This tutorial is taken directly from SWR Research Group discussions on the No Fee Boards

Safe Withdrawal Rates versus Valuations

There are many factors that affect the safety of withdrawals during retirement. Prices are one of them. Prices are special. You control the price that you pay for an investment.

Our first major advance at the SWR Research Group was to relate Safe Withdrawal Rates quantitatively with valuations.
Our most successful indicator of valuations so far is Professor Robert Shiller’s P/E10.
(P/E10 is the price or index value of the S&P500 divided by the average of the last ten years of earnings.)

We have had to draw some precise distinctions between what has happened (which we call Historical Surviving Withdrawal Rates or simply HSWR), what is likely to happen (which we call Calculated Rates) and the confidence limits about the Calculated Rates.

We refer to the lower confidence limit as the Safe Withdrawal Rate.
We refer to the higher confidence limit as the High Risk Rate (or the High Risk Withdrawal Rate).

Early researchers estimated the Safe Withdrawal Rate to be the same as the smallest Historical Surviving Withdrawal Rate. They did not provide confidence limits.

Historical Surviving Withdrawal Rates (HSWR) are based upon using an actual historical sequence of investment returns. A hypothetical portfolio would have survived for a specified number of years at the HSWR. Its balance would have fallen to zero or become negative at a slightly higher (0.1%) withdrawal rate. Unless we identify the first year of a historical sequence, the HSWR is the lowest among those that we investigate (typically, 1921-1980). We adjust withdrawals to match inflation (and deflation).

We have learned that today’s stock market is well outside of the range of the historical database.
(This is true using both P/E10 and dividend yields.)
We have made suitable adjustments.

We have found that earnings yield (using the average of the past decade’s earnings) does an excellent job when estimating future Safe Withdrawal Rates. It is better than using dividend yield (plus about 1%) as a lower bound. The earnings yield overcomes the problem of surprise dividend cuts.

Here we consider the Withdrawal rates that will reduce an initial portfolio to half its original value.
These Half Failure Withdrawal Rates we call HFWR.
It's assumed that:

  • Withdrawals are over a thirty year period.
  • Withdrawals were adjusted to match inflation in accordance with the CPI.
  • Annual expenses are 0.2%.
  • The portfolio is rebalanced annualy to maintain an 80% / 20% ratio of "the S&P 500" and "commercial paper".

When we plot Half Failure Withdrawal Rates HFWR80
(80% stocks + 20% commercial paper)
versus percentage earnings yield (or E10/P%) ),
it is immediately apparent that the two are related.
This relationship is strongest from the 1920s on.
It is not entirely clear why the relationship was not as strong in the earlier period covered by the available data. There are many possible reasons. One likely reason is related to falling prices at the end of the 19th century. In many of those years, commercial paper would have provided an inflation-adjusted (i.e. real) Safe Withdrawal Rate of 5% or 6% simply because interest rates always stayed positive! In any event, the data show good predictability using P/E10 from the 1920s on.

Many of our curves start with 1923 instead of 1921. The reason is to get a better curve fit with straight lines. They have a disproportionately large effect when plotting Historical Surviving Withdrawal Rates versus the percentage earnings yield. Years 1921 and 1922 were among those with the best prices ever. In addition, they exhibit saturation. Their higher earnings yields are associated with higher Historical Surviving Withdrawal Rates... but not as much higher as a straight line would indicate.

We use Excel for plotting HFWR, starting from 1921 or 1923 versus 100 / (P/E10) and for curve fitting. It works remarkably well. We have found that even a single decade of retirement starting years provides good fits. Using more data always helps. Using curves derived from data with earnings yields similar to those of interest is always best. That is, it is a good idea to minimize the amount of extrapolation.

It was mentioned above that E10/P is a good indicator of valuation. As a comparison, here are charts with HFWR80 vs E/P:

and the correlations (over a moving 36-year window) using both E/P and E10/P:

The above charts are for the HFWR, the Half Failure Withdrawal Rates (where the final portfolio is greater than Half the starting value).
For the HSWR, the Historical Survival Withdrawal Rates (where the final portfolio is slightly greater than $0), we get this chart:

Note that the correlation is very good at 81% over the period 1920 - 1980.
In order to see the changes over this period, we can look at this chart:

Suppose it is January, 2005 and we wish to estimate the Surviving Withdrawal Rate for the coming year.
We do this:

