In these examples, we use the equations for N = 10 years. In addition, the confidence levels are not 90%. They are better estimated as close to 86% (and no worse than 75%).

return0 = 0%

If we could reliably predict that the return0 would be 0% annualized after a decade, then 10WFAIL50 = 10.0% plus and minus 1.5% = from 8.5% through 11.5%. In addition, RETURN0 = ratio of the final balance to the initial balance after 10 years = 1. For any actual withdrawal rate w, the ten-year balance / the initial balance = RETURN0*(1 - w/WFAIL) = 1.000*(1 - w/WFAIL) = 1.000*(1 - w/10.0) with confidence limits of 1.000*(1 - w/8.5) and 1.000*(1 - w/11.5).

With a 3% withdrawal rate, that ratio would be (1 - 3/10) = 0.70 (most likely) with confidence limits of 0.65 to 0.74. That is, you would still have between 65% and 74% of your initial balance after making withdrawals for a decade.

With a 4% withdrawal rate, the ratio would be 0.60 (most likely) with confidence limits of 0.53 to 0.65. That is, you would still have between 53% and 65% of your initial balance after making withdrawals for a decade.

With a 5% withdrawal rate, the ratio would be 0.50 (most likely) with confidence limits of 0.41 and 0.57. That is, you would still have between 41% and 57% of your initial balance after making withdrawals for a decade.

All of this is consistent with our previous investigations. The curve for (30-year) Historical Database Rates using Return0 = 0% at the ten-year mark projects a HDBR50 of 4.0% plus and minus 1.2% (for the 90% confidence limits). Notice that the confidence limits dip below a 3% withdrawal rate. Even 3% can be hazardous if the portfolio's annualized real return absent of any withdrawals (return0) is flat for a decade.

Historically, portfolios that have fallen in half within the first decade have almost always failed.

The worst case historical value of return0 at the end of a decade was close to -1.5% (annualized real growth).

return0 = 2%

Let us suppose that the future is a little brighter and that the real, annualized return after ten years is +2%. Then RETURN0 = (1+return0)^(the number of years) = 1.02^10 = 1.219. Looking at our graphs, this annualized real growth rate corresponds to WFAIL = 11.0%. The confidence limits are still plus and minus 1.5%.

The ten-year balance / the initial balance = RETURN0*(1 - w/WFAIL) = 1.219*(1 - w/WFAIL) = 1.219*(1 - w/11.0) with confidence limits of 1.219*(1 - w/9.5) and 1.219*(1 - w/12.5).

With a 3% withdrawal rate, the ten-year balance to initial balance ratio would be 0.89 (most likely) with confidence limits of 0.83 to 0.93. That is, you would still have between 83% and 93% of your initial balance after ten years.

With a 4% withdrawal rate, the ratio would be 0.78 (most likely) with confidence limits of 0.71 to 0.83. That is, you would still have between 71% and 83% of your initial balance after ten years.

With a 5% withdrawal rate, the ratio would be 0.66 (most likely) with confidence limits of 0.58 and 0.73. That is, you would still have between 58% and 83% of your initial balance after ten years.

Our previous investigations indicated that a real annualized growth of 2% over a decade would have produced a 30-year Historical Database Rate of 4.8% plus and minus 1.2% using 90% confidence limits.

return0 = 4%

Let us suppose that the future is even brighter, but still not quite as good as the long-term return of the stock market. Let us suppose that the real, annualized return after ten years is +4%. Then RETURN0 = (1+return0)^(the number of years) = 1.04^10 = 1.480. Looking at our graphs, an annualized real growth rate corresponds to WFAIL = 12.2%. The confidence limits are still plus and minus 1.5%.

The ten-year balance / the initial balance = RETURN0*(1 - w/WFAIL) = 1.480*(1 - w/WFAIL) = 1.480*(1 - W/12.2) with confidence limits of 1.480*(1 - w/10.7) and 1.48*(1 - w/13.7).

With a 3% withdrawal rate, the ratio would be 1.12 (most likely) with confidence limits of 1.07 and 1.16. That is, your balance would have grown to between 107% and 116% of your initial balance after ten years.

With a 4% withdrawal rate, the ratio would be 0.99 (most likely) with confidence limits of 0.93 and 1.05. That is, your balance would have ended up between 93% and 105% of your initial balance after ten years.

With a 5% withdrawal rate, the ratio would be 0.87 (most likely) with confidence limits of 0.79 and 0.94. That is, your balance would have ended up between 79% and 94% of your initial balance after ten years.

Our previous investigations indicated that a real annualized growth of 4% over a decade would produce a 30-year Historical Database Rate of 5.6% plus and minus 1.2% using 90% confidence limits.

return0 = 6%

It is important to remember that return0 is the annualized real growth of the portfolio (in this case, over a decade). Unless your restrict yourself to an all-stock portfolio, return0 is not the annualized real growth of the stock market.

Let us suppose that the future is bright and that the real, annualized return after ten years is +6%. Then RETURN0 = (1+return0)^(the number of years) = 1.06^10 = 1.791. Looking at our graphs, an annualized real growth rate corresponds to WFAIL = 13.3%. The confidence limits are still plus and minus 1.5%.

The ten-year balance / the initial balance = RETURN0*(1 - w/WFAIL) = 1.791*(1 - w/WFAIL) = 1.791*(1 - W/13.3) with confidence limits of 1.791*(1 - w/11.Cool and 1.791*(1 - w/14.Cool.

With a 3% withdrawal rate, the ratio would be 1.39 (most likely) with confidence limits of 1.34 and 1.43. That is, your balance would have grown to between 134% and 143% of your initial balance after ten years.

With a 4% withdrawal rate, the ratio would be 1.25 (most likely) with confidence limits of 1.18 and 1.31. That is, your balance would have ended up between 118% and 131% of your initial balance after ten years.

With a 5% withdrawal rate, the ratio would be 1.12 (most likely) with confidence limits of 1.03 and 1.19. That is, your balance would have ended up between 103% and 119% of your initial balance after ten years.

With a 6% withdrawal rate, the ratio would be 0.98 (most likely) with confidence limits of 0.88 and 1.06. That is, your balance would have ended up between 88% and 106% of your initial balance after ten years.

With a 7% withdrawal rate, the ratio would be 0.85 (most likely) with confidence limits of 0.73 and 0.94. That is, your balance would have ended up between 73% and 94% of your initial balance after ten years.

With an 8% withdrawal rate, the ratio would be 0.71 (most likely) with confidence limits of 0.58 and 0.82. That is, your balance would have ended up between 58% and 82% of your initial balance after ten years.

Our previous investigations indicated that a real annualized growth of 6% over a decade would produce a 30-year Historical Database Rate of 6.5% plus and minus 1.2% using 90% confidence limits.

You can apply these results immediately as part of the design process. We have just seen how sensitive your portfolio's ten-year balance is to market returns in the first decade.

When applying these results, you can consider all possible final balances, not simply a final balance of zero.

You can use the other equations [for N = 4, 6 and 8 years] as well. They can tell you how well you are progressing. They can warn you, if necessary. They can tell you of your good fortune as well, if that is the case.

Have fun.

John R.

## An Application of the New Tool: Examples

**Moderator:**hocus2004