The Fed model emerged at the beginning of the 21st century as a stock valuation methodology used by Wall Street gurus and the financial press. The Fed model compares stock yield to bond yield. Proponents almost always cite the following three attributes as the reasons for its popularity:

- It is simple.
- It is backed by empirical evidence.
- It is backed by financial theory.

This article examines the basic concepts behind the Fed model: How it works, and how it was developed, and the article will also outline the challenges to its success and theoretical soundness.

## What Is the Fed Model?

The Fed model is a valuation methodology that recognizes a relationship between the forward earnings yield of the stock market (typically the S&P 500 Index,) and the 10-year Treasury bond yield to maturity (YTM).

The yield on a stock is the expected earnings over the next 12 months divided by the current stock price and is symbolized in this article as (E_{1}/P_{S}). This equation is the inverse of the familiar forward P/E ratio but, when shown in the same yield form, it highlights the same concept as the bond yield (Y_{B})—that is, the concept of a return on investment.

Some advocates of the Fed model think the yield relationship varies over time, so they use an average of each period's yield comparison. The more popular method is where the relationship is fixed at the particular value of zero. This technique is referred to as the strict form of the Fed model because it implies that the relationship is strictly equality.

In the strict form, the relationship is such that the forward stock yield equals the bond yield:

$\begin{aligned} &Y_B = \frac{E_1}{P_S}\\ &\textbf{where:}\\ &Y_B=\text{bond yield}\\ &\frac{E_1}{P_S}=\text{forward stock yield}\\ \end{aligned}$

Two conclusions can be drawn from this:

The difference in the forward stock yield equals 0.

$\frac{E_1}{P_S} - Y_B = 0$

Alternatively, the ratio of the forward stock yield divided by the bond yield equals 1:

$(\frac{E_1}{P_S}) \div Y_B = 1$

The premise behind the model is that bonds and stocks are competing investment products. An investor is constantly making choices between investment products as the relative prices between these products change in the market place.

## Origins

The name Fed Model was manufactured by Wall Street professionals in the late 1990s, but this system is not officially endorsed by the Federal Reserve Board. On July 22, 1997, the Fed's Humphrey-Hawkins Report introduced a graph of the close relationship between long-term Treasury yields and the forward earnings yield of the S&P 500 from 1982 to 1997.

**Equity Valuation and Long-term Interest Rate**

Shortly thereafter, in 1997 and 1999, Edward Yardeni, then at Deutsche Morgan Grenfell, published several research reports further analyzing this bond yield/stock yield relationship. He named the relationship the Fed's Stock Valuation Model, and the name stuck.

The original use of this type of analysis is not known, but a bond yield versus equity yield comparison has been used in practice long before the Fed graphed it out and Yardini began marketing the idea. For example, I/B/E/S has been publishing the forward-earnings yield on the S&P 500 versus the 10-year Treasury since the mid-1980s. Considering its simplicity, this type of analysis was probably in use some time before that as well. In their March 2005 paper titled "The Market P/E Ratio: Stock Returns, Earnings, and Mean Reversion," Robert Weigand and Robert Irons commented that empirical evidence suggests that investors began using the Fed model in the 1960s soon after Myron Gordon described the dividend discount model in the seminal paper "Dividends, Earnings, and Stock Prices" in 1959.

## Using the Model

The Fed model evaluates whether the price paid for the riskier cash flows earned from stocks is appropriate by comparing expected return measures for each asset: YTM for bonds and E_{1}/P_{S} for stocks.

This analysis is typically done by looking at the difference between the two expected returns. The value of the spread between (E_{1}/P_{S}) - Y_{B} indicates the magnitude of mispricing between the two assets. In general, the bigger the spread, the cheaper the stocks relative to bonds and vice versa. This valuation suggests that a falling bond yield dictates a falling earnings yield, which will ultimately result in higher stock prices. That is P_{S} should rise for any given E_{1 }when bond yields are below the stock yield.

Sometimes, financial market pundits carelessly (or ignorantly) claim that stocks are undervalued according to the Fed model (or interest rates). Although this is a true statement, it is careless because it implies that stock prices will go higher. The correct interpretation of a comparison between the stock yield and the bond yield is not that stocks are cheap or expensive but that stocks are cheap or expensive* relative* to bonds. It may be that stocks are expensive and priced to deliver returns below their average long-run returns, but bonds are even more expensive and priced to deliver returns far below their average long-run returns.

It could be possible that stocks could continuously be undervalued according to the Fed model while stock prices fall from their current levels.

## Observational Challenges

Opposition to the Fed Model has been based on both empirical, observational evidence, and theoretical shortcomings. To begin, although stock and long-term bond yields appear to be correlated from the 1960s forward, they appear to be far from correlated prior to the 1960s.

Also, there may be statistical issues in the way the Fed model has been calculated. Originally, statistical analysis was conducted using ordinary least-squares regression, but bond and stock yields may seem co-integrated, which would require a different method of statistical analysis. Javier Estrada wrote a paper in 2006 called "The Fed Model: The Bad, The Worse, And The Ugly" where he looked into the empirical evidence using the more appropriate co-integration methodology. His conclusions suggest that the Fed model may not be as good of a tool as originally thought.

## Theoretical Challenges

Opponents of the Fed model also pose interesting and valid challenges to its theoretical soundness. Concerns arise over comparing stock yields and bond yields because Y_{B} is the internal rate of return (IRR) of a bond and accurately represents the expected return on bonds. Remember that IRR assumes that all coupons paid over the life of the bond are reinvested at Y_{B,} whereas, E_{1}/P_{S} is not necessarily the IRR of a stock and does not always represent the expected return on stocks.

Furthermore, E_{1}/P_{S} is a real (inflation-adjusted) expected return while Y_{B} is a nominal (unadjusted) rate of return. This difference causes a breakdown in the expected return comparison.

Opponents argue that inflation does not affect stocks in the same way it affects bonds. Inflation is typically assumed to pass to stock holders via earnings, but coupons to bond holders are fixed. So, when the bond yield rises due to inflation, P_{S} is not affected because earnings rise by an amount that offsets this increase in the discount rate. In short, E_{1}/P_{S} is a real expected return and Y_{B} is a nominal expected return. Thus, in periods of high inflation, the Fed model will incorrectly argue for a high stock yield and depress stock prices and, in periods of low inflation, it will incorrectly argue for low stock yields and increase stock prices.

The above circumstance is called the illusion of inflation, which Franco Modigliani and Richard A. Cohn presented in their 1979 paper "Inflation, Rational Valuation, and the Market." Unfortunately, the inflation illusion is not as easy to demonstrate as it seems when dealing with corporate earnings. Some studies have shown that a great deal of inflation does pass through to earnings while others have shown the opposite.

## The Bottom Line

The Fed model may or may not be an effective investment tool. However, one thing is certain: If an investor considers stocks real assets that pass inflation through to earnings, they cannot logically invest their capital based on the Fed model.