Longer Sequences

This time, I examine longer sequences. I compare 1-year, 2-year and 4-year sequences covering 1921-1980.

I find that 2-year sequences make an excellent choice.

I have also discovered that these sequences have a predictive time frame of the order of 2 to 4 years. That is, creating 2500 simulated years of data does not narrow down the confidence limits more than what is already inherent in the data.

The Sequences

To construct sequences, I start by listing monthly returns. Then I select a month, typically January 1921, for beginning a group of sequences. I associate the first month’s P/E10 with the following 11 months to form a 1-year sequence, 23 months to form a 2-year sequence or 47 months to form a 4-year sequence. I repeat this process until I cover all of the sequences starting in 1921-1980.

Single-year (1-year) January sequences begin in January 1921, January 1922 and so forth until January 1980. Two-year January sequences begin in January 1921, January 1923 and so forth until January 1979. Four-year January sequences begin in January 1921, January 1925 and so forth until January 1978.

I also constructed a group of two-year sequences based upon the January 1920 value of P/E10. I used this value for all of 1921. I used the P/E10 value of January 1922 for the sequence, from January 1922 through December 1923. I used the P/E10 value from January 1924 for the sequence from January 1924 through December 1925. And so forth.

I truncated all sequences at December 1980. I only used the first year of the January 1980 two-year sequence (from the group that started with the January 1920 value of P/E10). I only used the first two years of the January 1978 four-year sequence. That is, I truncated part of these two-year and four-year sequences.

Data Collection

I collected an extensive amount of data. I used the Forsey-Sortino model. It comes with the book, Managing Downside Risk in Financial Markets, by Frank Sortino and Stephen Satchell. Their methods are clearly superior to and much more advanced than Mean-Variance Optimization.

I determined the mean, mean plus and minus one standard deviation, probability of exceeding the Minimum Acceptable Return (MAR) for levels of 0% and 2% (approximately), below target deviation, upside potential and upside ratio for all individual segments.

I have placed these data into my Yahoo Briefcase under the folder for Current Research E and the file 1921-1980 2-yr 4-yr Graphs D7.

Yahoo Briefcase

Low, Medium, High Results

I ordered the data by P/E10. Then I broke them into three equal parts to make three portfolios for the Forsey-Sortino model. I present both old and new results for making comparisons.

January Single-Year Ordering

Low P/E10: -5.8%, 17.4%, and 40.5%
Middle P/E10: -10.5%, 6.7%, and 23.9%
High P/E10: -11.4%, 3.9%, and 19.3%

Low P/E10: 5.12 to 11.44
Middle P/E10: 11.47 to 16.72
High P/E10: 17.09 to 27.08

January Two-Year Ordering (original)

The January 1921 P/E10 value was used for the first sequence.

Low P/E10: -0.9%, 15.4%, and 31.8%
Middle P/E10: -10.5%, 12.7%, and 35.8%
High P/E10: -16.8%, -1.0%, and 14.7%

Low P/E10: 5.12 to 11.47
Middle P/E10: 11.50 to 16.71
High P/E10: 16.72 to 27.08

January Two-Year Ordering (offset, new)

The January 1920 P/E10 value was used for the first sequence.

Low P/E10: -5.9%, 17.1% and 40.1%
Middle P/E10: -8.5%, 9.2% and 26.9%
High P/E10: -14.0%, 1.5%, and 17.0%

Low P/E10: 5.99 to 11.19
Middle P/E10: 11.34 to 17.09
High P/E10: 17.09 to 24.06

January Four-Year Ordering (new)

Low P/E10: 4.0%, 20.4% and 36.7%
Middle P/E10: -2.9%, 10.1% and 23.0%
High P/E10: -24.7%, -3.5%% and 17.7%

Low P/E10: 5.12 to 11.44
Middle P/E10: 11.96 to 18.47
High P/E10: 18.71 to 27.08

Low, Medium, High Comparisons

Previously, we had found that single-year and two-year sequences behaved similarly at low and high values of P/E10. Two-year sequences performed better at middle level valuations. The lowest values of P/E10 always led to the best returns. The highest values of P/E10 always led to the worst returns.

The two-year sequence retained these features when using the offset. That is, ordering in two-year increments starting with the January 1920 P/E10 value produces results similar to those starting with the January 1921 P/E10 value.

The four-year sequences behaved similarly to the two-year sequences.

Detailed Two-Year and Four-Year High Results

I plotted Means and the Probability>MAR for a Minimum Acceptable Return (MAR) of approximately 0% versus the percentage earnings yield 100E10/P of each segment. I plotted straight-line graphs using Excel.

Two-Year 1921-1980 Sequences (original)

Two-year P/E10 groupings of 1921-1980 data:
Portfolios: 2BJanL1 to 2BJanL10.
y = 2.7924x - 12.552 plus and minus 10%.
R-squared = 0.6545.
When x = 4% (P/E10 = 25), y = -1.38%.
When x = 10% (P/E10 = 10), y = 15.37%.

There was saturation of the Probability of exceeding the Minimum Acceptable Return MAR at an 8% earnings yield. The data were (roughly) flat above 8% (P/E10 = 12.5). They (roughly) fit a straight-line at lower percentage earnings yields.

Four-Year 1921-1980 Sequences (new)

This is the formula of the Mean return y as a function of x, the percentage earnings yield 100E10/P:
Portfolios: 4F21Jan1 to 4F21Jan10.
y = 2.8937x - 12.678 plus and minus 10%.
R-squared = 0.655.
When x = 4% (P/E10 = 25), y = -1.10%.
When x = 10% (P/E10 = 10), y = 16.26%.

Break point in MAR data is around 100E10/P = 9% or P/E10 = 11.

Detailed Two-Year and Four-Year High Comparisons

The results are very similar. The formulas are almost identical.

Comparison with 10-Year Stock Returns

From You Can't Count on 7%:

This is the regression equation for the 10-year stock return and the percentage earnings yield 100E10/P (using 1923-1972 data): y = 1.5247x-4.5509 where y is the annualized real return in percent and x is 100E10/P or 100/[P/E10]. The confidence limits are plus and minus 6%.

Comparisons

Ten-year confidence limits taken directly from the stock market are to plus and minus 6%. All of these single-year, two-year and four-year confidence limits are plus and minus 10% (very roughly).

The formulas differ significantly.

Assessment

These (single-year, two-year and four-year) data sets predict time intervals of 2 to 4 years. (If the variance obeyed a 1/N law, where N is the number of years, then the time interval would have been 2.79 years. It does not. But 1/N is OK as a first approximation.)

Conclusions

Two-year sequences make an excellent choice.

These sequences have a predictive time frame of the order of 2 to 4 years. They do not narrow down the confidence limits more than what is already inherent in the data.

This is consistent with our single-month findings. That is, using single-month data directly was appropriate for single-month predictions. Using single-year sequences is appropriate for single-year predictions. Using two-year and four-year sequences is appropriate for two to four year predictions.

Have fun.

John Walter Russell
December 18, 2005