From Chapter 2

Research on Safe Withdrawal Rates

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JWR1945
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Posts: 1697
Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

From Chapter 2

Post by JWR1945 »

In Chapter 2 of Common Sense on Mutual Funds, the book that ES has featured for us at the NoFeeBoards, John Bogle presented a modified version of the Dividend Discount Model (or Gordon Model, in a slightly different form) as the source of long-term stock market returns.

For those wishing to see a mathematical derivation of these models, I recommend visiting gummy's site.
http://home.golden.net/~pjponzo/dividend-discount.htm

John Bogle adds three terms:
1) The initial dividend yield,
2) The subsequent growth rate in earnings and
3) The effect of price to earnings ratio changes over the period of interest.

The first two terms are reasonably well behaved. They form the investment component of the total return. The third component varies greatly. It is the speculative component.

Those familiar with the Gordon equation will notice that John Bogle has replaced the dividend growth rate with the earnings growth rate. I find this to be reasonable because 1) dividends come out of earnings, 2) the earnings growth rate is much more predictable than the dividend growth rate and 3) the earnings growth rate does not depend upon the dividend payout percentage, which is largely a management policy decision. That having been said, I suspect that he chose to use the earnings growth rate simply because it provides a better curve fit.

In my earlier investigations into Safe Withdrawal Rates, I looked first at using dividend yields and valuations. The best choice for including valuations turns out to be Professor Robert Shiller's P/E10, which is the current (real) price divided by the average of the previous ten years of (real) earnings. When I first investigated using the earnings yield E10/P, which is 1/[P/E10], I subtracted the dividend yield from the Historical Database Rates. Later, when I plotted Historical Database Rates versus earnings yield, I found that I got a better curve fit. In essence, the earnings yield had captured all of the effects of the initial dividend amount and did even better. This makes sense because dividends come out of earnings and the ten-year average of (real) earnings is well behaved. Using earnings yield (using E10/P) gets around the problem of surprise cuts in dividends.

Because of my previous success with Safe Withdrawal Rates, I have modified John Bogle's model to incorporate earning yield E10/P. In addition, I use changes in E10/P for the speculative term as opposed to a single year's (trailing) price to earnings ratio.

I have made tables of the annualized, real total return of stocks with dividends reinvested and with an expense ratio of 0.20%. I have looked at a variety of different approaches using curve fits. The best so far is to plot the difference of the annualized, real total return and the price to earnings component versus the earnings yield. I have ended up using John Bogle's adjustment except that I use P/E10 (or E10/P) instead of P/E (or E/P). John Bogle refers to the price to earnings ratio at the beginning of a period and at the end of a period. Assuming that earnings remain constant, the increase in the total return that comes from multiple expansion is (1+r)^N = Price after N years / Initial Price. Normalizing according to earnings, (1+r)^N = [P/E after N years] / [P/E at the start]. Using P/E10 instead of P/E, (1+r)^N = [P/E10 after N years] / [P/E10 at the start]. Converting to earnings yield, (1+r)^N = [E10/P at the start] / [E10/P after N years].

Early Results

My first investigation looked at the annualized, real total return versus earnings yield. When I plotted the 5-Year returns for 1871-1980 (start years), I needed a plotting range from minus 10% to plus 30%. When I subtracted the multiple expansion term, the 5-Year plotting range was from 0% to plus 16%. If I subtracted both the multiple expansion term and the initial earnings yield, the 5-Year plotting range was from minus 20% to plus 10% (although minus 18% to plus 6% would have been sufficient). At the same time, there was a strong negative slope of (1.3222) times the percentage earnings yield 100E10/P with R Squared equal to 0.7276. Clearly, the best choice with 5-Year returns was to subtract only the multiple expansion term.

[Curves with 10-Year, 15-Year and 20-Year data had negative slopes of (1.0603), (1.0951) and (1.0449) and R Squared values of 0.7844, 0.8146 and 0.827 respectively. Because these curves had slopes very close to minus one, I chose not to complicate matters by adding a fraction of earnings yield back in.]

[I plotted the annualized, real total return minus earnings yield versus earnings yield for 10-Year, 20-Year and 30-Year returns (but not for 5-Years). A plotting range of minus 12% to plus 8% was sufficient. This is not as good as the corresponding range when subtracting the multiple expansion term instead (from 0% to plus 12%).]

Shorter Periods

I looked at 5-Year, 10-Year, 15-Year and 20-Year plots of the annualized, real total return minus the multiple expansion term versus earnings yield for 1871-1980 and 1921-1980. I stopped at 1980 (as a start year) to keep the final year at 2000 or earlier in all cases.

These are the equations for 1871-1980, where y is the annualized, real total return minus the multiple expansion term (both in percent) and x is the percentage earnings yield 100E10/P.
For 5-Years: y = -0.3917x+9.3605 and R Squared = 0.164.
For 10-Years: y = -0.2421x+8.086 and R Squared = 0.1231.
For 15-Years: y = -0.1785x+7.4661 and R Squared = 0.0991.
For 20-Years: y = -0.1416x+7.101 and R Squared = 0.0906.
Most often, the initial earnings yield is between 4% and 10%.

