 If the S&P 500s actual priceearnings ratio is less than the reciprocal of the 10year Treasury bond yield (1/Y), equities are priced attractively.
 If the S&P 500s actual priceearnings ratio is greater than the reciprocal of the 10year Treasury bond yield (1/Y), equities are relatively unattractive.
 If the actual earnings yield on the S&P 500 is greater than the yield on a 10year Treasury bond, equities are attractively priced.
 If the yield on a 10year Treasury bond exceeds the actual earnings yield on the S&P 500, equities are relatively unattractive.
Market Valuation Measures: Does the Fed Model Really Work?
by Stephen E. Wilcox
Has the steep market decline finally produced a significant undervaluation of stock prices?
Some market commentators in the media would have us believe so, based on evidence from the Fed Model.
The Fed Model is a measure that is often used to judge the relative attractiveness of stocks compared to other asset categories. This approach compares the yield on 10year Treasury bonds to the earnings yield on the S&P 500: Stocks become more attractive as earnings yields rise above those on 10year Treasury bonds.
Most individuals are more familiar with priceearnings ratios, using forward (expected) earnings as a measure of the relative attractiveness of stocks. However, priceearnings ratios are simply the reciprocal of earnings yields:
P/E = 1 ÷ Earnings Yield
And using this relationship allows investors to compare priceearnings ratios to various measures of yield.
So, where does the S&P 500 stand today relative to 10year Treasuries?

Thomson First Call (www.firstcall.com) reports the yield on the 10year Treasury bond at the close of trading on October 11, 2002, at 3.80%. This would be equivalent to a predicted or equilibrium S&P 500 priceearnings ratio of 26.3 or [1 ÷ 0.038], according to the Fed Model.
On October 11, the S&P 500 closed at 835, and Thomson First Call reports that industry analysts forecasted forward earnings (from fourth quarter 2002 to third quarter 2003) for the index of $54.67. This results in an earnings yield of 6.55% or [54.67 ÷ 835] and a pricetoforward earnings ratio of 15.3 or [835 ÷ 54.67].
These results suggest that the market is about 42% undervalued, at least according to the Fed Model! Given these results, should investors dramatically alter their portfolios to take advantage of the Fed Models buy signal?
This article shows the danger of placing too much emphasis on the Fed Model when determining your asset allocation, particularly during the current market environment. Not only is it fairly easy to find flaws in this model, but several other measures of valuation suggest that equities are not cheap relative to their underlying fundamentals.
Deriving the Model
Unfortunately, there are problems inherent in using the Fed Model as a valuation/timing tool. And these problems can easily be revealed if you understand how the Fed Model was derived.

The Fed Model is based on a wellknown multiplier model that can be found in any investment textbook. It simply states that the S&P 500s priceearnings ratio based on estimated earnings for next year (P/E) is a function of the S&P 500s dividend payout ratio (DPR), its return on equity (ROE), and a return investors require for bearing the risk of equity investment (R), in the following manner:
P/E = DPR/[R (1 DPR) × ROE]
The model provides a simple way to determine intrinsic value for the S&P 500 under the rigid assumption that all variables on the righthand side of the equation are constant over time.
The Fed Model is derived from this multiplier model by substituting the yield on a 10year Treasury bond (Y) for both the required return (R) and ROE. Simplifying the terms, you get:
P/E = 1/Y
This is the Fed Model and it implies that the reciprocal of the yield on a 10year Treasury bond provides an estimate of the S&P 500s equilibrium priceearnings ratio (based on forward or on estimated earnings):
E/P = Y
where the 10year Treasury bond yield is the predicted equilibrium earnings yield. This variation of the model compares the actual earnings yield for the S&P 500 to the 10year Treasury bond yield:
Problems
The effectiveness of the Fed Model as a valuation tool requires that the rigid assumptions and substitutions made in the previous section are valid in the financial markets.
In the previous section, producing the Fed Model from the multiplier model assumed that the required return and the S&P 500s return on equity are both equal to the yield on the 10year Treasury bond. Does this assumption hold true in the real world? Although the required return for bearing equity risk is not directly observable, historical data for the return on equity and the yield on the 10year Treasury bond are readily available.

