, for whole life and term, were chosen to approximate average markups found by Babbel and Staking (1983), who sampled insurance company pricing over a period of three decades. The figures are based on period interval lengths of 15 years, where the initial age of the insured is 35. Final age is 80. Beginning disposable income/wealth is $20,000. To reflect increasing productivity over time, income was assumed to grow at 2 percent per year, in real terms, until the second period, provided that the insured remains healthy. If the individual becomes unhealthy and uninsurable, income is assumed to grow only at the rate of inflation. Inflation rates range from 5 to 8 percent, where the probabilities of attaining either level are given in the appendix.

Below the focus is on the influence of changing insurance prices, wealth, wages, diminished ability income, pensions, risk aversion, and preference for bequest relative to consumption on the demand for term and whole life insurance. (10) Because these equilibrium values and comparative statics were obtained by performing numerical analyses over broad ranges of parameter values, a simple presentation of the partial derivatives would mask the richness of the interrelationships. Therefore, graphs are presented which depict these relationships over fairly broad ranges of parameter values. The graphs are presented in pairs, where the first always relates to the actuarially fair premium base case and the second reflects typical premium loadings. A comparison within each pair reveals the great extent to which standard premium loadings reduce the demand for insurance.

Figure 2 shows the impact of increasing whole life rates on the demand for both term life insurance (Term) and whole life insurance (WL). Note in both graphs the wide range of WL rates for which there is coexisting demand for both kinds of insurance. The left graph reveals how sharply the demand for WL declines as its price rises above actuarially fair levels, while Term's price is held constant at the fair rate. Demand patterns shown are characteristic of goods that have some substitutability. Total insurance in force (the sum of WL and Term) declines modestly as WL prices are increased. The graph to its right, which shows demand under loaded premiums, exhibits a similar pattern to the fair prices case, except with lower total insurance in force at each price interval.

Figure 3 depicts the analogous case for changing Term premiums, with WL premiums held constant at either their fair value or standard markup. Note again how in both graphs there is a wide range of Term rates for which there is concurrent demand for both kinds of insurance. These graphs reconfirm that the consumer will treat WL and Term as substitutes, at least to a limited degree. Of particular note is that when WL is priced fairly, the demand for term insurance evaporates quickly when small loadings are added to term prices. However, when both kinds of insurance have premium loadings, the demand for term does not fall off so quickly as term prices are raised. Total insurance in force goes from around $30,000 to $20,000 when premiums go from fair to loaded.

Figure 4 identifies the relationship between increasing (initial) wealth and insurance demand. Endowed wealth is inheritable in every bequest state and is therefore a partial substitute for the savings element of WL, so one would expect that WL purchases decrease as wealth increases, and this is indeed the case. The effect on Term is ambiguous, however. Depending on the level of wealth increase, and whether premiums are fair or loaded, Term purchases can either increase or decrease. This ambiguous relationship is consistent with virtually every theoretical study on insurance demand and wealth.

In Figures 5a and 5b we see the influence of increasing future wages on the demand for insurance. Under fair insurance, more of both WL and Term is sought as wages rise. When premiums are loaded, almost all of the increased purchases are directed toward Term. This is to be expected. Term guards only against lost wages, whereas WL protects also against lost pension income. When WL is priced fairly, increased wages can lead to increased insurance demand through either vehicle, and any excess WL can be offset by reduced savings. But when insurance prices are marked up, the attractiveness of saving through WL is diminished, and the consumer seeks incremental protection against potential

loss of increased expected wages during earning the periods mostly through Term.

None of the numerical analyses performed revealed any positive demand for second-period Term. Only first-period Term was ever bought. In examining the model, the reason becomes readily apparent. Income is received at the beginning of each period, and insurance protects against a loss of future income. Thus, second-period Term protects only against a loss of retirement income. In practice, Term is rarely available to consumers beyond middle age, and consumers seek guaranteed insurability for their retirement years through WL. The same pattern held true in the three-period model.

In contrast to the previous pair of figures, which showed the influence of increasing good health wages upon insurance demand, Figure 6 illustrates the effect of changing levels of income during periods of poor health on insurance purchases. Recall that under poor health, the consumer is uninsurable via new Term purchases, and is assumed not to survive beyond the end of the second 15-year period. Thus, the only way of providing for bequest is through WL and savings. The figures for both fair and loaded insurance show demand for WL increasing and Term decreasing as disability income becomes a lower percentage of healthy wages. In the loaded premium case, however, the demand for WL reaches a plateau as disability income sinks below a threshold level where the need to maintain adequate consumption precludes devoting additional resources for bequest.

Figure 7 focuses on increasing retirement income's effect on insurance demand. For an individual with a positive bequest motive, an increasing pension or social security benefit would allow for a potentially larger legacy to be made, provided that he or she lives through a retirement period of typical length. Loss of retirement income is compensated only by WL, and as this source of earnings becomes more prominent, the individual seeks to protect it via greater WL purchases.

Figures 8 and 9 show how increased risk aversion with respect to adequate consumption or desired bequest alters the mix of desired insurance. As the minimum standard of living, [gamma], is raised, the individual seeks to provide for bequest progressively with more Term and less WL. Current consumption levels are impaired more by costly WL than by Term expenditures. However, when the insured increases his or her risk aversion for bequest-states wealth, [beta], more WL and less Term insurance are purchased. This observation underscores the different properties of these two kinds of insurance, where WL provides for all bequest states, while also supplementing savings available for use in every state, whereas Term provides only for bequests in states that begin with a healthy individual.

Finally, Figure 10 notes the impact of increasing the relative weight, [rho], on utility for bequest versus consumption. Whether premiums are fair or loaded, more WL and less Term are sought. Although costly WL encroaches more on consumption levels than Term, the diminished relative importance attached to consumption fosters the increased attractiveness of WL.

Several other relationships were uncovered using this framework. For example, increasing together by equal percentages initial wealth, wages, and pensions resulted in higher purchases of Term and WL over all ranges of income studied. This pattern held true when only future wages and pensions were increased together, except that more WL and less Term were sought.

Another useful insight can be obtained by varying the cash value accumulation patterns , [Q.sub.t] and policy loan interest rates, G. This exercise will show the extent to which policy provisions can be modified before the insurer elicits adverse selection from lapsing policyholders, where those branches of the decision tree in Figure 1 that were dominated in utility terms by branches linked to policy persistency become, in turn, the dominant branches.

