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Abstract
In contrast to single-period mean-variance (MV) portfolio allocation, multi-period MV optimal portfolio allocation can be modified slightly to be effectively a down-side risk measure. With this in mind, we consider multi-period MV optimal portfolio allocation in the presence of periodic withdrawals. The investment portfolio can be allocated between a risk-free investment and a risky asset, the price of which is assumed to follow a jump diffusion process. We consider two wealth management applications: optimal de-accumulation rates for a defined contribution pension plan and sustainable withdrawal rates for an endowment. Several numerical illustrations are provided, with some interesting implications. In the pension de-accumulation context, Bengen (1994)’s [J. Financial Planning, 1994, 7, 171–180], historical analysis indicated that a retiree could safely withdraw 4% of her initial retirement savings annually (in real terms), provided that her portfolio maintained an even balance between diversified equities and U.S. Treasury bonds. Our analysis does support 4% as a sustainable withdrawal rate in the pension de-accumulation context (and a somewhat lower rate for an endowment), but only if the investor follows an MV optimal portfolio allocation, not a fixed proportion strategy. Compared with a constant proportion strategy, the MV optimal policy achieves the same expected wealth at the end of the investment horizon, while significantly reducing the standard deviation of wealth and the probability of shortfall. We also explore the effects of suppressing jumps so as to have a pure diffusion process, but assuming a correspondingly larger volatility for the latter process. Surprisingly, it turns out that the MV optimal strategy is more effective when there are large downward jumps compared to having a high volatility diffusion process. Finally, tests based on historical data demonstrate that the MV optimal policy is quite robust to uncertainty about parameter estimates.
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Acknowledgements
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Notes
No potential conflict of interest was reported by the authors.
1 Obtained from data from Historical Indexes 2015 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of Business.
2 This ballpark estimate is conservative in the following sense. The Treasury does not actually issue 15-year bonds. If the investor had bought newly issued 30-year bonds for their par value at the start of 2000, the same $50 000 of coupon income would have been collected each year, but there would also have been a significant capital gain due to declining interest rates between 2000 and 2015. In other words, as of 2015 the investor would have owned bonds with a remaining maturity of 15 years that were worth substantially more than par.
3 Unlike previous work (e.g. Björk Citation2009, Vigna Citation2014), we do not assume that the portfolio is continuously rebalanced. As a result, we cannot specify the total wealth process in terms of a single stochastic differential equation. Consequently, it is simpler to define S(t) and B(t) in terms of dollar amounts invested, rather than prices of a unit investment in each assets, as it is typically done with continuous rebalancing.
4 One may argue that it would be preferable to include stochastic volatility effects in the S process. However, recent tests indicate that stochastic volatility has little effect on long-term dynamic MV optimal strategies (Ma and Forsyth Citation2016).
5 We are careful to distinguish the expected wealth target from the quadratic wealth target
in equation (Equation2.18
(2.18) ).
6 While similar numerical results are obtained for the pension de-accumulation case, the one–off withdrawal is unlikely to be practically feasible in that setting since pensioners rely on periodic withdrawals.
7 An alternative perspective on this strategy is to consider it as a form of debt restructuring. The commitment to withdraw cash on an annual basis for spending purposes has similar effects to those generated by incurring annual interest payments on a loan. The strategy outlined here effectively substitutes long-term zero-coupon debt for the leverage inherent in having yearly withdrawals.
8 The same effect also occurs in the base case.
9 Of course, if controlled wealth is ever greater than the amount needed to be invested in the risk-free asset which ensures that all remaining withdrawals can be made and the final wealth target can be reached for certain, the excess is a free cash flow.
10 We obtain qualitatively similar results for the endowment scenario.
11 See www.federalreserve.gov/releases/h15/data.htm. This data series is only available starting in 1934.
12 In particular, we use the annual average of the all urban consumers index (CPI-U), see http://www.bls.gov/cpi.
13 This table is reproduced from Table 8.1 of Dang and Forsyth (Citation2016).
14 A different type of robustness test for long-term MV optimal strategies has recently been reported by Ma and Forsyth (Citation2016). They compare a stochastic volatility model with GBM, and find that the two models produce very similar results.
