Is the Fed Model Useful?

Quantitative investment methods are useful provided the abstractions

and reasoning relate as directly as possible to actual investor concepts

and behaviors. The ratio Bond Yields/Stock Earnings Yields is a statistically significant means of comparing the prices of stocks and bonds, the resulting changes in asset allocation causing a flow of funds between the stock and bond markets. We also mention event exceptions and the role of judgment.

In its July, 1997 Humphrey-Hawkins report to Congress, the Fed hinted how it might value the stock market. The “Fed Model” compared 1982-1997 ten year treasury bond yields to the earnings yield of the S&P 500, showing a very close correlation between the two. This model is a simple way to value stocks and is in accord with standard economic equilibrium theory. A paper titled “Fight the Fed Model,”(Asness, 2002) calls into substantial question its logical and statistical validity because of its contradictory treatment of inflation in an actual economy.

We discuss the Fed model, econometrics in general, and the valuation of stocks.

Is the Fed Model Illogical?

The paper’s primary objection to the model Bond Yields/Stock Earnings Yields is that it is illogical, comparing apples to oranges. To digress just a moment. Economists, considering money to be a veil upon real activity, prefer to analyze real (inflation adjusted) quantities rather than nominal (quoted) quantities. The paper contends that the comparison: Bond Yields (a nominal quantity)/Stock Yields (a real quantity) is illogical. The consequence in this illogic is that statistical tests will fail to validate the Fed Model.

Consider, however, what the Fed model really compares. A direct comparison of bond yields to stock earnings yields compares a nominal quantity (bond yields) against the E/P ratio, where inflation in the numerator and in the denominator apparently cancel out. There are theoretical and practical problems in calling this ratio “real”.

First the theoretical: Assuming a constant inflation which cancels out, E/P is not “real” in the conventional economic sense; you can’t meaningfully calculate a “nominal” E/P by adding back inflation. This ratio specifies a certain kind of analytic behavior; it is not economically “real” in the sense of real GDP.

Now the practical: the ratio might cancel out a constant inflation; but the past inflation applied to earnings is not the same as the future expected inflation applied to stock and bond prices; so inflation can’t cancel out in an actual economy. In fact, our econometric model demonstrates that the Fed ratio responds negatively to increased expectations of future inflation; therefore this ratio is nominal.

In nominal terms:

Bond Yields = Y__

Stock Earnings Yields E/P

Where: Y = Bond Yields (affected by expectations of inflation)

E = Earnings of the S&P 500 (affected by past inflation). We use

trailing twelve month operating earnings; at this writing, the

quality of these earnings is likely improving.

P = Price of the S&P 500 (affected by expectations of inflation)

Statistics (If your utility function does not include statistics, skip to the next section.)

The illogical nominal/real comparison of bond yields/stock earnings yields, according to the paper, leads to a major statistical problem. The econometric Ordinary Least Squares (OLS) model will fail to falsify the null hypothesis of no difference from random effects. This is, indeed, what the paper shows on Table 2. Using the Fed Model and (1955-2001) 10 year rolling average real earnings to forecast future 10 year rolling average real S&P 500 returns – the study found that the model has an adjusted R-Squared of only .014 and no statistical significance in the independent variables. Compare this with the alternative low P/E model, where the adjusted R-Squared is .296; and the independent P/E variable is statistically significant. An R-Squared of one means the model fits the data perfectly.

The reason for the low performance of the Fed model is not the problem nominal/real, as we have shown, but the model’s specification. The study tests the statement, in difference form: future real stock market returns = f(E/P -Interest Rates). In all but extreme cases, there is no reason to suppose that future stock market returns should depend on present interest rates in any form, whether real or nominal. The equivalent bond market statement would be that future bond market returns depend on present interest rates. The Random Walk model is not a long-term model.

The ordinary least squares model (OLS) model is a complicated scientific laboratory in a computer that seeks to discover the coefficients of the true econometric model. It makes a difference whether that true econometric model is y=mx + b or y=mx. In the ordinary econometric case, the constant b represents economic variables not included in the analysis. To analyze ratios, we think the zero intercept form is appropriate because the ratio: Bond Yields/Stock Earnings Yields cancels out all common macroeconomic variables (not of course the independent variables) relevant to both stocks and long term bonds.

Here is an example of what happens when you use ratio analysis, as suggested earlier. In company analysis, Cost of Goods Sold and Revenues are affected by a common factor, say business demand, and other management factors. A constant ratio of CGS/Revenues usually indicates that, regardless of demand which cancels out, management has its costs and pricing under control.

To return to our econometric discussion, which model is appropriate? As we have indicated in an earlier footnote, the mean of the data Bond Yields/Stock Earnings Yields falls almost exactly upon the regression line. Therefore the model form of y=mx is appropriate.

Our regression accomplishes two tasks. It fits how the dependent and independent variables covary and also fits the mean value of the dependent variable. If a zero intercept model is the true model, then our model using nominal and unsmoothed data from (1968-1999):

 
Bond Yields(t)
----------- = .022 * cap util(t+1) - .079 * infl(t+1)
Stock Earnings Yields(t)
 
 where: Bond Yields are the long term AA utility rate.
 Stock Earnings Yields are trailing S&P 500 operating 
 earnings divided by the current level of the S&P 500.
 

has an adjusted R-Squared of .95 and statistically significant coefficients.

Consequences

The Fed ratio is logical, and it is statistically significant due to the resulting flow of funds between the stock and bond markets. The paper makes a useful distinction between prediction and explanation. The P/E model discussed above is a predictive model. The Fed model, and our modification of it, is an explanatory model. This model analyzes the equilibrium level of the present to near-term stock market, not far future returns.

 If the current corporate bond rate of 5.91% increases by 10% in the future to 6.50%, our model calculates an equilibrium value of 1125 for the S&P 500. The current 3/31/03 stock market of 848 is undervalued by 24%, an undervaluation which will likely be corrected in time with the favorable resolution of the war in Iraq and as economic growth absorbs the current slack in the labor and industrial markets. In contrast, a P/E based valuation, which the paper presents, has an adjusted R-Squared of only .296, leaving more than 70% of the market’s return unexplained.

So what use is our near-term explanatory model? Its useful for calculating the current stock market equilibrium value, and its also useful for asking what-if economic questions for the near future. Furthermore, its measured residuals usefully track the impacts of identifiable macro events upon the willingness of investors to buy stocks rather than bonds - the OPEC crisis in the 1970s, Internet exuberance; and the current combination of 9/11, accounting scandals, and the uncertainties accompanying the war in Iraq, which have affected general confidence. Thus the real world correlates to both our model’s calculated equilibrium (that we have earlier discussed), and identifiable shorter-term deviations from this equilibrium which denote the events of history.

What is the investment constancy that our econometric analysis has discovered? It is that the valuation ratio Bond Yields/Stock Earnings Yields is statistically significant, that the stock market is inherently cyclical because the economy is, and that after considering the statistical residuals of the model, the events of history do matter. We have therefore developed, not a model of prediction, but a model of explanation – requiring the exercise of judgment. Value investors, emphasizing equilibrium valuation, and momentum traders, emphasizing the news and price trends, use different information sets.

This result is entirely consistent with life in liberal market societies. John Stuart Mill (1859) wrote, “…use observation to see, reasoning and judgement to foresee, activity to gather materials for decision, discrimination to decide… .”

RETURN TO PREVIOUS PAGE