While it is true that the model presented here has not uncovered any surprising relationships, this is the strongest endorsement of the model. It is consistent with observed patterns of consumer behaviour [see Babbel (1985) and Han, Babbel, and Outreville (1988)], as well as with economic intuition.

Summary

By using a richer model, as was done here, rational consumers are shown to desire both whole life and term insurance under various sets of feasible pricing systems and economic environments. This contrasts with the results of prior economic models, which do not show both term and whole life insurance coexisting in an optimum portfolio. This difference arises from the introduction of stochastic health status and interest rates into the model, which together impute value to some of the options that are part of the whole life insurance product. A further and perhaps more fruitful use of the model is in illustrating how demand would shift among these two products, along with desire for savings, by changing some of the cash value accumulation patterns, premium levels, income, initial weatlh, interest rates, and risk-preference parameters. At certain prices, the demand for insurance (of one type or another) disappears altogether.

Many of the options usually associated with whole life policies were not included as part of the model. These other options could only add to the attractiveness of the whole life policy, other things equal. Finally, while the results were achieved using a class of utility functions that is theoretically attractive, and fulfills most of the rationality requirements that economists feel are important, they nonetheless are valid only within that class of utility functions, and have empirical counterparts only to the extent that consumer preferences can indeed be accurately reflected by utility functions and that these same consumers act in a manner consistent with them.

Many further extensions of the basic model have already been tested, as alluded to in various places throughout this manuscript. However, several extension that have not yet been undertaken can be suggested. The inclusion of uncertain future policy dividends, like those provided by participating whole life policies, would be an interesting addition. Likewise, the introduction of variable policy loan rates would contribute to an understanding of this relatively recent practice. Also, the allowance for partial policy loans would never rationally enter the model, although in practice such partial loans are sometimes observed. Because all cash flows and interest rates were modeled on an after-tax basis, the present model has not captured the effects of changing tax rates explicitly. Techniques for taking full advantage of pre-Tax Reform Act 1986 provisions through a prescribed pattern of policy loans were not modeled. Nonetheless, the model proposed takes some important steps towards enhancing the ability to analyze more fully various types of insurance within a consumption-investment framework.

(1) During most of the past decades, approximately half of new individual (ordinary) life insurance in force has been whole life (see Life Insurance Fact Book, various issues).

(2) While the insurance industry has often maintained that this distinction should be made, the academic literature has been slow to incorporate this view into formal modeling, perhaps due to the much greater analytical complexities introduced by so doing.

(3) The value of a policy loan option has been studied independently of such a framework by Bykerk and Thompson (1979) and several others, most recently by Babbel and Godin (1989). Additionally, the notion of guaranteed reinsurability without incurring a changing premium is present in the work by Venezia and Levy (1982), although there the focus is on the optimal timing of claims. In the case of life insurance, the timing issue is moot, as only a single claim is made for a given policy.

Berkovitch and Venezia (1987), in a paper that simplifies the framework employed here, revisit the multiperiod term life insurance problem treated earlier by Fischer. They find that by introducing stochastic health status, the multiperiod level-premium term policy dominates the single-period variety under equilibrium pricing for initial insurance purchases. Their simplified version of the model first set forth here allows them to obtain closed-form solutions to the optimal insurance problem, although the level of model simplicity does not allow them to examine the common whole life policy.

(4) Although logarithmic utility possesses theoretical properties that are unsurpassed for many multiperiod situations, experimentation was also conducted for a 30-state model using quadratic and exponential utility forms. The results, which can be viewed as approximations for the preferred (logarithmic) functional form, tended to confirm those attained here based on the logarithmic utility.

(5) In an extended version of this model, part of first period income could be expended for a policy rider that provided for a disability income supplement in the second period in the event of bad health. The disability income was then added to the reduced wages, savings accumulations and policy loans, thereby expanding the sources of money available for second-period consumption. Additionally, unlike in the present version, it was assumed that the insured could possibly survive beyond the second period, even if his or her health rendered him or her uninsurable. Furthermore, it allowed for first-period income to vary not only with health status, but also with evolving interest rates. These extensions were dropped from the present version to achieve greater simplicity. They were not deemed essential elements of the model for examining the attractiveness of whole life insurance. The extended model also permitted purchase of high cost term insurance if health was poor. When the assumption was later added that a consumer in poor health would survive only one more period, such a policy became actuarially indistinguishable from saving and so this feature was also deleted.

(6) Under the Tax Reform Act of 1986, interest paid by the insured on policy loans is no longer tax deductible. This treatment strips much of the economic value of the policy loan option away from individuals. See Babbel and Godin (1989) for a full treatment.

(7) Health status, if the individual is alive, is not considered in the third period. Since, whether in good or bad health, the model operates under the assumption that no individuals will survive beyond the end of the third period, and since retirement income is received irrespective of future earning capacity, the variables of the model cannot distinguish between good and bad health, although the state-dependent utility function could make this distinction. For simplicity, this consideration was not included.

(8) A majority of states in the U.S. have recently enacted legislation permitting insurers to change variable policy loan rates tied to market conditions. Even so, many policies continue to feature fixed policy loan rates of interest. The advent of variable policy loan rates and the removal of tax deductibility of interest payments on policy loans, have greatly decreased the value of the policy loan option, a consideration which reduces the attractiveness of the whole life policy, other things equal.

(9) To solve the maximization problem, the authors began with the last period and worked backwards using the procedure of backwards optimization. When the second period was reached, the solutions were becoming very complex and increasingly difficult to obtain. A computer program called "MACSYMA" was enlisted to obtain results and ensure accuracy. This program enables a user to solve algebraic and calculus problems symbolically, and is especially adept at symbolically solving systems of simultaneous equations. Additional nonnegativity constraints on some of the decision variables were not imposed at this stage because the problem had already reached the practical limits of the computer. Rather, solutions were obtained and then experimentation was done with the parameters to determine what levels would result in the nonnegativity criteria being met, without imposing the constraints.

Although it was possible to find closed-form analytical solutions for each of the decision variables for each of the last three time-points, the nature of the results rendered the solutions of little immediate practical value. Solution values for the coefficients of interest each required between 60 to 80 lines of algebraic notation to state. Clearly, further analysis as to whether the solutions values for [T.sub.0], [T.sub.1], and W were globally nonnegative was practically prohibitive. Worse, when attempting to solve the problem for the first time-point, the model involved taking roots of equations of the fifth order and higher, for which no mathematical solutions have yet been discovered.

