Some advisors remain critical of Monte Carlo simulations, instead preferring to use analysis based on rolling historical periods or specific pre-defined scenarios. We believe Monte Carlo is a superior tool for measuring the uncertainties in long-term financial planning.
Lynn Hopewell implored the financial advisory profession to adopt Monte Carlo simulation tools in his seminal 1997 article1 in the Journal of Financial Planning. He argued forcefully against merely developing spreadsheets for financial plans based on average input assumptions, or by testing the robustness of plans with worst-case scenarios. Those approaches do not provide probabilities for outcomes, and one must struggle to figure out how to save, spend and develop a financial plan using such limited analysis.
In the past 17 years, the widespread use of Monte Carlo simulations has led to much progress. In our previous article, we provided a brief refresher about Monte Carlo to help ensure that readers are up to speed. In this column, we extend our analysis to illustrate what Monte Carlo simulations show relative to other methodologies when developing a retirement-income plan.
As an example, we use Monte Carlo simulations to predict the likelihood of a successful 4% withdrawal rate under today’s market conditions.
Comparing Monte Carlo simulations to rolling historical periods
The 1998 “Trinity” study, by Cooley, Hubbard, and Walz, is one of the classics in the field of retirement income planning. The authors concluded that the 4% rule (that is, withdraw 4% of retirement-date assets and adjust this amount for inflation in subsequent years) has a 95% chance for success over 30 years based n investing in a portfolio with 50% large-capitalization U.S. stocks and 50% long-term corporate bonds (i.e., a 50/50 portfolio).
The phrase “95% chance for success” might lead one to conclude that the Trinity Study is based on Monte Carlo simulations. It wasn’t. It was based on rolling historical returns.
With data from 1926 through 2013, we used Monte Carlo simulations to calculate sustainable withdrawal rates for hypothetical 30-year retirements starting between 1926 and 1984. (We don’t yet know how matters will work out for post-1984 retirees.) This represents 59 retirement periods, each lasting 30 years. Our results are shown in Figure 1, for stock allocations ranging from 0% to 100%.
Figure 1 compares the historical portfolio success rates based on the Trinity study’s rolling-period analysis to our success rates using Monte Carlo simulations. We parameterized our analysis to the same annual historical data as was used with historical simulations, the Stocks, Bonds, Bills, and Inflation dataset from Ibbotson and Morningstar.
To calculate the probability of success for the 4% rule, the Trinity approach adds up the percentage of these historical retirement periods where the target withdrawal amount was achieved. When using the long-term corporate bond series, retirements beginning in 1965 and 1966 would have just missed being sustainable over 30 years with a 4% initial withdrawal rate. William Bengen’s initial research from 1994 used less-volatile intermediate-term government bonds, and he found that 4% would have always worked for a 50/50 portfolio (100% success rate).
Monte Carlo simulations have a number of advantages over their historical-simulation counterparts. First, Monte Carlo allows for a wider variety of scenarios than the rather limited scenarios that historical data can provide. In this case, we simulated 10,000 retirements, compared to the 59 available with the historical data. (Those 59 are not even independent of each other, because they share data points.) This provides an opportunity to observe a much wider variety of return sequences that support a deeper perspective about possible retirement outcomes.
Because of the way that historical simulations form overlapping periods, the middle part of a historical record plays a disproportionately important role in the analysis. In our data set, 1926 appears in only one rolling historical simulation, while 1927 appears in two (for the 1926 and 1927 retirees). This pattern continues until 1955, which appears in 30 simulations (the last year for the 1926 retiree through the first year for the 1955 retiree). The years 1955 through 1984 all appear in 30 simulated retirements. Then a decline occurs as 1985 appears in 29 simulations, through 2013 which only appears in one simulation (as the final year of retirement for the 1984 retiree). The over-weighted portion (1955-1984) of the data coincides with a severe bear market for bonds. For the years 1955 to 1981, the real arithmetic return on intermediate-term government bonds was -0.1%, compared to an average of 3.6% for the combined years prior to 1955 and subsequent to 1981. The differences are even more severe for long-term government and corporate bonds.
On the other hand, Monte Carlo simulations treat each data point equally. The middle years do not play a disproportionate role in determining outcomes. Monte Carlo simulations of the 4% rule based on the same underlying data as historical simulations show greater relative success for bond-heavy strategies and less relative success for stock-heavy strategies. The optimal stock allocation is lower as well.
The 4% rule did not fail in the historical period for stock allocations between 40% and 70%. More bond-heavy portfolios experienced much lower success rates, though, with a bonds-only portfolio succeeding just in 41% of the historical simulations. With Monte Carlo simulations based on the same historical data, retirees would be encouraged to hold some stocks, as success rates of over 90% are possible with stock allocations of only 20%. The highest success rates occurred in the range between 30% and 50% stocks.
Monte Carlo simulations count each year of the historical data equally. It is not subject to the bias against bonds in the historical simulations resulting from an over-weighting of the middle historical years.