These are the equations for 1921-1980, where y is the annualized, real total return minus the multiple expansion term (both in percent) and x is the percentage earnings yield 100E10/P.
For 5-Years: y = -0.3222x+8.5338 and R Squared = 0.1369.
For 10-Years: y = -0.0603x+6.6135 and R Squared = 0.0116.
For 15-Years: y = -0.0951x+6.7181 and R Squared = 0.032.
For 20-Years: y = -0.0449x+6.2772 and R Squared = 0.0088.
Most often, the initial earnings yield is between 4% and 12%.

For the longer period, 1871-1980, the line has a downward slope that is loosely related to the percentage earnings yield. Over the more recent period, 1921-1980, multiple expansion explains all of the effects related to earnings yield except in the shortest period (of 5-Years). The 5-Year line has a downward slope that is loosely related to the earnings yield.

Here are the ranges of variation:
For 5 -Years: y ranges from 0% to 15% in the 1871-1980 period and from 0% to 11% in the 1921-1980 period.
For 10-Years: y ranges from 1% to11% in the 1871-1980 period and from 3% to 11% in the 1921-1980 period.
For 15 -Years: y ranges from 3% to 9% in the 1871-1980 period and from 3% to 9% in the 1921-1980 period.
For 20-Years: y ranges from 3% to9% in the 1871-1980 period and from 3% to 9% in the 1921-1980 period.

Longer Periods

I looked at 30-Year, 40-Year, 50-Year and 60-Year plots of the annualized, real total return minus the multiple expansion term versus earnings yield for 1871-1940. I stopped at 1940 (as a start year) to keep the final year at 2000 or earlier in all cases.

These are the equations for 1871-1940, where y is the annualized, real total return minus the multiple expansion term (both in percent) and x is the percentage earnings yield 100E10/P.
For 30-Years: y = -0.1034x+7.154 and R Squared = 0.1889.
For 40-Years: y = -0.0035x+6.3069 and R Squared = 0.0005.
For 50-Years: y = -0.0126x+6.3263 and R Squared = 0.0177.
For 60-Years: y = -0.0381x+6.4374 and R Squared = 0.1218.
Most often, the initial earnings yield is between 4% and 12%.

Over the longer periods starting in 1871-1940, there is little variation. The 30-Year and 60-Year lines have downward slopes that are loosely related to the earnings yield. Particular decades may influence these results. The 30-Year data end in 1970. The 60-Year data includes the 1990s. [We may be seeing an artificial result of my choosing to plot more than one set of data on the same graph.]

Here are the ranges of variation:
For 30 -Years: y ranges from 4.7% to 7.7% (in the 1871-1940 period).
For 40-Years: y ranges from 5.3% to 7.5% (in the 1871-1940 period).
For 50-Years: y ranges from 5.4% to 7.1% (in the 1871-1940 period).
For 60-Years: y ranges from 5.5% to 6.9% (in the 1871-1940 period).

Equations and Long-Term Returns

We have ended up with equations that look like this:
Total Return-Multiple Expansion Term = a constant - a small fraction of the earnings yield.

The importance of the earnings yield term is seen in R Squared. It is consistently small. This leaves us with the equation:
Total Return = Multiple Expansion Term + a constant + randomness only loosely related to earnings yield.

In most cases, we have an even simpler equation:
Total Return = Multiple Expansion Term + a constant + randomness (with almost no influence from earnings yield at all).

Something Significant

We can replace the three terms in John Bogle's equation with only one: the multiple expansion term.

Notice that our constant terms are very close to 6.3%. Our return is a little bit lower than what is reported because our calculated returns include 0.20% in expenses. But that is not all. The 7% annualized, real total return that John Bogle reports apparently includes some effects from a long-term expansion of multiples.

Have fun.

John R.
Mike
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Posts: 278
Joined: Sun Jul 06, 2003 4:00 am

Post by Mike »

The 7% annualized, real total return that John Bogle reports apparently includes some effects from a long-term expansion of multiples.
Along with multiple expansion, earnings and return exhibit a slight upward trend. This effect is visible when you plot earnings 10 versus date. A regression line slopes upward, rather than being a flat average number. Although we are currently far above the trend line. If you plot yield and return 10 on the same chart, with the date as the x axis, you can see how yield and return follow the same rollar coaster pattern (lagged a bit). In the late 80s, the lines diverge from their historic pattern, but there is not enough data to see clearly where they are going.

It is possible that multiples could continue their slow upward trek, after reverting closer to their historic upward trend line. Of course, history shows that multiples regularly go below the trend line for extended periods also.
JWR1945
***** Legend
Posts: 1697
Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

Post by JWR1945 »

From Chapter 2 tables

Here are some relevant tables in a form suitable for copying. The stock allocation is 100%, expenses are 0.20% and there are no withdrawals. All dividends are reinvested. All returns are adjusted for inflation.

An annualized return r over N years satisfies this equation: (1+r)^N = Final Balance/Initial Balance.

