Road-testing the "Fed" model of valuing US stocks

• Realised equity returns;
• Dividends;
• Earnings yields;
• Surveyed returns; and
• Modelled returns.

The "Fed" model relies on the earnings yield, which is the inverse of the price-earnings ratio. The earnings yield can be calculated using either actual ("trailing") or, more usually, forecast ("forward") earnings, where there is a long history of actual earnings and a short history of forecast earnings.   

As for the risk-free rate, it is approximated by either the 3-month government bill yield or, more commonly, the 10-year government bond yield. 

However, the decision whether to use the nominal or real risk-free rate depends on the choice of expected equity returns.  

For example, an equity risk premium based on surveyed investor expected returns would be matched with a nominal risk-free rate.    

In the case of the "Fed" model, the earnings yield is a real concept, which means that a real government bond yield should be used, even though many market participants rely on a nominal government bond yield.   

There are different ways to demonstrate that the earnings yield is a real concept, but in simple terms the earnings yield equals both nominal earnings divided by the nominal stock price and real earnings divided by the real stock price.[1]  

There is no actual research paper that spells out the "Fed" model, but the common mistake of using the nominal bond yield likely stems from the fact that the Federal Reserve's first reference to this measure of the premium compared the earnings yield with the nominal bond yield. 

In recent years, though, the Federal Reserve has consistently compared the earnings yield with the real bond yield (see, for example, the latest published briefing papers for the Federal Open Markets Committee and the most recent Financial Stability Report).  

The real government bond yield can be measured on either an ex-ante or ex-post basis:

These different measures broadly track each other, but can diverge sharply at times, contributing to the variation in estimates of the equity risk premium.   

The "Fed" model version of the equity risk premium suggests the S&P500 is overvalued.

Using the available data for the S&P500, we constructed a range of earnings yields using both trailing and forward earnings.  

• The trailing earnings yields are highly correlated, but diverge sharply at times, most notably in the global financial crisis.  
• The forward earnings yields cover a 1- and 2-year forecast horizon are also highly correlated, albeit where the history of the 2-year-ahead forecasts is very short. 

Comparing these earnings yields with a range of real government bond yields, the measured equity risk premiums broadly track each other, but there are large and sometimes persistent deviations. 

The other feature of the estimated risk premiums is that they do not quick revert to their historic averages, which means that there are long periods where equities are either overvalued or undervalued relative to bonds.  

The “Fed” model estimates of the equity risk premium currently suggest that:

Judging which of these two valuations to place more weight on, a simple regression suggests that forward earnings are the better choice, which points to stocks being overvalued.  

That is, while forward earnings regularly fail to miss turning points by a large margin, they are better than the current level of earnings in predicting earnings.  

Keeping in mind that the estimated forward earnings premiums can persistently diverge from "fair value" historical averages, this suggests that shares are about 30% overvalued relative to bonds.  

The predictive power of the "Fed" model estimates of the equity risk premium rests with the price-earnings ratio, with a high price-earnings ratio pointing to poor returns over the long term.

Given the forward earnings version of the "Fed" model suggests that stocks are overvalued, we tested  how well it does in forecasting real equity returns compared with the common alternative of price-earnings ratios.    

This horse race involved estimating a set of simple regressions from the early 1990s to now to judge how much of the variation in future equity returns was explained by these different stock market indicators over 1-, 5- and 10-year forecast horizons.     

The results showed that both the equity risk premium and most measures of the price-earnings ratio helped forecast real equity returns over the medium to long term, but performed poorly as short-term valuation tools.  

The Shiller cyclically-adjusted price-earnings ratio (aka CAPE) and the forward price-earnings ratio both explained more of the variation in future returns than the different measures of the the forward earnings risk premium, although the risk premium still did well over the longer term.[2]

Delving deeper to identify what explains the long-term forecasting performance of the forward earnings equity risk premium, we ran an additional set of regressions where the equity risk premium was split into the earnings yield and different measures of the real bond yield.  

This analysis showed that the ability of the risk premium to forecast returns over the long term wholly reflects the earnings yield - or in other words the price-earnings ratio - as the different measures of the real bond yield were not statistically significant.  

This means that the "Fed" model equity risk premium does not possess any predictive power beyond the price-earnings ratio, such that it is more straightforward to use measures of the price-earnings ratio as medium- to long-term valuation tools for the stock market. 

Comparing the various measures of the price-earnings ratio, the most robust long-term valuation measure with the longest history is the Shiller cyclically-adjusted price-earnings ratio.   

This ratio is still extremely high, not far from its highest level in history, which suggests that total real equity returns are likely to be poor over the next ten years.   

The current earnings yield seems too low relative to bond yields and bond/stock volatility. 

While this analysis suggests that real bond yields - as captured in the "Fed" model of the equity risk premium - do not seem to add value in forecasting stock returns over the medium to long term. this leaves open the question of whether bond yields influence current returns.   

This issue was raised in an AQR research note some years ago, where we have replicated their model of the current earnings yield, which was specified as a function of the nominal government bond yield, realised bond volatility, and realised stock volatility.  

The idea behind this model was that a higher bond yield should have a positive relationship with the earnings yield and hence a negative relationship with the price-earnings ratio.

More volatile stock returns also had the same relationship with the earnings yield, while more volatile bond returns would reduce the earnings yield, equating to a higher price-earnings ratio.  

Inverting the model results to derive the price-earnings ratio, it does a reasonable job at estimating the trend in the post-WW2 period.  

At present, this yardstick also suggests that stocks are overvalued, with the earnings yield too low and the price-earnings ratio too high based on current bond yields and bond/stock volatility.  

Notes:

[1] For example, the earnings yield = nominal earnings/nominal price = (real earnings*CPI)/(real price*CPI) = real earnings/real price, where the CPI = consumer price index.
[2] The price-earnings ratio based on the current Shiller earnings series failed at forecasting returns over any horizon because the ratio spiked during the global financial crisis when  there was an extremely steep decline in earnings.