(10) In an extended version of this paper, available from the authors upon request, we report the influence of these changing parameters on all decision variables in the consumer's problem including the consumption, savings, and bequest decision for each state of nature at every point in time.

REFERENCES

[1.] Arnott, Richard, 1982. "The Structure of Multiperiod Employment Contracts with Incomplete Insurance Markets," Canadian Journal of Economics 15, 51-76.

[2.] Babbel, David F., 1985, "The Price Elasticity of Demand for Whole Life Insurance," Journal of Finance 40, 225-239.

[3.] Babbel, David F. and Kim B. Staking, 1983, "" Capital Budgeting Analysis of Life Insurance Costs in the United States: 1950-1979," Journal of Finance 38, 149-170.

[4.] Babbel, David F. and Marc Godin, 1989, "An Options Approach Toward Valuing Policy Loans," Working Paper, University of Pennsylvania.

[5.] Brown, David P. and Michael R. Gibbons, 1984, "A Simple Econometric Approach for Utility-Based Asset Pricing Models," Research Paper #685R, Graduate School of Business, Stanford University.

[6.] Berkovitch, Elazar and Itzhak Venezia, 1987, "Term vs. Whole Life Insurance: An Equilibrium Approach," Working Paper, University of Michigan.

[7.] Bykerk, Cecil D. and A. Frank Thompson, 1979, "Economic Analysis of the Policy Loan Privilege," Transactions of the Society of Actuaries 31, 261-281.

[8.] Fischer, Stanley, 1973, "A Life Cycle Model of Life Insurance Purchases," International Economic Review 14, 132-152.

[9.] Fortune, Peter, 1973, "A Theory of Optimal Life Insurance: Development and Tests," Journal of Finance 278 587-600.

[10.] Grauer, Robert R. and Nils H. Hakansson, 1982, "Higher Return, Lower Risk: Historical Returns on Long-Run Actively Managed Portfolios of Stocks, Bonds and Bills," Financial Analysts Journal, (March/April), 1-16.

[11.] Hakansson, Nils H., 1969, "Optimal Investment and Consumption Strategies Under Risk, an Uncertain Lifetime, and Insurance," International Economic Review 10, 443-366.

[12.] Hakansson, Nils H., 1970, "Optimal Investment and Consumption Strategies under Risk for a Class of Utility Functions," Econometrica 38, 587-607.

[13.] Hakansson, Nils H., 1971a, "On Optimal Myopic Portfolio Policies, With and Without Serial Correlation of Yields," Journal of Business 44, 324-334.

[14.] Hakansson, Nils H., 1971b, "Multiperiod Mean-Variance Analysis: Toward a Theory of Portfolio Choice," Journal of Finance 26, 857-884.

[15.] Hakansson, Nils H., 1974, "Convergence to Isoelastic Utility and Policy in Multiperiod Choice," Journal of Financial Economics 1, 201-224.

[16.] Hakansson, Nils H., 1979, "A Characterization of Optimal Multiperiod Portfolio Policies," in Portfolio Theory, 25 Years Later, eds. Edwin Elton and Martin Gruber (Amsterdam: North-Holland).

[17.] Han, Li-Ming, David F. Babbel, and J. Francois Outreville, "The Demand

for Life Insurance: A Portfolio Choice," Working Paper, The Wharton School, 1988.

[18.] Jones-Lee, M. W., 1975, "Optimal Life Insurance: Comment," Journal of Finance 30, 902-903.

[19.] Leland, Hayne, 1972, "On Turnpike Portfolios," in Mathematical Methods in Investment and Finance, eds. Karl Shell and G. P. Szego (Amsterdam: North-Holland).

[20.] Klein, Michael R., 1975, "Optimal Life Insurance: Comment," Journal of Finance 30, 904-908.

[21.] Merton, Robert C., 1981, "On the Role of Social Security as a Means for Efficient Risk-Bearing in an Economy where Human Capital is Not Tradeable," Working Paper, Massachusetts Institute of Technology.

[22.] Moffet, Denis, 1979, "An Analysis of the Demand for Life Insurance," Journal of Risk and Insurance 46, 87-112.

[23.] Mossin, Jan, 1968, "Optimal Multiperiod Portfolio Policies," Journal of Business 41, 215-229.

[24.] Outreville, Jean-Francois, 1981, "The Declining Role of Life Insurance Business as a Saving Institution," Geld Banken and Versicherungen, 864-879.

[25.] Richard, Scott F., 1975, "Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain Lived Individual in a Continuous Time Model," Journal of Financial Economics 2, 187-203.

[26.] Ross, Stephen, 1974, "Portfolio Turnpike Theorems for Constant Policies," Journal of Financial Economics 1, 171-198.

[27.] Rubinstein, Mark, 1976, "The Strong Case for the Generalized Logarithmic Utility Model as the Premier Model of Financial Markets," Journal of Finance 31, 551-571.

[28.] Rubinstein, Mark, 1977, "The Strong Case for the Generalized Logarithmic Utility Model as the Premier Model of Financial Markets," in Financial Decision Making Under Uncertainty, eds. H. Levy and M. Sarnat (Academic Press). (This paper is an extended version of that listed directly above and contains many results only alluded to in the shorter version.)

[29.] Samuelson, Paul, 1969, "Lifetime Portfolio Selection by Dynamic Stochastic Programming," Review of Economics and Statistics 51, 239-246.

[30.] Smith, Michael L., 1982, "The Life Insurance Policy as an Options Package," Journal of Risk and Insurance 49, 583-601.

[31.] Townsend, Robert M., 1982, "Optimal Multiperiod Contracts and the Gain from Enduring Relationships under Private Information," Journal of Political Economy 90, 1166-1186.

[32.] Venezia, Itzhak and Haim Levy, 1982, "Optimal Multiperiod Insurance Contracts," Insurance: Mathematics and Economics 2, 199-208.

[33.] Walden, Michael L., 1985, "The Whole Life Policy as an Options Package: An Empirical Investigation," Journal of Risk and Insurance 52, 44-58.

[34.] Winter, Ralph A., 1981, "Financial Securities with the Option of Withdrawal: Market Equilibrium and Regulation," Working Paper, Institute for Policy Analysis, University of Toronto.

