Risk Measures for DC Pension Plan Decumulation

TL;DR

The paper tackles decumulation of DC pension plans by framing it as an optimal stochastic control problem, comparing tail-risk measures (ES, LS, PS) against the reward of total withdrawals. It develops a parametric market model, solves the control problem via dynamic programming, and validates strategies with block bootstrap on historical data. A key theoretical finding is that, under certain conditions, the optimal EW–ES frontier coincides with the EW–LS frontier, making LS a practical, time-consistent risk measure. The results consistently favor EW–LS as a robust, implementable approach that outperforms the traditional 4% rule in out-of-sample tests, providing actionable guidance for retirement planning with explicit wealth targets and feasible withdrawal rules.

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.

Figures (9)

• Figure 5.1: EW-LS convergence test. Real stock index: deflated real capitalization weighted CRSP, real bond index: deflated 30 day T-bills. Scenario in Table \ref{['base_case_1']}. Parameters in Table \ref{['fit_params']}. The optimal control is determined by solving the PIDEs as described in Appendix \ref{['Numerical_Appendix']}. Grid refers to the grid used in the algorithm in Appendix \ref{['Numerical_Appendix']}: $n_x \times n_b$, where $n_x$ is the number of nodes in the $\log s$ direction, and $n_b$ is the number of nodes in the $\log b$ direction. Units: thousands of dollars (real). The controls are stored, and then the final results are obtained using a Monte Carlo method, with $2.56 \times 10^6$ simulations. Target wealth $\mathbb{W}=0.0$.
• Figure 5.2: EW-LS efficient frontier. Real stock index: deflated real capitalization weighted CRSP, real bond index: deflated 30 day T-bills. Scenario in Table \ref{['base_case_1']}. Parameters in Table \ref{['fit_params']}. Synthetic market. Controls computed using EW-PS and EW-ES, and results plotted in terms of EW-LS criteria. The EW-LS frontier plots above the the controls computed using EW-PS and EW-ES objective functions. The Bengen control withdraws 40 per year, and rebalances annually to 50% bonds and 50% stocks. The wealth target level $\mathbb{W} = 0$ for both $\text{EW-LS}_{t_0}$ and $\text{EW-PS}_{t_0}$.
• Figure 5.3: EW-PS and EW-ES efficient frontiers. Real stock index: deflated real capitalization weighted CRSP, real bond index: deflated 30 day T-bills. Scenario in Table \ref{['base_case_1']}. Parameters in Table \ref{['fit_params']}. Synthetic market. The Bengen control withdraws 40 per year, and rebalances annually to 50% bonds and 50% stocks. Target wealth $\mathbb{W}=0$ for EW-LS and EW-PS.
• Figure 5.4: Optimal controls computed using the synthetic market model. These controls tested using bootstrapped historical data. Expected blocksizes (years) shown. $10^6$ bootstrap resamples. Real stock index: deflated real capitalization weighted CRSP, real bond index: deflated 30 day T-bills. Scenario in Table \ref{['base_case_1']}. Parameters in Table \ref{['fit_params']}. The Bengen control withdraws 40 per year, and rebalances annually to 50% bonds and 50% stocks. The Bengen results are also shown for expected blocksizes of $0.5, 1.0, 2.0$ years.
• Figure 6.1: CDF curves, all strategies have the same average EW $\simeq 53$. Optimal controls computed using the synthetic market model. Tests in the synthetic market Figure \ref{['CDF_synth_fig']} and the historical market, Figure \ref{['CDF_boot_fig']} shown. Expected blocksize: two years. Real stock index: deflated real capitalization weighted CRSP, real bond index: deflated 30 day T-bills. Scenario in Table \ref{['base_case_1']}. Parameters in Table \ref{['fit_params']}.

Theorems & Definitions (14)

• Remark 4.1: Pre-commitment policy
• Proposition 4.1: Optimal EW-ES strategy solves EW-LS
• proof
• Corollary 4.1
• Remark 4.2: EW-ES $\rightarrow$ EW-LS
• Remark 4.3: Construction of $\alpha_{\hat{\kappa}}^*(\mathbb{W})$
• Proposition 4.2: Relationship between $\text{EW-LS}_{t_0}$ and $\text{EW-LS}_{t_0}$ for general $\mathbb{W}$
• proof
• Corollary 4.2
• proof