I recently released the HARS historical simulation tool. I have been using it to explore the world of asset allocation and in particular stock / bond allocation.
HARS lives here: Historical Actuarial Retirement Simulator
The conventional wisdom as I understand it and which I wish to explore is that when you are young you should favor stocks but as you grow older you should shift towards bonds. "Age in bonds" is one common rule of thumb. Another strategy I wish to explore I will call "go to bonds" and involves switching your entire portfolio to bonds if you are ever sufficiently fortunate to have enough assets to cover your entire future needs. Go to bonds goes to bonds when the total portfolio value equals or exceeds future needs computed at an indicated percentile value for life expectancy.
Baseline case: Couple retiring both aged 65 with $1m in assets and annual living expenses of $40k (4% withdrawal rate). Projected after inflation returns on stocks (domestic) / bonds (TIPS) reflecting historical data of 5.2% / 2.0% and are tax free. Rebalance annually. Simulated returns data taken at random from 1926-2008.
best static allocation 20% age in bonds go to bonds success prob.
success prob. (% stocks) stock success prob. 95pctl 98pctl 99pctl
with 3% withdrawal rate 100% (0-50%) 100% 100% 100% 100% 100%
Baseline (4% w.r.) 100% (10-20%) 100% 98% 100% 100% 100%
with 5% withdrawal rate 91% (30-40%) 90% 90% 91% 91% 91%
with 6% withdrawal rate 76% (60-70%) 65% 72% 77% 77% 76%
with 7% withdrawal rate 63% (70%) 40% 51% 63% 63% 63%
both age 60 and retiring 98% (10-20%) 98% 96% 98% 98% 98%
no rebal. deplete equity first 99% (0-30%) 99% n/a 99% 99% 99%
What can be learned from such simulations? First. Age in bonds never performs better than fixed stock / bond ratios, although the $64 question then becomes what is the right ratio. This lack of out performance is surprising. The dynamic nature of age in bonds is supposed to give it an advantage over static allocations. Thinking about the life of a portfolio, it would seem to make sense to take gains early on because then they can compound, and this is what age in bonds is intended to do. This though is a fallacy, 105% x 105% x 105% x 102% x 102% gives the same as 102% x 102% x 105% x 105% x 105. How does age in bonds stack up against its static cousin? A couple aged 65 have a median lifespan of an additional 25 years where one or the other of them will still be alive. Over their remaining lifespan their mean age will be (65 + 90) / 2 = 77.5 years. We thus declare holding 77.5% bonds the static cousin of age in bonds for this couple. As can be seen from the 20% equity column (the closest we can get to 22.5% equity) for portfolios with a high success probability (90-100%), the static cousin does as well or better than age in bonds. For the two low success rate portfolios the static cousin does significantly worse than age in bonds.
Second. Go to bonds rarely helps and rarely harms. This too makes sense. Any portfolio that is strong enough to take advantage of this strategy is well on its way to portfolio success, and has little need for this strategy.
Third. This is the most contentious part of this entire essay. For maximal odds of portfolio success the optimal static stock / bond ratio is larger the smaller the portfolio. This can be seen from the above table using withdrawal rate as a proxy that is inverse to effective portfolio size. This finding flies in the face of conventional wisdom. So to be clear, we are not talking about creating a legacy, donating to charity, or leaving something for the kids, we are solely concerned with portfolio success. What we will call the relatively wealthy middle class can afford not to take risks on the stock market, and so they shouldn't. While the poor have no other alternative for achieving portfolio success, so they should take a gamble on the stock market. These gambles frequently fail, and so the poor have worse odds of portfolio success, but it is better than not gambling at all. Turning now to the very rich, they can, but need not, take gambles on the stock market with impunity, such as if they wish to maximize their legacy. So optimal asset allocation now has the poor and very rich favoring stocks, while the relatively wealthy middle class sits comfortably in between favoring bonds.
Fourth. For portfolios with a high success probability but excluding the very rich, say 90-99% success probability, the optimal proportion of stocks held is surprisingly low, typically something like 20-30%.
Baseline case: Couple both aged 30 with $0 in retirement assets, saving $20k per year tax free. Plan to retire aged 60 and annual retirement living expenses of $40k. Projected after inflation returns on stocks (domestic) / bonds (TIPS) reflecting historical data of 5.2% / 2.0%. Rebalance annually. Simulated returns data taken at random from 1926-2008.
best static allocation 40% age in bonds go to bonds success prob.
success prob. (% stocks) stock success prob. 95pctl 98pctl 99pctl
with $30k withdrawal 100% (10-30%) 99% 97% 100% 100% 100%
Baseline ($40k withdrawal) 94% (30-50%) 94% 93% 94% 94% 94%
with $50k withdrawal 86% (50%) 85% 86% 86% 86% 86%
with $60k withdrawal 78% (70-80%) 72% 79% 79% 79% 79%
with $70k withdrawal 72% (80,100%) 56% 72% 74% 73% 73%
retire 5 years earlier 78% (70%) 73% 79% 79% 79% 79%
no rebal. deplete equity first 91% (50%) 90% n/a 91% 91% 91%
The first thing that should be striking about these numbers is how bland they are. A static stock / bond ratio performs as well as age in bonds, which performs as well as go to bonds.