The multiple expansion term with a return of r over N years satisfies this equation: (1+r)^N = [Initial value of E10/P] / [Final value of E10/P], which is also equal to [Final value of P/E10] / [Initial value of P/E10]. When r is presented in percent (as opposed to a decimal fraction), r is called the multiple expansion percentage.

Here are the values of P/E10 from 1871-1920.

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13.3
14.5
15.3
13.9
13.6
13.3
10.6
9.7
10.7
15.3
18.5
15.7
15.3
14.4
13.1
16.7
17.5
15.4
15.8
17.2
15.4
19.0
17.7
15.7
16.5
16.6
17.0
19.2
22.9
18.7
21.0
22.3
20.3
15.9
18.5
20.1
17.2
11.9
14.8
14.5
14.0
13.8
13.1
11.6
10.4
12.5
11.0
6.6
6.1
6.0
Here are the values of P/E10 from 1921-2002.

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5.1
6.3
8.2
8.1
9.7
11.3
13.2
18.8
27.1
22.3
16.7
9.3
8.7
13.0
11.5
17.1
21.6
13.5
15.6
16.4
13.9
10.1
10.2
11.1
12.0
15.6
11.5
10.4
10.2
10.7
11.9
12.5
13.0
12.0
16.0
18.3
16.7
13.8
18.0
18.3
18.5
21.2
19.3
21.6
23.3
24.1
20.4
21.5
21.2
17.1
16.5
17.3
18.7
13.5
8.9
11.2
11.4
9.2
9.3
8.9
9.3
7.4
8.8
9.9
10.0
11.7
14.9
13.9
15.1
17.0
15.6
19.8
20.3
21.4
20.2
24.8
28.3
32.9
40.6
43.8
37.0
30.3
Here are the values of percentage earnings yield 100E10/P from 1871-1920.

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7.52
6.90
6.54
7.19
7.35
7.52
9.43
10.31
9.35
6.54
5.41
6.37
6.54
6.94
7.63
5.99
5.71
6.49
6.33
5.81
6.49
5.26
5.65
6.37
6.06
6.02
5.88
5.21
4.37
5.35
4.76
4.48
4.93
6.29
5.41
4.98
5.81
8.40
6.76
6.90
7.14
7.25
7.63
8.62
9.62
8.00
9.09
15.15
16.39
16.67
Here are the values of percentage earnings yield 100E10/P from 1921-2002.

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19.61
15.87
12.20
12.35
10.31
8.85
7.58
5.32
3.69
4.48
5.99
10.75
11.49
7.69
8.70
5.85
4.63
7.41
6.41
6.10
7.19
9.90
9.80
9.01
8.33
6.41
8.70
9.62
9.80
9.35
8.40
8.00
7.69
8.33
6.25
5.46
5.99
7.25
5.56
5.46
5.41
4.72
5.18
4.63
4.29
4.15
4.90
4.65
4.72
5.85
6.06
5.78
5.35
7.41
11.24
8.93
8.77
10.87
10.75
11.24
10.80
13.53
11.42
10.11
10.00
8.54
6.70
7.20
6.63
5.87
6.41
5.06
4.92
4.67
4.95
4.04
3.53
3.04
2.46
2.28
2.70
3.30
Here are the values of the annualized, real total return minus the multiple expansion percentage after one year from 1871-1920.

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3.90
2.88
11.76
14.35
13.39
7.15
23.70
17.91
(21.64)
11.84
8.44
7.97
9.05
8.09
4.02
6.28
7.11
5.15
3.01
2.48
1.89
5.08
6.46
2.46
2.59
3.50
2.78
6.57
7.04
10.59
9.52
7.38
9.52
10.37
11.59
10.96
10.40
7.72
6.35
7.28
5.84
4.73
5.02
4.21
5.28
8.28
10.70
5.88
2.72
3.84
Here are the values of the annualized, real total return minus the multiple expansion percentage after one year from 1921-1990.

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(1.19)
(1.99)
3.21
5.05
3.46
(3.37)
(5.88)
2.53
9.33
10.61
12.55
8.68
(0.36)
1.35
0.89
2.60
8.17
0.48
(1.12)
5.91
10.26
9.19
10.03
7.60
3.99
1.55
2.99
10.07
14.05
11.28
9.64
8.96
9.31
11.63
12.69
12.16
8.72
5.80
4.57
3.09
2.97
4.97
6.41
6.36
5.54
5.94
5.94
6.88
5.85
4.80
4.81
4.84
5.09
5.92
2.55
3.61
5.40
4.49
6.25
7.15
6.64
4.45
1.56
2.63
3.03
0.67
1.09
3.35
3.12
2.41
Here are the values of the annualized, real total return minus the multiple expansion percentage after 5 years from 1871-1920.

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9.40
9.87
13.78
14.96
9.22
8.80
9.12
5.69
4.21
9.00
7.77
7.30
7.13
6.32
5.26
4.83
4.08
3.60
3.94
3.82
3.87
4.11
3.66
3.51
4.60
6.13
7.35
8.21
8.78
9.48
9.67
9.98
10.69
10.53
9.70
8.89
7.83
6.34
5.82
5.39
5.02
5.56
7.23
7.41
7.19
6.73
5.30
2.28
1.67
2.05
Here are the values of the annualized, real total return minus the multiple expansion percentage after 5 years from 1921-1990.

