Faster and Easier SWR Calculations: Excursion

I received a letter asking whether collecting more data would reduce the errors of the new Faster and Easier approach. After all, collecting more data almost always improves statistical descriptions.

I do not know the answer to the question. I think that it is NO. The source of errors is not from of a lack of data points. It is in the linearity approximation. Dollar balances at year 30 do not always translate into Safe Withdrawal Rates, Calculated Rates and High Risk Rates as well as we might want. Still, the new approach is very fast and it does amazingly well.

We can take advantage of the linearity between withdrawal rates, the percentage earnings yield 100E10/P and portfolio balances to speed up data collection and to conduct a detailed analysis.

This special procedure is very fast.

It is less accurate than the standard process.

In this excursion I examine the relationship between the new procedure’s accuracy and the selection of withdrawal rates. My intent is to learn about the nature of the errors.

Recap of My Standard Procedure

Here is my standard procedure for calculating Safe Withdrawal Rates.

1) I collect 30-year Historical Surviving Withdrawal Rate HSWR data on one of my calculators.
2) I construct a table in an Excel spreadsheet. It lists the year, the percentage earnings yield 100E10/P in January of that year and the Historical Surviving Withdrawal Rate HSWR for that year.
3) I make a scatter plot of HSWR versus the percentage earnings yield 100E10/P. I have Excel fit a straight line (i.e., make a linear fit) to the data, display its equation and report its goodness of fit (R-squared). The straight line is the Calculated Rate.
4) I construct confidence limits. Although I can calculate the confidence limits, I am usually satisfied with eyeball estimates. I place my greatest emphasis on the data with low percentages of earnings yields (below 8% or 10%).
5) The Safe Withdrawal Rate is the lower confidence limit. The High Risk Rate is the upper confidence limit.
6) The most likely outcome associated with a specified percentage earnings yield is the Calculated Rate. The odds are (roughly) 95% that the actual outcome will be higher than the Safe Withdrawal Rate. The odds are (roughly) 95% that the actual outcome will be lower than the High Risk Rate.

Recap of My Alternative Procedure: First Part

To make this faster and easier, I take advantage of linearity.

I fit curves to portfolio ending balances at year 30. Each curve requires only a single application of the calculator. I do not calculate Historical Surviving Withdrawal Rates directly.

If the linearity were perfect, two curves would be sufficient. But linearity is not perfect. In this case, I collected data at withdrawal rates of 2.0%, 3.0%, 4.0%, 5.0% and 6.0%.

I copied the real balances at year 30 from the calculator and pasted them into a spreadsheet. I listed each start year and its percentage earnings yield 100E10/P in that year. I pasted the final balances in five columns on the right.

I made Excel charts of balances at year 30 versus earnings yield at withdrawal rates of 2.0%, 3.0%, 4.0%, 5.0% and 6.0%. I had Excel report the formulas and R-squared values.

I estimated confidence limits visually.

The Example with 50% Stocks

I used the HDBR50/HSWR50 portfolio. It consists of 50% stocks and 50% commercial paper. Its expenses are set at 0.20%. It is rebalanced annually.

The initial balances were $100000.

Here are the equations at year 30 along with eyeball estimates of the confidence intervals. They are the equations for the Calculated Rates. The balance is y and the percentage earnings yield 100E10/P is x.

When the withdrawal rate is 2.0%, the Calculated Rate equation is:
y = 9343.3x+130109.
When the withdrawal rate is 3.0%, the Calculated Rate equation is:
y = 12711x+47014.
When the withdrawal rate is 4.0%, the Calculated Rate equation is:
y = 16078x-36082.
When the withdrawal rate is 5.0%, the Calculated Rate equation is:
y = 19483x-119533.
When the withdrawal rate is 6.0%, the Calculated Rate equation is:
y = 23021x-204381.

