Decomposition of CAPE variable into frequency components and stock market prediction

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Introduction

In the 1990s Robert Shiller, a 2013 Nobel laureate in economics, and John Campbell developed the “cyclically adjusted price-earnings ratio”, or CAPE (Shiller, 2000). One of the most valuable insights which this ratio brought to the academic field is that it is supposed to be a good predictor of future stock market returns. R. Shiller and J. Campbell investigated data going back to 1872 and found that when CAPE was high, future trend on the stock market tended to be low, and vice versa.

The CAPE ratio is a better measure of publicly held company's long-term financial performance than the normal P/E ratio as it considers the effects of the economic cycles on the company's earnings. CAPE ratio divides the current price by inflation-adjusted earnings. It differs from the normal P/E ratio by the inflation component in the formula. Average inflation-adjusted profits of the previous 10 years are used because it ensures that the earnings are averaged out more than one earnings cycle. Rather than pick the current year's earnings, which may be abnormally high or low, due to a temporary boom or recession, taking an average over the past 10 years, creates a smoother series for earnings. The adjustment on inflation ensures the comparability of profits even at times of high inflation. That is the reason why CAPE is considered a better predictor of long-term trends of the stock market returns. Periods of high CAPE are associated with low future returns for the S&P Composite Index and periods of low CAPE are associated with high future returns (Schiller and Campbell, 1998).

Prediction of stock market returns is an important issue in finance. One of the main questions for the equity investors is how they can most accurately forecast long-term stock market price fluctuations. Based on CAPE's peaks and troughs Robert Shiller often makes predictions that the market is likely to decline or increase substantially in value over the succeeding ten years. These kinds of predictions are very difficult to test on historical data. What is more realistic to test is whether cyclically adjusted earnings to price ratio can predict out-of-sample future equity returns.

The results of previous researches are similar whether normal or cyclically adjusted P/E ratio is used - variables have shown poor historical ability to forecast returns. Campbell and Thompson (2007) investigated the predictability power of cyclically adjusted earnings to price ratio and reported weak OOS performance of this variable - it does not significantly beat a simple forecast based on the historical average of stock returns.

We re-evaluated the out-of-sample forecasting performance of the cyclically adjusted earnings to price ratio by isolating and analyzing the its different frequencies' ERP predictive power - like Faria and Verona (2018) did with the term spread (one of popular ERP fundamental predictors). Considered in the literature frequency decomposition using wavelet approach has become a popular technic to analyze the financial time series. Wavelet transform is a promising signal processing technique that simultaneously analyzes the time domain and frequency domain. Faria and Verona (2018) found that the methodology used to extract the frequencies components of the TMS is useful for the quality of a forecasting exercise. In their paper, they show that the low frequency component of the term spread - extracted through wavelets decomposition - have higher predictability power than the simple TMS (-0.72 - 15.0 R-squares for simple TMS depending on the forecasting period compared to 2.09 - 31.9 R-squares for the low frequency TMS component. Phikhodko (2018) in his master thesis also decomposed E/P and D/P ratios into different frequency components to analyze whether they can forecast stock market return.

Similarly, we build a model with different cyclically adjusted earnings to price ratio frequency components to test its ERP forecasting power. The main hypothesis is that some of the components might be able to be good predictors of the EPR. We are building this model for different forecasting periods to test whether some components of the cyclically adjusted earnings to price ratio (CAPE_inv - multiplicative inverse CAPE) predict long-term returns better than short-term returns, and vice versa. Faria and Verona (2018) provided evidence of this theory. They found that the forecasting power of the ERP (versus the historical mean benchmark) increases as the forecasting horizon increases. CAPE reflects and, supposedly, signalize about possible equity market peaks and troughs in long terms. Another advantage of the CAPE_inv and its components obtained from wavelets decomposition is that this parameters are easy to monitor and to compute from publicly available data.

1. Data and methodology

We use monthly data from January 1977 to December 2018 and the main data sources are the Robert Shiller's website (S&P 500 price and Cyclically Adjusted Price Earnings Ratio) and Federal Reserve Economic Data (FRED) database (for treasury rates with different maturities).

The ERP was estimated as a difference between log monthly returns of the S&P 500 index and monthly treasury bills rates with the maturity which equals the forecasting periods (1 month, 3 months, 6 months, 1 year, 2 years). It is important to take respectful risk-free rates as we are willing to get a robust forecast for different forecasting periods. Using different risk-free rates allows us to make the results of the predictability power comparable.

As a predictor we use the components of CAPE_inv. CAPE_inv is calculated as a ratio of the monthly real (corrected for inflation using the Consumer Price Index) S&P 500 earnings and S&P 500 Price Index.

As a predictand we take the average monthly ERP on the forecasting period. Firstly, it is consistent with Faria and Verona (2017). Secondly, if we just take monthly ERP (not the average), the model will automatically be better on the longer forecasting periods.

