Management of Portfolio Depletion Risk through Optimal Life Cycle Asset Allocation

Abstract

Members of defined contribution (DC) pension plans must take on additional responsibilities for their investments, compared to participants in defined benefit (DB) pension plans. The transition from DB to DC plans means that more employees are faced with these responsibilities. We explore the extent to which DC plan members can follow financial strategies that have a high chance of resulting in a retirement scenario that is fairly close to that provided by DB plans. Retirees in DC plans typically must fund spending from accumulated savings. This leads to the risk of depleting these savings, that is, portfolio depletion risk. We analyze the management of this risk through life cycle optimal dynamic asset allocation, including the accumulation and decumulation phases. We pose the asset allocation strategy as an optimal stochastic control problem. Several objective functions are tested and compared. We focus on the risk of portfolio depletion at the terminal date, using such measures as conditional value at risk (CVAR) and probability of ruin. A secondary consideration is the median terminal portfolio value. The control problem is solved using a Hamilton-Jacobi-Bellman formulation, based on a parametric model of the financial market. Monte Carlo simulations that use the optimal controls are presented to evaluate the performance metrics. These simulations are based on both the parametric model and bootstrap resampling of 91 years of historical data. The resampling tests suggest that target-based approaches that seek to establish a safety margin of wealth at the end of the decumulation period appear to be superior to strategies that directly attempt to minimize risk measures such as the probability of portfolio depletion or CVAR. The target-based approaches result in a reasonably close approximation to the retirement spending available in a DB plan. There is a small risk of depleting the retiree’s funds, but there is also a good chance of accumulating a buffer that can be used to manage unplanned longevity risk or left as a bequest.

Notes

1 Note that industry surveys suggest that retirees are extremely concerned about possibly exhausting their savings. For example, see https://www.allianzlife.com/about/news-and-events/news-releases/Generations-Ahead-Study-2017.

2 As mentioned above, we ignore labor income risk. Many studies assume that real earnings are expected to follow a hump-shaped pattern, rising rapidly until about age 35, then more slowly until around age 45–50, and slowly declining thereafter (see, e.g., Cocco, Gomes, and Maenhout 2005; Blake, Cairns, and Dowd  2014). It is common to add diffusive shocks to this trend, though Cocco, Gomes, and Maenhout (2005) calculate that the utility costs of assuming labor income has no risk are not high, absent a very large negative shock to income, which would be highly unlikely in a diffusive model. It is also worth noting here that the hump-shaped pattern described above has been questioned recently by Rupert and Zanella (2015), who find that although wage rates do rise rapidly in the early years of a typical employee’s career, they do not decline before retirement. Average income does fall on average during those years, but this is because of a reduction in hours worked by some employees transitioning into retirement.

3 Investment Company Fact Book (2018), available at www.ici.org.

4 Donnelly et al. (2017) conduct some resampling experiments, but only for the equity market (not the bond market) and over a relatively short period.

5 A possible extension would be to incorporate stochastic volatility. However, previous work has shown that stochastic volatility effects are small for the long-term investor (Ma and Forsyth 2016). This can be traced to the fact that stochastic volatility models are mean-reverting, with typical mean-reversion times of less than one year.

6 We have also carried out tests using a 10-year U.S. treasury as the bond asset (Forsyth and Vetzal 2017a). The results are qualitatively similar to those reported in this article.

7 More frequent rebalancing has little effect for long-term (> 20 years) investors (Forsyth and Vetzal 2017b).

8 See Appendix C for a precise definition of CVAR as used in this work.

9 Results for the linear glide path are again similar to the constant proportion case with p = 0.40 and have been excluded from Table 3 to save space.

10 Experiments with larger values of W^min increased Pr$[W_T<0]$ in the bootstrap tests.

11 Recall that units are thousands of dollars, so this corresponds to real terminal wealth of $1,000,000.

12 This is the time average of the median value of the equity weight p.

13 We experimented with other ways of specifying $W^{*}$. For example, rather than using the value that resulted in $E[W_T]=1000$, we determined the value which minimized Pr$[W_T<0]$. Although this looked promising in the synthetic market, its performance in the historical market tests was worse compared to the strategy that set $E[W_T]=1000$.

14 Recall that the optimal strategy for minimizing Pr$[W_T<0]$ was not very robust in terms of the bootstrap stress tests.

15 Table 7 excludes some strategies that performed relatively poorly, such as minimizing the probability of ruin with $W^{\mathit{min}}=0$ and the mean-CVAR strategy with $\kappa =-{10}^{-8}$.

16 Recall that we define CVAR as the mean of the worst α fraction of terminal wealth, not the losses, so we want to maximize CVAR to minimize risk.

17 This parameter has the intuitive interpretation that if the absolute value of the $\text{log}\hspace{0.17em}$ return in a period is larger an α standard deviation Brownian motion return, then it is identified as a jump.

18 This approach has also been used in other tests of portfolio allocation problems recently (e.g. Dichtl, Drobetz, and Wambach 2016).

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. The authors have no conflicts of interest to report.

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