## An Application of the New Tool: Tables

Research on Safe Withdrawal Rates

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JWR1945
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An Application of the New Tool: Tables
I will start with some tables. These are the values of 10WFAIL50 for a portfolio consisting of 50% stocks and 50% commercial paper, re-balanced annually. The expense ratio is 0.20%. The portfolio survives for 10 years when the amount withdrawn is 10WFAIL50 times the initial balance (plus adjustments that match inflation). It fails when the amount withdrawn is increased by 0.1%. This is identical to what I normally refer to as HDBR50 except that the portfolio lasts for 10 years instead of 30 years.

These numbers are used with Gummy's formula. For any actual withdrawal rate w and specified number of years, the final balance / the initial balance = RETURN0*(1 - w/WFAIL). RETURN0 is the ratio of the balance (after the specified number of years) to the initial balance when there are no withdrawals. WFAIL is the withdrawal rate that lasts for exactly the prescribed period of time (in this case, ten years).

Values of 10WFAIL50 for 1921-1980.

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``Year   10WFAIL501921   15.71922   16.11923   15.61924   16.31925   16.11926   15.01927   15.11928   13.31929   11.11930   11.01931   11.21932   12.71933   13.71934   11.41935   12.31936   10.11937   8.71938   9.91939   9.01940   9.01941   10.41942   11.91943   11.51944   10.91945   10.61946   10.91947   13.31948   14.21949   14.21950   15.31951   14.71952   14.41953   14.51954   15.21955   13.31956   12.61957   13.11958   14.21959   12.61960   12.61961   12.61962   11.91963   12.51964   11.61965   10.71966   10.21967   10.61968   10.01969   9.61970   10.01971   9.71972   9.21973   9.01974   10.41975   12.21976   11.01977   11.41978   13.11979   14.11980   14.8``

Here are the values of 10WFAIL50 in a single column for the years 1921-1980.

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``15.716.115.616.316.115.015.113.311.111.011.212.713.711.412.310.18.79.99.09.010.411.911.510.910.610.913.314.214.215.314.714.414.515.213.312.613.114.212.612.612.611.912.511.610.710.210.610.09.610.09.79.29.010.412.211.011.413.114.114.8``

I made scatter plots of return0 (which is the annualized percentage return when there are no withdrawals and RETURN0 = (1+return0)^N after N years) and 10WFAIL50. I made four plots using values of return0 at 4, 6, 8 and 10 years. I fit each plot with a linear equation return0 = mx+b = slope*10WFAIL50+b (and equivalently 10WFAIL50 = (y-b)/m = (return0-b)/m).

Here are the equations:

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``N   slope m      b      b/10     R Squared4     2.111   -21.646   -2.1646   0.75416    1.8255   -18.226   -1.8226   0.83368    1.6273   -15.847   -1.5847   0.832510   1.3926   -13.075   -1.3075   0.8429``

We see from the values of R Squared that we have good curve fits.

Here are the slopes m in a single column.

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``2.1111.82551.62731.3926``

Here are the intercepts b in a single column.

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``-21.646-18.226-15.847-13.075``

Here is the necessary statistical information. What I have identified as being 90% confidence levels are better estimated as being close to 86% (and no worse than 75%).

N = 4 Years
Slope = m = 2.111. A 1% change in 10WFAIL50 corresponds to a 2.111% change in return0. A 1% change in return0 corresponds to a 1/m =1/2.111 = 0.47% change in 10WFAIL50.
R Squared = 0.7541
return0 Standard Deviation = 2.578976%
10WFAIL50 Standard Deviation = 1.221684%
return0 90% Confidence limits = + and - 1.64* 2.578976% = 4.23%
10WFAIL50 Standard Deviation = + and - 1.64*1.221684% = 2.00%

N = 6 Years
Slope = m = 1.8255. A 1% change in 10WFAIL50 corresponds to a 1.8255% change in return0. A 1% change in return0 corresponds to a 1/m =1/1.8255 = 0.55% change in 10WFAIL50.
R Squared = 0.8336
return0 Standard Deviation = 1.745206%
10WFAIL50 Standard Deviation = 0.956016%
return0 90% Confidence limits = + and - 1.64*1.745206% = 2.86%
10WFAIL50 Standard Deviation = + and - 1.64*0.956016% = 1.57%

N = 8 Years
Slope = m = 1.6273. A 1% change in 10WFAIL50 corresponds to a 1.6273% change in return0. A 1% change in return0 corresponds to a 1/m =1/1.6273 = 0.61% change in 10WFAIL50.
R Squared = 0.8325
return0 Standard Deviation = 1.561522%
10WFAIL50 Std Dev = 0.959579%
return0 90% Confidence limits = + and - 1.64*1.561522% = 2.56%
10WFAIL50 Standard Deviation = + and - 1.64*0.959579% = 1.57%

N = 10 Years
Slope = m = 1.3926. A 1% change in 10WFAIL50 corresponds to a 1.3926% change in return0. A 1% change in return0 corresponds to a 1/m =1/1.3926 = 0.72% change in 10WFAIL50.
R Squared = 0.8429
return0 Standard Deviation = 1.286267%
10WFAIL50 Std Dev = 0.923644%
return0 90% Confidence limits = + and - 1.64*1.286267% = 2.11%
10WFAIL50 Standard Deviation = + and - 1.64*0.923644% = 1.51%

Next, I will provide some examples.

Have fun.

John R.