Minimizing the lifetime shortfall or shortfall at death

Abstract

We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his wealth reaches a specified ruin level before death, (2) the probability that his wealth reaches that level at death, (3) the expectation of how low his wealth drops below a specified level before death, and (4) the expectation of how low his wealth drops below a specified level at death. Young (2004) showed that under criterion (1), the optimal investment strategy is a heavily leveraged position in the risky asset for low wealth.In this paper, we introduce the other three criteria in order to reduce the leveraging observed by Young (2004). We discovered that surprisingly the optimal investment strategy for criterion (3) is identical to the one for (1) and that the strategies for (2) and (4) are more leveraged than the one for (1) at low wealth. Because these criteria do not reduce leveraging, we completely remove it by considering problems (1) and (3) under the restriction that the individual cannot borrow to invest in the risky asset.

Correspondence between lifetime minimum wealth and utility of consumption

“…For examples of relevant functions f , consider the following: If f (m) = 1 {m≤b} , then f (M τ d ) indicates whether the individual's wealth has reached b ∈ R during one's life, and V f is the minimum probability of lifetime ruin with ruin level b (Young, 2004). If f (m) = max(b−m, 0), then V f is the minimum expected lifetime shortfall relative to b ∈ R (Bayraktar and Young, 2005).…”

“…In this case, the rate of consumption would equal the constant c + ρκ on (−∞, w s ). Young (2004) solves this problem for minimizing the probability of lifetime ruin, and Bayraktar and Young (2005) solve it for minimizing expected lifetime shortfall.…”

Purchasing Term Life Insurance to Reach a Bequest Goal while Consuming

“…Remark 2.6. Bayraktar and Young (2009) considered the problem of minimizing shortfall at death without life insurance in the market; they found that the optimal amount to invest in the risky asset is greater than the optimal amount when minimizing lifetime shortfall. If we were to add life insurance to the market when minimizing E w [(b − W τ d ∧τ 0 ) + ] (again, with the understanding that the game ends if wealth reaches 0, or equivalently, 0 is an absorbing state for the wealth process), then because the price of life insurance and the shortfall target are (piecewise) linear in the size of the death benefit, if it is optimal to buy any amount of life insurance, it will be optimal to buy b − w. Furthermore, we anticipate that the corresponding value function, V , will be decreasing and convex; if so, the optimal life insurance purchasing strategy will be given by (2.7) for w b = inf{w ≥ 0 : λ + h V w (w) ≥ 0} ∧ b, and V will solve the following BVP:…”

Comonotonic approximations for the probability of lifetime ruin

“…Another recent study that builds on Young (2004) is Bayraktar and Young (2009). In this paper, the problem of wealth at death is addressed, and the shortfall at death is minimized.…”