Optimal Annuitization with stochastic mortality: Piecewise Deterministic Mortality Force

2025-09-16

AI Summary

This paper investigates the optimal timing for retirees to annuitize their savings under joint market and longevity risk. Wealth is invested in a dividend-paying asset following a geometric Brownian motion, while mortality evolves according to a piecewise-deterministic Markov process (PDMP), capturing nonlinear mortality-rate shifts induced by health shocks. By reducing the original three-dimensional optimal stopping problem to lower dimensions and combining analytical derivation with numerical analysis, the study derives, for the first time, an explicit, globally valid optimal strategy across the full parameter space. Key contributions include: (i) introducing a PDMP-based dynamic mortality framework that reveals a non-monotonic effect of health deterioration on annuitization timing; (ii) proving that the optimal policy exhibits a two-region “delay–immediate” structure; and (iii) demonstrating robustness to key parameters via numerical experiments—providing actionable, theoretically grounded guidance for longevity risk management.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Life-Long and Continual Learning

Application Category

• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems
• Security and Privacy: Large-scale security measurements

Abstract

This paper addresses the problem of determining the optimal time for an individual to convert retirement savings into a lifetime annuity. The individual invests their wealth into a dividend-paying fund that follows the dynamics of a geometric Brownian motion, exposing them to market risk. At the same time, they face an uncertain lifespan influenced by a stochastic mortality force. The latter is modelled as a piecewise deterministic Markov process (PDMP), which captures sudden and unpredictable changes in the individual's mortality force. The individual aims to maximise expected lifetime linear utility from consumption and bequest, balancing market risk and longevity risk under an irreversible, all-or-nothing annuitization decision. The problem is formulated as a three-dimensional optimal stopping problem and, by exploiting the PDMP structure, it is reduced to a sequence of nested one-dimensional problems. We solve the optimal stopping problem and find a rich structure for the optimal annuitization rule, which cover all parameter specifications. Our theoretical analysis is complemented by a numerical example illustrating the impact of a single health shock on annuitization timing, along with a sensitivity analysis of key model parameters.

Problem

• Determining optimal timing for irreversible lifetime annuity conversion
• Balancing market risk and uncertain lifespan in retirement planning
• Modeling stochastic mortality as piecewise deterministic Markov process

Innovation

• Piecewise deterministic Markov process mortality modeling
• Optimal stopping problem reduction technique
• Balancing market and longevity risk annuitization