1.1 The predictors

We are estimating the predictability power of CAPE_inv frequency components on the ERP. Wavelets allow to define over a finite window in the time domain according to the frequency which is best suited for the aims of the research. We perform the frequency decomposition of CAPE using multiresolution analysis (MRA) based on maximal overlap discrete wavelet transform method (MODWT). It decomposes the time series into a sum of simpler time series, named Smooth and Detail . Being X(t) a time series, it can be written as:

where is the wavelets details and is the wavelet smooth.

We got components from the original time series of CAPE_inv. For small j, the wavelets components represent higher frequency, or short-term, dynamics and vice versa. The wavelet smooth component represents the trend (or long-term dynamics of the variable). Sum of all the components equals the initial variable value.

We choose to perform a =6 level MODWT on the CAPE series using Haar wavelet filter.

We apply MODWT-based multiresolutional analysis to the CAPE time series. The original data series is decomposed using Haar wavelet transform up to 6 levels in consistency with Faria and Verona (2017). They showed that the concrete methodology used to extract the component is relevant for ERP forecasting.

In accordance with Faria and Verona (2017), we call the high frequency CAPE_inv component the sum of components which captures oscillations of the short term CAPE_inv trend - between 3 and 16 months (as , , captures 2-4, 4-8, 8-16 month oscillations, respectively):

The business cycle frequency component captures oscillations of the CAPE_inv ratio with a period of 16-128 month (, , corresponds to 16-32, 32-64, 64-128 months, respectively).

The low frequency component captures oscillations of the CAPE_inv ratio with a period of more than 128 months. It corresponds exactly to the smooth component.

This decomposition reveals the hidden dynamics of the different CAPE_inv frequency components' volatility in time - Figure 1. As expected, the lower the frequency, the smoother the component through the time.

1.2 Out-of-sample OLS regression

The aim of the article is to build a model to test the out-of-sample (OOS) performance in predicting the ERP. We are building the forecasting model based on the method which Welch and Goyal (2007) applied to various variables to test whether they are good or poor predictors of the equity premium. They investigated the OOS performance of linear regressions that predict the equity premium with prominent fundamental variables from earlier academic research. The OOS performance is not only a useful model to investigate the IS regressions with stable results, but is also interesting itself for an investor who is looking for a good predictive method to use as a trading strategy (Welch and Goyal, 2007). The bad OOS is enough to say that the model is bad for the aims mentioned above. And vice versa, the OOS model might perform well independently from the IS model - the source of strong OOS correlation of two parameters might not be economically sound and sensible and, therefore, the IS regression is acceptable to be statistically insignificant. The OOS forecast is based on the OLS regression and uses the data available up to the time at which the forecast is made - using a sequence of expanding windows (to be consistent with Faria and Verona, 2018).

Each step of the OOS regression is can be written as:

 (1)

where is average monthly ERP for the period from to , is the first period of the test sample (different for each prediction).

Each prediction of the ERP is can be written as:

(2)

where is predicted monthly average return for the period from to ( is the forecasting horizon), and are the OLS estimated coefficients in step (1) on the training dataset with expanding window.

The initial sample to test the model starts from January 1997 to December 1989 and then expands up to December 2018 by one month for each forecast. The data starts in January 1977 due to the treasury rates data limit. This estimation period is considered to be enough. Firstly, we would like to obtain results that remain relevant today. Secondly, the sample size is close to what Faria and Verona (2018) and Phikhodko (2018) used in their studies.

The predictive power is measured using the OOS (Faria and Verona, 2018), which can be written as:

where is the fitted value from a predictive regression, and is the historical average return. If the is positive, then the predictive regression has a lower average mean squared prediction error than the historical average return (HM).

To investigate the predictability power we would analyze not only the total statistics. We would also look at the floating statistic with 15 and 30 months rolling windows and the trend of the cumulative . Investigating the time series pattern allows diagnosis of periods with good or bad performance of the model. When the line is above zero, the prediction based on CAPE is better than the historical mean, and visa versa - historical mean performs better than the model when the line is below zero. The results would be of most interest for potential investors if the floating was positive over the most recent several years.

We build the model for different forecasting periods from being equal to 1 month to 24 months. Some models might predict long-term returns better than short-term returns, and vise versa. Due to its nature, the CAPE ratio is supposed to be a good long-term predictor of the future stock returns, therefore, we would expect better results in on higher forecasting periods.

2. Results

Table 1 contains estimations of the OOS regression statistics. for frequency components are all negative, which means the CAPE_inv_lf, CAPE_inv_bcf, CAPE_inv_hf are performing worse than the historical mean through the whole test period (1990.01 - 2018.12) cumulatively. Some of them are substantially negative with values being less than -100%. We might have expected this result because the cyclically adjusted earnings to price ratio itself has a poor predictive power (Campbell and Thompson, 2007).

To investigate the model performance through the time, we can look at the rolling and cumulative for each of the components and forecasting horizon on the figures 2-15. There are several interesting conclusions we can make from further review of the graphs.

Firstly, low and business cycle components on 1, 3, 6 months forecasting periods represent the highest total R-squares comparing to other models. If we investigate the movements of its cumulative and floating R-squares, we can see that the components perform poor cumulatively (below zero) during almost all the test sample except for the 1995-2001 period, where it could bring investors positive profits. The floating R-squares significantly outperforms the HM on some periods but this trend does not last longer than 1-3 years.

