Valuing Equities without Discounting

In the diagram above, each time a firm generates profits (earnings), a portion are paid out as a dividend, and the remainder are re-invested, adding to the firm's capital stock. Assuming that the firm is on a stable growth trajectory, then the various multiples (price to earnings or price to dividend) should be constant over time, while the earnings, dividend, and market cap should grow with some real growth rate, g. If this were not the case, these multiples would go to zero or to infinity over time.

By "real growth rate", I mean the logarithmic derivative (i.e. (1/x)dx/dt = g). Now let r be the re-investment ratio:

r*earnings = retained earings = earnings - dividend

In particular, if r = 1, then all earnings are re-invested, and if r is negative, then the firm is selling off capital, since it cannot meet the owners' required rate of return.

Then the growth rate of capital (dC/dt) is just rE -- the amount re-invested. Therefore the "real growth rate" is rE/C, so that we have

g = rE/C = r*(E/C) = r*(earnings yield)

Which we can re-arrange to read:

dividend yield = ((1-r)/r)*g= (1/r-1)*g

earnings yield = g/r

earnings yield – dividend_yield = g

earnings yield = dividend yield + g ("dividend capitalization model")

• g = growth rate of earnings, dividends, market-cap
• r = re-investment ratio

This allows us to value equities without discounting infinite sums -- by defining steady-state ratios. The assumption is only one of stability (which the discounted sums approach also requires). In order to value a company, you need to estimate the earnings growth rate as well as the dividend yield (or the re-investment ratio). This gives the standard "dividend discount" valuation approach, although the method here is a bit more general, in that no dividends need to be paid -- full re-investment is also allowed.

Now, in the real world, the market doesn't know how much to value capital -- that is a difficult problem and so the market value oscillates around the "true" value, as investors constantly adjust their expectations based on optimism and pessimism, excessively marking down the value of capital during busts, and over-valuing capital during booms. To see these mood swings, consider the following:

This chart graphs the stable growth path, g, obtained by taking the 10 year average earnings-yield and the present dividend yield. It then graphs the 10-year trailing average of the observed earnings growth.

You can see that there are turning points, in which the actual earnings growth (blue) wildly overshoot and undershoot. These are equivalent to "earnings booms" and "earnings busts", in which the earnings grow much faster or slower than the required growth rate as determined from a 10 year backwards-looking window. These correspond to secular booms and busts in corporate profits, and are influenced not just by changes in output, but by debt-fueled demand, tax law changes, access to new markets, etc.

In general, expectations formation is backwards looking but there are real turning points -- changes in mood about the future (such as the Great Depression or the post-war boom) that a purely backward looking metric will not detect. Also note that the booms and busts are secular -- they last much longer than simple recessions, so that large multiples seem normal, until the next collapse and re-pricing, or until the next boom occurs.