  1. Look up E10/P from Professor Shiller’s database at the start each year of the historical sequences that you are going to use. Typically, this is 1921-1980 or 1923-1980. Sometimes, it is necessary to limit the historical sequences to completed sequences. In such cases, the final year of each sequence must be 2002 or earlier.
    (The Retire Early Safe Withdrawal Calculator and its modified versions available from the website use dummy data with heavy losses for 2003-2010.)
    Sometimes, you will be interested in starting from 1871, typically using start years of 1871-1980
    (or 1881-1980 because of uncertainty regarding how the 1871-1880 values of P/E10 were determined).
  2. Calculate the HSWR over the past N years.
  3. Plot HSWR vs E10/P for the N years of data and determine the equation of the "best fit" line.
  4. If the equation is (for example) :
          y = 0.490x + 0.029
    and the most recent
          E10/P = 0.035 (or 3.5%)
    then we get
          y = 0.49*0.035 + .029 = 0.046 or 4.6% as the Calculated Rate for the coming year. This is our best estimate of what history will record in 2035 as the 2005 30-year Historical Surviving Withdrawal Rate.
  5. To calculate the Safe Withdrawal Rate, we must identify the lower confidence limit. (We use a 90% confidence level.)
  6. A quick and easy way to approximate the 90% confidence interval from our existing plot is to remove some of the data points that are farthest away from the regression line.
  7. The line has 61 data points corresponding to the 61 years in the interval of 1920-1980. Ten percent of this is 6.1. Rounded, this is 6. We remove the 6 data points farthest from the regression line.
  8. Using our eyeball to make our choices, we remove the five data points above the line with Historical Surviving Withdrawal Rates of 9% and greater. We remove the single data point below the line (for the year 1921) which has the highest earnings yield and a Historical Surviving Withdrawal Rate close to 9.8%.
  9. The data that remain range between +1.4% or +1.5% above the regression line and –1.8% below the regression line. The total range of the data above and below the regression line is 3.2% or 3.3%.
  10. Our approximate confidence interval is plus and minus 1.6% to 1.7% (for a 90% level of confidence).
  11. We choose to use the bigger number, 1.7%.
  12. The Safe Withdrawal Rate equals the Calculated Rate minus 1.7%. The High Risk Rate equals the Calculated Rate plus 1.7%.
  13. In our example, the January 2005 Calculated Rate is 4.6%. The January 2005 Safe Withdrawal Rate is 2.9%. The January 2005 High Risk Rate is 6.3%.
  14. The amount one actually withdraws is his Personal Withdrawal Rate. Most people would choose something slightly above the Safe Withdrawal Rate of 2.9% but well below the Calculated Rate of 4.6%. Some might limit withdrawals to less than 2.9% based upon their assessment of the future, especially for the stock market. Very few people knowledgeable of these findings would withdraw 6% or more.

Statistical Approximations

Handling the statistics is a more difficult task. There are no standard formulas available for us to use.

We are content to apply the Central Limit Theorem. We restrict ourselves to very coarse levels of precision. We act as if the actual distribution were Gaussian (i.e., normal or bell shaped). We limit ourselves to 90% confidence limits. We reject outright any claims to high statistical precision (such as 98% or 99%). We hope that the actual precision is in the right ballpark (such as between 80% and 95%).

The underlying issue with statistics is that our historical sequences overlap.

One way of handling such a situation is to demand almost complete independence of all data sets. Typically, one calculates an autocorrelation function and determines the number of years that it takes to reach 70% to 90% of the total area (or energy). This kind of approach is helpful when one is most interested in eliminating false alarms.

We look at the problem differently. We want high sensitivity. We are willing to accept a reasonable amount of error.

We observe that prices swing radically from one year to the next but that E10, which is the average of a decade’s earnings, varies slowly. When looking at historical sequences and overlapping data, we focus on the independent data points: the first (or last year) of a sequence. If we have two historical sequences, they will differ by two (or more) points: the first year of one sequence and the last year of the other. Next, we look at the ability of prices to shift enough in those two years to cause the observed variation in Historical Surviving Withdrawal Rates. The answer turns out to be that they easily come close to doing just that. The issue of overlapping sequences turns out to be a reduction in the effective number of data points (degrees of freedom) by something less than one half.

Keep in mind that, if Historical Surviving Withdrawal Rates were all the same number, there would be a lot of variation in a plot of them versus the percentage earnings yield (100% / [P/E10] ). This scatter would be caused by price changes.

There is the loss of a degree of freedom because we calculate the slope of the line. There is a loss of a degree of freedom because we estimate variance from the data.

In this way, we have converted our nonstandard statistical problem to something very close to a standard curve-fitting problem along with a reasonable adjustment.