Care must be taken with portfolio return assumptions, as small differences compound into dramatically different financial outcomes over a lifetime. My research shows just how big those differences are and how they vary in the pre and postretirement phases.
Though Monte Carlo simulations are the methodology of choice for projecting most financial plans, advisors often use simple spreadsheets for lifetime financial plans. Those spreadsheets typically use a fixed portfolio return assumption. After reviewing some of the basics for making the portfolio return assumption — which can provide valuable insights for clients — I will explore relatively new ground with respect to how sequenceofreturns risk should impact the assumptions.
I find empirical support for the idea that portfolio return assumptions for the postretirement period should be more conservative than for the preretirement period. In turn, assumptions for the preretirement period should be more conservative than when simply applying a compounded return to a lump sum investment.
Basics for choosing a portfolio return assumption
Table 1
Summary Statistics for U.S. Market Returns, 1926  2011



S&P 500 
Intermediate Term
Government Bonds (ITGB) 

Average (Arithmetic) Return 
11.8% 
5.5% 
volatility (stocks: 20.3%; bonds: 5.7%) > 


Compounded Return 
9.8% 
5.4% 
inflation (arith. mean: 3.1%, volatility: 4.2%) > 


Real Compounded Return 
6.6% 
2.3% 
asset allocation (client uses a 50/50 portfolio) > 


50/50 Portfolio Real Arithmetic Return 
5.6% 
50/50 Portfolio Real Compounded Return 
5.0% 
50/50 Annual Volatility 
11.0% 

Source: own calculations from Stocks, Bonds, Bills, and Inflation data provided by Morningstar and Ibbotson Associates. The U.S. S&P 500 index represents the stock market, and intermediateterm U.S. government bonds represent the bond market. 
For a lifetime financial plan, the most intuitive way to express a portfolio return assumption is as the inflationadjusted compounded portfolio return. Unfortunately, this is generally not the most common way returns are expressed. It is worth a quick review of the steps needed to arrive at a real compounded return.
For someone seeking to develop portfolio return assumptions based on U.S. historical data, Table 1 shows the basic steps. This table is based on the Stocks, Bonds, Bills, and Inflation data since 1926 from Morningstar and Ibbotson Associates. Historically, the S&P 500 provided an arithmetic return of 11.8%. This is the number one gets by adding up all the annual returns from the historical data and then dividing by the number of years in the data set. For intermediateterm government bonds, the arithmetic return was 5.5%.
While these numbers can arguably provide the best estimate of what the next year's return will be based on historical data, they are not good return assumptions for a longterm financial plan. Returns are volatile, and high and low returns do not have a symmetric impact on wealth. Negative returns must be followed by even larger positive returns to get back to where one started. For this reason, wealth will grow at a lower compounded rate than arithmetic averages. In the case of this historical data, the historical compounded returns were 9.8% for the S&P 500 and 5.4% for intermediateterm government bonds.
Another important step is to remove the effects of inflation. This puts the analysis on a consistent basis over time so that the advisor and client can discuss financial plans in terms of today's purchasing power. Even low inflation can compound over time into a big impact on purchasing power. Not removing inflation from the calculations can lead to additional confusion for the client. The next step in Table 1 is to remove the effects of inflation from the compounded returns. Historically, the S&P 500 provided an inflationadjusted compounded return of 6.6%, and it was 2.3% for intermediateterm government bonds.
A further step is to consider the asset allocation for the client portfolio. Let's consider a 50/50 asset allocation between stocks and bonds, rebalanced annually. In real terms, the 50/50 portfolio historically earned 5.6% as an arithmetic return and 5% as a compounded return. Its annual measure of volatility (the standard deviation) was 11%.
There are a number of further adjustments we could make, such as incorporating taxes, accounting for the fact that today's interest rates are much lower than the historical averages, removing the impacts of advisory and investment fees and considering the possibility of outperformance or underperformance with respect to the underlying market indices. While these are all important considerations, I will focus specifically on how advisors should adjust their assumptions to account for sequenceofreturns risk.
The impact of sequence risk
The motivation for this analysis is based on two discussions I recently enjoyed. First, Ramiro Sanchez, a retired feeonly certified financial planner, sent an email asking whether sequence risk could be quantified as the difference between the geometric mean for a series of returns and the underlying internal rate of return for a series of distributions based on a portfolio earning those returns. Second, Daniel Flanscha, a Coloradobased financial planner, noted that when he uses a fixed return assumption, he subtracts 12.5% from the return during the accumulation phase and 2.54% from the return during the distribution phase to account for the effects of randomness.
I will test these ideas using 100,000 Monte Carlo simulations with a return distribution for a 50/50 portfolio based on Table 1. The arithmetic mean return and the standard deviation feeding the simulations are 5.6% and 11%, respectively. With this volatility, the simulations will naturally provide outcomes with a compounded return centered on 5%.
Consider three scenarios: an individual making a lumpsum investment for 30 years, an individual saving a fixed percentage of a constant inflationadjusted salary over a 30year accumulation period and an individual withdrawing a constant inflationadjusted amount from a portfolio over a 30year retirement period. In retirement, the amount withdrawn represents the highest sustainable withdrawal amount over the 30year period, such that in each simulation the investor is left with $0 at the end of the 30th year.
Table 2 provides the distribution of results for the simulations. For the lumpsum investment, the numbers represent the distribution of average compounded returns over the 100,000 30year periods. For the accumulation phase, the distribution of outcomes is for the internal rate of return when making 30 annual contributions at the end of each year relative to the final accumulated wealth at the end. For the retirement phase, the distribution of results is for the internal rates of return on the portfolio when withdrawing the maximum sustainable amount over a 30 year period. In retirement, withdrawals are made at the beginning of each year.
Table 2
Distribution of Real Portfolio Returns Over 30 Years 