Here are my eyeball estimates of the confidence limits:
With a withdrawal rate of 2.0%, the confidence limits are minus 40000 and plus 80000.
With a withdrawal rate of 3.0%, the confidence limits are minus 50000 and plus 60000.
With a withdrawal rate of 4.0%, the confidence limits are minus 50000 and plus 60000.
With a withdrawal rate of 5.0%, the confidence limits are minus 50000 and plus 50000.
With a withdrawal rate of 6.0%, the confidence limits are minus 70000 and plus 50000.

Subtract the lower confidence limit to calculate Safe Withdrawal Rates. These are the equations for Safe Withdrawal Rates.

When the withdrawal rate is 2.0%, the Safe Withdrawal Rate equation is:
y = 9343.3x+90109.
When the withdrawal rate is 3.0%, the Safe Withdrawal Rate equation is:
y = 12711x-2986.
When the withdrawal rate is 4.0%, the Safe Withdrawal Rate equation is:
y = 16078x-86082.
When the withdrawal rate is 5.0%, the Safe Withdrawal Rate equation is:
y = 19483x-169533.
When the withdrawal rate is 6.0%, the Safe Withdrawal Rate equation is:
y = 23021x-274381.

Add the upper confidence limit to calculate High Risk Rates. Here are the equations for High Risk Rates.

When the withdrawal rate is 2.0%, the High Risk Rate equation is:
y = 9343.3x+210109.
When the withdrawal rate is 3.0%, the High Risk Rate equation is:
y = 12711x+107014.
When the withdrawal rate is 4.0%, the High Risk Rate equation is:
y = 16078x+23918.
When the withdrawal rate is 5.0%, the High Risk Rate equation is:
y = 19483x-69533.
When the withdrawal rate is 6.0%, the High Risk Rate equation is:
y = 23021x-154381.

Recap of My Alternative Procedure: Second Part

We use all of these equations. We use them to tell us the percentage earnings yield that would meet their criteria at their withdrawal rates, which are 2.0% and 3.0% and 4.0% and 5.0% and 6.0%. We have written separate equations so that the criteria is that the final balance y equals zero.

For example, we start with the 2.0% (withdrawal rate) equation for Calculated Rates, which is y = 9343.3x+130109. It is zero when x equals -13.93%. This gives us a pair of points, a withdrawal rate of 2.0% and a percentage earnings yield 100E10/P of –13.93%.

We continue collecting pairs of points for the other withdrawal rates (3.0%, 4.0%, 5.0% and 6.0%) and their corresponding percentage earnings yields. We make a graph with withdrawal rates on the y axis and with percentage earnings yields on the x axis. We use Excel to fit a straight line to these points. It tells us the Calculated Rate as a function of the earnings yield.

Today’s earnings yield 100E10/P is 3.5% (roughly). We use 3.5% for the value of x in the equation of the straight line. The value of y is the Calculated Rate.

We can calculate a host of rates using a similar procedure. They are critically dependent upon the linearity approximation.

Continuing the Example with 50% Stocks

Now we determine the Calculated Rates.

When we use the equation derived from a 2.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=-13.93%.
When we use the equation derived from a 3.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=-3.70%.
When we use the equation derived from a 4.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=2.24%.
When we use the equation derived from a 5.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=6.14%.
When we use the equation derived from a 6.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=8.88%.

Now we determine the Safe Withdrawal Rates.

When we use the equation derived from a 2.0% withdrawal rate, the final balance [for the Safe Withdrawal Rate equation] is zero when the percentage earnings yield is x=-9.64%.
When we use the equation derived from a 3.0% withdrawal rate, the final balance [for the Safe Withdrawal Rate equation] is zero when the percentage earnings yield is x=0.23%.
When we use the equation derived from a 4.0% withdrawal rate, the final balance [for the Safe Withdrawal Rate equation] is zero when the percentage earnings yield is x=5.35%.
When we use the equation derived from a 5.0% withdrawal rate, the final balance [for the Safe Withdrawal Rate equation] is zero when the percentage earnings yield is x=8.70%.
When we use the equation derived from a 6.0% withdrawal rate, the final balance [for the Safe Withdrawal Rate equation] is zero when the percentage earnings yield is x=11.92%.

