Risk Measures for DC Pension Plan Decumulation

Paper: arXiv 2502.16364

Abstract

As the developed world replaces Defined Benefit (DB) pension plans with Defined Contribution (DC) plans, there is a need to develop decumulation strategies for DC plan holders. Optimal decumulation can be viewed as a problem in optimal stochastic control. Formulation as a control problem requires specification of an objective function, which in turn requires a definition of reward and risk. An intuitive specification of reward is the total withdrawals over the retirement period. Most retirees view risk as the possibility of running out of savings. This paper investigates several possible left tail risk measures, in conjunction with DC plan decumulation. The risk measures studied include (i) expected shortfall (ii) linear shortfall and (iii) probability of shortfall. We establish that, under certain assumptions, the set of optimal controls associated with all expected reward and expected shortfall Pareto efficient frontier curves is identical to the set of optimal controls for all expected reward and linear shortfall Pareto efficient frontier curves. Optimal efficient frontiers are determined computationally for each risk measure, based on a parametric market model. Robustness of these strategies is determined by testing the strategies out-of-sample using block bootstrapping of historical data.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper uses advanced stochastic control theory and establishes theoretical equivalence between risk measures, indicating high mathematical density. It also employs computational methods, parametric model calibration, and out-of-sample testing with block bootstrapping on historical data, demonstrating significant empirical and implementation effort.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Develop optimal DC plan<br>decumulation strategies"] --> B["Formulate Stochastic Control Problem<br>Define Reward: Total Withdrawals<br>Define Risk: Left Tail Measures"]
  B --> C{"Define Risk Measures"}
  C --> C1["Expected Shortfall"]
  C --> C2["Linear Shortfall"]
  C --> C3["Probability of Shortfall"]
  C1 & C2 & C3 --> D["Compute Efficient Frontiers<br>via Stochastic Control<br>using Parametric Market Model"]
  D --> E["Key Findings<br>1. Optimal controls for ES & LS<br>are identical<br>2. Out-of-sample robustness<br>verified via Bootstrapping"]

Cited References (223)

1. Equitable Retirement Income Tontines: Mixing Cohorts Without Discriminating (2016)
2. Tonuity: A Novel Individual-Oriented Retirement Plan (2019)
3. A Stochastic Control Approach to Defined Contribution Plan Decumulation: The Nastiest, Hardest Problem in Finance (2022)
4. Machine Learning and Hamilton-Jacobi-Bellman Equation for Optimal Decumulation: a Comparison Study (2023)
5. Time inconsistent control theory with finance applications (2021)
6. On the Distribution of Terminal Wealth under Dynamic Mean-Variance Optimal Investment Strategies (2021)
7. Modern Life Care Tontines (2022)
8. Mortality Credits within Large Survivor Funds (2022)

0 of 223 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.