[35.] Yaari, Menahem E., 1965, "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," Review of Economic Studies 32, 137-150.

David F. Babbel is Associate Professor at The Wharton School, University of Pennsylvania and currently serves as Insurance Strategist for Goldman, Sachs & Company. Eisaku Ohtsuka is Assistant Professor at Yokohama University. The authors are indebted to Nils Hakansson, John Harsani, David Klock, Robert Merton, Scott Richard, Richard Roll, Steve Ross, Mark Rubinstein, Clifford Smith, and anonymous referees for many helpful suggestions.

ABSTRACT

A multiperiod model is used to analyze aspects of whole life and term insurance contracts within the context of the lifetime consumption-investment problem. Not only the life status, but also the state of health of the consumer is considered in examining optimal insurance purchases. Particular attention is given to the policy loan and guaranteed reinsurability options of the whole life insurance contract. When viewed in this context, whole life insurance, term insurance, and savings are shown likely to coexist in an optimal consumption-investment plan rather than act as substitutes for each other. In this paper it is shown how the bounds of the insurance pricing parameters can be derived to ensure that both kinds of insurance will continue to be sought by rational consumers, while avoiding adverse selection and lapsation.

Introduction

Over the past three decades there has been a conundrum in the economic and business literature. Why do so many people purchase whole life insurance when it has been shown that a combination of term insurance and savings dominates a whole life policy? Equally puzzling is that many people hold both (individual or group) term life and whole life insurance. Are consumers irrational or simply poorly informed?

This article demonstrates that rational, well-informed consumers will choose to hold both term and whole life policies. This result is achieved by considering aspects of whole life insurance which have heretofore escaped formal modeling--features which render the whole life contract distinguishable from linear combinations of term life insurance and a savings program. The simultaneous demand for both kinds of insurance is shown to occur under standard actuarial pricing practices, both with actuarially fair pricing and with typical premium loadings.

This article comes at a juncture characterized by a pause in the demand for whole life insurance. During the past variations of life insurance (e.g., universal life) and new investment vehicles (e.g., interest rate options), which effectively unbundle features of the whole life policy that had rendered it unique, were actively marketed. Consequently, consumers have been afforded the opportunity of seeking only those policy options they deem useful while avoiding payment for the others.

The past several years have raised doubts, however, that these newer insurance policies will ever return a profit to insurers, notwiethstanding a remarkable consumer demand for them. Actuaries claim the problem lies in pricing that is so competitive as to preclude their ability to properly charge for the options granted policyholders. While this situation cannot persist in the long run, it has created a strong desire among some insurers to be more circumspect in the kinds and amounts of policies they write. Many insurers are returning to more traditional products, where the bundling together of options leaves them less susceptible to simultaneous utilization. Most have attempted to offer revised policy provisions in their old book of business that strip the policies of some of their more troublesome options. For example, consumers can opt for a revised contract with a variable-rate policy loan feature and in return receive higher dividends or cash value crediting rates reflecting their utilization of policy loans. Some newly offered whole life policies feature dual premium schedules--a low-premium schedule for insureds who periodically provide evidence of insurability and another rate for those who do not provide such evidence. Other policies link the cash value accumulation patterns to actual market values of the assets supporting these policies.

Most of the innovations in life insurance policies sold today represent some combination of elements in the traditional whole life policy with term insurance and/or a variable savings program. Thus, traditional products remain the starting point in a comprehensive analysis.

Review of the Literature

Among the important advances in financial economics in recent years is the development of multiperiod investment theory. [See, for example, Hakansson (1970, 1971a, 1971b, 1974, 1979), Leland (1971), Mossin (1968), Ross (1974), and Samuelson 1969).] In the wake of this development have come numerous studies focusing on the structure of optimal multiperiod contracts in the presence of informational asymmetries [e.g., Townsend (1982)]. Various types of multiperiod contracts have been investigated. To give a few examples, Arnott (1982) studied the structure of multiperiod employment contracts with incomplete insurance markets, Merton (1981) viewed the role of social security as a means for efficient inter-generational risk bearing in an economy where human capital is not tradable, and Venezia and Levy (1982) derived optimal multiperiod automobile insurance contract.

The ubiquitous multiperiod life insurance policy has also been a subject of theoretical inquiry. Typically the approach taken has been to posit the life insurance policy as a series of single-period or instantaneous term contracts [e.g., Yaari (1965, Hakansson (1969), Richard (1974)]. This approach has meant that some of the complexities associated with the ever popular whole life policy (1) have escaped formal analysis within the context of multiperiod consumption-investment models. The degree of abstraction customarily employed in these models has rendered the whole life policy indistinguishable from a constructible linear combination of single-period term insurance and a savings program of some sort [Richard (1974)], with the latter usually adjudged superior to the former [e.g., Fischer (1973, Moffitt (1979, and Winter (1981)]. Fischer explicitly considered a multiperiod term insurance contract and concluded that it either presents the individual with arbitrage possibilities, or makes no difference to his welfare. Moffitt has extended the analysis to the whole life policy and reached a similar conclusion under well specified conditions. Winter (1981), who took into account the option of withdrawal, also concluded that the fair cash value whole life policy will not be purchased in equilibrium. When fortune (1973) introduced (with a two-period, three time-point model) a policy featuring some aspects usually associated with a whole life policy [e.g., maturity or surrender cash value, level premiums, and (implicitly) guaranteed reinsurability], he was criticized by Klein (1975) and others on the grounds that his whole life policy was little more than a combination of pure insurance and savings, and therefore was dominated by a strategy of purchasing single-period term insurance and investing the difference.

More recently the whole life policy has been perceived as a package of options and not simply a linear combination of single-period term insurance and a savings program [e.g., Smith (1982), Walden (1985), Outreville (1981), and Berkovitch and Venezia (1987)]. (2) Hence, the notion that term and whole life forms of insurance are substitute goods has been called into question. It should be emphasized here that the earlier analyses are correct within the frameworks they propose. Only by adding new dimensions to the framework can the multiperiod life insurance policy be more fully analyzed.

The following model extends the work of Peter Fortune and Stanley Fischer by allowing for an analysis that includes the two most prominent options of a whole life contract--the policy loan option and guaranteed reinsurability option--within the context of a multiperiod consumption-investment framework. (3) Basic features of the whole life policy, such as level premiums and surrender cash values, are also incorporated into the analysis. Other options that are often part of the contract [see Smith (1982) for a list] are not modeled here.

