Optimal annuitization with labor income under age-dependent force of mortality

TL;DR

This work addresses optimal annuitization with labor income under an age-dependent force of mortality by formulating a stochastic control problem solved via dynamic programming. The authors derive a closed-form value function and explicit, wealth-dependent policies for consumption, labor, and portfolio allocation, along with an optimal retirement threshold x^*. A key finding is the competition between an increasing personal discount rate ρ_t with age and mortality credits offered by insurers; the model shows that mortality credits may be outweighed by higher discounting, delaying annuitization, while post-retirement labor income acts as a useful substitute. The results extend prior literature by incorporating Gompertz mortality and an explicit annuitization decision, delivering practical retirement-planning insights under longevity risk and market risk. These theoretical insights have implications for how individuals balance liquidity, longevity insurance, and labor income across the retirement horizon.

Abstract

We consider the problem of optimal annuitization with labour income, where an agent aims to maximize utility from consumption and labour income under age-dependent force of mortality. Using a dynamic programming approach, we derive closed-form solutions for the value function and the optimal consumption, portfolio, and labor supply strategies. Our results show that before retirement, investment behavior increases with wealth until a threshold set by labor supply. After retirement, agents tend to consume a larger portion of their wealth. Two main factors influence optimal annuitization decisions as people get older. First, the agent's perspective (demand side); the agent's personal discount rate rises with age, reducing their desire to annuitize. Second, the insurer's perspective (supply side); insurers offer higher payout rates (mortality credits). Our model demonstrates that beyond a certain age, sharply declining survival probabilities make annuitization substantially optimal, as the powerful incentive of mortality credits outweighs the agent's high personal discount rate. Finally, post-retirement labor income serves as a direct substitute for annuitization by providing an alternative stable income source. It enhances the financial security of retirees.

Figures (9)

• Figure 1: Evolution of the effective discount rate ($\rho_t$) for an agent starting at age 60, shown for different mortality dispersion parameters ($\lambda$). Age-dependent mortality causes $\rho_t$ to rise sharply, reflecting higher mortality risk at advanced ages.
• Figure 2: Agent survival probability from age 60 ($\px[t]{60}$) under different mortality scenarios, defined by the modal age of death ($m$) and the dispersion parameter ($\lambda$).
• Figure 3: Wealth level and the contribution of consumption to the retiree's period utility.
• Figure 4: Optimal consumption and wealth level. The solid blue line represents the pre-retirement optimal consumption path. The dashed lines show post-retirement consumption paths at different time horizons ($t=5, 10, 15$ years post-retirement, corresponding to ages 65, 70, and 75). The vertical dotted lines indicate key thresholds: the subsistence wealth level ($\tilde{x}$) and the retirement wealth level ($x^*$). Notably, at the point of retirement ($x^*$), there is a discrete upward jump in consumption, reflecting a regime shift in the retirees' spending behavior.
• Figure 5: Optimal labor income and wealth level. The red dashed line represents the threshold wealth level corresponding to the labor supplied. The green dashed line represents the threshold wealth level corresponding to the optimal retirement time $\tau$. The discontinuities at $\tilde{x}$ and ${x^*}$ represent sudden shifts in the retiree's optimal labor supply as their wealth crosses critical thresholds. These shifts are linked to retiree decisions about annuitization (in the broader sense of securing future income streams) and the transition into a phase of reduced or no labor (retirement).

Theorems & Definitions (24)

• Remark 2.1: Admissibility Conditions
• Definition 2.1
• Remark 2.2
• Theorem 2.1: Dynamic Programming Principle
• Remark 2.3
• proof : Proof of Theorem \ref{['thm:Dynamic_programming_principle']}: Dynamic Programming Principle (DPP)
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.2