Variable Withdrawal Showdown

We attempt to realistically compare 6 asset allocation strategies x 7 consumption strategies over the complete asset accumulation-decumulation life cycle. Making the comparison realistic means we consider the importance of Social Security income to retirement consumption, and we struggle over the appropriate utility (well being) function associated with different consumption levels. We might not pin things down perfectly, but we feel the framework we use is a major improvement over failing to consider Social Security income, or using a more simple mathematically pure utility function.

At some point you are better off purchasing a Single Premium Immediate Annuity (SPIA), but if you choose not to, the table below indicates how you would be expected to fare under each of the strategies relative to age in bonds / 4% rule.

Improvement in Portfolio Consumption consumption strategy
best constant
1/life SDP
asset allocation strategy age in bonds - 20% 30% 33% 7% 34% 37%
age minus 10 in bonds 5% 27% 38% 42% 14% 42% 46%
target date 8% 29% 41% 45% 17% 45% 49%
best fixed 12% 32% 53% 55% 28% 55% 61%
best fixed GI-bond 17% 42% 60% 62% 32% 62% 69%
SDP n/a n/a 62% 64% 34% 63% 70%
  • Asset allocation:
    • Age in bonds appears a very poor asset allocation strategy for this realistic scenario. This scenario has a best fixed / SDP asset allocation of 90/10. This high stock allocation is plausible remembering that for many retirees Social Security income is their primary asset and acts as a surrogate bond.
    • All of the glide path rules perform poorly.
    • An asset allocation of best fixed with guaranteed income treated like a bond does very well. Unfortunately there is no easy way to compute the appropriate best fixed asset allocation (30/70 is the right answer for best fixed GI-bond / SDP).
  • Consumption:
    • 4% rule and best% rule appear ill-suited to this scenario. The 4% rule was designed not to deplete the portfolio over a 30 year period, but in the presence of guaranteed income doing so isn't the end of the world.
    • IRS RMD does poorly. This is to be expected. It employs a life table for a couple with a spouse 10 years younger.
    • Consumption strategies best constant percentage, VPW, and 1/life all do well, but fall short of SDP. In VPW's case this is presumably because of the lack of stochastic mortality, while in 1/life's case it might be because of the lack of consideration of the portfolio's potential for growth.
  • SDP offers the best asset allocation and consumption strategies. This isn't too surprising. SDP is optimal if returns are independent over time. The long term time-wise dependence of returns (reversion to the mean and/or momentum) is relatively weak.
  • There is a big gap between what may be computed and easily implemented by an individual using rules of thumb (age minus 10 in bonds / 1/life) and the best results (SDP / SDP) - a difference of 28%. This is unfortunate and significant. It points to a role for professional advisors and/or presenting SDP in a more accessible fashion.


