[Paper Review] Optimal retirement consumption with a stochastic force of mortality

Huaxiong Huang, Moshe A. Milevsky+1 more|arXiv (Cornell University)|May 10, 2012

Global Health Care Issues31 references4 citations

TL;DR

This paper extends Yaari's lifecycle consumption model to a stochastic force of mortality, where mortality rates follow a diffusion process and consumers update consumption based on new health information. Under CRRA preferences, stochastic mortality increases consumption when risk aversion is low (coefficient < 1) and decreases it when risk aversion is high (coefficient > 1), though the effect is relatively minor numerically for most utility functions.

ABSTRACT

We extend the lifecycle model (LCM) of consumption over a random horizon (a.k.a. the Yaari model) to a world in which (i.) the force of mortality obeys a diffusion process as opposed to being deterministic, and (ii.) a consumer can adapt their consumption strategy to new information about their mortality rate (a.k.a. health status) as it becomes available. In particular, we derive the optimal consumption rate and focus on the impact of mortality rate uncertainty vs. simple lifetime uncertainty -- assuming the actuarial survival curves are initially identical -- in the retirement phase where this risk plays a greater role. In addition to deriving and numerically solving the PDE for the optimal consumption rate, our main general result is that when utility preferences are logarithmic the initial consumption rates are identical. But, in a CRRA framework in which the coefficient of relative risk aversion is greater (smaller) than one, the consumption rate is higher (lower) and a stochastic force of mortality does make a difference. That said, numerical experiments indicate that even for non-logarithmic preferences, the stochastic mortality effect is relatively minor from the individual's perspective. Our results should be relevant to researchers interested in calibrating the lifecycle model as well as those who provide normative guidance (a.k.a. financial advice) to retirees.

Motivation & Objective

• To extend the classical lifecycle consumption model to incorporate a stochastic force of mortality, moving beyond deterministic mortality assumptions.
• To analyze how optimal consumption strategies adapt in real time to new information about an individual’s mortality rate (health status).
• To compare the impact of stochastic mortality versus deterministic mortality on optimal consumption, especially in the retirement phase.
• To assess the sensitivity of consumption decisions to relative risk aversion and the structure of the mortality process.
• To provide normative guidance for retirees and financial advisors under realistic, stochastic longevity risk.

Proposed method

• Formulates the optimal consumption problem using a Hamilton-Jacobi-Bellman (HJB) equation with two state variables: current wealth and current mortality rate.
• Models the force of mortality as a diffusion process, inspired by actuarial models such as those in Milevsky & Promislow (2001) and Cairns et al. (2006).
• Solves the resulting partial differential equation (PDE) numerically to derive the optimal consumption rate under stochastic mortality.
• Derives a closed-form solution for consumption under deterministic Gompertz mortality as a benchmark for comparison.
• Applies constant relative risk aversion (CRRA) utility preferences and assumes no bequest motives and equal subjective discount rate and interest rate.
• Uses calculus of variations to re-derive the deterministic mortality case, establishing a foundation for the stochastic extension.

Experimental results

Research questions

• RQ1 How does allowing the force of mortality to follow a stochastic diffusion process affect optimal consumption paths compared to deterministic mortality?
• RQ2 Under what conditions does stochastic mortality lead to higher or lower initial consumption relative to deterministic mortality?
• RQ3 How does the coefficient of relative risk aversion (CRRA) influence the magnitude and direction of the consumption response to mortality uncertainty?
• RQ4 To what extent does the impact of stochastic mortality on consumption differ from the effect of subjective discount rate adjustments in traditional models?
• RQ5 How do real-time updates to mortality rates (e.g., health news) affect consumption decisions in a dynamic, information-adaptive framework?

Key findings

• For logarithmic utility (CRRA = 0), initial consumption rates are identical under both stochastic and deterministic mortality, despite different underlying dynamics.
• When CRRA > 1, stochastic mortality leads to lower initial consumption compared to deterministic mortality, reflecting increased precautionary saving due to higher longevity risk.
• When CRRA < 1, stochastic mortality results in higher initial consumption, as the consumer becomes more willing to spend today due to reduced perceived longevity risk.
• Numerical experiments show that the quantitative impact of stochastic mortality on consumption is relatively minor across most CRRA values, even though the qualitative structure changes.
• A positive shock to mortality (e.g., improved health) reduces consumption immediately, while a negative shock (e.g., terminal diagnosis) increases consumption beyond prior expectations.
• The model suggests that optimal financial product allocation in a stochastic mortality world may involve a mix of participating tontines and guaranteed annuities, reflecting systematic mortality risk pricing.