Now we determine the High Risk Rates.

When we use the equation derived from a 2.0% withdrawal rate, the final balance [for the High Risk Rate equation] is zero when the percentage earnings yield is x=-22.49%.
When we use the equation derived from a 3.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=-8.42%.
When we use the equation derived from a 4.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=-1.49%.
When we use the equation derived from a 5.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=3.57%.
When we use the equation derived from a 6.0% withdrawal rate, the final balance is zero when the percentage earnings yield is x=6.71%.

Applying Excel For the Final Calculations

The final step is to use Excel to calculate results in the form that we want. The most useful form is to report withdrawal rates as a function of earnings yield. This is exactly the opposite of the data that we have collected.

The Excursions

I selected various combinations of withdrawal rates this time. Previously, I used withdrawal rates of 3.0%, 4.0%, 5.0% and 6.0%.

I compare those earlier results with the combination of 2.0%, 3.0%, 4.0% and 5.0%.

Finally, I look at the combination 2.0%, 3.0% and 4.0% and the combination 3.0%, 4.0% and 5.0%.

Calculated Rates

Our data points for Calculate Rates are:
1) A 2.0% withdrawal rate and a –13.93% Calculated Rate.
2) A 3.0% withdrawal rate and a –3.70% Calculated Rate.
3) A 4.0% withdrawal rate and a 2.24% Calculated Rate.
4) A 5.0% withdrawal rate and a 6.14% Calculated Rate.
5) A 6.0% withdrawal rate and an 8.88% Calculated Rate.

Excel calculates equations, where x is the percentage earnings yield (which corresponds to the Calculated Rate) and y is the corresponding withdrawal rate. Here are the equations.

Using [the original combination of] 3.0%, 4.0%, 5.0% and 6.0% withdrawal rates, the equation is:
y = 0.2332x+3.7096
R-squared = 0.9709

Using 2.0%, 3.0%, 4.0% and 5.0% withdrawal rates, the equation is:
y = 0.1444x+3.8339
R-squared = 0.9552

Using 2.0%, 3.0% and 4.0% withdrawal rates, the equation is:
y = 0.1209x+3.62
R-squared = 0.9771

Using 3.0%, 4.0% and 5.0% withdrawal rates, the equation is:
y = 0.2004x+3.6874
R-squared = 0.9859

Here are the results using today’s earnings yield of (approximately) 3.5%:
1) Using the 3-4-5-6 data points, today’s Calculated Rate is 4.53%.
2) Using the 2-3-4-5 data points, today’s Calculated Rate is 4.34%.
3) Using the 2-3-4 data points, today’s Calculated Rate is 4.04%.
4) Using the 3-4-5 data points, today’s Calculated Rate is 4.39%.

Safe Withdrawal Rates

Our data points for Safe Withdrawal Rates are:
1) A 2.0% withdrawal rate and a –9.64% Safe Withdrawal Rate.
2) A 3.0% withdrawal rate and a 0.23% Safe Withdrawal Rate.
3) A 4.0% withdrawal rate and a 5.35% Safe Withdrawal Rate.
4) A 5.0% withdrawal rate and an 8.70% Safe Withdrawal Rate.
5) A 6.0% withdrawal rate and an 11.92% Safe Withdrawal Rate.

Excel calculates the following equation, where x is the percentage earnings yield (which corresponds to the Safe Withdrawal Rate) and y is the corresponding withdrawal rate.