Model Development

The consumer is hypothesized to possess state-dependent utility for consumption, U(C), and bequest V(B). The underlying functional form of consumer utility across all states is assumed to be of the generalized logarithmic class. this class was chosen due to its many attractive theoretical properties [see Rubinstein (1977, 1978) and Hakansson (1979)] as well as its consistency with observed empirical facts [see, for example, Brown and Gibbons (1984) and Grauer and Hakansson (1982)]. (4)

The model proposed here is comprised of three periods (four points in time): two productive earning periods, and a retirement period covered by retirement plan benefits and savings, after which survivors are assumed to reach the end of their lifetimes. Three full periods are the minimum number necessary for adequately modeling the salient features of whole life vs. term insurance.

At the beginning of the first period, the consumer is endowed with wealth and earns a certain income, which sum to [Y.sub.0]. This total wealth is then divided among comsumption, [C.sub.0], and whole life insurance W at a fixed premium rate of [pi] per unit of W, term life insurance [T.sub.0] at a first period rate of [[pi].sub.0], and saving, [S.sub.0] (which may be positive or negative--borrowing), which are placed in a one-period bond (or taken in a single-period loan) that will return (or cost) [R.sub.0.S.sub.0] at the conclusion of the first period. Whenever a new insurance policy is desired, the state of health of the consumer is assessed by a medical expert and reported to both the insurer and the consumer. The premium rates stated above are based upon a clean bill of health.

At the end of the first period, the consumer could be in one of three states of health: good (insurable), bad (uninsurable via new policies), or dead, with probabilities [p.sub.1g.,p.sub.1b.,p.sub.1d.], where the number on the subscript denotes end of the first period and the letter denotes health status. If the insured is dead, no additional income is earned. However, the estate receives a bequest [B.sub.1], comprised of proceeds from any life insurance policies in force, W and [T.sub.0], and savings with interest, [R.sub.0.S.sub.0]. If the consumer survives the first period but is in poor health, he is ineligible for new term life insurance, but is assumed to maintain in force whatever amount of whole life insurance had earlier been purchased by making an additional premium payment of [pi]. The consumer has available for consumption reduced wages [Y.sub.1b], if any, as well as [R.sub.0.S.sub.0] plus the proceeds from any policy loan undertaken, [Q.sub.1.W]. (5) The latter is assumed to be incurred whenever the second-period (after-tax) market rate of interest exceeds that charged on policy loans (after tax). (6) This loan can be used as a low cost source of funds for current consumption or for additional investment at the favorable, higher market interest rate. Second-period consumption and saving in the event of poor health will depend, in part, upon investment opportunities, captured in the model by (one plus) the interest rate, [R.sub.1l] (l for low rate) or [R.sub.1h] (h for high rate). Subscripts are placed on the interest, savings, consumption and bequest variables throughout to indicate point in time (0, 1, 2, 3).

The individual who retains goods health until the start of the second period may choose whether to purchase a new term policy at the revised rates for newly issued policies, [[pi].sub.1h] of [[pi].sub.1l]. These rates will reflect the current interest rate levels, [R.sub.1h] or [R.sub.1l], as well as any updated information about the mortality experience obtained from period 1. The individual can either surrender or continue the whole life policy. If he or she elects to maintain in force the whole life insurance, he or she may choose to obtain a policy loan either for consumption or investment provided that market rates of interest are high. (If markt rates are low, the consumer would do better to access funds at the lower market rates.) The individual will again need to allocate available funds from earned income ([Y.sub.1g], physical capital ([R.sub.0.S.sub.0]}, and any policy loans ([Q.sub.1.W]) among current consumption, insurance and savings.

Individuals who were uninsurable at the start of the second period are assumed not to survive beyond the end of the period. At this time, their estates are settled, with any remaining funds going toward their bequests. If a policy loan had been taken out earlier, the loan plus interest ([GQ.sub.1.W]) is subtracted from the proceeds of the whole life policy.

Individuals who started the second period in good health may either be alive or dead (a or d) at the start of the third period. (7) If they are not alive, their estates are settled along similar lines to that described earlier, where the amount available for request will depend, in part, upon prior decisions about what kind and how much insurance to purchase. If alive, they once again are faced with allocating retirement income and wealth among consumption, savings, and whole life insurance (provided that the policy has continued in force). The option of policy surrender at this late date was not modeled explicitly, because during the last period, the amount available for a policy loan should equal the present value of a sure payment of W one period hence, or arbitrage opportunities present themselves. If higher, the loan could be taken and later repaid at the bargain rates. If lower, it would be better to obtain desired funds through an outside loan loan, because the cash values are accumulating at more favorable rates than those available outside. In neither case would the policy be surrendered.

AT the end of the retirement period, all available funds net of any loan repayments become the bequest available for heirs. The amount of these funds will depend upon prior decisions, and is influenced by the level of interest rates during the final period. In Figure 1 on the following page, the cash flows and decision variables associated with each of the 30 states described above are presented. To assist in deciphering the diagram, the notation is reviewed below:

[Y.sub.t]: Wages (after-tax) received by individual at time t. When t = 0, Y includes accumulated physical wealth.

[S.sub.t]: Savings undertaken or (if negative) loans incurred at time t.

[C.sub.t]: Consumption at time t.

W: Face amount of whole life insurance policy.

[T.sub.t]: Face amount of term life insurance policy at time t.

[B.sub.t]: Amount of bequest at time t.

[R.sub.t]: One plus the after-tax market rate of interest for saving or borrowing (assumed to be equal) at time t.

[Q.sub.t]: The proportion of the face amount of the whole life policy available for policy loan at time t.

G: One plus the after-tax rate of interest charged on policy loans (assumed to be fixed throughout life of policy). (8)

[P.sub.t]: Probability of occurrence of stated event at time t.

[pi]: The whole life insurance premium rate, per dollar of insurance in force. This periodc premium rate is fixed throughout the life of the policy.

[[pi].sub.0]: The term insurance premium rate, per dollar of insurance in force during the first period.

[[pi].sub.1]: The term insurance premium rate, per dollar in insurance in force during the second period, available only to individuals who have retained their good health.

