The Demand for Life Insurance: An Application of the Economics of Uncertainty

IN HIS THEORETICAL STUDY on the demand for life insurance R.A. Campbell [I], has derived simple demand-for-insurance equations in an attempt to relate households' optimal responses to human capital uncertainty. One major conclusion of his paper is that for households characterized by risk aversion, the optimal amount of human capital insurance is a decreasing function of the "load factor," A, which is defined as a percentage markup from the actuarially fair value of insurance. Campbell's conclusions follow directly from a Taylor Series expansion of his objective function. In this comment, the demand-for-insurance equations are derived once again, but this time the derivation will be direct, without use of the Taylor approximation. It is demonstrated that Campbell's approximate functions do not converge in the limit to the exact solutions for logarithmic utility functions. For this class of utility functions, Campbell's approximate solution dictates nonoptimal holdings of life insurance for virtually all policies whose load factors are greater than zero. The reason for this is that Campbell's approximations contain utility derivatives only up to the second order. As will be shown, for the logarithmic utility function, third and higher derivatives are significant. The organization of this note is as follows. First, the optimal amount of insurance coverage, I* , will be derived and compared with Campbell's alternative, INS. Second, it is demonstrated that INS does not approximate I* as the time horizon, A t , becomes infinitely small; i.e., as At + 0, lim I* # INS. Third,' it is shown that a t one point, where the bequest and utility functions are identical and where insurance is sold with no load (A = 0 ) , INS and I* coincide. Fourth, the optimal ( I * ) and approximate (INS) life insurance coverage (and their difference) are calculated for some realistic parameter values. Finally, a general result on approximations is given. At the outset, it is useful to reintroduce the problem, notation, and assumptions of Campbell. A wage earner of age x and retirement age R has present value of wage income from x to R R HI = H and current marketable assets of W, = W. At instant x he faces a lottery. With probability 1 q A t he survives a t time x + At and has wealth W + Hand utility function V(.) and with probability q A t he dies, has wealth W, and bequest utility function B( . ) . The assumption is made that both the utility and the bequest functions are increasing and concave, i.e. V' > 0,