Secondly, we see that the low frequency component, which has not the best R-squares statistics on the 12 and 24 months forecasting periods, have positive cumulative R-squares during almost all the test sample until 2008. The model performs significantly worse during the crisis in the test sample. The rolling fell on tens and even hundreds of percent in the result of both tech bubble and the world financial crisis. It starts to decrease significantly in 1999 and 2008 and is returning back to the previous positive trend during 5-7 years. However, the first crisis does not bring the cumulative statistics below zero and it remains to be positive until the training period of the forecast does not include data from the 2008-2009 period. On 2008 the cumulative R-squares statistics fells on the levels below zero and never becomes positive again. That means that the variable cannot predict the ERP well under quite unusual periods of time such as the world financial crisis. However, in fact, it might be called the best result among all other models because it means that before the 2008 an investor could earn positive results on the trading strategy based on the low frequency CAPE_inv component.

We decided to investigate this particular two models - low frequency components predictability power on the 12- and 24-month forecasting periods - further. We recalculated total R-squares without the errors which the model generated during the crisis years - 2008 and 2009. We continue to use these years in the training set. The component CAPE_inv_lf* generates positive total R-squares statistics on both forecasting periods (Figure 1). It performs better on the longest forecasting period.

Conclusion

Our primary finding is that all the frequency components of the cyclically adjusted earnings to price ratio have shown some poor historical ability to forecast both short run and long run returns. Even though the model generates positive cumulative and floating R-squares during some periods, the conclusion remains the same. Our aim was to test whether we could rely on some parameter to generate positive returns on the stock market in the future. That is why we were looking for sustainable results which remain positive through the whole sample. The best result which the frequency components of the cyclically adjusted earnings to price ratio can generate - the low frequency component on the long terms would bring investor positive results if he would not use this strategy during the two years of the world financial crisis. We still consider it a poor result for a potential investor's interest as nobody could predict such recessions in advance.

Literature

1. Campbell, J. Y., & Thompson, S. B. (2007). Predicting excess stock returns out of sample: Can anything beat the historical average?. The Review of Financial Studies, 21(4), 1509-1531.

2. Faria, G., & Verona, F. (2018). The equity risk premium and the low frequency of the term spread.

3. Prikhodko, L. (2018). Frequency decomposition of E/P and D/P and stock market return predictability. New Economic School

4. Schiller, R., & Campbell, J. (1998). Valuation ratios and the long-run stock market outlook. National bureau of economic research.

5. Shiller, R. C. (2000). Irrational exuberance. Philosophy and Public Policy Quarterly, 20(1), 18-23.

6. Welch, I., & Goyal, A. (2007). A comprehensive look at the empirical performance of equity premium prediction. The Review of Financial Studies, 21(4), 1455-1508.

Appendix

Table 1. Out-of-sample R-squares

Figure 1. CAPE_inv and its frequency components time series from 1977.01 to 2018.12

Figure 2. Rolling and cumulative R-square CAPE_inv low frequency component with 1 months forecasting horizon

Figure 3. Rolling and cumulative R-square CAPE_inv business cycle frequency component with 1 months forecasting horizon

Figure 4. Rolling and cumulative R-square CAPE_inv high frequency component with 1 months forecasting horizon

Figure 4. Rolling and cumulative R-square CAPE_inv low frequency component with 3 months forecasting horizon

Figure 5. Rolling and cumulative R-square CAPE_inv business cycle frequency component with 3 months forecasting horizon

Figure 6. Rolling and cumulative R-square CAPE_inv high frequency component with 3 months forecasting horizon

financial forecasting price

Figure 7. Rolling and cumulative R-square CAPE_inv low frequency component with 6 months forecasting horizon

Figure 8. Rolling and cumulative R-square CAPE_inv business cycle frequency component with 6 months forecasting horizon

Figure 9. Rolling and cumulative R-square CAPE_inv high frequency component with 6 months forecasting horizon

Figure 10. Rolling and cumulative R-square CAPE_inv low frequency component with 12 months forecasting horizon

Figure 11. Rolling and cumulative R-square CAPE_inv business cycle frequency component with 12 months forecasting horizon

Figure 12. Rolling and cumulative R-square CAPE_inv high frequency component with 12 months forecasting horizon

Figure 13. Rolling and cumulative R-square CAPE_inv low frequency component with 24 months forecasting horizon

Figure 14. Rolling and cumulative R-square CAPE_inv business cycle frequency component with 24 months forecasting horizon

Figure 15. Rolling and cumulative R-square CAPE_inv high frequency component with 24 months forecasting horizon

Figure 16. Rolling and cumulative R-square CAPE_inv low frequency component with 12 months forecasting horizon (omitted errors from 2008.01 to 2009.12)

Figure 17. Rolling and cumulative R-square CAPE_inv low frequency component with 24 months forecasting horizon (omitted errors from 2008.01 to 2009.12)

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