Additional subscripts are attached to certain variables to clarify the state paths over time of these variables, or to indicate present state. Where additional subscripts are present, they refer to states of consumer health, interest rate levels, and consumer action as follows:

g,b,d,a: g and b refer to state of consumer health at time 1, where g indicates good, and b indicates bad (hence, uninsurable); d indicates dead and a indicates alive.

h,l: Market interest rate is high or low.

s,p,c: Refers to consumer choice at time 1 to surrender his policy, take out a policy loan, or continue keeping his or her insurance in force without taking out a policy loan, respectively.

Utility is assumed to be of the generalized logarithmic class, such that U([C.sub.t..]) = ln([C.sub.t.] - [[gamma].sub.t.]) and U([B.sub.t.]) = ln([B.sub.t.] - [[beta].sub.t.]), where [[gamma].sub.t.] and [[beta].sub.t.] can be interpreted as the minimum tolerable standard of living and minimum acceptable bequest provision, respectively, for period t and state ".". As noted by Rubinstein, the assumption of generalized logarithmic utility is not very restrictive. While all consumers are required by this functional form to have decreasing absolute risk aversion (theoretically preferred), the heterogeneity of the model permits consumers to have separate taste parameters [[gamma].sub.t.] and [[beta].sub.t.]for their consumption and bequest preference functions for each state and at each point in time, and tolerates quite diverse attitudes of proportional risk aversion. [gamma] (and/or [beta]) may be positive, zero, or negative, implying decreasing, constant or increasing proportional risk aversion, respectively. Presumably, in the context of this model, [gamma] must be positive to provide for at least some consumption, but [beta] could be either zero or positive. The higher the [gamma] (and/or [beta]), the more risk averse the consumer. Furthermore, utility is influenced by a separate state-dependent risk preference parameter [rho.] as well as a time- preference factor [[rho].sub.t]. Thus, the model accommodates a wide variety of individual time and state preference forms. (To simplify notation and computation, without any loss of generality, the state probabilities [rho][.sub.t.] can be combined with the state-department risk preference parameter [rho.] into a single parameter [[rho].sub.t.].)

The problem for the consumer, then, is to maximize exptected utility [Omega] of consumption and bequest across all periods and states by selecting appropriate levels of consumption, insurance, and investment, while making the optimal choice at time 1: surrender policy, persist with policy and incur a policy loan, or persist without incurring a policy loan. The formal problem is presented in the following equations:

[Omega] = U([C.sub.0])

[Mathematical Expression Omitted]

Observe that there are fewer arguments in the multiperiod utility function than there are states shown in Figure 1. This is because beginning with the second period, two branches of the decision tree will be dominated by their alternatives; in particular, the consumer in good health will choose either to surrender or take out a policy loan if interest rates are high in the second period, or either continue with his policy or surrender it if interest rates are low in the second period. The decision depends upon which is associated with higher utility for the consumer. Thus, the first task was to compute the utilities associated with each branch and eliminate those branches with lower utility over which the consumer could exercise control. As one might expect, the two branches entailing policy surrender are only dominated by those entailing persistence if the insurer has priced his policies within certain bounds. Thus, this stage becomes important to the insurer that wishes to avoid the problem of adverse selection. Otherwise, it may end up retaining only the high risks beyond the first period, as the low-risk individuals will be able to get insurance cheaper elsewhere. The consumer then maximizes expected utility subject to the solvency constraints given in the boxes of Figure 1.

Consumer Demand Equilibria and Comparative Statics

To solve a maximization problem of this complexity, dynamic programming would seem appropriate. Unfortunately, an algebraic solution was implausibe. (9) Hence, the authors were compelled to rely on numerical analysis to gain insights into the dynamics of the decision variables over time. Numerical methods were employed to solve the equations for widely ranging values. In selecting ranges of values, several plausible values were substituted for the variables, where the values chosen adhered to certain known relationships. For example, it was assumed that the probability of death increased over time for a healthy individual, that retirement income was lower than income during working years, that term insurance rates increase with age ([[pi].sub.0] [is less than] [[pi].sub.1]) and begin at levels lower than whole life insurance ([[pi].sub.0] [is less than] [pi]), that surrender cash values increase over time ([Q.sub.1] [is less than] [Q.sub.2] [is less than] 1), and that policy loan rates are as high as current market rates but lower than an arbitrarily high level of potential future market interest rates (G [iss less than] [R.sub..h]). All interest rates are assumed to be positive. A further technical requirement arising from the use of logarithmic utility was that C [is greater than] [gamma] and B [is greater than] [beta], to ensure that utility is always defined. Another constraint imposed was that both term and whole life insurance were priced at or above their actuarially fair cost. This article reports the results for a representative range of the many values used in experiments. Details of the numerical analyses are provided in the appendix.

Whole life demand was positive for a wide range of values, as was the demand for term insurance. Many levels of feasible parameter values were found for which there was a simultaneous, positive demand for both types of insurance. This contrasts with other studies, in which only one form of insurance would appear in an optimal portfolio of a rational individual. When loading factors became excessive on the premiums of one type of insurance relative to the other, however, it was not uncommon for an optimal portfolio to include only one kind of insurance.

In the figures below, equilibrium values of whole life and term insurance purchases are shown for two cases: actuarially fair insurance, and standard loaded premiums. Loadings of 27.5 percent and 110 percent, for whole life and term, were chosen to approximate average markups found by Babbel and Staking (1983), who sampled insurance company pricing over a period of three decades. The figures are based on period interval lengths of 15 years, where the initial age of the insured is 35. Final age is 80. Beginning disposable income/wealth is $20,000. To reflect increasing productivity over time, income was assumed to grow at 2 percent per year, in real terms, until the second period, provided that the insured remains healthy. If the individual becomes unhealthy and uninsurable, income is assumed to grow only at the rate of inflation. Inflation rates range from 5 to 8 percent, where the probabilities of attaining either level are given in the appendix.

Below the focus is on the influence of changing insurance prices, wealth, wages, diminished ability income, pensions, risk aversion, and preference for bequest relative to consumption on the demand for term and whole life insurance. (10) Because these equilibrium values and comparative statics were obtained by performing numerical analyses over broad ranges of parameter values, a simple presentation of the partial derivatives would mask the richness of the interrelationships. Therefore, graphs are presented which depict these relationships over fairly broad ranges of parameter values. The graphs are presented in pairs, where the first always relates to the actuarially fair premium base case and the second reflects typical premium loadings. A comparison within each pair reveals the great extent to which standard premium loadings reduce the demand for insurance.

