See A New Tool from Wed Apr 28, 2004 at 4:41 pm CDT.

http://nofeeboards.com/boards/viewtopic.php?t=2427

The equations for the straight lines that I produced have the form y = mx+b or return0 = slope*HDBR + intercept. Both return0 and HDBR are in percent. I have examined two portfolios, HDBR50 with a 50% stock allocation and HDBR80 with an 80% stock allocation. The Historical Database Rates are for 30 years. Expenses were 0.20% of the portfolio's current balance. Portfolios consisted of stocks and commercial paper with annual re-balancing.

In the tables: return0 = y. m = slope. HDBR50 or HDBR80 = x. b = intercept.

These days, we would refer to the HDBR terms in the equations as 30-year Calculated rates (i.e., the best estimate of future Zero Balance Rates). We would call the lower confidence limit the Safe Withdrawal Rate and we would call the higher confidence limit the High Risk Withdrawal Rate. To be very precise, these are conditional rates, not true Safe Withdrawal Rates since they depend upon your ability to estimate the portfolio's (real, annualized) total return (return0) over a specified number of years.

Throughout this investigation, I looked into developing a simple rule of thumb adjustment of a portfolio's real, annualized, total return without withdrawals (return0) to estimate the conditional Safe Withdrawal Rate. My calculations show that this cannot work. You cannot take return0 over 10-years, for example, and add or subtract a single, fixed amount to come up with a conditional Safe Withdrawal Rate.

This is the relevant HDBR50 information:

For HDBR50, which has 50% stocks and 50% commercial paper and an expense ratio of 0.20%, the equations for the 30-year Calculated rates (i.e., the most likely Zero Balance Rates) and return0 are:

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Years R Squared Slope Intercept Equation

6 0.471 2.6452 -9.7732 y = 2.6452x - 9.7732

10 0.7381 2.341 -9.2439 y = 2.341x - 9.2439

14 0.9027 2.1245 -8.0652 y = 2.1245x - 8.0652

and the statistics are:

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Std. Dev. HDBR50 return0

6 years 1.242862 3.287619

10 years 0.709388 1.660678

14 years 0.391109 0.830911

These are the standard deviations for HDBR50 and its annualized return (return0) for 6, 10 and 14 years. All values are percentages. (The slope also equals ratio of the standard deviations.) The 30-Year Intercept is 3.8169%, which is very close to the long-term annualized return of this portfolio.

This is the relevant HDBR80 information:

For HDBR80, which has 80% stocks and 20% commercial paper and an expense ratio of 0.20%, the equations for the 30-year Calculated rates (i.e., the most likely Zero Balance Rates) and return0 are:

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Years R Squared Slope Intercept Equation

6 0.5241 2.2778 -9.5372 y = 2.2778x - 9.5372

10 0.7478 1.9526 -7.6448 y = 1.9526x - 7.6448

14 0.9045 1.8432 -6.8875 y = 1.8432x - 6.8875

and the statistics are:

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Std. Dev. HDBR80 return0

6 years 1.813831 4.792738

10 years 1.084223 2.529336

14 years 0.669129 1.418626

These are the standard deviations for HDBR80 and its annualized return (return0) for 6, 10 and 14 years. All numbers are percentages. (The slope also equals ratio of the standard deviations.) The 30-Year intercept is 4.3784%, which is very close to the long-term annualized return of this portfolio.

APPLYING THE EQUATIONS:

You should specify return0 at ten or fourteen years. Six years is too short. Fourteen years may be a little long, but it has the best curve fit.

Conditional Safe Withdrawal Rates are at the lower 90% confidence limits relative to Calculated Rates. The very short sequence at six years results in very wide confidence limits. It takes a very strong return during those first six years to bring the lower confidence limit up to 4%. Better outcomes are likely, but not guaranteed.

SIX YEARS AND HDBR50:

For HDBR50 and with its return0 [the real, annualized total return of the 50% stock portfolio with dividends reinvested, no withdrawals and 0.20% expenses] treated as if it were known exactly, the Calculated rates at six years for 0%, 2%, 4%, 6% and 8% are:

1) The equation is y = 2.6452x-9.7732. Solving for x: x = [y+9.7732] / [2.6452].

2) When y = 0, [where y = return0], x = the Calculated rate for HDBR50 = 3.69%.

3) When y = 2, x = 4.45%.

4) When y = 4, x = 5.21%.

5) When y = 6, x = 5.96%.

6) When y = 8, x = 6.72%.

[Every increase in the total return0 of 2% results in an increase in the Calculated rate of [2] / [2.6452] = 0.756%, which corresponds to increments of either 0.75 or 0.76 depending upon rounding.]

We read that the standard deviation of the HDBR50 term x (i.e., the scatter around the Calculated rate) is 1.242862% at 6 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 2.04%.

For the conditional Safe Withdrawal Rates, we subtract 2.04% from the Calculated rates at six years.

1) When return0 (or y) = 0%, the conditional SWR = 1.65%.

2) When return0 (or y) = 2%, the conditional SWR = 2.41%.

3) When return0 (or y) = 4%, the conditional SWR = 3.17%.

