The comparison is seductive because it looks like arbitrage. Equities yield 4 percent in earnings, bonds yield 7 percent in coupons, therefore bonds are better, therefore equities are expensive. Every part of that sentence is grammatical and the conclusion is wrong, because the two yields are denominated in different things.
An earnings yield is approximately real. Corporate earnings are a claim on nominal output, and over long horizons nominal output carries inflation with it, so an earnings yield of 4 percent is a claim on about 4 percent plus whatever inflation does. A nominal government bond yield is nominal by construction: 7 percent means 7 percent of currency, and inflation eats it. Subtracting a nominal number from a real one produces a quantity with no interpretation.
The arithmetic makes the size of the error visible. Take two years, both illustrative and shaped after real episodes.
| Year H (high inflation) | Year L (low inflation) | |
|---|---|---|
| Earnings yield | 12.00% (multiple 8.33) | 3.40% (multiple 29.41) |
| Nominal ten-year government yield | 13.90% | 1.50% |
| Expected inflation | 9.00% | 2.40% |
| Exact real yield | 4.4954% | −0.8789% |
| Fed-model spread | −1.90 pp | +1.90 pp |
| Correct real spread | +7.50 pp | +4.28 pp |
| Error introduced | 9.40 pp | 2.38 pp |
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How big is the Fed model's error?
The Fed model ranks Year L above Year H by 3.80 percentage points, calling the 29-times market the better buy. The real-yield comparison ranks Year H above Year L by 3.23 points, calling the 8-times market the better buy. The ordering reverses, which is the sharpest possible demonstration that a measurement error is not a nuisance here but a sign error.
The error has a closed form. Since a nominal yield equals the real yield plus expected inflation plus their cross term, the Fed-model spread equals the correct spread minus expected inflation minus that cross term. In Year H the error is 9.40 points; in Year L it is 2.38 points. The difference between the two errors, 7.03 points, is set almost entirely by the 6.60-point gap in expected inflation between the two years. The Fed model is a measurement of inflation wearing a valuation costume.
Look at what it implies as a fair multiple and the absurdity is complete. Invert the nominal yield and Year H's fair multiple is 7.19, while Year L's is 66.67. A model that says a market deserves a multiple of 67 because the government can borrow at 1.5 percent has stopped being a valuation framework. Using the real yield instead, and adding a 4 percent equity risk premium, gives fair multiples of 11.77 and 32.04, which are at least in the space of things a business could deliver.
Is the Fed model valid, and can it be fixed?
Two honest defences of the model exist and neither rescues it. The first is that it forecasted returns tolerably during one thirty-year window of falling inflation, which is what you would expect of any indicator that is essentially a transformation of the inflation trend during a period when the inflation trend was the dominant variable. The second is that some investors genuinely choose between the two assets in nominal terms and are therefore moved by the nominal spread, which makes it a sentiment indicator rather than a valuation one. Keep it if you like as a description of how a certain kind of allocator behaves, but do not let it price anything.
The repair costs one line. Use a real yield: either an inflation-linked government bond yield where a liquid one exists, or a nominal yield deflated by a stated expected-inflation figure, with the source and date attached to the expectation. Then compare it to the earnings yield and call the difference what it is, a spread between two real quantities.