Using [the original combination of] 3.0%, 4.0%, 5.0% and 6.0% withdrawal rates, the Safe Withdrawal Rate equation is:
y = 0.2567x+2.8188
R-squared = 0.9861

Using 2.0%, 3.0%, 4.0% and 5.0% withdrawal rates, the Safe Withdrawal Rate equation is:
y = 0.1567x+3.3182
R-squared = 0.9423

Using 2.0%, 3.0% and 4.0% withdrawal rates, the Safe Withdrawal Rate equation is:
y = 0.1291x+3.1747
R-squared = 0.9676

Using 3.0%, 4.0% and 5.0% withdrawal rates, the Safe Withdrawal Rate equation is:
y = 0.2327x+2.8922
R-squared = 0.9857

Here are the results using today’s earnings yield of (approximately) 3.5%:
1) Using the 3-4-5-6 data points, today’s Safe Withdrawal Rate is 3.72%.
2) Using the 2-3-4-5 data points, today’s Safe Withdrawal Rate is 3.87%.
3) Using the 2-3-4 data points, today’s Safe Withdrawal Rate is 3.63%.
4) Using the 3-4-5 data points, today’s Safe Withdrawal Rate is 3.71%.

High Risk Rates

Our data points for High Risk Rates are:
1) A 2.0% withdrawal rate and a –22.49% High Risk Rate.
2) A 3.0% withdrawal rate and a -8.42% High Risk Rate.
3) A 4.0% withdrawal rate and a –1.49% High Risk Rate.
4) A 5.0% withdrawal rate and a 3.57% High Risk Rate.
5) A 6.0% withdrawal rate and a 6.71% High Risk Rate.

Excel calculates the following equation, where x is the percentage earnings yield (which corresponds to the High Risk Rate) and y is the corresponding withdrawal rate.

Using [the original combination of] 3.0%, 4.0%, 5.0% and 6.0% withdrawal rates, the High Risk Rate equation is:
y = 0.1928x+4.4822
R-squared = 0.9726

Using 2.0%, 3.0%, 4.0% and 5.0% withdrawal rates, the High Risk Rate equation is:
y = 0.1109x+4.299
R-squared = 0.9435

Using 2.0%, 3.0% and 4.0% withdrawal rates, the High Risk Rate equation is:
y = 0.0917x+3.9904
R-squared = 0.9629

Using 3.0%, 4.0% and 5.0% withdrawal rates, the High Risk Rate equation is:
y = 0.1655x+4.3497
R-squared = 0.992

Here are the results using today’s earnings yield of (approximately) 3.5%:
1) Using the 3-4-5-6 data points, today’s High Risk Rate is 5.16%.
2) Using the 2-3-4-5 data points, today’s High Risk Rate is 4.69%.
3) Using the 2-3-4 data points, today’s High Risk Rate is 4.31%.
4) Using the 3-4-5 data points, today’s High Risk Rate is 4.93%.

Comparisons

The Safe Withdrawal Rate, Calculated Rate and High Risk Rate for HDBR50/HSWR50 with an earnings yield 100E10/P of 3.5% were 3.03%, 4.04% and 5.05% using the older, more accurate method of calculation.

The newer approach Safe Withdrawal Rates range from 3.63% to 3.87%. The older, more accurate value was 3.03%.

The newer approach Calculated Rates range from 4.04% to 4.53%. The older, more accurate value was 4.04%.

The new approach High Risk Rates range from 4.31% to 5.16%. The older, more accurate value was 5.05%.

Conclusions

It is easy to be deceived by the high values of R-squared in the final calculations. R-squared tells us how well the straight line fits the three or four data points used in the final steps. It does not tell us about the accuracy of our final results.

Picking and choosing a combination of withdrawal rates does not tell us much about the accuracy of the final calculations.

A major source of error might be how well the amount of scatter of dollar values at year 30 translates into the scatter about the Calculated Rates. Remember that I use the confidence limits about the Calculated Rate to define the Safe Withdrawal Rate and the High Risk Rate.

Otherwise, I found no simple, new insights regarding errors.

Have fun.

John Walter Russell
I wrote this on June 18, 2005.