A Stochastic Control Approach to Defined Contribution Plan Decumulation: “The Nastiest, Hardest Problem in Finance”

Abstract

We pose the decumulation strategy for a defined contribution (DC) pension plan as a problem in optimal stochastic control. The controls are the withdrawal amounts and the asset allocation strategy. We impose maximum and minimum constraints on the withdrawal amounts, and impose no-shorting no-leverage constraints on the asset allocation strategy. Our objective function measures reward as the expected total withdrawals over the decumulation horizon, and risk is measured by expected shortfall (ES) at the end of the decumulation period. We solve the stochastic control problem numerically, based on a parametric model of market stochastic processes. We find that, compared to a fixed constant withdrawal strategy, with minimum withdrawal set to the constant withdrawal amount the optimal strategy has a significantly higher expected average withdrawal, at the cost of a very small increase in ES risk. Tests on bootstrapped resampled historical market data indicate that this strategy is robust to parametric model misspecification.

CONFLICT OF INTEREST

The author has no conflicts of interest to report.

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Notes

1 In Australia, DC plans have 86% of pension assets, compared with 14% in DB assets (Towers-Watson 2020).

2 An implementable strategy has the property that the investor has no incentive to deviate from the strategy computed at time zero at later times (Forsyth 2020a).

3 In practice, the negative of $W_{\alpha }^{*}$ is often the reported VAR.

4 This is the same as noting that a finite value at risk exists. This easily shown, assuming $0<\alpha <1,$ because our investment strategy uses no leverage and no shorting.

5 More specifically, results presented here were calculated based on data from Historical Indexes, [copyright] 2020 Center for Research in Security Prices, The University of Chicago Booth School of Business, Chicago, IL. Wharton Research Data Services was used in preparing this article. This service and the data available therein constitute valuable intellectual property and trade secrets of Wharton Research Data Services and/or its third-party suppliers.

6 The 10-year Treasury index was constructed from monthly returns from CRSP back to 1941. The data for 1926–1941 were interpolated from annual returns in Homer and Sylla (2005).

7 “If we have a good year, we take a trip to China,…if we have a bad year, we stay home and play canasta.” Retired Professor Peter Ponzo, discussing his DC plan withdrawal strategy, https://www.theglobeandmail.com/report-on-business/math-prof-tests-investing-formulas-strategies/article22397218/

Additional information

Funding

P. A. Forsyth’s work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant RGPIN-2017-03760.