Figure 2 shows the impact of increasing whole life rates on the demand for both term life insurance (Term) and whole life insurance (WL). Note in both graphs the wide range of WL rates for which there is coexisting demand for both kinds of insurance. The left graph reveals how sharply the demand for WL declines as its price rises above actuarially fair levels, while Term's price is held constant at the fair rate. Demand patterns shown are characteristic of goods that have some substitutability. Total insurance in force (the sum of WL and Term) declines modestly as WL prices are increased. The graph to its right, which shows demand under loaded premiums, exhibits a similar pattern to the fair prices case, except with lower total insurance in force at each price interval.

Figure 3 depicts the analogous case for changing Term premiums, with WL premiums held constant at either their fair value or standard markup. Note again how in both graphs there is a wide range of Term rates for which there is concurrent demand for both kinds of insurance. These graphs reconfirm that the consumer will treat WL and Term as substitutes, at least to a limited degree. Of particular note is that when WL is priced fairly, the demand for term insurance evaporates quickly when small loadings are added to term prices. However, when both kinds of insurance have premium loadings, the demand for term does not fall off so quickly as term prices are raised. Total insurance in force goes from around $30,000 to $20,000 when premiums go from fair to loaded.

Figure 4 identifies the relationship between increasing (initial) wealth and insurance demand. Endowed wealth is inheritable in every bequest state and is therefore a partial substitute for the savings element of WL, so one would expect that WL purchases decrease as wealth increases, and this is indeed the case. The effect on Term is ambiguous, however. Depending on the level of wealth increase, and whether premiums are fair or loaded, Term purchases can either increase or decrease. This ambiguous relationship is consistent with virtually every theoretical study on insurance demand and wealth.

In Figures 5a and 5b we see the influence of increasing future wages on the demand for insurance. Under fair insurance, more of both WL and Term is sought as wages rise. When premiums are loaded, almost all of the increased purchases are directed toward Term. This is to be expected. Term guards only against lost wages, whereas WL protects also against lost pension income. When WL is priced fairly, increased wages can lead to increased insurance demand through either vehicle, and any excess WL can be offset by reduced savings. But when insurance prices are marked up, the attractiveness of saving through WL is diminished, and the consumer seeks incremental protection against potential

loss of increased expected wages during earning the periods mostly through Term.

None of the numerical analyses performed revealed any positive demand for second-period Term. Only first-period Term was ever bought. In examining the model, the reason becomes readily apparent. Income is received at the beginning of each period, and insurance protects against a loss of future income. Thus, second-period Term protects only against a loss of retirement income. In practice, Term is rarely available to consumers beyond middle age, and consumers seek guaranteed insurability for their retirement years through WL. The same pattern held true in the three-period model.

In contrast to the previous pair of figures, which showed the influence of increasing good health wages upon insurance demand, Figure 6 illustrates the effect of changing levels of income during periods of poor health on insurance purchases. Recall that under poor health, the consumer is uninsurable via new Term purchases, and is assumed not to survive beyond the end of the second 15-year period. Thus, the only way of providing for bequest is through WL and savings. The figures for both fair and loaded insurance show demand for WL increasing and Term decreasing as disability income becomes a lower percentage of healthy wages. In the loaded premium case, however, the demand for WL reaches a plateau as disability income sinks below a threshold level where the need to maintain adequate consumption precludes devoting additional resources for bequest.

Figure 7 focuses on increasing retirement income's effect on insurance demand. For an individual with a positive bequest motive, an increasing pension or social security benefit would allow for a potentially larger legacy to be made, provided that he or she lives through a retirement period of typical length. Loss of retirement income is compensated only by WL, and as this source of earnings becomes more prominent, the individual seeks to protect it via greater WL purchases.

Figures 8 and 9 show how increased risk aversion with respect to adequate consumption or desired bequest alters the mix of desired insurance. As the minimum standard of living, [gamma], is raised, the individual seeks to provide for bequest progressively with more Term and less WL. Current consumption levels are impaired more by costly WL than by Term expenditures. However, when the insured increases his or her risk aversion for bequest-states wealth, [beta], more WL and less Term insurance are purchased. This observation underscores the different properties of these two kinds of insurance, where WL provides for all bequest states, while also supplementing savings available for use in every state, whereas Term provides only for bequests in states that begin with a healthy individual.

Finally, Figure 10 notes the impact of increasing the relative weight, [rho], on utility for bequest versus consumption. Whether premiums are fair or loaded, more WL and less Term are sought. Although costly WL encroaches more on consumption levels than Term, the diminished relative importance attached to consumption fosters the increased attractiveness of WL.

Several other relationships were uncovered using this framework. For example, increasing together by equal percentages initial wealth, wages, and pensions resulted in higher purchases of Term and WL over all ranges of income studied. This pattern held true when only future wages and pensions were increased together, except that more WL and less Term were sought.

Another useful insight can be obtained by varying the cash value accumulation patterns , [Q.sub.t] and policy loan interest rates, G. This exercise will show the extent to which policy provisions can be modified before the insurer elicits adverse selection from lapsing policyholders, where those branches of the decision tree in Figure 1 that were dominated in utility terms by branches linked to policy persistency become, in turn, the dominant branches.

While it is true that the model presented here has not uncovered any surprising relationships, this is the strongest endorsement of the model. It is consistent with observed patterns of consumer behaviour [see Babbel (1985) and Han, Babbel, and Outreville (1988)], as well as with economic intuition.

Summary

By using a richer model, as was done here, rational consumers are shown to desire both whole life and term insurance under various sets of feasible pricing systems and economic environments. This contrasts with the results of prior economic models, which do not show both term and whole life insurance coexisting in an optimum portfolio. This difference arises from the introduction of stochastic health status and interest rates into the model, which together impute value to some of the options that are part of the whole life insurance product. A further and perhaps more fruitful use of the model is in illustrating how demand would shift among these two products, along with desire for savings, by changing some of the cash value accumulation patterns, premium levels, income, initial weatlh, interest rates, and risk-preference parameters. At certain prices, the demand for insurance (of one type or another) disappears altogether.