Do we learn the same lessons as for the retirement portfolio? This time for the $60k withdrawal strategy age in bonds performs better than a fixed stock / bond ratio, but only by 1%, which could be noise. Additionally, age in bonds doesn't suffer the same drop off in performance as the withdrawal rate is increased. Overall, this time age in bonds matches rather than lags the fixed stock / bond ratios. But once again it certainly doesn't outperform the alternatives. How does the static cousin compare this time (58 years of additional life expectancy, giving 59 as the average age, and 41% a the percentage stocks)? The static cousin just slightly outperforms age in bounds on the high success rate portfolios (90-100%), while significantly lagging on the low success rate portfolios. Additional data would be good here.
As before, go to bonds rarely helps and rarely harms.
And for maximal odds of portfolio success the optimal stock / bond ratio is larger the larger the withdrawal amount. A higher proportion of stocks is needed with a smaller effective portfolio size.
Lastly, for portfolios with a 90-99% success probability, the optimal proportion of stocks held is probably something like 30-50%. (There is a paucity of data supporting this claim). For the retirement portfolio we found a lower optimal proportion of stocks held, 20-30%. This would seem to support age in bonds. The young new 90-99% chance success portfolio (relatively) high in stocks mutating into the 90-99% chance of success retirement portfolio high in bonds. Age in bonds dynamic nature should thus have an advantage. This though overlooks the harm age in bonds can do. If the portfolio slips in size then age in bonds prevents the increased stock allocation that is then warranted. Or if the portfolio grows above its expected size, age in bonds does no, harm, but nor does it do good. A new strategy of asset allocation based in projected portfolio size seems warranted.
Can we trust the simulator itself? Like any software the simulator could contain bugs. The core has engine has been validated against Milevsky's probability formula, as well as a Monte-Carlo simulator written especially for the purpose that doesn't contain any code in common with the core simulator. In both cases they were found in good agreement. This though only goes as far as the core simulator engine. The simulator contains additional code for handling varying stock bond percentages, age in bonds, go to bonds, rebalancing, and all the rest. These components could contain bugs. The likelihood of this is reduced by these components being simple, typically only a few lines of code. The one exception being go to bonds, which is quite complex to implement. I've tested some of the corner cases for go to bonds and they appear to be working successfully, and in addition go to bonds is observed to cause a drastic reduction in mean assets at death, as would be expected. So it is likely, but by no means certain that go to bonds is working correctly.
What then remains of asset allocation? Asset allocation still exists and still matters. I tentatively suggest what has to happen is to think in terms of relative projected portfolio size, which is the size or the portfolio including future projected contributions relative to future withdrawals. The relative projected portfolio size of a young couple who have just started saving $50k a year for retirement would be large, while that of someone with $1m that retired at 50 could be small. Then asset allocation builds off of relative projected portfolio size as measured by the odds of portfolio success. A small projected portfolio demands mostly stocks. A medium projected portfolio demands mostly bonds. And a large projected portfolio could do either, but may as well do stocks. Exact values to use for asset allocation might be informed by simulation. With the asset allocation decisions revisited not over time, but in response to changes in relative projected portfolio size.
Combining the previous data gives the following table:
success prob. % stocks
100% 0-50%
100% 10-20%
100% 10-30%
99% 0-30%
98% 10-20%
94% 30-50%
91% 30-40%
91% 50%
86% 50%
78% 70%
78% 70-80%
76% 60-70%
72% 80,100%
63% 70%
Which can be boiled down into the following rough rules of thumb:
current
projected
success prob. % stocks
100% any - up to risk limitations
98-99% 0-30%
90-98% 30-50%
60-90% 50-80%
Samuelson (1963, 1969) and Merton (1969) show that a rational risk-averse investor's optimal bond-stock allocation for a fixed horizon of length T does not depend on the value of T. Samuelson (1994) writes that "it is an exact theorem that investment horizons have no effect on your portfolio proportions." Nonetheless, many financial advisors believe that investors should hold more bonds as they grow older [...]So my results are hardly new, except perhaps the suggestion that the poor take more risk with equities and the relatively more wealthy less. But even for that I have a feeling Jeremy Siegel might have once said something similar.