Purchasing Term Life Insurance to Reach a Bequest Goal while Consuming

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 Bayraktar and Young (2015) 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 Bayraktar and Young (2015).