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1.83
1.35
0.74
0.42
1.97
4.09
8.14
9.95
9.64
8.00
6.44
3.06
3.59
3.55
3.10
4.00
5.60
5.45
7.06
8.70
8.65
6.22
4.93
5.03
6.19
7.34
9.36
10.78
10.61
10.13
10.39
10.97
10.88
10.40
8.96
7.12
5.23
4.28
4.41
4.76
5.27
5.84
6.05
6.15
6.06
5.90
5.69
5.49
5.17
5.27
5.00
4.77
4.88
4.74
4.63
5.40
6.02
5.91
5.41
4.66
3.88
2.49
1.81
2.17
2.25
2.15
2.12
2.13
2.08
2.26
Here are the values of the annualized, real total return minus the multiple expansion percentage after 10 years from 1871-1920.

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9.11
9.54
10.04
9.74
9.11
8.27
8.19
6.44
5.27
7.20
6.29
5.67
5.37
5.11
4.52
4.36
4.09
3.63
3.73
4.21
4.98
5.66
5.91
6.31
7.05
7.93
8.73
9.52
9.63
9.60
9.27
8.90
8.64
8.12
7.50
6.98
6.69
6.84
6.65
6.30
5.95
5.43
4.95
4.77
4.87
4.71
3.56
1.56
1.11
2.01
Here are the values of the annualized, real total return minus the multiple expansion percentage after 10 years from 1921-1990.

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3.00
5.07
6.01
5.88
5.43
5.31
5.90
7.13
6.84
5.79
5.19
4.57
4.59
5.38
6.05
6.23
5.91
5.20
6.06
7.46
7.98
7.79
7.76
7.72
8.06
8.78
10.15
10.83
10.51
9.53
8.69
8.05
7.62
7.31
6.86
6.21
5.55
5.18
5.30
5.45
5.62
5.77
5.77
5.65
5.65
5.45
5.22
5.18
4.96
4.98
5.19
5.39
5.38
5.07
4.65
4.67
4.46
3.95
3.83
3.50
3.02
2.30
1.97
2.12
2.26
2.69
3.36
4.16
4.33
4.43
Have fun.

John R.
JWR1945
***** Legend
Posts: 1697
Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

Post by JWR1945 »

From Chapter 2 tables (continued)

Here are some relevant tables in a form suitable for copying. The stock allocation is 100%, expenses are 0.20% and there are no withdrawals. All dividends are reinvested. All returns are adjusted for inflation.

An annualized return r over N years satisfies this equation: (1+r)^N = Final Balance/Initial Balance.

The multiple expansion term with a return of r over N years satisfies this equation: (1+r)^N = [Initial value of E10/P] / [Final value of E10/P], which is also equal to [Final value of P/E10] / [Initial value of P/E10]. When r is presented in percent (as opposed to a decimal fraction), r is called the multiple expansion percentage.

Here are the values of the annualized, real total return minus the multiple expansion percentage after 15 years from 1871-1920.

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8.65
8.80
9.06
8.60
7.86
7.08
6.77
5.48
4.81
6.05
5.50
5.13
4.80
4.60
4.55
4.93
5.13
5.14
5.51
5.95
6.56
7.15
7.61
7.71
7.96
8.28
8.43
8.50
8.34
8.14
7.85
7.78
8.14
7.87
7.40
6.90
6.20
5.40
5.13
5.03
4.81
4.26
3.74
3.58
4.01
4.52
5.13
4.78
4.47
4.35
Here are the values of the annualized, real total return minus the multiple expansion percentage after 15 years from 1921-1985.

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4.23
4.46
5.24
5.10
4.68
4.85
5.81
6.56
6.91
6.77
6.30
5.11
4.70
5.26
6.10
6.61
6.99
6.95
7.49
8.29
8.74
8.82
8.80
8.57
8.36
8.22
8.50
8.66
8.42
7.91
7.54
7.29
7.09
6.91
6.58
6.11
5.60
5.29
5.26
5.39
5.40
5.42
5.46
5.34
5.34
5.43
5.49
5.41
5.10
4.88
4.79
4.58
4.31
4.17
3.89
3.85
3.69
3.35
3.25
3.08
3.09
3.09
3.38
3.60
3.71
Here are the values of the annualized, real total return minus the multiple expansion percentage after 20 years from 1871-1920.

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7.66
7.60
7.69
7.41
6.82
6.28
6.07
5.01
4.49
5.69
5.65
5.67
5.64
5.71
5.78
6.11
6.38
6.64
6.76
6.91
7.18
7.32
7.31
7.22
7.29
7.45
7.69
8.16
8.10
7.89
7.55
7.13
6.78
6.43
6.21
5.86
5.17
4.38
4.19
4.36
4.65
5.27
5.47
5.31
5.14
5.00
4.69
4.50
4.24
4.04
Here are the values of the annualized, real total return minus the multiple expansion percentage after 20 years from 1921-1980.

