Purchasing Term Life Insurance to Reach a Bequest Goal while Consuming

Abstract

We determine the optimal strategies for purchasing term life insurance and for investing in a risky financial market in order to maximize the probability of reaching a bequest goal while consuming from an investment account. We extend by allowing the individual to purchase term life insurance to reach her bequest goal. The premium rate for life insurance, h, serves as a parameter to connect two seemingly unrelated problems. As the premium rate approaches 0, covering the bequest goal becomes costless, so the individual simply wants to avoid ruin that might result from her consumption. Thus, as h approaches 0, the problem in this paper becomes equivalent to minimizing the probability of lifetime ruin, which is solved in Young (2004). On the other hand, as the premium rate becomes arbitrarily large, the individual will not buy life insurance to reach her bequest goal. Thus, as h approaches infinity, the problem in this paper becomes equivalent to maximizing the probability of reaching the bequest goal when life insurance is not available in the market, which is solved in .

Minimizing lifetime poverty with a penalty for bankruptcy

“…More recently, Bayraktar et al [4] and Bayraktar and Young [7] determined the optimal investment strategy to maximize the probability of reaching a bequest goal with and without life insurance, respectively. In the wealth regions for which it is optimal not to buy life insurance (specifically, at lower wealth levels), then the individual invested as if life insurance were not available.…”

“…Proposition 4. 4 The optimal amount to invest in the risky asset π * (w) (weakly) increases with l and (weakly) decreases with ρ for w ∈ (a, d) and is independent of l and ρ for w ∈ d, c−A r .…”

“…Mathematically, our problem is closely related to those in the goal-seeking literature, such as minimizing the probability of lifetime ruin, 2 maximizing the probability of reaching a bequest goal, 3 minimizing expected lifetime occupation (that is, the time that wealth stays below a given level), 4 , or minimizing the probability of lifetime drawdown or expected lifetime spent in drawdown. 5 In fact, this paper generalizes the problems of minimizing the probability of lifetime ruin and minimizing expected lifetime occupation, with the poverty function serving as a bridge between the two, as we discuss in Remark 2.1 below.…”

“…For an early reference, see Young[14], and for a more recent reference, see Bayraktar and Zhang[8] 3. Bayraktar and Young[7] and Bayraktar et al[4] 4. Bayraktar and Young[6] 5.…”

Minimizing the expected lifetime spent in drawdown under proportional consumption

“…Under that constraint, when the constraint did not bind (specifically, at greater wealth levels), then the individual invested as if the constraint did not exist. More recently, Bayraktar et al [2] and Bayraktar and Young [5] determined the optimal investment strategy to maximize the probability of reaching a bequest goal with and without life insurance, respectively. In the wealth regions for which it is optimal not to buy life insurance (specifically, at lower wealth levels), then the individual invested as if life insurance were not available.…”