The role of annuities and life insurance

6. The role of annuities and life insurance

A frequent concern in the economic literature on insurance is the potentially adverse effect of insurance on self-protection, commonly termed ‘‘moral hazard’’. Do actuarially fair markets for annuities and life insurance create such a hazard in Ž . connection with life protection? Following the approach pursued in EB 1972 , the question could be answered by assessing how the introduction of complete insurance markets affects the shadow price of the conditional mortality risk, or Ž . value of life saving, in its general form, as specified in Eq. 5.6 . To make things comparable, assume that individuals have a bequest motive, and that all the parameters of the model are invariant to the opening of insurance Ž . markets in period 0, including the initial assets’ level i.e., A s A . In the 10 Ž . U Ž .w Ž .x absence of any insurance, Eq. 5.6 changes to Õ s 1rJ J y V A , while A U Ž .w Ž U .x under actuarially fair insurance markets it becomes Õ s 1rJ J y V B q 1 1 A 1 U Ž . Ž . B t y A t . The difference between these two value-of-life-saving levels can be assessed as follows: Õ U y Õ U s D q D , 6.1 Ž . 1 1 2 Ž . Ž . w Ž Ž . . x w Ž Ž U . where D J rJ y JrJ , and D V A rJ y A y V B r 1 1 1 A A 2 A X Ž U .. U x X Ž U . V B y B after replacing J with V B in the expression for Õ , based 1 A 1 Ž . on the optimality condition for bequest in Eq. 5.8 . Ž . The term D in Eq. 6.1 is expected to be positive in sign, essentially because 1 the opportunity to trade in the market for actuarial notes enables individuals to choose optimal paths of future bequest and consumption independently of each other. Moreover, this market expands the effective wealth constraint at any given Ž . Ž . Ž . Ž . period from A t G 0 see Section 3 to A t q L t G 0. The resulting added efficiency in inter-temporal resource allocation implies that D G 0. 1 Ž . The term D summarizes two additional outcomes: 1 A change in actual 2 bequest level, because of the opportunity to select an optimal amount of this Ž . Ž commodity; 2 A reduction in the opportunity cost of bequest the life insurance . premium but also in the extra return the insurer can offer on annuities, because of the fall in mortality risk. The strict concavity of the utility function implies that D 2 is positive if A B U and negative if A - B U . 15 A fall in actual bequest relative to accidental bequest — i.e., A B U — occurs if a person takes advantage of the opening of insurance markets to purchase annuities as well as, possibly, life insurance as a substitute for regular savings. 15 To prove this proposition, recall the familiar property of a strictly concave function that w Ž . X Ž .x 4 w Ž . X Ž .x 4 Ž . V x r V x y x V y r V y y y whenever x y. Since V x is unique up to a linear Ž . transformation, one may set V A s 0 without loss of generality. Consequently, one may substitute in Ž . Ž . X Ž . Ž . Eq. 6.1 V A r V A s 0 for V A r J , and the concavity property of V just cited implies that A Ž . Ž . U D - 0 as A - B . 2 This outcome is possible since in the pre-insurance period, accumulated savings serves as a means of self-insuring both future consumption needs and bequest. The level of accidental bequest implied by savings may exceed the desired level of bequest, given the ability to set it independently. The purchase of annuities eliminates the inefficiency implicit in such accidental bequest, and in this case, as Ž . is formally proven see Footnote 15 , there is an unambiguous increase in the value of life saving. The intuitive reason is that if the optimally chosen bequest is Ž . smaller than the previously uncontrolled accidental bequest, the disparity be- tween the utility levels in the states ‘‘alive’’ and ‘‘dead’’ becomes larger, and this increases the private value of life protection. It is also possible, however, that for people with a high bequest preference, the previously accumulated amount of regular savings falls short of the desired level of bequest they can now secure by borrowing against their future earnings. The inequality A - B U , signaling an increase in the optimal level of bequest, is made possible through the selling of actuarial notes; i.e., if individuals purchase only life insurance, but no annuities. 16 Utilizing the insurance system exclusively to promote the welfare of future generations is expected to lower the value of one’s own life saving, as our analysis of the distinct role of bequest preferences has shown. What if individuals have no bequest motive before or after the introduction of fair insurance? By the preceding analysis, Õ Õ , when A B U 0. But by Eq. 1 Ž . 6.1 , the value of life saving is a monotonically decreasing function of relative bequest preferences, so that when the latter approaches zero, Õ peaks By 1 transitivity we conclude, therefore, that persons with no bequest motive stand to benefit the most from the opening of insurance markets by specializing in annuity purchasing. The interesting conclusion of this analysis is that the emergence of actuarially fair insurance markets will necessarily raise the value of life saving, and hence the demand for life protection, as long as individuals take advantage of these markets to purchase annuities as well as life insurance. This generalizes the analysis of Ž . ‘‘moral hazard’’ in EB 1972 , where the introduction of fair insurance had a more ambiguous effect on self-protection. Furthermore, these results are independent of any specific utility function or age-earning profile, and they may apply even under actuarially unfair insurance 16 Defining the stock values of ordinary saving notes, actuarial notes, and life insurance by S, Q, and N, respectively, one can write B U s Sq N and A s Qq B U . In the present case where A s A 10 10 and B U 0 by assumption, if A y B U 0, then Q 0 and Sq N 0. Both annuities and life insurance may be purchased in this case. But if A y B U - 0 then Q- 0 and Sq N 0: To effect an increase in bequest above initial assets a person must sell an actuarial note to buy more regular saving notes, i.e., purchase only a conventional life insurance policy but no annuities. terms, provided that insurance premiums are responsive to individuals’ true odds of mortality. 17 If the price of insurance is not responsive to such odds, inefficient Ž . self-protection may be an inevitable outcome of insurance cf. EB, op. cit. , but its direction in this case will be ambiguous. Whereas the purchase of life insurance may depress investment in life protection, that of annuities creates an incentive to increase it. Insurance companies may seek to minimize such inefficiencies by offering customers a package of life insurance and lifetime annuities or pension funds.

7. Insurance, life protection, and life saving: concluding remarks