Changing capital-market expectations
Another fundamental problem with historical simulations is that a limited historical period does not provide sufficient variability in outcomes to understand what might happen in the future. Some investors may be comfortable with the idea that because the U.S. historical experience includes a wide variety of market environments, the worst-case scenario from history should be fairly reflective of what can be expected to happen in the future.
To put it bluntly, though, who cares if something worked in the past? The problem with attempting to gain insights from historical outcomes is that future market returns and withdrawal-rate outcomes are connected to the current values of the sources for market returns. Writing in his 2009 book Enough, John Bogle made this point:
My concern is that too many of us make the implicit assumption that stock market history repeats itself when we know, deep down, that the only valid prism through which to view the market’s future is the one that takes into account not history, but the sources of stock returns.2
For equity markets, those sources include income, growth and changing valuation multiples. Returns on bonds, meanwhile, depend on the initial and subsequent bond yields. Low bond yields will translate into lower returns due to less income and the heightened interest-rate risk associated with capital losses if interest rates rise.
Today we are dealing with a situation in which Shiller’s cyclically adjusted price-to-earnings ratio (CAPE) is well above historical averages, while bond yields are at historical lows. Retirees are also exposed to sequence-of-returns risk, as the returns experienced in early retirement will have a disproportionate impact on their final retirement outcomes. When spending down their principal in the meantime, it will be unfortunate even if interest rates and market valuations “normalize” in the next 5-10 years.
Today’s high-valuation/low-yield situation has been quite rare in U.S. history, indicating we are in uncharted territory when trying to determine if the 4% rule will remain a safe strategy. Historical simulations don’t analyze this possibility, but with Monte Carlo simulations we can adjust our capital-market expectations to better account for the types of returns that are more likely to be experienced in the future.
Here is an example to demonstrate the vulnerability of the 4% rule. On Aug. 15, five-year Treasury Inflation-Protected Securities were yielding -0.28%. This is 2.76 percentage points less than the historical 2.48% real return experienced by intermediate-government bonds. Today’s retirees will be more strained to spend principal to achieve a 4% sustainable withdrawal rate. If we assume that the historical risk premium for stocks and other asset characteristics remains the same, but we adjust the average return on stocks and bonds downward by 2.76% to reflect today’s lower bond yields, we obtain higher failure rates for the 4% rule.
Figure 2 repeats Figure 1 but includes portfolio success rates for Monte Carlo simulations based on today’s low yields. The ability to make these adjustments is unique to Monte Carlo simulations. These adjustments question the safety of 4% as a sustainable withdrawal strategy. The highest portfolio success rates are seen with stock allocations between 80% and 100%, and the success rate is only 73% in these cases. With low bond yields, it is extremely difficult for a bond-heavy portfolio to sustain 4% withdrawals, and success rates fall as low as 10% for an all-bond strategy (based on bond funds, not laddered individual bonds).
The international perspective and uncertainty of our assumptions
How realistic are our underlying assumptions for mean returns, standard deviations and correlations, which guide our Monte Carlo simulations? While it seems reasonable to focus on U.S. historical data, who is to say whether the future experience of American retirees will be similar to the past or whether it will be more reflective of situations experienced in other countries?
Wade has written before at Advisor Perspectives about the international experience of safe withdrawal rates. Table 1 summarizes a key point about the discussion by showing the historical success rates for the 4% rule using financial-market data from different countries since 1900. Results vary a bit when using different datasets. With a fixed allocation of 50% stocks, 40% bonds and 10% bills, the U.S. enjoyed a 95% historical success rate for the 4% rule using rolling periods. This number was equally high in Canada.
In other countries, results varied dramatically. The historical success rate for the 4% rule was as low as 36% for France and 25% for Italy. Care must be taken when choosing the assumptions to guide Monte Carlo simulations.
Table 1
International Success Rates for the 4% Rule
For a 50/40/10 Stocks/Bonds/Bills Asset Allocation |
Australia |
81% |
Austria |
54% |
Belgium |
45% |
Canada |
95% |
Denmark |
92% |
Finland |
60% |
France |
36% |
Germany |
46% |
Ireland |
62% |
Italy |
25% |
Japan |
64% |
Netherlands |
71% |
New Zealand |
91% |
Norway |
47% |
South Africa |
91% |
Spain |
51% |
Sweden |
81% |
Switzerland |
74% |
United Kingdom |
76% |
United States |
95% |
Note: Assumptions include a 30-year retirement duration, no administrative fees, constant inflation-adjusted withdrawal amounts and annual rebalancing. |
Source: Own calculations from Dimson, Marsh, and Staunton (1900 - 2013) Global Returns Data. |
Mean reversion in the historical data
While Monte Carlo simulations can incorporate mean reversion in market returns for both stocks and bonds (for example, see our article with Michael Finke in the Retirement Management Journal), most financial-planning software and research is based on an assumption that while asset returns may be correlated with each other in the current period, they are otherwise “independent and identically distributed.” What this means is that basic Monte Carlo will not develop a link between past returns and current returns. Each return is simulated independently of the previous return.
