JWR1945 ***** Legend
Joined: 26 Nov 2002 Posts: 1697 Location: Crestview, Florida

Posted: Sun May 02, 2004 3:34 pm Post subject: 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, rebalanced 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 The portfolio survives for 10 years when the amount withdrawn is 10WFAIL50 times the initial balance (plus adjustments that match inflation). It fails when the amount withdrawn is increased by 0.1%.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 19211980.
Code: 
Year 10WFAIL50
1921 15.7
1922 16.1
1923 15.6
1924 16.3
1925 16.1
1926 15.0
1927 15.1
1928 13.3
1929 11.1
1930 11.0
1931 11.2
1932 12.7
1933 13.7
1934 11.4
1935 12.3
1936 10.1
1937 8.7
1938 9.9
1939 9.0
1940 9.0
1941 10.4
1942 11.9
1943 11.5
1944 10.9
1945 10.6
1946 10.9
1947 13.3
1948 14.2
1949 14.2
1950 15.3
1951 14.7
1952 14.4
1953 14.5
1954 15.2
1955 13.3
1956 12.6
1957 13.1
1958 14.2
1959 12.6
1960 12.6
1961 12.6
1962 11.9
1963 12.5
1964 11.6
1965 10.7
1966 10.2
1967 10.6
1968 10.0
1969 9.6
1970 10.0
1971 9.7
1972 9.2
1973 9.0
1974 10.4
1975 12.2
1976 11.0
1977 11.4
1978 13.1
1979 14.1
1980 14.8 
Here are the values of 10WFAIL50 in a single column for the years 19211980.
Code: 
15.7
16.1
15.6
16.3
16.1
15.0
15.1
13.3
11.1
11.0
11.2
12.7
13.7
11.4
12.3
10.1
8.7
9.9
9.0
9.0
10.4
11.9
11.5
10.9
10.6
10.9
13.3
14.2
14.2
15.3
14.7
14.4
14.5
15.2
13.3
12.6
13.1
14.2
12.6
12.6
12.6
11.9
12.5
11.6
10.7
10.2
10.6
10.0
9.6
10.0
9.7
9.2
9.0
10.4
12.2
11.0
11.4
13.1
14.1
14.8 
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 = (yb)/m = (return0b)/m).
Here are the equations:
Code: 
N slope m b b/10 R Squared
4 2.111 21.646 2.1646 0.7541
6 1.8255 18.226 1.8226 0.8336
8 1.6273 15.847 1.5847 0.8325
10 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.
Code: 
2.111
1.8255
1.6273
1.3926 
Here are the intercepts b in a single column.
Code: 
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.