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4.17
4.82
5.30
5.63
5.74
5.78
5.91
6.16
6.45
6.63
6.57
6.16
6.16
6.53
7.04
7.48
7.92
7.91
8.17
8.45
8.32
7.92
7.69
7.52
7.46
7.50
7.81
7.99
7.83
7.42
7.10
6.87
6.67
6.44
6.23
5.82
5.38
5.19
5.13
5.21
5.40
5.58
5.57
5.36
5.18
5.08
4.86
4.61
4.44
4.29
4.16
3.99
3.80
3.67
3.48
3.71
3.92
4.05
4.08
3.96
Here are the values of the annualized, real total return minus the multiple expansion percentage after 25 years from 1871-1920.

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6.90
6.87
6.88
6.66
6.38
6.25
6.32
5.65
5.40
6.44
6.45
6.56
6.73
6.67
6.60
6.70
6.68
6.58
6.57
6.59
6.74
6.96
7.30
7.27
7.27
7.30
7.17
7.05
6.91
6.84
6.60
6.14
5.75
5.47
5.49
5.54
5.78
5.64
5.42
5.17
5.00
4.90
5.13
4.96
4.75
4.79
4.90
4.71
4.85
5.04
Here are the values of the annualized, real total return minus the multiple expansion percentage after 25 years from 1921-1975.

Code: Select all

5.07
5.10
5.23
5.51
5.83
6.11
6.58
7.02
7.23
7.27
7.27
7.08
7.09
7.25
7.42
7.41
7.40
7.21
7.43
7.73
7.73
7.49
7.36
7.24
7.16
7.16
7.36
7.47
7.25
6.94
6.65
6.42
6.28
6.08
5.92
5.73
5.52
5.33
5.18
5.10
5.12
5.05
4.89
4.78
4.66
4.53
4.34
4.15
4.00
3.89
3.99
4.11
4.26
4.23
4.10
Here are the values of the annualized, real total return minus the multiple expansion percentage after 30 years from 1871-1920.

Code: Select all

6.77
6.95
7.10
7.04
6.90
6.82
6.96
6.58
6.25
7.01
6.88
6.78
6.67
6.53
6.39
6.41
6.49
6.71
6.73
6.70
6.75
6.66
6.50
6.39
6.48
6.53
6.33
6.15
6.06
6.15
6.21
6.49
6.53
6.26
5.96
5.69
5.40
5.33
5.12
4.84
4.83
5.04
5.18
5.33
5.45
5.42
5.11
4.74
4.88
5.24
Here are the values of the annualized, real total return minus the multiple expansion percentage after 30 years from 1921-1970.

Code: Select all

5.48
5.80
6.11
6.33
6.50
6.77
7.27
7.64
7.71
7.54
7.25
6.77
6.64
6.78
6.98
7.06
7.14
7.02
7.21
7.44
7.40
7.17
7.03
6.87
6.82
6.77
6.90
6.98
6.80
6.56
6.44
6.35
6.22
5.97
5.71
5.44
5.08
4.79
4.72
4.68
4.65
4.59
4.46
4.35
4.27
4.34
4.39
4.47
4.41
4.34
Here are the values of the annualized, real total return minus the multiple expansion percentage after 35 years from 1871-1920.

Code: Select all

7.19
7.40
7.66
7.54
7.31
7.14
7.09
6.54
6.19
6.76
6.61
6.60
6.76
6.67
6.51
6.48
6.31
6.10
6.04
6.10
6.15
6.01
5.79
5.71
5.93
6.20
6.60
6.76
6.64
6.45
6.25
6.08
6.15
5.88
5.58
5.44
5.44
5.34
5.41
5.41
5.36
5.20
5.15
5.29
5.55
5.70
5.69
5.57
5.67
5.90
Here are the values of the annualized, real total return minus the multiple expansion percentage after 35 years from 1921-1965.

Code: Select all

6.14
6.51
6.78
6.87
6.85
6.82
6.99
7.19
7.25
7.16
6.97
6.64
6.55
6.69
6.84
6.89
6.93
6.79
6.90
7.10
7.03
6.80
6.69
6.54
6.50
6.57
6.76
6.83
6.60
6.29
6.08
5.87
5.63
5.46
5.27
4.99
4.68
4.43
4.36
4.34
4.46
4.60
4.70
4.66
4.61
Here are the values of the annualized, real total return minus the multiple expansion percentage after 40 years from 1871-1920.

Code: Select all

7.42
7.46
7.50
7.32
7.06
6.87
6.89
6.65
6.36
6.82
6.64
6.43
6.23
6.08
6.00
5.99
5.77
5.52
5.49
5.66
5.91
6.29
6.38
6.27
6.22
6.23
6.23
6.40
6.26
6.06
5.95
6.02
6.06
6.04
5.98
5.82
5.54
5.29
5.36
5.50
5.62
5.71
5.81
5.92
6.08
6.24
6.30
6.22
6.22
6.28
Here are the values of the annualized, real total return minus the multiple expansion percentage after 40 years from 1921-1960.