4) When return0 (or y) = 6%, the conditional SWR = 3.92%.

5) When return0 (or y) = 8%, the conditional SWR = 4.68%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 1.65%.

2) If return0 = 2%, you add 0.41%.

3) If return0 = 4%, you subtract 0.83%.

4) If return0 = 6%, you subtract 2.08%.

5) If return0 = 8%, you subtract 3.32%.

TEN YEARS AND HDBR50:

For HDBR50 and with its return0 [the real, annualized total return of the 50% stock portfolio with dividends reinvested, no withdrawals and 0.20% expenses] treated as if it were known exactly, the Calculated rates at ten years for 0%, 2%, 4%, 6% and 8% are:

1) The equation is y = 2.341x-9.2439. Solving for x: x = [y+9.2439] / [2.341].

2) When y = 0, [where y = return0], x = the Calculated rate for HDBR50 = 3.95%.

3) When y = 2, x = [2+9.2439] / [2.341] = 4.80%.

4) When y = 4, x = [4+9.2439] / [2.341] = 5.66%.

5) When y = 6, x = [6+9.2439] / [2.341] = 6.51%.

6) When y = 8, x = [8+9.2439] / [2.341] = 7.37%.

[Every increase in the total return0 of 2% results in an increase in the Calculated rate of [2] / [2.341] = 0.854%, which corresponds to increments of either 0.85 or 0.86 depending upon rounding.]

We read that the standard deviation of the HDBR50 term x (i.e., the scatter around the Calculated rate) is 0.709388% at 10 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 1.16%.

For the conditional Safe Withdrawal Rates, we subtract 1.16% from the Calculated rates at ten years.

1) When return0 (or y) = 0%, the conditional SWR is 3.95%-1.16% = 2.79%.

2) When return0 (or y) = 2%, the conditional SWR is 4.80-1.16 = 3.63%.

3) When return0 (or y) = 4%, the conditional SWR is 5.66-1.16 = 4.50%.

4) When return0 (or y) = 6%, the conditional SWR is 6.51-1.16 = 5.35%.

5) When return0 (or y) = 8%, the conditional SWR is 7.37-1.16 = 6.21%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 2.79%.

2) If return0 = 2%, you add 1.63%.

3) If return0 = 4%, you add 0.50%.

4) If return0 = 6%, you subtract 0.65%.

5) If return0 = 8%, you subtract 1.79%.

FOURTEEN YEARS AND HDBR50:

Looking at fourteen years and HDBR50 (50% stocks), the equation is y = 2.1245x-8.0652. Solving for x, x = [y+8.0652] / 2.1245. The Calculated rates with return0 (or y) of 0%, 2%, 4%, 6% and 8%:

1) If return0 = 0%, the Calculated rate for HDBR50 is 3.80%.

1) If return0 = 2%, the Calculated rate for HDBR50 is 4.74%.

1) If return0 = 4%, the Calculated rate for HDBR50 is 5.68%.

1) If return0 = 6%, the Calculated rate for HDBR50 is 6.62%.

1) If return0 = 8%, the Calculated rate for HDBR50 is 7.56%.

We read that the standard deviation of the HDBR50 term x (i.e., the scatter around the Calculated rate) is 0.391109% at 14 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 0.64%.

For the conditional Safe Withdrawal Rates, we subtract 0.64% from the Calculated rates at 14 years.

1) When return0 (or y) = 0%, the conditional SWR is 3.16%.

2) When return0 (or y) = 2%, the conditional SWR is 4.10%.

3) When return0 (or y) = 4%, the conditional SWR is 5.04%.

4) When return0 (or y) = 6%, the conditional SWR is 5.98%.

5) When return0 (or y) = 8%, the conditional SWR is 6.92%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 3.16%.

2) If return0 = 2%, you add 2.10%.

3) If return0 = 4%, you add 1.04%.

4) If return0 = 6%, you subtract 0.02%.

5) If return0 = 8%, you subtract 1.08%.

SIX YEARS AND HDBR80:

For HDBR80 and with its return0 [the real, annualized total return of the 80% stock portfolio with dividends reinvested, no withdrawals and 0.20% expenses] treated as if it were known exactly, the Calculated rates at six years for 0%, 2%, 4%, 6% and 8% are:

1) The equation is y = 2.2778x-9.5372. Solving for x: x = [y+9.5372] / [2.2778].

2) When y = 0, [where y = return0], x = the Calculated rate for HDBR80 = 4.19%.

3) When y = 2, x = 5.07%.

4) When y = 4, x = 5.94%.

5) When y = 6, x = 6.82%.

6) When y = 8, x = 7.70%.

[Every increase in the total return0 of 2% results in an increase in the Calculated rate of 0.878%, which corresponds to increments of either 0.87 or 0.88 depending upon rounding.]

We read that the standard deviation of the HDBR80 term x (i.e., the scatter around the Calculated rate) is 1.813831% at 6 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 2.97%.

For the conditional Safe Withdrawal Rates, we subtract 2.97% from the Calculated rates at 6 years.

1) When return0 (or y) = 0%, the conditional SWR is 1.22%.