Figure 1 shows the relationship between the S&P 500s ROE and the 10year Treasury bond, based on S&P data from the Barra Web site (www.barra.com) and yield data for the 10year Treasury bond from the FRED® database of the St. Louis Federal Reserve Bank at research.stlouisfed.org/fred/. The data are monthly observations from January of 1977 through September of 2002.
Clearly, ROE and the yield on the 10year Treasury bond have never been equal, and the former has exceeded the latter over this entire time period. The implication of this relationship is that the Fed Model provides an upwardly biased estimate of the equilibrium S&P 500 priceearnings level and a downwardly biased estimate of the equilibrium earnings yield for the S&P 500.
The Fed Model also assumes that the required return for equity risk and ROE are equal. Financial theory suggests that this should be true in the long run, but the shortrun relationship varies over time. Although the current relationship between required return and ROE is indeterminate, a reasonable argument can be made that recent market events reflect the situation where the required return is increasing.
Equities are a risky asset class and longterm investors have earned a significant equity risk premium above riskfree assets. Although the future equity risk premium cannot be directly observed, risk premiums in the corporate bond market are readily available.
Figure 2 presents the risk premium on Baarated bonds computed as the yield on Baarated bonds minus the yield on 10year Treasury bonds over the January of 1977 through September of 2002 time period. Yields on Baarated bonds were gathered from the FRED® database.

The risk premium on Baarated bonds has increased significantly in recent years and is very close to its highwater mark during this time period. On a relative basis, it is actually at its highest point because current market interest rates are considerably lower than those in the early 1980s. It is logical to assume that this type of market reaction is also reflected in equity prices and that the required return for equities has recently increased.
Given the recent decline in ROE, illustrated in Figure 1, and the recent increase in risk premiums, illustrated in Figure 2, it is fairly easy to make the argument that priceearnings ratios should continue to decline. This is inconsistent with the current buy signal produced by the Fed Model.
Another inherent problem with the Fed Model is that the relationship between the reciprocal of the 10year Treasury bond yield to the predicted equilibrium priceearnings ratio is not linear. For example, at a 4% Treasury bond yield, the Fed Model predicts a priceearnings ratio of 25, but at a 1% yield, the Fed Model predicts a priceearnings ratio of 100. As Treasury bond yields approach zero, the predicted priceearnings ratio approaches infinitybasically, the Fed Model blows up at low levels of interest rates! For this reason, the Fed Model may not be appropriate at lower levels of market interest rates.
Finally, comparing the predicted equilibrium values from the Fed Model to pricetoforwardearnings estimates is problematic as these estimates are constantly being revised by analysts. Most recently, the majority of these revisions have been downward!
A Flawed Model
The Fed Model has received considerable acclaim in the financial media as a market timing/valuation tool. However, investors need to be aware that such a simple model requires rigorous assumptions that do not always hold in the marketplace.
The Fed Model was derived as a special case of the wellknown multiplier model, but a review of the rigorous assumptions needed to produce the Fed Model and anecdotal evidence suggests that the model is somewhat undeserving of its popularity. The Fed Model may also be less reliable in an era of low market interest rates and suffers from comparability problems if earnings estimates are inaccurate.

Although no one can consistently predict shortterm market movements, investors need to understand that many other valuation measures suggest that equities are still relatively expensive despite the recent market selloff. Figure 3, Figure 4 and Figure 5 present three other measures of market valuation. Monthly data from Barra for the S&P 500 is again used for the time period of January 1977 to September of 2002. Means (averages) for each of these valuation levels are also presented.
It would be incorrect to assume that these measures must eventually return to some preordained level, such as their average, during the time period examined. One major implication of the Fed Model is that low rates of interest result in higher priceearnings ratios, which also implies higher pricetosales and pricetofreecashflow ratios. However, these ratios are coming off of their historic highs following the stock market bubble of the late 1990s even when compared to other time periods when market interest rates were low. So, the figures do suggest that equities are still rather expensive relative to various underlying fundamentals.
Given this evidence and the Fed Models limitations, investors should be cautious in making major changes to their investment strategies based on the current buy signal provided by the Fed Model.
Academic and professional research also suggests that market timing strategies harm the average investors performance. Most investors are best served if they avoid overtrading their accounts and invest for the long term.
Stephen E. Wilcox, Ph.D., CFA, is a professor of finance at Minnesota State University, Mankato, MN.