January 19, 2010
Completing the model requires another equation to generate the end-of-year PE's. As one might expect, each annual change in PE is closely related to that year's rate of return. Based on this relationship and the historic data, I developed the following equation to generate the ending PE's:
PE(t+1) = PE(t)*(1+.90*Return(t)-.084+N(0,.07))
These equations can be combined in an Excel model, and repeated simulations can be run using Excel’s random number generator.
Table 1 provides a comparison of actual versus test results. Each of the tests involved 10 runs of 81-year (1928 – 2008) versions of the model, so the average of all tests is based on 50 simulations. Overall, the model produced a good fit with the actual data. The average PE from the model came in slightly higher than the actual average PE, but it's unclear whether we need to fine-tune the model or accept that the actual numbers will contain some statistical variation. (The actual historic numbers are really just a sample, and may or may not represent a best estimate of numbers going forward.)
What is perhaps most interesting in the comparison is that the correlations and R-squares are higher for the actual than for the average of the tests. My guess is that there are likely closer ties between PE's and subsequent performance than are captured by my simple model, but I have no way of proving this. In general, it looks like this model provides a good start toward building a workable simulation model to show the effect of PE's on performance.
Table 2 provides a comparison of test results with the asset allocation results I developed in the first article referenced above. Based on a small sample of just 10 cases, the results for the test cases are similar to the actuals, although the actuals show a bigger impact from incorporating PE's in the asset allocation. This result is consistent with the lower correlations for the test results shown in Table 1.
The behavioral influence on stock prices has been described in the popular press as an excess of greed when stock prices are high, and an excess of fear when prices are low. The result is stock price movements that do not follow a random walk. This model also rejects the random walk assumption, but it is based on a different characterization of investor behavior – some fear (but not enough) when prices are high, and some greed (but not enough) when prices are low. To the extent we can improve our understanding of investor behavior and build simulation models that reflect its effect on investment returns, we can provide more useful tools for financial planners and their clients.
Table 1 - Shiller PE's and Returns--Actual versus Test Results
Averages |
PE |
Return |
Std Dev of Return |
Lowest PE |
Highest PE |
Correlation PE and 10-Yr Returns |
R-Square |
Test #1 |
19.46 |
10.74% |
20.47% |
7.9 |
43.2 |
-0.57 |
0.34 |
Test #2 |
16.44 |
11.21% |
21.09% |
6.1 |
33.8 |
-0.53 |
0.32 |
Test #3 |
18.15 |
11.07% |
20.84% |
7.1 |
40.2 |
-0.64 |
0.42 |
Test #4 |
16.32 |
11.41% |
19.87% |
6.8 |
33.8 |
-0.61 |
0.39 |
Test #5 |
21.08 |
11.42% |
20.41% |
7.2 |
44.9 |
-0.61 |
0.38 |
Average All Tests |
18.29 |
11.17% |
20.54% |
7.0 |
39.2 |
-0.59 |
0.37 |
Actual 1928-2008 |
17.37 |
11.09% |
20.39% |
7.4 |
42.5 |
-0.69 |
0.47 |
Table 2 - Asset Allocation Results--Test versus Actual
|
50/50 Bond/Stock Buy and Hold |
50/50 Annual Rebalance |
PE-Based Allocation Buy and Hold |
PE-Based Allocation Annual Rebalance |
Average of 10 test Cases |
7.84% |
7.99% |
8.40% |
8.71% |
Difference vs. 50/50 Buy and Hold |
0.00% |
0.15% |
0.56% |
0.87% |
Based on Actual 1928-2008 |
8.30% |
8.41% |
9.45% |
9.65% |
Difference vs. 50/50 Buy and Hold |
0.00% |
0.11% |
1.15% |
1.35% |
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