Paper: arXiv 2604.09342

Authors: Matteo Buttarazzi

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs and closed-form solutions for optimal stopping problems using stochastic calculus and stochastic control theory. However, it relies entirely on synthetic calibration and numerical illustrations rather than empirical backtests using real market or health datasets.

Research Flowchart

  flowchart TD
    A["Research Question: Optimal Annuitization Timing Under Mortality Shock"] --> B["Inputs & Assumptions:<br>Geometric Brownian Motion, Exponential Health Shock, Bequest Motives"]
    B --> C["Methodology: Formulate Optimal Stopping Problem Across Two Health States"]
    C --> D["Computational Process: Derive Explicit Closed-Form Value Functions & Boundaries"]
    D --> E["Key Finding: Strategy Driven by Money's Worth, Fund Returns & Bequest Needs"]
    D --> F["Key Finding: Health Shock Likelihood & Severity Alter Annuitization Timing vs Constant Baseline"]

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2 of 100 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.