Real Earnings Yield (REY) Model Details
What is a fair price for the real return on equity investments?
The following discussion provides the rationale, design and outputs of the Real Earnings Yield (REY) stock market model, constructed by the CXO Advisory Group LLC as a potential decision aid for timing equities investments and trades. The REY model is similar to the Fed Model , but it uses the inflation rate (a wealth discount rate) rather than Treasury instrument yields (competitors to stocks) as a benchmark for the stock operating earnings yield. It is a simple model, constructed from the viewpoint of an investor (rather than an economist).
The 2002 paper entitled "Market Timing Strategies that Worked" by Pu Shen, summarized in our blog entry of 11/8/04, provided stimulus for development of this model. Blog Synthesis: Valuation Based on Fundamentals and Blog Synthesis: The Economy and the Stock Market identify some other formal research on the validity of earnings-inflation valuation hypotheses. Blog Synthesis: Gunning for the Fed Model? offers some pros and cons on the Fed Model.
We intend that this model offer visualization of such models rather than a proof or rigorous test.
We will update this discussion whenever we change the model, and every few weeks as new inputs accumulate.
To discuss this model, go to the REY and RTV Models Forum.
Rationale for Model - Constructing the Model - Using the Model to Project S&P 500
This section offers some empirical justification the REY Model framework, as defined by the opening question above. It suggests that, since March 1989, aggregate stock prices (the S&P 500 index) adjust to the inflation rate more consistently than to either the 90-day Treasury bill (T-bill) rate or the 10-year Treasury note (T-note) rate. Other models, including the Fed Model, use the T-bill or T-note rate as the benchmark for the aggregate stock market earnings yield (earnings divided by index level or price).
The following chart shows the behaviors from March 1989 through November 2008 of the following:
- The S&P 500 operating earnings yield (E/P) calculated using monthly closes of the S&P 500 index and trailing 12-month operating earnings based on data from Standard and Poor's. We spread each quarterly earnings increment over the next subsequent quarter to reflect the gradual release of actual earnings, as follows: one sixth the first month, two thirds the second month and one sixth the third month.
- The monthly closes of T-bills.
- The monthly closes of T-notes.
- The monthly total and core inflation rates based on the 12-month trailing change in the non-seasonally adjusted Consumer Price Index (all items and all items excluding food and energy) as published by the Bureau of Labor Statistics (BLS). Since BLS releases data for each month about the middle of the following month, we introduce new inflation rate data with a one-month lag (for example, the inflation rate for January affects the stock market in February).
Visual inspection of this chart suggests that the T-bill yield, which is most directly affected by Federal Reserve Board actions, is the least connected of the four rates. It appears also that the most consistent relationship may be that between E/P and the inflation rates. To test this observation, we calculate the Pearson correlation between each pair of rates and the standard deviation of the gap between each pair of rates over the entire sample period, with results as follows:
This table confirms that the S&P 500 E/P is more closely related to the inflation rates than to the yields of Treasury instruments over this period. In fact, with both higher correlations and smaller variations in gap, E/P relates to inflation rates more strongly than does the T-bill yield. A possible interpretation is that most equity investors do not closely track T-bills and T-notes as competing investments, and that they instead focus on the "real earnings yield" of stocks in deciding whether to buy, hold or fold. In other words, investors use the inflation rate as a discount rate for corporate earnings (or for that part of their wealth invested in stocks). This interpretation differs from the Fed Model, which nonetheless would work via the stocks-Treasuries intermediating effect of inflation. Buyers of Treasuries also generally want a positive real return. A corollary of this interpretation is that stock investors/traders are interested in Federal Reserve Board monetary policy actions mostly because these actions link to the Board's inflation outlook. (See our blog entry of 11/4/08 for a test of the direct relationship between stock prices and the Federal Funds Rate.)
The above period includes the Internet bubble, unusual by many financial market measures. We therefore repeat the analysis for the recent post-bubble subperiod of December 2004 through November 2008. We choose a four-year window of analysis because of evidence of a four-year cycle in U.S. stock market behavior related to the U.S. national election cycle (see Blog Synthesis: Politics and the Stock Market). The following chart depicts the behaviors of the four rates over this shorter interval. This chart illustrates clearly, via the T-bill yield, the recent rapid reintroduction of stimulative policy by the Federal Reserve Board.
