Post
by **JWR1945** » Mon Jan 17, 2005 12:59 pm

This is my first cut. The graph of Historical Database Rates (for HDBR50 and HDBR80 portfolios) from the SWR Research Section goes along with this. This presents the technical issues. The fun part is taking advantage of the pictures (i.e., using Excel to plot curves). The hard part is getting a handle on the statistics.

**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 Historical Database Rates), 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 (or Historical Database Rates) are based on using an actual historical sequence of investment returns. A hypothetical portfolio would have survived for a specified number of years at the Historical Surviving Withdrawal Rate. 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 Historical Surviving Withdrawal Rate 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 of predicting 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.

When we plot Historical Surviving Withdrawal Rates (which are also known as Historical Database Rates) versus the percentage earnings yield (or 100% / [P/E10] ), 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 (1871-2002). 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, just not as much higher as a straight line would indicate.]

We use Excel for plotting Historical Surviving Withdrawal Rates (starting from 1921 or 1923) versus the percentage earnings yield (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.

**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.

Have fun.

John R.