This is how our results have come about.

1. We made an extensive search to identify the best way to link valuations and Historical Surviving Withdrawal Rates. So far this has turned out to be Professor Robert Shiller's P/E10. Other measures have included the standard (one-year trailing) price to earnings ratio P/E, Tobin's Q, short-term interest rates and many combinations.

2. This link has credibility since Professors Shiller and Campbell have demonstrated convincingly that P/E10 has predictive power for estimating S&P500 returns. P/E10 was adapted from Benjamin Graham's recommendation to use 5 to 10 years of earnings when evaluating companies. With P/E10, the price is the value of the S&P500 index and E10 is the average of the previous decade of earnings for the S&P500. [Both P and E10 are in real dollars. The adjustment is needed for the calculation of E10.]

3. To those who say that the future is unknowable and then proceed to act as if nothing at all about future stock returns can be predicted, don't argue with me. Take up your arguments with Professors Shiller and Campbell. They have laid down a rigorous foundation.

4. Notice that the future is not known completely. There is a whole lot of difference between having less than perfect knowledge and not having any knowledge at all.

5. The data tell us that there is a difference between the early relationship between Historical Surviving Withdrawal Rates and more recent results.

6. We have drawn the line at 1920-1921 so as to use round numbers (i.e., we place the break at the beginning and end of a decade instead of extracting the best location from the data).

7. Upon examination, there are many possible causes for the change in the relationship. We have simply noted that there has been a change and that there are many credible explanations as to why it has taken place. We add that alternative data sources such as those collected by Professors Fama and French reach back only into the mid-1920s. They cannot provide illumination. There is no reason to believe that they would behave differently from Professor Shiller's 1871-2003 S&P500 data.

8. We looked at the link between Historical Surviving Withdrawal Rates and P/E10.

9. Examination of Gummy's Magic Sum formula shows that stock returns and Safe Withdrawal Rates are intimately connected.

10. Gummy was able to prove that if stock returns have a lognormal distribution, then Gummy's Magic Sum has a lognormal distribution as well. The Safe Withdrawal Rate and Calculated Rate and High Risk Rate can be found from that distribution. [A lognormal distribution means that the percentage gains are losses are normally distributed. If the distribution were normal, the dollar gains and losses of stocks would have a normal distribution, not their percentages.]

11. This lends credence to the hypothesis that higher valuations (as seen in P/E10) lead to lower returns and, therefore, to lower Safe Withdrawal Rates.

12. This does not automatically prove the hypothesis because stock returns are not lognormal. They are close enough to being lognormal, however, to support the hypothesis except under very unusual circumstances.

13. We looked at the years with the lowest Historical Surviving Withdrawal Rates. Earlier investigations had focused on finding a lower bound. We looked for cause and effect related to those particular years.

14. We isolated the years of the Great Depression and the decade of the 1960s (immediately followed by the Stagflation of the 1970s) as the worst case years.

15. We found that the lower bound was slightly above dividend yields. If dividend amounts track inflation, then having a final balance of zero means that one can withdraw a little bit more than the dividend yield. This was our first major cause and effect relationship.

16. We introduced the effects of changing valuations (with P/E10) in the usual manner, which is to annualize it.

17. This was our second major result. The Surviving Withdrawal Rate was estimated to equal the dividend yield plus a price expansion or compression term. The fit with Historical Surviving Withdrawal Rates was surprisingly good.

18. This is consistent with how long-term stock market returns are estimated. There is a dividend yield term and a dividend or earnings growth term and a speculative term for changing price to earnings multiples. Our calculation assumes that the growth term is constant.

19. Later investigations for comparing Calculated Rates and Historical Surviving Withdrawal Rates included the effects of dividend yields and P/E10 and earnings yield E10/P.

20. Plots of Historical Surviving Withdrawal Rates versus the percentage earnings yield 100E10/P showed a strong relationship, even better than those that included dividend yields (in addition to earnings yield).

21. Careful examination revealed that earnings yield is better behaved than dividend yield. Dividends have been cut abruptly at times, especially during the Great Depression. Since dividends come out of earnings, focusing on smoothed earnings reduces the likelihood of surprise dividend cuts.

22. This is another cause and effect relationship. The foundation is solid and it is based on dividends (via the Dividend Discount Model and the Gordon Equation).

23. We have looked at the issue of monitoring how well one's portfolio is doing.

24. Early investigations showed that whether a portfolio will last is usually known within the first eleven years. Either the portfolio has grown enough so as to assure success or it is likely to be in danger.

25. This observation leads to another cause and effect relationship. Growth is needed to extend a portfolio's lifetime but excessive volatility can cause an early failure.

26. This is consistent with another cause and effect observation: retirement portfolios fail because of excessive selling when stock prices are low.

27. The New Tool helps us monitor portfolio safety.

28. It allows us to separate the Total Return component and the Gummy's Magic Sum component of Gummy's Safe Withdrawal Rate equation.

29. This allows us to separate the effects of the sequence of returns from the total return over a specified number of years.

30. We have included statistical estimates whenever we can.

Here is the logical priority of our results.

1. Our first priority is determining what the data have to say, not forcing them to fit into any particular model.

2. Our next step is to identify cause and effect relationships. We always lend higher credibility to the logic behind those relationships than in numbers by themselves.

3. We include statistical estimates whenever we can.

4. We have included pure mathematical calculations when appropriate.

5. We place our least confidence in numbers without statistics. We place very little confidence in numbers in isolation. We place our highest confidence when there is mathematical proof. We rely more on the cause and effect relationships that we are able to identify than in numbers alone.

Have fun.

John R.

## The Logical Sequence

**Moderator:** hocus2004