The degree of relatedness between pairs of rates is less obvious by visual inspection for this shorter period, and we turn as before to calculations of Pearson correlations and gap standard deviations. The following table summarizes results:
Over this shorter period, equity investors are paying little attention to inflation rates, and Treasury yields are moving in the opposite direction from E/P. Differences between these subperiod statistics and those for the overall sample period above may reflect the volatility of both operating earnings and the total inflation rate, and they cast doubt on use of the E/P-inflation relationship over the short term.
See our blog entry of 12/18/08 for a look at the E/P-inflation relationship over the very long term (January 1871 through March 2008).
In summary, stock investors/traders can reasonably consider stock prices, corporate earnings and inflation as indicators of overall stock market behavior, but investors may now be paying more attention to earnings than to inflation and more attention to core than total inflation. Movements of T-bills and T-notes may relate to stock prices principally through the shared influence of inflation.
Our next step is to develop a model of stock market behavior by quantifying the gap between E/P and the inflation rate.
The following chart focuses on the behaviors of the S&P 500 E/P and the inflation rate from March 1989 through November 2008. Blue notations highlight two unusual intervals: Gulf War I (GWI), and the Internet bubble (seemingly ended by the 9/11 terrorist attack). During these two intervals, the gap between E/P and inflation rate shrank considerably. For GWI, investors may have believed that the inflation spike driven by higher oil prices would subside quickly after the war. During the Internet bubble, investors may have held irrationally exuberant expectations for earnings growth based on infusion of Internet technology. For Gulf War II, unlike for Gulf War I, investors may have feared the long-term consequences of "nation building" and therefore reacted normally to an inflation spike from an oil price surge. Energy price fluctuations and the credit bubble burst have driven high volatility for total inflation over recent years.
The average gap between the S&P 500 E/P and inflation rate (the "real earnings yield") over the entire period March 1989 through November 2008 is 2.26% with standard deviation 0.92%. Excluding the GWI and Internet bubble intervals, we find the average gap is 2.60% (standard deviation 0.71%) for 1992-1998 and 2.56% (standard deviation 0.96%) for December 2004 through November 2008. The next chart is a close up of the recent four-year subperiod. As noted, crude oil price fluctuations and the credit bubble burst have arguably driven high volatility for the total inflation rate during this subperiod.
The following scatter plot depicts the linear best-fit relationship between the S&P 500 E/P and the total and core inflation rates since March 1989. It shows that, over the entire sample period (and as previously used), E/P responds to roughly 55-65% of the variation in the inflation rate. We can then forecast the level of the S&P 500 index using four inputs: an inflation forecast, an S&P 500 earnings forecast, and the slope and y-intercept for this best-fit line. New data may affect the slope and y-intercept. The full historical sample includes the very unusual Internet bubble, and this unusual data might generate atypical results. We therefore consider also a recent four-year subperiod.
The next scatter plot depicts the exact linear relationship between the S&P 500 E/P and the total and core inflation rates for a subsample starting at December 2004. It shows that, over this shorter period, E/P responds minimally to the variation in the total inflation rate but to about one fifth of the variation in the core inflation rate. We can use these different relationships to construct alternate short-term S&P 500 index projections. This subperiod represents a fairly small sample size, so a little new data might affect its statistics substantially.
The preceding two charts show that the E/P response to inflation rate changes is muted (E/P is less volatile than the inflation rate), and the level of E/P response to inflation rate shocks varies.
Why might investors not respond fully to inflation rate shocks? Perhaps investors deal with relatively high inflation rate volatility by responding only incrementally to inflation rate shocks in anticipation of near-term reversals. In other words, they view wealth discount (inflation) rate shocks as less reliable than cash flow (earnings) shocks. This interpretation is reasonable given that earnings shocks are far more granular (via 500 company earnings releases spread out over each quarter) than inflation rate shocks (one announcement each month).
