Optimal Annuitization Time under a Mortality Shock

TL;DR

This paper derives explicit solutions for the optimal timing of annuitization in the presence of a mortality shock, considering investment dynamics and health deterioration, providing insights into decision-making under health risks.

Contribution

It introduces analytical expressions for the value function and optimal stopping boundaries in a model with mortality shocks, advancing understanding of annuitization timing under health risks.

Findings

Optimal strategy depends on annuity attractiveness, investment returns, and bequest motives.

Health shocks significantly influence annuitization timing.

Numerical analysis shows health shocks alter optimal annuitization compared to constant mortality.

Abstract

In this paper, we derive explicit closed-form solutions for the value function and the associated optimal stopping boundaries in an optimal annuitization problem under a mortality shock. We consider an individual whose retirement wealth is invested in a financial fund following the dynamics of a geometric Brownian motion and has the option at any time to irreversibly convert their wealth into a life annuity. The individual faces a sudden, permanent health deterioration occurring at a random, exponentially distributed time, and the annuitization decision is modelled as an optimal stopping problem across two health states. Our analytical expressions characterise both the value function and the optimal timing of annuitization. The results provide clear economic intuition: the optimal strategy is governed by the critical interplay between the relative attractiveness of the annuity (money's worth), the financial returns from the investment fund, and bequest motives across different health states. A numerical analysis compares the optimal annuitization strategy of an individual facing a health shock against a benchmark case with constant mortality, highlighting how the likelihood and severity of a health shock significantly alter optimal annuitization behaviour.