2) When return0 (or y) = 2%, the conditional SWR is 2.10%.

3) When return0 (or y) = 4%, the conditional SWR is 2.97%.

4) When return0 (or y) = 6%, the conditional SWR is 3.85%.

5) When return0 (or y) = 8%, the conditional SWR is 4.73%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 1.22%.

2) If return0 = 2%, you add 0.10%.

3) If return0 = 4%, you subtract 1.03%.

4) If return0 = 6%, you subtract 2.15%.

5) If return0 = 8%, you subtract 3.27%.

TEN YEARS AND HDBR80:

For HDBR80 and with its return0 [the real, annualized total return of the 80% stock portfolio with dividends reinvested, no withdrawals and 0.20% expenses] treated as if it were known exactly, the Calculated rates at ten years for 0%, 2%, 4%, 6% and 8% are:

1) The equation is y = 1.9526x-7.6448. Solving for x: x = [y+7.6448] / [1.9526].

2) When y = 0, [where y = return0], x = the Calculated rate for HDBR80 = 3.92%.

3) When y = 2, x = 4.94%.

4) When y = 4, x = 5.96%.

5) When y = 6, x = 6.99%.

6) When y = 8, x = 8.01%.

[Every increase in the total return0 of 2% results in an increase in the Calculated rate of 1.024%, which corresponds to increments of either 1.02 or1.03 depending upon rounding.]

We read that the standard deviation of the HDBR80 term x (i.e., the scatter around the Calculated rate) is 1.084223% at 10 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 1.78%.

For the conditional Safe Withdrawal Rates, we subtract 1.78% from the Calculated rates at 10 years.

1) When return0 (or y) = 0%, the conditional SWR is 2.14%.

2) When return0 (or y) = 2%, the conditional SWR is 3.16%.

3) When return0 (or y) = 4%, the conditional SWR is 4.18%.

4) When return0 (or y) = 6%, the conditional SWR is 5.21%.

5) When return0 (or y) = 8%, the conditional SWR is 6.23%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 2.14%.

2) If return0 = 2%, you add 1.16%.

3) If return0 = 4%, you add 0.18%.

4) If return0 = 6%, you subtract 0.79%.

5) If return0 = 8%, you subtract 1.77%.

FOURTEEN YEARS AND HDBR80:

For HDBR80 and with its return0 [the real, annualized total return of the 80% stock portfolio with dividends reinvested, no withdrawals and 0.20% expenses] treated as if it were known exactly, the Calculated rates at 14 years for 0%, 2%, 4%, 6% and 8% are:

1) The equation is y = 1.8432x-6.8875. Solving for x: x = [y+6.8875] / [1.8432].

2) When y = 0, [where y = return0], x = the Calculated rate for HDBR80 = 3.74%.

3) When y = 2, x = 4.82%.

4) When y = 4, x = 5.91%.

5) When y = 6, x = 6.99%.

6) When y = 8, x = 8.08%.

[Every increase in the total return0 of 2% results in an increase in the Calculated rate of 1.085%, which corresponds to increments of either 1.08 or1.093 depending upon rounding.]

We read that the standard deviation of the HDBR80 term x (i.e., the scatter around the Calculated rate) is 0.669129% at 14 years. We multiply this by 1.64 to get the 90% confidence limits of plus and minus 1.10%.

For the conditional Safe Withdrawal Rates, we subtract 1.10% from the Calculated rates at 14 years.

1) When return0 (or y) = 0%, the conditional SWR is 2.64%.

2) When return0 (or y) = 2%, the conditional SWR is 3.72%.

3) When return0 (or y) = 4%, the conditional SWR is 4.81%.

4) When return0 (or y) = 6%, the conditional SWR is 5.89%.

5) When return0 (or y) = 8%, the conditional SWR is 6.98%.

Now let us make a rule of thumb:

1) If return0 = 0%, you add 2.64%.

2) If return0 = 2%, you add 1.72%.

3) If return0 = 4%, you add 0.81%.

4) If return0 = 6%, you subtract 0.11%.

5) If return0 = 8%, you subtract 1.02%.

Final Remark

Refer to the Summary and to the original post about the New Tool for many details.

I wrote this for the optimist:

We can summarize this by saying that 7% real returns at year 10 result in Safe Withdrawal Rates between 5.21% and 6.23%. At year 14, they result in Safe Withdrawal Rates between 5.89% and 6.98%.

I wrote this for the pessimist:

We can summarize this by saying that a 1% real return at 10 years results in Safe Withdrawal Rates between 2.14% and 3.63%. At year 14, they result in Safe Withdrawal Rates between 2.64% and 4.10%.

The actual numbers depend upon the stock allocation (50% or 80%) and they allow return0 to vary by plus and minus one percent. That is, the 7% result allows for return0 values ranging from 6% to 8% and the 1% result allows for return0 values ranging from 0% to 2%.

I expect the latter numbers to be the better prediction because I expect multiples to contract. It is very easy for me to envision the real, annualized total return as falling below 1% for over a decade. [William Bernstein's numbers do not include this component, known as the speculative return.]

Have fun.

John R.