Why might investors vary their responses to inflation rate shocks? Perhaps when the total inflation rate is especially volatile (as it has recently been), they view total inflation rate shocks as even less reliable than usual and shift attention to the core inflation rate. Or, perhaps they follow the lead of the Federal Reserve, as discussed in our blog entry of 7/17/07. Perhaps they shift focus from inflation rate to earnings when earnings appear to be in extraordinary jeopardy.
Inferring a general investor requirement for a predictable "real earnings yield" with variable muted response to inflation rate shocks, we define the following model:
This model is reasonably simple, and backtesting it using historical data, earnings projections and an inflation rate forecast is reasonably straightforward. The recent subperiod of analysis is adaptive in that we drop old data as new data accrues and thereby adjust the E/P response to inflation (thereby reflecting persistent shifts in the E/P-inflation rate relationship).
See our blog entry of 6/28/07 for discussion of the drawbacks of limiting long-term analysis to less than 20 years, and to using a rolling short-term analysis. We intend that our focus on relatively recent data accommodate structural breaks in the relationship between investors and valuations. Such breaks arguably make "ancient" data of low relevance.
In summary, stock investors/traders apparently pay close attention to the E/P-inflation rate gap and require a reasonably constant "real earnings yield" over the period of analysis.
USING THE REY MODEL TO ESTIMATE S&P 500
Our next step is to construct both long-term and short-term mockups of the S&P 500 index using the above model (including projections through December 2009), and then compare them to actual S&P 500 index behavior. We build the mockups based on the belief that the historical roughly constant gap between E/P and the inflation rate will persist.
The following chart shows long-term (March 1989 through December 2009) mockups of the S&P 500 index based on this approach using both total and core inflation rates, along with the actual S&P 500 index. The average monthly difference between actual and modeled data is 0.0% for both, and the standard deviation of daily differences based on total (core) inflation is 15.9% (18.1%). As a model input, total inflation outperforms core inflation across the entire sample. The projection to December 2009 is sensitive to errors in both earnings predictions and (especially) the inflation rate forecast.
The large jump up in the mockup based on total inflation in October-November 2006 results from an historic two-month drop in the total inflation rate which roughly reverses a year later as the drop exits the 12-month trailing inflation rate calculation.
As shown by the above chart, the underlying model does not explain the Internet bubble, and because the model uses an overall linear E/P-inflation rate relationship as a prediction engine, pre-bubble and post-bubble mockup values are likely higher than they should be. This bubble effect argues for using only more recent data for stock market prediction. Even with the muting inherent in the linear relationship described above, the mockup values are particularly volatile the past few years, with changes in very low inflation rates producing large fractional shifts in the E/P-inflation rate gap. This sensitivity applies to projection of future stock market behavior, with fairly small errors in inflation rate estimates generating relatively large errors in S&P 500 index estimates.
The following log version of the above chart depicts more clearly the percentage differences among the three series.
The next chart applies the linear E/P-inflation rate relationship over a shorter, recent period (December 2004 through December 2009) to construct short-term mockups of the S&P 500 index for both total and core inflation rates. The average monthly difference between actual and modeled data is 0.0% (0.8%) for total (core) inflation, and the standard deviation of daily differences based on total (core) inflation is 5.6% (5.7%). The projection to December 2009 is sensitive to errors in both earnings predictions and (especially) the inflation rate forecast.
In summary, using the E/P-inflation rate gap to establish an investor/trader-required "real earnings yield" for stocks, with inflation rate volatility muted, generates a pretty good fit between modeled and actual behaviors of the S&P 500 index.
We find that inflation rates based on inputs that are seasonally adjusted generate almost identical statistics with respect to stock prices as do inflation rates based on non-seasonally adjusted inputs.
We have tested whether other measures of the inflation rate outperform the total and core inflation rates from BLS as stock market indicators. See our blog entries of 7/6/06, 12/15/06 and1/20/07 for details. We have also tried using the Producer Price Index instead of the Consumer Price Index to define inflation for this model, but this alternate approach produces a very erratic relationship between E/P and inflation.
We have also investigated the use of second order inflation effects (trend and volatility) in the model, but every attempt has degraded backtest statistics. See our blog entry of 6/21/07.
