*Photo by Standard Travel Fotos*

In the last article we evaluated the CER (Constant Expected Return) Model. In this article, we evaluate the CAPE Model.

**CAPE Model**

CAPE, Cyclically-Adjusted Price Earnings Ratio, is the P/E developed by Robert Shiller of Yale University. CAPE uses the average of the past ten years worth of earnings, P/E10. The forecasting model that we will evaluate actually uses the inverse E10/P, which is the smoothed earnings yield.

This webspage shows the latest value of CAPE: Shiller PE Ratio. For example, when this article was written, P/E10 = 25.22 and the the inverse E10/P = 0.0396511.

The CAPE Model accounts for valuations when predicting stock market returns. The earnings yield varies over time, so the expected return of stocks varies over time. The idea that expected returns, or discount rates, vary over time is more in line with current thinking by financial economists than the Constant Expected Return model. See Presidential Address: Discount Rates by Professor John H. Cochrane.

**Validating the CAPE Model**

The model is typically presented as the statement: *the long-term expected real return of the stock market is the smoothed earnings yield. *

Applying the six grading rubrics:

- The CAPE model makes a prediction of the discount rate. 1 pt.
- The prediction does NOT specify a definite time frame. 0 pts.
- There is NO measure of uncertainty or precision. 0 pts.
- It does not use future data for earlier predictions, 1 pts.
- The prediction is reproducible. 1 pt.
- No computational errors. 1 pt.

The total score is only 4 of 6 points, so the CAPE model as typically stated fails validation. What is lacking is a definite time frame and a level of precision.

We will add time frame and precision to make a valid model. To start, choose a time frame of 20 years and add error bars.

**Measuring CAPE Model Performance**

The CAPE model was coded up in R and the tests run.

**20-Year Forecast Performance**

For 20-year forecasts, we chose +/-4 percent prediction interval.

*Measured Accuracy*

`Number of Samples = 786 , Trueness = 0.800452 , Precision = 1.64144`

*Measured Confidence Level*

`At interval width 8 , passed 412 out of 424 tests. Confidence Level 97.1698 per cent`

*Figure of Merit*

`Confidence Level = 97.1698 , Interval Width = 8 , F.O.M = 12.1462`

A plot of showing predicted and actual 20-year returns with lower and upper error bars for the period 1950-2014

Over 20-year periods, 97% of the outcomes fell within the +/-4% prediction interval. The performance is fairly good considering the great uncertainty in stock returns. The good performance is reflected in the Figure-of-Merit of 12.15.

**10-Year Forecast Performance**

Since no time frame is typically specified for CER Model, we’ll test both 10 years and 30 years.

For 10-year forecasts, we kept the same +/-4 percent prediction interval.

*Measured Accuracy*

`Number of Samples = 786 , Trueness = 0.956727 , Precision = 4.49009`

*Measured Confidence Level*

`At interval width 8 , passed 317 out of 544 tests. Confidence Level 58.2721 per cent`

*Figure of Merit*

`Confidence Level = 58.2721 , Interval Width = 8 , F.O.M = 7.28401`

A plot showing predicted and actual 10-year returns with lower and upper error bars for the period 1950-2014

The 10-year forecast is much worse than the 20-year forecast. Only 58% of the outcomes fall with in the prediction interval. Many outcomes were far below the prediction. For forecasts made in 1990, many outcomes far exceeded the predictions thanks to the stock market bubble of the late 1990s.

The lower Figure-of-Merit=7.28 shows the poor performance of the CAPE Model over 10-year periods.

**30-Year Forecast Performance**

For the 30-year forecast, we used the same +/- 4 percent error bars.

*Measured Accuracy*

`Number of Samples = 786 , Trueness = 0.949083 , Precision = 2.15035`

*Measured Confidence Level*

`At interval width 8 , passed 268 out of 304 tests. Confidence Level 88.1579 per cent`

*Figure of Merit*

`Confidence Level = 88.1579 , Interval Width = 8 , F.O.M = 11.0197`

A plot showing predicted and actual 30-year returns with lower and upper error bars for the period 1950-2014.

The CAPE Model performs worse over 30-year periods than over 20 year periods. Only 88% of the outcomes fell in the +/-4 percent prediction interval.

**Summary and Conclusion**

The CAPE Forecasting Model was evaluated. The model performed poorly over 10-year periods, but performance improved over 20-year periods. Over 30-year periods, the forecasting model was less successful.

It would appear that taking into account valuations works best at around a 20-year horizon. At shorter periods, the outcomes can vary widely; at longer periods the outcome is less dependent on the initial earnings yield.

*Applying the CAPE model to today’s valuation: P/E10 = 25.22, E10/P = 3.97%. With about 97% confidence, CAPE forecasts a 20-year real return of stocks from about 0% to 8% per annum.*