Improvement in Portfolio Consumption (IPC)
This is the increase in Certainty Equivalent consumption relative to age in bonds / 4% rule after factoring out $15,000 in guaranteed Social Security income. Age in bonds / 4% rule has a CE of $22,442, SDP / SDP has a CE of $27,667, so the IPC is 70%.
Certainty Equivalent (CE) consumption
This is the constant consumption over retirement that would have the same discounted expected utility as the range of simulated consumption sequences.
Asset allocation strategies
age in bonds age divided by 100 in bonds
age minus 10 in bonds age minus 10 divided by 100 in bonds
target date the consensus target date fund as tracked by the S&P Target Date indexes and reported by iShares as of June 30, 2012 but with all equities treated as U.S. stocks, and all fixed income treated as bonds
best fixed the best fixed percentage of stocks/bonds
best fixed GI-bond best fixed with future expected guaranteed income treated as bonds
SDP SDP with goal of maximizing expected retirement CE
Consumption strategies
4% rule 4% of the portfolio size at retirement adjusted for inflation
best% rule the best constant real dollar amount
best constant percentage the best constant percentage of the current portfolio size
VPW consume portfolio equally over 30 years assuming growth rate of 4% real per year based on current portfolio size (4% was used rather than the more commonly used 3% to bring return expectations closer to historical market realities)
IRS RMD the formula the IRS uses to compute Required Minimum Distributions extended back to age 65
1/life current portfolio size / remaining life expectancy
SDP SDP with goal of maximizing expected retirement CE
Stochastic Dynamic Programming (SDP)
SDP is a general purpose mathematical technique that can be applied to asset allocation to compute the optimal asset allocation and consumption as a function of the current age and current portfolio size. Correlations between asset classes are allowed. Asset class returns are assumed to come from a distribution that is independent over time. Another name for SDP is backward induction. It computes the optimal strategy at age 119, uses that information to then compute the optimal strategy at age 118, and so on.
Asset classes
Stocks and bonds. Based on historical returns (1927-2013). Stocks adjusted to 19 country average. Bonds adjusted to U.S. investment grade bond universe. After expenses:
  • stocks - 5.1% geometric real return; 19.4% std. dev.
  • bonds - 2.6% geometric real return; 9.7% std. dev.
Rebalancing to the target asset allocation is performed annually. No attempt is made to constrain asset allocations to a risk tolerance. Leverage is not allowed.
This indicates the desirability of a particular consumption level. Constant relative risk aversion (CRRA) utility functions have a number of nice mathematical properties, but they are not especially realistic. We attempt to remedy this. Below $30,000 we use a CRRA utility function with a coefficient of relative risk aversion of 4. Highly risk averse. Above $40,000 we use a CRRA utility function with a coefficient of relative risk aversion of 1, and with a slope of 1/20th of the first utility segment. Very non-risk averse, and surplus money is valued very little. In between we interpolate using a cubic polynomial. In layman's terms, this all means that we will be largely oblivious to high consumption values, but we will punish a strategy that results in potentially low consumption very harshly.
The future is valued less than the present for a variety of reasons. We discount future consumption at a rate of 3% per year.
Bequest motive
Taxes were not considered.
Accumulation begins at age 25 of $500/year and grows by 7% real per year, reaching $6,997 real/year at age 64. A more stochastic accumulation process might be possible, but it would be difficult to get the parameters right.
Retirement begins at age 65, and the length of retirement is random (stochastic). We analyze a male born in 1949 in accordance with the Social Security Actuaries AS 120 cohort life tables.
Guaranteed income
We assume an inflation indexed $15,000/year in guaranteed Social Security income, which will probably provide a majority of the retiree's income.
An internal command line version of was used to generate and validate the strategies. 50,000 synthetic returns sequences were produced by bootstrapping the historical record with a block size of 20 years. Each sequence was evaluated for the full range of longevity possibilities.


The scenario above results in a portfolio size of around $150,000-400,000 at retirement. Doubling the amount of all retirement contributions increases the expected retirement portfolio to $250,000-$800,000 and yields the following table:

Improvement in Portfolio Consumption consumption strategy
best constant
1/life SDP
asset allocation strategy age in bonds - 5% 13% 17% 2% 14% 22%
age minus 10 in bonds 4% 9% 18% 22% 6% 18% 27%
target date 5% 10% 20% 23% 8% 19% 28%
best fixed 5% 11% 26% 27% 12% 24% 34%
best fixed GI-bond 11% 16% 30% 30% 16% 28% 38%
SDP n/a n/a 29% 29% 16% 25% 36%

The absolute magnitude of the improvements are reduced, but the conclusions are largely unchanged.


Reducing all retirement contributions by a factor of 2.5 yields the following table:

Improvement in Portfolio Consumption consumption strategy
best constant
1/life SDP
asset allocation strategy age in bonds - 38% 45% 45% 18% 46% 48%
age minus 10 in bonds 5% 46% 54% 54% 26% 56% 58%
target date 9% 50% 59% 59% 30% 61% 63%
best fixed 18% 58% 82% 82% 50% 84% 87%
best fixed GI-bond 21% 68% 86% 86% 53% 88% 91%
SDP n/a n/a 86% 86% 53% 88% 91%

Once again the conclusions are largely unchanged.


Joe Tomlinson has provided many helpful comments along the way.