[Paper Review] Optimal Deferred Life Annuities to Minimize the Probability of Lifetime Ruin

Erhan Bayraktar, Virginia R. Young|ArXiv.org|Mar 29, 2007

Insurance, Mortality, Demography, Risk Management9 references3 citations

TL;DR

This paper determines the optimal strategy for an investor to minimize the probability of lifetime ruin by investing in a risky and riskless asset while deferring purchase of a life annuity. The key finding is that individuals will only buy deferred annuities when they have sufficient wealth to cover all consumption until the annuity begins, ensuring they do not face ruin during the deferral period.

ABSTRACT

We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset and can purchase a deferred annuity. Although we let the admissible set of strategies of annuity purchasing process to be increasing adapted processes, we find that the individual will not buy a deferred life annuity unless she can cover all her consumption via the annuity and have enough wealth left over to sustain her until the end of the deferral period.

Motivation & Objective

• To determine the optimal investment and deferred annuity purchase strategy that minimizes the probability of lifetime wealth depletion before death.
• To analyze how the presence of deferred annuities affects optimal portfolio allocation and ruin avoidance in a stochastic financial market.
• To establish conditions under which an individual will purchase a deferred annuity, given the ability to invest in risky and riskless assets.
• To extend prior work on immediate annuities to the case of deferred annuities, particularly in the context of minimizing ruin probability.
• To demonstrate that the optimal strategy involves a threshold wealth level beyond which deferred annuity purchase becomes optimal.

Proposed method

• Formulate the problem as a stochastic control problem where wealth evolves under a risky asset (geometric Brownian motion) and a riskless asset earning rate r.
• Model the individual’s consumption at a constant net rate c and lifetime as an exponential random variable with hazard rate λ^S.
• Use a verification lemma to characterize the minimal probability of lifetime ruin as the solution to a free-boundary problem.
• Transform the ruin minimization problem into an equivalent optimal stopping problem via Legendre transformation.
• Solve the resulting variational inequality using numerical methods such as projected SOR, with boundary conditions defined at the end of the deferral period.
• Derive the optimal investment policy in the risky asset as a function of the derivative of the value function, showing discontinuity at the free boundary.

Experimental results

Research questions

• RQ1 Under what conditions will an investor purchase a deferred life annuity to minimize the probability of lifetime ruin?
• RQ2 How does the optimal investment strategy in risky and riskless assets change when deferred annuities are available?
• RQ3 What is the threshold wealth level at which it becomes optimal to purchase a deferred annuity?
• RQ4 How does the presence of deferred annuities affect the safe level of wealth required to avoid ruin during the deferral period?
• RQ5 Can the minimal probability of lifetime ruin be computed via transformation to an optimal stopping problem?

Key findings

• An individual will only purchase a deferred life annuity when her wealth reaches a specific threshold, denoted as w̄(A,t), which ensures she can cover all consumption until the annuity payout begins.
• The optimal strategy involves not purchasing any deferred annuity if wealth is below the threshold, as the individual prefers to retain exposure to the risky asset’s return potential.
• The minimal probability of lifetime ruin is convex in wealth up to the threshold w̄(A,t), and drops to zero for wealth above this level.
• The optimal investment in the risky asset is discontinuous at the threshold w̄(A,t), reflecting a shift in risk exposure strategy.
• Numerical results confirm that the probability of ruin is lower when deferred annuities are available, especially for investors with moderate wealth levels.
• The safe level w̄(A,t) decreases over time as the annuity payout date approaches, indicating increasing urgency to purchase when wealth is sufficient.