Code: Select all

6.26
6.35
6.48
6.56
6.59
6.63
6.84
7.05
7.12
7.01
6.83
6.51
6.41
6.49
6.63
6.64
6.64
6.53
6.61
6.79
6.82
6.70
6.59
6.40
6.28
6.25
6.29
6.24
6.08
5.83
5.61
5.41
5.22
5.06
4.90
4.79
4.67
4.65
4.63
4.62
Have fun.

John R.
JWR1945
***** Legend
Posts: 1697
Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

Post by JWR1945 »

From Chapter 2 tables (continued)

Here are some relevant tables in a form suitable for copying. The stock allocation is 100%, expenses are 0.20% and there are no withdrawals. All dividends are reinvested. All returns are adjusted for inflation.

An annualized return r over N years satisfies this equation: (1+r)^N = Final Balance/Initial Balance.

The multiple expansion term with a return of r over N years satisfies this equation: (1+r)^N = [Initial value of E10/P] / [Final value of E10/P], which is also equal to [Final value of P/E10] / [Initial value of P/E10]. When r is presented in percent (as opposed to a decimal fraction), r is called the multiple expansion percentage.

Here are the values of the annualized, real total return minus the multiple expansion percentage after 45 years from 1871-1920.

Code: Select all

7.14
7.24
7.47
7.33
7.08
6.86
6.70
6.17
5.86
6.34
6.18
5.94
5.70
5.58
5.62
5.79
6.05
6.08
6.01
5.96
5.97
5.99
6.10
5.98
5.90
5.97
6.16
6.29
6.36
6.36
6.24
6.04
5.93
5.92
6.01
6.00
5.95
5.87
5.91
5.97
6.10
6.25
6.35
6.38
6.39
6.34
6.18
6.01
6.02
6.11
Here are the values of the annualized, real total return minus the multiple expansion percentage after 45 years from 1921-1955.

Code: Select all

6.16
6.29
6.43
6.52
6.53
6.55
6.71
6.87
6.88
6.80
6.62
6.30
6.23
6.28
6.40
6.50
6.57
6.46
6.48
6.56
6.51
6.29
6.10
5.96
5.87
5.81
5.84
5.81
5.65
5.43
5.36
5.33
5.33
5.23
5.10
Here are the values of the annualized, real total return minus the multiple expansion percentage after 50 years from 1871-1920.

Code: Select all

7.11
7.03
6.97
6.79
6.62
6.43
6.23
5.70
5.42
5.96
5.99
6.17
6.18
6.04
5.89
5.86
5.80
5.85
5.77
5.69
5.77
5.95
6.03
6.09
6.19
6.23
6.17
6.15
6.22
6.34
6.36
6.36
6.39
6.37
6.38
6.40
6.42
6.35
6.32
6.25
6.20
6.15
6.16
6.19
6.24
6.24
6.14
6.02
6.04
6.11
Here are the values of the annualized, real total return minus the multiple expansion percentage after 50 years from 1921-1950.

Code: Select all

6.13
6.23
6.33
6.37
6.39
6.38
6.50
6.65
6.66
6.59
6.49
6.28
6.20
6.19
6.23
6.25
6.21
6.03
6.08
6.17
6.09
5.88
5.72
5.59
5.51
5.57
5.73
5.85
5.75
5.57
Here are the values of the annualized, real total return minus the multiple expansion percentage after 55 years from 1871-1920.

Code: Select all

6.69
6.58
6.48
6.33
6.25
6.23
6.42
6.15
5.85
6.18
6.03
5.93
5.97
5.82
5.65
5.68
5.78
5.81
5.90
5.97
6.02
5.97
5.93
5.99
6.19
6.34
6.45
6.55
6.60
6.66
6.69
6.75
6.78
6.71
6.61
6.46
6.31
6.17
6.15
6.12
6.12
6.12
6.15
6.18
6.22
6.21
6.10
5.97
5.95
6.02
Here are the values of the annualized, real total return minus the multiple expansion percentage after 55 years from 1921-1945.

Code: Select all

6.02
6.09
6.19
6.21
6.23
6.29
6.45
6.58
6.55
6.42
6.26
5.97
5.82
5.85
5.90
5.89
5.86
5.69
5.73
5.82
5.85
5.77
5.77
5.68
5.63
Here are the values of the annualized, real total return minus the multiple expansion percentage after 60 years from 1871-1920.

Code: Select all

6.49
6.72
6.81
6.65
6.42
6.25
6.18
5.95
5.66
5.94
5.85
5.90
5.92
5.93
5.92
5.92
5.82
5.74
5.82
5.99
6.14
6.25
6.31
6.36
6.49
6.65
6.80
6.90
6.89
6.85
6.73
6.63
6.58
6.52
6.46
6.37
6.27
6.16
6.15
6.11
6.10
6.09
6.09
6.09
6.13
6.10
5.98
5.87
5.85
5.91
Here are the values of the annualized, real total return minus the multiple expansion percentage after 60 years from 1921-1940.

