Paper: arXiv 2510.10371

Authors: Criscent Birungi, Cody Hyndman

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly theoretical, relying on advanced stochastic calculus, dynamic programming (HJB), and analytical proofs for closed-form solutions, but it lacks any empirical backtesting, real-world data, or implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Optimal annuitization with labor income<br>under age-dependent mortality"] --> B["Methodology: Dynamic Programming Approach"]
  
  B --> C["Inputs / Model Setup"]
  C --> C1["Age-dependent<br>force of mortality"]
  C --> C2["Labor income process"]
  C --> C3["Utility functions<br>Consumption & Labor"]
  
  B --> D["Computational Process"]
  D --> D1["Derive Bellman Equation"]
  D --> D2["Solve Value Function<br>(Closed-form solutions)"]
  D --> D3["Optimal Strategies<br>Consumption / Portfolio / Labor"]
  
  D2 --> E["Key Findings & Outcomes"]
  E --> E1["Pre-retirement:<br>Investment ↑ with wealth until threshold"]
  E --> E2["Post-retirement:<br>Higher consumption from wealth"]
  E --> E3["Annuitization Decision Drivers:<br>Demand: Personal discount rate ↑<br>Supply: Mortality credits ↑"]
  E --> E4["Post-retirement labor income<br>substitutes for annuitization"]