Many of the options usually associated with whole life policies were not included as part of the model. These other options could only add to the attractiveness of the whole life policy, other things equal. Finally, while the results were achieved using a class of utility functions that is theoretically attractive, and fulfills most of the rationality requirements that economists feel are important, they nonetheless are valid only within that class of utility functions, and have empirical counterparts only to the extent that consumer preferences can indeed be accurately reflected by utility functions and that these same consumers act in a manner consistent with them.

Many further extensions of the basic model have already been tested, as alluded to in various places throughout this manuscript. However, several extension that have not yet been undertaken can be suggested. The inclusion of uncertain future policy dividends, like those provided by participating whole life policies, would be an interesting addition. Likewise, the introduction of variable policy loan rates would contribute to an understanding of this relatively recent practice. Also, the allowance for partial policy loans would never rationally enter the model, although in practice such partial loans are sometimes observed. Because all cash flows and interest rates were modeled on an after-tax basis, the present model has not captured the effects of changing tax rates explicitly. Techniques for taking full advantage of pre-Tax Reform Act 1986 provisions through a prescribed pattern of policy loans were not modeled. Nonetheless, the model proposed takes some important steps towards enhancing the ability to analyze more fully various types of insurance within a consumption-investment framework.

(1) During most of the past decades, approximately half of new individual (ordinary) life insurance in force has been whole life (see Life Insurance Fact Book, various issues).

(2) While the insurance industry has often maintained that this distinction should be made, the academic literature has been slow to incorporate this view into formal modeling, perhaps due to the much greater analytical complexities introduced by so doing.

(3) The value of a policy loan option has been studied independently of such a framework by Bykerk and Thompson (1979) and several others, most recently by Babbel and Godin (1989). Additionally, the notion of guaranteed reinsurability without incurring a changing premium is present in the work by Venezia and Levy (1982), although there the focus is on the optimal timing of claims. In the case of life insurance, the timing issue is moot, as only a single claim is made for a given policy.

Berkovitch and Venezia (1987), in a paper that simplifies the framework employed here, revisit the multiperiod term life insurance problem treated earlier by Fischer. They find that by introducing stochastic health status, the multiperiod level-premium term policy dominates the single-period variety under equilibrium pricing for initial insurance purchases. Their simplified version of the model first set forth here allows them to obtain closed-form solutions to the optimal insurance problem, although the level of model simplicity does not allow them to examine the common whole life policy.

(4) Although logarithmic utility possesses theoretical properties that are unsurpassed for many multiperiod situations, experimentation was also conducted for a 30-state model using quadratic and exponential utility forms. The results, which can be viewed as approximations for the preferred (logarithmic) functional form, tended to confirm those attained here based on the logarithmic utility.

(5) In an extended version of this model, part of first period income could be expended for a policy rider that provided for a disability income supplement in the second period in the event of bad health. The disability income was then added to the reduced wages, savings accumulations and policy loans, thereby expanding the sources of money available for second-period consumption. Additionally, unlike in the present version, it was assumed that the insured could possibly survive beyond the second period, even if his or her health rendered him or her uninsurable. Furthermore, it allowed for first-period income to vary not only with health status, but also with evolving interest rates. These extensions were dropped from the present version to achieve greater simplicity. They were not deemed essential elements of the model for examining the attractiveness of whole life insurance. The extended model also permitted purchase of high cost term insurance if health was poor. When the assumption was later added that a consumer in poor health would survive only one more period, such a policy became actuarially indistinguishable from saving and so this feature was also deleted.

(6) Under the Tax Reform Act of 1986, interest paid by the insured on policy loans is no longer tax deductible. This treatment strips much of the economic value of the policy loan option away from individuals. See Babbel and Godin (1989) for a full treatment.

(7) Health status, if the individual is alive, is not considered in the third period. Since, whether in good or bad health, the model operates under the assumption that no individuals will survive beyond the end of the third period, and since retirement income is received irrespective of future earning capacity, the variables of the model cannot distinguish between good and bad health, although the state-dependent utility function could make this distinction. For simplicity, this consideration was not included.

(8) A majority of states in the U.S. have recently enacted legislation permitting insurers to change variable policy loan rates tied to market conditions. Even so, many policies continue to feature fixed policy loan rates of interest. The advent of variable policy loan rates and the removal of tax deductibility of interest payments on policy loans, have greatly decreased the value of the policy loan option, a consideration which reduces the attractiveness of the whole life policy, other things equal.

(9) To solve the maximization problem, the authors began with the last period and worked backwards using the procedure of backwards optimization. When the second period was reached, the solutions were becoming very complex and increasingly difficult to obtain. A computer program called "MACSYMA" was enlisted to obtain results and ensure accuracy. This program enables a user to solve algebraic and calculus problems symbolically, and is especially adept at symbolically solving systems of simultaneous equations. Additional nonnegativity constraints on some of the decision variables were not imposed at this stage because the problem had already reached the practical limits of the computer. Rather, solutions were obtained and then experimentation was done with the parameters to determine what levels would result in the nonnegativity criteria being met, without imposing the constraints.

Although it was possible to find closed-form analytical solutions for each of the decision variables for each of the last three time-points, the nature of the results rendered the solutions of little immediate practical value. Solution values for the coefficients of interest each required between 60 to 80 lines of algebraic notation to state. Clearly, further analysis as to whether the solutions values for [T.sub.0], [T.sub.1], and W were globally nonnegative was practically prohibitive. Worse, when attempting to solve the problem for the first time-point, the model involved taking roots of equations of the fifth order and higher, for which no mathematical solutions have yet been discovered.

(10) In an extended version of this paper, available from the authors upon request, we report the influence of these changing parameters on all decision variables in the consumer's problem including the consumption, savings, and bequest decision for each state of nature at every point in time.

REFERENCES

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David F. Babbel is Associate Professor at The Wharton School, University of Pennsylvania and currently serves as Insurance Strategist for Goldman, Sachs & Company. Eisaku Ohtsuka is Assistant Professor at Yokohama University. The authors are indebted to Nils Hakansson, John Harsani, David Klock, Robert Merton, Scott Richard, Richard Roll, Steve Ross, Mark Rubinstein, Clifford Smith, and anonymous referees for many helpful suggestions.

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Author: | Babbel, David F.; Ohtsuka, Eisaku |
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Publication: | Journal of Risk and Insurance |

Date: | Sep 1, 1989 |

Words: | 7191 |

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