Code: Select all

5.97
6.08
6.16
6.15
6.10
6.10
6.17
6.22
6.21
6.10
5.93
5.66
5.53
5.55
5.60
5.68
5.76
5.74
5.81
5.90
Have fun.

John R.
JWR1945
***** Legend
Posts: 1697
Joined: Tue Nov 26, 2002 3:59 am
Location: Crestview, Florida

Post by JWR1945 »

Here are some tables related to Mike's comments. I extracted them from some of Professor Shiller's tables that I had thinned down to include only January data. They cover the years 1871-2000 or 1881-2000, as applicable. They include the nominal and real prices for the S&P500 index, real dividend amounts, real earnings, averaged real earnings E10 and P/E10.

I have made some curve fits [using 1880-2000 data]. The year is the value of x and it is very large. The values of y are in real dollars except for P/E10, which is dimensionless.

For the real dividend amount, y = 0.0829x-151.99 and R Squared is 0.6523.
For the real (single year) earnings, y = 0.2143x-399.75 and R Squared is 0.6553.
For the ten year average of real earnings E10, y = 0.205x-382.78 and R Squared is 0.8334.

It is possible to make an exponential fit to the E10 data. The equation is y = (3E-11)*exp(0.0137) and R Squared is improved slightly at 0.873.
It was not possible to make an exponential fit to either the dividend data or the earnings data because I entered dummy values of $0.00 for 2000. [My error.]

I made linear and exponential curve fits of P/E10, but they were very poor. Their R Squared values were 0.0293 and 0.0091, respectively.

These are the nominal prices of the S&P500 index from 1871-1920.

Code: Select all

4.44
4.86
5.11
4.66
4.54
4.46
3.55
3.25
3.58
5.11
6.19
5.92
5.81
5.18
4.24
5.20
5.58
5.31
5.24
5.38
4.84
5.51
5.61
4.32
4.25
4.27
4.22
4.88
6.08
6.10
7.07
8.12
8.46
6.68
8.43
9.87
9.56
6.85
9.06
10.08
9.27
9.12
9.30
8.37
7.48
9.33
9.57
7.21
7.85
8.83
These are the nominal prices of the S&P500 index from 1921-2000.

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7.11
7.30
8.90
8.83
10.58
12.65
13.40
17.53
24.86
21.71
15.98
8.30
7.09
10.54
9.26
13.76
17.59
11.31
12.50
12.30
10.55
8.93
10.09
11.85
13.49
18.02
15.21
14.83
15.36
16.88
21.21
24.19
26.18
25.46
35.60
44.15
45.43
41.12
55.62
58.03
59.72
69.07
65.06
76.45
86.12
93.32
84.45
95.04
102.00
90.31
93.49
103.30
118.40
96.11
72.56
96.86
103.80
90.25
99.71
110.90
133.00
117.30
144.30
166.40
171.60
208.20
264.50
250.50
285.40
339.97
325.49
416.08
435.23
472.99
465.25
614.42
766.22
963.36
1248.77
1425.59
These are the real prices of the S&P500 index from 1871-1920.

Code: Select all

60.13
64.83
66.66
63.60
66.57
69.41
54.77
59.44
73.00
86.34
110.93
98.16
98.17
94.74
86.46
109.83
117.85
107.05
110.67
119.31
104.72
126.95
119.91
106.45
109.28
108.22
110.10
123.68
151.92
130.39
154.85
173.56
164.93
136.22
168.04
196.75
182.37
133.55
171.00
171.95
169.55
168.54
160.19
141.29
125.01
151.43
138.07
86.93
80.31
77.23
These are the real prices of the S&P500 index from 1921-2000.

Code: Select all

63.17
72.91
89.42
86.16
103.23
119.29
129.25
171.04
245.40
214.31
169.65
97.97
92.77
134.78
114.93
168.31
210.58
134.45
150.71
149.37
126.30
96.01
100.78
114.96
127.93
167.13
119.42
105.62
108.03
121.25
140.95
154.09
166.13
159.76
225.07
278.08
277.85
242.69
323.75
334.32
338.28
388.63
361.25
417.63
465.93
495.36
433.29
470.46
483.64
403.29
396.51
424.26
469.15
348.14
235.09
294.06
299.51
243.75
246.43
240.62
258.05
209.97
249.06
275.65
274.56
320.66
401.51
365.47
397.82
450.45
408.19
508.58
515.20
546.11
522.52
671.72
812.93
1006.28
1282.97
1425.59
These are the real dividend amounts of the S&P500 index from 1871-1920.

Code: Select all

3.52
3.51
3.95
4.50
4.80
4.67
4.49
3.46
3.71
3.46
4.75
5.31
5.42
6.00
6.20
5.03
4.70
5.01
4.84
4.88
4.76
5.11
5.15
6.08
5.36
4.80
4.70
4.61
5.02
4.65
6.61
6.86
6.47
7.07
6.21
6.69
7.69
8.51
7.61
7.55
8.60
8.70
8.27
8.02
7.03
7.15
8.24
8.20
5.80
4.62
These are the real dividend amounts of the S&P500 index from 1921-2000.

