In his seminal work from 1965, Yaari shows that under uncertainty of lifetime a utility maximizing decision-maker purchases life insurance to make the most of her lifetime consumption. Yaari distinguishes between two approaches:
In the Fisherian model, an individual maximizes her utility from lifetime consumption under the constraint that her wealth is 0 at the time of her death. Under these assumptions life insurance enables the individual to increase her consumption choices by means of borrowings. At the point of death, life insurance ensures that the individual’s wealth constraint is fulfilled by covering any remaining debt, thus, allowing the consumer to maximize her expected utility over lifetime. This setting is often the initial situation in the context of credit life micro insurance. The sum insured equals the financial obligations of the policyholders and the lumpsum payout covers for her remaining debt in case of death. Also for non-credit linked term life micro insurance, providing for debt could be one motivation to insure.
The Marshallian model is different insofar as an individual’s utility from consumption is constrained by her need for bequest. In this model, a consumer’s utility over lifetime depends on his discounted consumption rate at every moment in time and her final wealth at the time of death, that is her bequeath. Yaari assumes that the bequest is always larger or equal to zero. Like in the Fisherian model, life insurance allows the consumer to increase current consumption. In the Marshallian model, however, life insurance is bought up to the point where the utility from current consumption is equal to its utility of bequest. As the utility of bequest depends on timing and weighted subjectively life insurance demand further depends on a so-called intensity for bequest. Similar to the purpose of protecting against the decedents’ loan obligations, micro life insurance could be purchased to provide financial support to the dependents and hence serves the intention to bequest. Contrary to traditional retail life insurance though, the maximum sum insured is much smaller and the policy duration shorter in term life micro insurance. For example, in Sri Lanka, as explicated in Section 1.6.3, the available products cover up to LKR 4 million (approx. EUR 22,280) but average sum insured is LKR 50,000 (approx. EUR 278,50) and are either annually renewable or run up to 5 years. The author would expect that an individual who is motivated by a bequest motive to repeatedly renew his term life policy. Because of the limited average sum insured, which still covers the average monthly income of a household (LKR 7,666; approx. EUR 278,50) for more than six months, she expects an effect of smaller scale of a bequest motive in term life micro insurance.
Following the approach of Yaari, Fischer (1973) explicates the effect and individual importance of a bequest motive. He shows that an individual with a large weighting on the bequest function would buy even heavily loaded insurance, whereas an individual with a low intensity for bequest would abstain from purchasing life insurance, even if it is actuarially fair or sold at a subsidized premium rate. He further demonstrates that insurance demand increases, the higher the weighting attached to the bequest motive, which in turn reduces current consumption. On the other hand, if the weighting attached to bequest is low and the probability of death increases, insurance demand decreases as current consumption grows. For the case of micro term life insurance, it is expected that even though premiums are loaded with operational and distribution expenses, a bequest motive could support its uptake.
Uncertainty about labor income
Introducing labor income to his model, Fischer (1973) illustrates that a wage earner who plans to leave an inheritance is more likely to buy life insurance than an individual living off her wealth and that the amount of coverage positively relates to the future labor income. Life insurance is purchased to compensate for a loss of income the policyholder would share with her family. Further, a wage earner is more likely to purchase life insurance early in her lifetime. Campbell (1980) takes up the point of Fischer (1973) that some individual cares more about uncertainty related to her income streams from labor than from her assets. In his model, the main and sole breadwinner of the household is maximizing her utility from current wealth and future labor income. Until the retirement of the breadwinner, the utility from future human capital income is uncertain as it depends on the survival of the breadwinner. To transfer the uncertainty of future labor income streams, the breadwinner can purchase term life insurance. Insurance premium is calculated based on the expected loss and a loading factor. Due to the law of large numbers, the insurance company knows the expected loss, whereas the probability of death assumed by the risk averse household might be biased. The optimal amount of term life insurance is then determined by the future expected income of the household, reduced by a proportion of her total wealth, whereby the proportion of wealth is dependent on the households’ intensity for bequest, its risk aversion and perception of the insurer’s loading fee. Campbell finds that the demand for term life insurance of a risk averse household with an interest in leaving an inheritance increases
if (i) her intensity of bequest or level of risk aversion increases, or (ii) the perceived loading factor of the insurer decreases, or (iii) she is overestimating her death probability and at the same time underestimating the loading factor of the insurer. In sum, the models of Fischer and Campbell emphasize the relevance of a bequest motive by introducing a second factor that defines the scope of a bequest motive: the protection against lost income flows of the breadwinner in case she dies. In the case of micro term life insurance, it could be argued that a bequest motive related to lost labor income is irrelevant because the target group does not have regular, monthly income flows or because the total sum insured is relatively small and hence is not able cover lost labor income over a long period. Even though, income flows are unsteady they guarantee a household’s survival and with the average sum insured of term life micro insurance products available in the region under study, a payout of LKR 50,000 (approx. EUR 278.50) compensates the average monthly income of LKR 7,666 (approx. EUR 42.44) for six and a half months.
Campbell further shows that the optimal amount of insurance is less than full insurance coverage, that is the present value of the wage earners future income flows. Up to the amount of her accumulated assets, the household can self-insure its human capital income. Full insurance is only optimal if the breadwinners’ utility from consumption, conditional on her survival, is equal to her utility from bequest, and if the actuarially fair premium charged by the insurer is perceived as such by the household. The possibility of self-insurance further explains the fact that older households demand less insurance. Over their lifetime, these households have built up a higher stock of assets and their expected income from human capital decreases. This theoretical approach is of special interest to the field study because it introduces the possibility of self-insurance and a life-cycle effect, the latter meaning that life insurance consumption is related to a person’s age and working years. If self-insurance is a suitable instrument for low- income households must still be assessed, as their capacity to save and accumulate assets is usually limited.
Preferences of the dependents
In 1989, Lewis published another modification of Yaari’s idea of a bequest motive in a life cycle model under uncertainty. Lewis directly incorporates the preferences of the dependents in the consumption function of the breadwinner, and provides a first
analysis of the demand for life insurance according to the preferences of dependents. He assumes that the breadwinner of a family regularly conducts income transfers to her dependents. Hence, the utility function of the children and spouse depends on the breadwinner’s uncertain income. A utility maximizing dependent will prefer that some income is allocated to insure the ability of the breadwinner to generate and transfer income beyond the breadwinner’s lifetime. In his model the amount of life insurance increases with the breadwinner’s probability of death, present value of the dependents consumption and their degree of risk aversion while it decreases with the premium loading factor and the household’s wealth. Similar to Fischer’s model, insurance is understood as a measure to secure future income flows. However, Lewis incorporates that the decision not only depends on the breadwinner’s utility, but on the utility of the family.
Overall, the models presented in this section, and summarized in Table 7 (p. 48), assume that life insurance allows an individual, who does not know how long she will live, to increase her utility either from consumption or a bequeath motive. In the context of term life micro insurance the models provide the theoretical foundations for the relevance of a a bequest motive (relative to the premium loadings), labor income (in dependence of a bequest motive, risk aversion, death probability, accumulated assets, and premium loadings) and a lifecycle effect for micro life insurance consumption. These expected utility theory models assume complete life insurance markets. The following section describes life insurance demand behavior if this assumption is released.