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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.
Rationale for Model - Constructing the Model - Using the Model to Project S&P 500
This section empirically justifies the framework of the REY Model, as defined by the opening question above. It shows that, since the beginning of 1990, aggregate stock prices (the S&P 500 index) adjust to the inflation rate more precisely than 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 1/2/90 through 5/13/08 of the following:
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 these observations, we calculate the Pearson correlation between each pair of rates and the standard deviation of the gap between each pair of rates, 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 a higher correlation and smaller variation in gap, E/P follows the inflation rate more tightly 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 through the stocks-Treasuries intermediating effect of inflation. (Buyers of Treasuries also 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 3/15/07 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 period of 4/1/05 through 5/13/08. The following chart depicts the behaviors of the four rates over this shorter interval. This chart illustrates clearly, via the relationship between the inflation rate and the T-bill yield, the elimination and reintroduction of stimulative policy by the Federal Reserve Board. Note the elevated shocks to the total inflation rate that disrupt the relationship between the total inflation rate and the S&P 500 E/P within this window.
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, core inflation leads in matching the overall direction of the aggregate stocks earnings yield, while core inflation and the T-note yield outperform in tracking short-term variations of E/P. The differences between these short-term statistics and those for the entire sample above reflect the above-noted shocks to the total inflation rate. Investors apparently recognize that such shocks are likely transitory.
In summary, stock investors/traders can reasonably focus on stock prices, corporate earnings and inflation as the critical indicators of overall stock market behavior, but investors may be paying 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 1/2/90 through 5/13/08. Four unusual intervals are highlighted in blue: Gulf War I (GWI), the Internet bubble (seemingly ended by the 9/11 terrorist attack), Gulf War II (GWII) and the aftermath of Hurricane Katrina. During two of these four 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. Hurricane Katrina had a smaller, shorter effect on E/P, with a reversal echo 12 months later.
The average gap between the S&P 500 E/P and inflation rate (the "real earnings yield") over the entire period 1/2/90-5/13/08 is 2.26% with standard deviation 0.91%. Excluding the GWI and Internet bubble intervals, we find the average gap is 2.59% (standard deviation 0.70%) for 1/2/92-12/31/98 and 2.79% (standard deviation 0.80%) for 4/1/05-5/13/08. The next chart is a close up of the most recent period. The elevated 12-month total inflation rate during late 2005 through late 2006 is associated with effects of Hurricane Katrina, with the recent increase in the total inflation rate an echo.
The above chart suggests a relationship between the S&P 500 E/P and the inflation rates, complicated by the late 2005-late 2006 elevation of the total inflation rate. The inflation rate has "stair steps" because, as noted above, BLS establishes a new rate only once per month. The total inflation rate has been unusually volatile during this shorter period.
The following scatter plot depicts the exact linear relationship between the S&P 500 E/P and the total and core inflation rates for the full sample starting at 1/2/90. It shows that, over the entire sample period (and as previously used), E/P responds to roughly 50%-60% 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 shorter, more current subsample.
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 4/1/05. It shows that, over this shorter period, E/P responds little to the variation in the total inflation rate but to about 35% of the variation in the core inflation rate. We can use these different relationships to construct alternate short-term S&P 500 index projections. Again, new data might affect the linear fit parameters.
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.
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. It can be somewhat adaptive in that we can drop old data as new data accrues and adjust E/P response to inflation to reflect persistent shifts in the E/P-inflation rate relationship. However, the model's forecasts are difficult to test because the forecasts change daily, weekly and monthly as new stock price, earnings forecast and inflation forecast data become available.
See our blog entry of 6/28/07 for discussion of the drawbacks of limiting long-term analysis to 1990 and later, and to using a rolling short-term analysis. We intend that our focus on relatively recent data avoid 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."
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 3/31/09), 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 (1/2/90-3/31/09) 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 daily difference between actual and modeled data is (by design) 0.0% for both, and the standard deviation of daily differences based on total (core) inflation is 16.0% (17.7%). As a model input, total inflation slightly outperforms core inflation across the entire sample. As noted above, we spread each quarterly earnings increment evenly over the next subsequent quarter to reflect the gradual release of actual earnings. We insert each monthly inflation rate change on the date of the BLS public release back to 1994, and on the 15th of each month for prior years.
The large jump up in the mockup based on total inflation in October-November 2006 result from the historic two-month drop in the total inflation rate as effects of Hurricane Katrina exit 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 percentage 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 (4/1/05-3/31/09) to construct short-term mockups of the S&P 500 index for both total and core inflation rates. The average daily difference between actual and modeled data is 0.0% for both, and the standard deviation of daily differences based on total (core) inflation is 3.1% (2.9%). For a recent subsample, core inflation outperforms total inflation as a model input. The projection to 3/31/09 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.
