## A Teaching Note: The Gordon Equation, Earnings Yields, and Stock Returns

Why can't I find this anywhere? And I could have sworn that I--personally--had written this down before:

A Teaching Note: The Gordon Equation, Earnings Yields, and Stock Returns

J. Bradford DeLong, U.C. Berkeley and NBER
May 2007

At the level of the stock market as a whole, the ratio of cyclically-adjusted and normal earnings to stock prices--the permanent earnings yield--should be a good guide to the market's expectation of the real rate of return.

Think of it as a Modigliani-Miller result. Firms could pay out all their earnings in dividends and still keep their companies' productive potential unimpaired--that's what "earnings" means, after all. In that case, as long as the companies' productive potential is truly unimpaired the expected return is the earnings yield. When firms retain and reinvest earnings it is because managers see it as advantageous (to them or to shareholders) to do so. Thus with the appropriate caveats--as long as there are no big gaps between the required market and firms' internal rates of return, and as long as current accounting earnings do not grossly understate or overstate true Haig-Simons permanent earnings--the earnings yield on an index should be a good guide to the market's expectation of the long-run real rate of return on equities.

To fix this intuition, consider the Gordon equation in a steady-state context. The Gordon equation relates the long-run rate of return r on a company's stock--or a stock index--to the long-run dividend yield D/P and to the long-run rate of capital gain g:

r = D/P + g

In steady-state there is a constant dividend yield D/P and a constant share of earnings paid out as dividends:

D/E = θ

Assume also a simple generating process determining earnings. Assume a constant internal rate of return r+ρ on reinvested earnings. Earnings are then:

E = (r+ρ)K

where K is the comprehensively-defined productive capital stock assets of the firm (or index), and where the capital stock increases over time as retained earnings are used to purchase new plant and equipment or acquire already-existing productive assets:

dK/dt = E-D

In steady-state, the rate of capital gain g will be the same as the rate of growth of the capital stock:

g = (1/K)(dK/dt)

And then it is a simple matter of algebra:

(1/K)(dK/dt) = (1-θ)E/K = (1-θ)(r+ρ)

r = D/P + r - θr + ρ - θρ

θr = θE/P + ρ - θρ

r = E/P + (1/θ - 1)ρ

For ρ=0--when there is no difference between the market and the internal rate of return--then simply:

r = E/P

For θ=1--when there are no retained earnings--then simply:

r = E/P

When θ is less than one--when there are retained earnings--then earnings yields will understate returns to the extent that superior managerial knowledge of projects creates a positive gap ρ between internal and market rates of return; earnings yields will overstate returns to the extent that managerial entrenchment allows the dissipation of retained earnings on low-return projects--a negative gap ρ.

When firm managers classify investments as operating expenses to reduce tax liability, there will be a further positive wedge between r and the accounting earnings yield. Should firm managers overstate earnings to try to boost stock prices to increase the value of their options, this will drive a negative wedge between r and the accounting earnings yield.

And, of course, these are permanent earnings yields we are talking about: they must be properly adjusted for the state of the business cycle and for windfalls and non-recurring expenses.

But with these caveats--accounting earnings a good approximation to true Haig-Simons permanent earnings, managerial entrenchment offsetting superior managerial information to keep internal close to market rates of return, and so forth--the same logic that leads us to the Gordon equation conclusion that the dividend yield should be equal to the difference between the required rate of return and the dividend growth rate leads, if taken one step further, to the conclusion that the earnings yield should be the required rate of return.