Optimal Annuitization, Consumption, and Investment with Costly Capital Injections

Paper: arXiv 2611.02558

Authors: Yijie Huang, Xiaoqing Liang, Jie Xiong, Xiang Yu

Abstract

We study a joint optimal decision of portfolio, consumption, and annuitization for a pension-plan manager who balances retirement consumption benefits against the cost of maintaining the solvency of pension account. The pension sponsor provides costly capital injection to prevent negative wealth. Annuity purchases are irreversible, and injected funds remain in the liquid account and earn interest. Using the Legendre transform, we derive a semi-explicit solution to the associated free-boundary problem and verify the optimality of the resulting strategies. We identify an endogenous annuity-income threshold separating injection-active and injection-inactive regions and characterize six policy regimes arising from the interplay between capital injections and consumption constraints. Compared with the no-injection benchmark, capital injections delay annuitization, alter the structure of the annuitization boundary, and permit positive holdings in risky assets even at zero liquid wealth. Numerical results further reveal that cheaper annuities may increase the expected capital contributions required from the pension sponsor.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper features advanced stochastic control theory, free-boundary problems, and Legendre transforms for its mathematical derivations. While it includes numerical experiments, the empirical rigor is based on simulations rather than real-world backtesting with extensive datasets. The novelty lies in integrating costly capital injections with irreversible annuitization and consumption decisions.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Annuitization, Consumption, & Investment with Costly Capital Injections] --> B{Methodology: Free-Boundary Problem & Legendre Transform};
    B --> C{Inputs: Pension Plan Parameters, Costly Capital Injection Model, Annuity Market Data};
    C --> D[Computational Process: Derivation of Semi-Explicit Solution, Optimality Verification];
    D --> E[Key Findings: Endogenous Annuity-Income Threshold, Six Policy Regimes, Capital Injections Delay Annuitization & Alter Boundary Structure];
    E --> F[Further Findings: Positive Risky Asset Holdings at Zero Liquid Wealth, Cheaper Annuities May Increase Expected Capital Contributions];