In reality, many asset classes will exhibit returns that relate to their past returns. In particular, with stocks a large positive return (which increases the value of a measure such as Shiller’s CAPE) will lower the expected future returns from that point forward. Basic Monte Carlo misses this.
The implication is that Monte Carlo simulations will exhibit wider tails on both the upside and downside relative to what is observed using rolling periods from historical data. Portfolio Pathfinder’s Dick Purcell has a way to illustrate this. For this analysis, we use Shiller’s dataset with large-cap stock returns and inflation since 1871. For rolling periods with lengths ranging from 1 to 40 years, we calculated the compounded real returns for 10,000 Monte Carlo simulations based on the historical data, as well as for as many rolling historical periods as are available with Shiller’s stock data from 1871 through 2013 (for example, there 124 rolling 20-year periods in the data). Figure 3 plots the median compounded return, as well as the 5th percentile and 95th percentile from the distribution. With the historical data, the distribution of outcomes is narrower. This illustrates mean reversion in the historical data.
Another criticism about basic Monte Carlo simulations is that the underlying distribution of returns is generally a normal distribution, which does not exhibit the types of fat tails observed in historical data. However, this concern is offset by the lack of mean reversion in the simulations.
Overall, the basic approach to Monte Carlo simulations does a good enough job to help clients understand whether their plans are on track. Monte Carlo simulations will illustrate a wide range of potential outcomes with greater dispersion than seen with the historical data. The lack of mean reversion serves as a more conservative assumption that can help counteract other effects such as observable fat tails in outcomes.
Using Monte Carlo to develop straight-line return assumptions
Using a fixed rate-of-return assumption in a spreadsheet for financial planning calls for caution about one’s choice of returns. It is not appropriate to simply use one’s expected compounded return, since it will probably leave a plan exposed to a 50% chance that the realized compounded return experienced by the client could fall below the expectation.
In a recent Advisor Perspectives column, Wade addressed how Monte Carlo simulations could be used to develop conservative fixed rate-of-return assumptions for a straight-line analysis in an Excel spreadsheet. The answers differ based on whether one is investing a lump sum, making period contribution during the accumulation phase or spending down assets in retirement. Indeed, Monte Carlo simulations provide a powerful tool to quantify what advisors may intuitively grasp but would have difficulty to illustrate because of the analytical complexity of the problem.
Wade found empirical support for the idea that portfolio-return assumptions for the post-retirement period should be more conservative than for the pre-retirement period. In turn, assumptions for the pre-retirement period should be more conservative than when simply applying a compounded return to a lump-sum investment.
For the example in that article, an advisor’s assumptions for a portfolio were an arithmetic average return of 5.6% with a standard deviation of 11%. The volatility drag means that the portfolio can be expected to grow at 5%. But that is the median outcome. With Monte Carlo simulations, at the 10th percentile of the distribution, a lump-sum investment compounds at 2.5% over 30 years, while the internal rate-of-return on the growth of a portfolio with new accumulations is 2.3%, and the internal rate of return when taking distributions (which exposes the client to even greater sequence risk) is only 1.9%. This means, for example, that a client seeking a 90% success rate who is comfortable with these capital market expectations, but who is building a financial plan with a fixed return assumption in a spreadsheet, would not want to use the 5% compounding average (and certainly not the 5.6% arithmetic average). Instead, the client would rather use 2.5%, 2.3% or 1.9%, depending on where he or she is in the lifecycle.
The power of Monte Carlo is that it lets us quantify this. There is an implied rate of return on the underlying portfolio connected to a given probability of success, though Monte Carlo simulations generally do not express their output in this way. Higher rates of success would be connected with lower portfolio returns, since this return hurdle must be exceeded by the portfolio for the financial plan to be successful. This example tackles Monte Carlo from a different direction – first using Monte Carlo simulations to get a rate of return for the portfolio, then to simulate a financial plan using a fixed rate of return.
The bottom line
Monte Carlo simulations are valuable for advisors to assist their clients in understanding the probabilities relating to different possible outcomes for their financial and retirement plans. Any deficiencies in Monte Carlo are solely based on how the tools are used, not with the underlying concept of Monte Carlo. Overall, the advantages of Monte Carlo simulations more than make up for any deficiencies with respect to historical simulations or fixed-scenario analyses.
Wade D. Pfau, Ph.D., CFA, is a professor of retirement income in the Ph.D. program in financial services and retirement planning at the American College in Bryn Mawr, PA. He is also the director of retirement research for inStream Solutions and McLean Asset Management. He actively blogs about retirement research. See his Google+ profile for more information.
David M. Blanchett, CFA, CFP® is the head of retirement research for Morningstar Investment Management in Chicago, IL.
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