Code: Select all

4.49
4.64
5.14
5.19
5.41
5.73
6.72
7.58
8.49
9.58
10.26
9.36
6.48
5.64
5.59
5.87
8.74
9.43
6.19
7.57
8.06
7.56
5.89
5.95
6.10
6.18
5.60
6.01
6.66
8.26
9.88
9.00
8.95
9.14
9.78
10.52
10.62
10.53
10.23
10.75
11.03
11.40
11.86
12.55
13.62
14.54
14.78
14.50
14.60
14.13
13.27
12.61
12.51
12.32
11.74
11.18
11.82
12.73
12.64
12.37
12.03
11.92
11.88
11.79
12.12
12.23
12.60
12.92
13.68
14.76
15.18
14.96
14.69
14.57
14.80
15.19
15.86
16.24
16.73
0.00
These are the real (single year) earnings of the S&P500 index from 1871-1920.

Code: Select all

5.42
5.37
5.64
6.28
6.62
5.50
4.35
5.50
6.44
6.58
8.71
7.28
7.22
7.18
6.25
5.81
7.02
7.09
5.56
6.64
6.37
7.89
7.71
6.20
4.31
6.25
5.70
7.94
9.02
10.26
10.55
10.92
12.12
10.74
10.07
13.51
14.34
12.74
11.23
12.92
13.14
11.07
11.96
10.48
9.19
15.16
21.77
15.14
10.08
8.04
These are the real (single year) earnings of the S&P500 index from 1921-2000.

Code: Select all

6.73
3.23
7.18
9.52
9.33
11.78
11.85
11.05
13.81
15.37
9.98
7.00
5.40
5.68
7.07
9.42
12.57
12.80
8.00
11.29
12.61
12.04
10.42
9.09
8.91
8.72
8.87
11.70
16.32
16.78
18.85
15.46
15.29
15.83
17.91
22.95
20.83
19.44
17.25
19.53
18.18
18.29
20.45
22.25
24.85
27.81
28.30
26.57
27.41
25.59
21.88
23.56
25.94
29.80
28.33
24.87
28.76
29.44
31.27
32.55
28.60
27.17
21.69
23.92
26.49
22.46
22.29
26.06
33.68
29.80
26.57
19.61
22.89
25.58
35.10
37.16
41.63
41.43
38.97
0.00
These are the real ten-year averages of earnings of the S&P500 index from 1881-1920.

Code: Select all

5.77
6.10
6.29
6.45
6.54
6.50
6.53
6.80
6.96
6.87
6.88
6.64
6.70
6.75
6.65
6.46
6.50
6.37
6.46
6.80
7.16
7.58
7.89
8.33
8.78
9.36
10.08
10.95
11.43
11.65
11.91
12.17
12.19
12.17
12.14
12.06
12.22
12.97
13.21
13.09
These are the real ten-year averages of earnings of the S&P500 index from 1921-2000.

Code: Select all

12.60
11.96
11.18
10.70
10.60
10.62
10.28
9.29
8.88
9.25
9.99
10.31
10.69
10.51
10.13
9.90
9.66
9.74
9.91
9.33
8.92
9.18
9.69
10.19
10.53
10.72
10.65
10.28
10.17
11.00
11.55
12.17
12.51
13.00
13.67
14.57
16.00
17.19
17.97
18.06
18.33
18.27
18.55
19.07
19.71
20.40
20.89
21.64
22.35
23.36
23.97
24.34
24.87
25.42
26.17
26.52
26.23
26.27
26.56
26.94
27.64
28.31
28.67
28.25
27.66
27.48
27.23
26.59
26.25
26.49
26.22
26.01
25.26
25.38
25.54
26.40
27.87
29.81
31.34
31.87
These are the values of P/E10 of the S&P500 index from 1881-1920.

Code: Select all

19.23
16.10
15.61
14.69
13.23
16.89
18.04
15.74
15.91
17.37
15.23
19.11
17.89
15.77
16.42
16.75
16.93
19.41
23.53
19.17
21.61
22.89
20.92
16.36
19.14
21.03
18.09
12.20
14.97
14.76
14.23
13.85
13.14
11.61
10.29
12.56
11.30
6.70
6.08
5.90
These are the values of P/E10 of the S&P500 index from 1921-2000.

Code: Select all

5.01
6.10
8.00
8.05
9.73
11.23
12.57
18.41
27.64
23.16
16.99
9.50
8.68
12.82
11.35
17.00
21.79
13.81
15.21
16.01
14.16
10.45
10.40
11.28
12.15
15.60
11.22
10.28
10.63
11.02
12.21
12.66
13.28
12.29
16.46
19.08
17.37
14.12
18.02
18.51
18.45
21.27
19.47
21.90
23.64
24.28
20.74
21.74
21.64
17.26
16.54
17.43
18.87
13.70
8.98
11.09
11.42
9.28
9.28
8.93
9.34
7.42
8.69
9.76
9.93
11.67
14.74
13.75
15.15
17.00
15.57
19.55
20.40
21.52
20.46
25.44
29.16
33.76
40.93
44.73
Have fun.

John R.
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