Lump Sum

Accumulation 
Retirement 

Minimum 
4.2% 
5.3% 
5.2% 
1st Percentile 
0.4% 
0.2% 
0.4% 
5th Percentile 
1.8% 
1.5% 
1.1% 
10th Percentile 
2.5% 
2.3% 
1.9% 
25th Percentile 
3.7% 
3.6% 
3.3% 
Median 
5.0% 
5.1% 
4.9% 
75th Percentile 
6.4% 
6.6% 
6.6% 
90th Percentile 
7.6% 
7.9% 
8.3% 
95th Percentile 
8.4% 
8.8% 
9.3% 
99th Percentile 
9.8% 
10.3% 
11.4% 
Maximum 
13.7% 
13.7% 
18.5% 

Std. Deviation 
2.0% 
2.2% 
2.5% 

Source: own calculations with 100,000 Monte Carlo Simulations for 30year periods using 5.6% arithmetic mean and 11% standard deviation. 
There are two issues to consider. First, riskaverse investors may not wish to assume that their wealth will grow at the median compounded return of 5%. In half of the simulations, the compounded returns and internal rates of returns will be less than 5%. When choosing a number to plug into a spreadsheet, a conservative investor might be more comfortable using something like the return in the 25th percentile, or even the 10th percentile, of the distribution. With the lumpsum investment, the compounded return at the 25th percentile is 3.7% over 30 years. An investor could expect to earn a compounded return of less than 3.7% in 25% of cases, but more than 3.7% in 75% of cases. The standard deviation for these compounded returns across the range of outcomes is 2%.
Monte Carlo simulations generally present results in terms of a probability for success. Clients may aim for 90% success or higher. There is an implied rate of return on the 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. In this discussion, I am tackling 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. Conservative investors will want to work with lower assumed returns, implying a need to save more today.
The second issue is that investors who are accumulating or spending their assets will have different experiences than implied by a lumpsum investment. Accumulation effectively places greater importance on the returns earned late in one’s career when one’s wealth is greater. For instance, a 10% return implies growth of $100,000 for a $1 million portfolio, but growth of $1,000 for a $10,000 portfolio. With new contributions each year, the timing of returns matters. This is sequenceofreturns risk as it applies in the accumulation phase. With greater importance placed on a shorter sequence of returns, we should expect a wider distribution of outcomes. With this 50/50 portfolio, the standard deviation of internal rates of return from the accumulation plan is 2.2% instead of 2%. This difference is relatively minor, but it does imply that a conservative investor will want to use a lower return assumption for a lifetime financial plan with annual contributions than if he or she is just investing a lump sum.
As for retirement, the impacts are even bigger. Retirees experience heightened sequenceofreturns risk when funding a constant spending stream from a volatile portfolio, as portfolio declines imply needing to spend a greater percentage of remaining assets. This digs a hole for the client's portfolio that can be difficult to overcome. The distribution of internal rates of return during retirement will be even wider because of the heightened importance placed upon the shorter sequence of postretirement returns. The standard deviation for this distribution increased from 2% for the lumpsum case to 2.5% for retirement. A conservative retiree seeking a return assumption at the 25th percentile would want to lower his or her assumption from 3.7% to 3.3% to account for sequenceofreturns risk in retirement. An even more conservative retiree looking at the 10th percentile of outcomes would want to lower his or her return assumption from 2.5% to 1.9% to account for retirement sequence risk, relative to the lumpsum investment case.
Not only does sequence risk widen the distribution of outcomes in retirement, but retirees also experience less risk capacity. With less time and flexibility to make adjustments to their financial plans, portfolio losses can have a bigger impact on remaining lifetime standards of living once retirees have left the workforce. For this reason, clients may want to use different return assumptions pre and postretirement. Extending my earlier discussion, a conservative client might be willing to use the 25th percentile return during accumulation but only the 10th percentile during retirement. If the client were comfortable with the arithmetic real return and volatility of 5.6% and 11%, this would suggest using a 3.6% compounded real return assumption for accumulation and a 1.9% compounded real return assumption for retirement. The haircuts of 2% and 3.7%, respectively, fit into the guidelines developed by Dan Flanscha.
The bottom line
I have quantified the conclusions reached by both Ramiro Sanchez and Dan Flanscha. Because of sequenceofreturns risk, conservative investors will want to use lower fixed return assumptions than just the compounded return assumed for a lumpsum investment. Sequence of returns risk is relevant for both the accumulation and retirement phases. Assumed returns should be lower in both cases. The impact is even greater for retirement. Conservative clients will not want to use the “expected return” for their portfolios when developing lifetime financial plans.
Wade D. Pfau, Ph.D., CFA, is a professor of retirement income in the new 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.
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