Paper: arXiv 2402.17164

Authors: Hayden Brown

Abstract

Consider a closed pooled annuity fund investing in n assets with discrete-time rebalancing. At time 0, each annuitant makes an initial contribution to the fund, committing to a predetermined schedule of withdrawals. Require annuitants to be homogeneous in the sense that their initial contributions and predetermined withdrawal schedules are identical, and their mortality distributions are identical and independent. Under the forementioned setup, the probability for a particular annuitant to complete the prescribed withdrawals until death is maximized over progressively measurable portfolio weight functions. Applications consider fund portfolios that mix two assets: the S&P Composite Index and an inflation-protected bond. The maximum probability is computed for annually rebalanced schedules consisting of an initial investment and then equal annual withdrawals until death. A considerable increase in the maximum probability is achieved by increasing the number of annuitants initially in the pool. For example, when the per-annuitant initial contribution and annual withdrawal amount are held constant, starting with 20 annuitants instead of just 1 can increase the maximum probability (measured on a scale from 0 to 1) by as much as .15.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a formal stochastic control framework for optimizing withdrawal success probabilities in pooled annuity funds, utilizing discrete-time dynamic programming over measurable portfolio weights. Empirical rigor is moderate, relying on historical S&P Index and TIPS asset mixes rather than exhaustive, out-of-sample backtesting across multiple market regimes.

Research Flowchart

  flowchart TD
    A[Research Goal: Maximize annuitant withdrawal success probability in a pooled annuity fund] --> B[Inputs: Homogeneous annuitants, predetermined withdrawals, S&P 500 & Inflation-Protected Bond]
    B --> C[Methodology: Discrete-time rebalancing & mortality risk modeling]
    C --> D[Optimization: Dynamic programming over progressively measurable portfolio weights]
    D --> E[Outcome: Increasing pool size significantly improves withdrawal success probability]
    E --> F[Key Metric: Expanding pool from 1 to 20 annuitants increases success probability by up to 0.15]

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