Let's review the most popular approach, which is to build a supply-side model. There are three steps:
1. Estimate the expected total return on stocks.
2. Estimate the expected risk-free return (bond).
3. Find the difference: expected return on stocks minus risk-free return equals the equity risk premium.
We'll keep it simple and sidestep a few technical issues. Specifically, we are looking at expected returns that are long-term, real, compound and pre-tax. By long-term, we mean something like 10 years, as short horizons raise questions of market timing. (That is, it is understood that markets will be over or under-valued in the short run.)

By "real," we mean net of inflation. Even if we estimated the stock and bond returns in nominal terms, inflation would fall out of the subtraction anyhow. And by "compound," we mean to ignore the ancient question of whether forecasted returns ought to be calculated as arithmetic or geometric (time-weighed) averages.

Finally, although it is convenient to refer to pre-tax returns (as virtually all academic studies do), individual investors should care about after-tax returns. Taxes make a difference. Let's say the risk-free rate is 3% and the expected equity premium is 4%; we therefore expect equity returns of 7%. Say we earn the risk-free rate entirely in bond coupons taxed at ordinary income tax rates of 35%, whereas equities may be deferred entirely into a capital gains rate of 15% (i.e., no dividends). The after-tax picture in this case makes equities look even better.

Step One: Estimate the Expected Total Return on Stocks

Dividend-Based Approach
The two leading supply-side approaches start with either dividends or earnings. The dividend-based approach says that returns are a function of dividends and their future growth. Consider an example with a single stock that today is priced at \$100, pays a constant 3% dividend yield (dividend per share divided by stock price), but for which we also expect the dividend - in dollar terms - to grow at 5% per year.

In this example, you can see that if we grow the dividend at 5% per year and insist on a constant dividend yield, the stock price must go up 5% per year too. The key assumption is that the stock price is fixed as multiple of the dividend. If you like to think in terms of P/E ratios, it is the equivalent to assuming that 5% earnings growth and a fixed P/E multiple must push the stock price up 5% per year. At the end of five years, our 3% dividend yield naturally gives us a 3% return (\$19.14 if the dividends are reinvested). And the growth in dividends has pushed the stock price to \$127.63, which gives us an additional 5% return. Together, we get a total return of 8%.

That's the idea behind the dividend-based approach: the dividend yield (%) plus the expected growth in dividends (%) equals the expected total return (%). In formulaic terms, it is just a re-working of the Gordon Growth Model, which says that the fair price of a stock (P) is a function of the dividend per share (D), growth in the dividend (g) and the required or expected rate of return (k):

Earnings-Based Approach
Another approach looks at the price-to-earnings (P/E) ratio and its reciprocal: the earnings yield (earnings per share ÷ stock price). The idea is that the market's expected long-run real return is equal to the current earnings yield. For example, at the end of 2003, the P/E for the S&P 500 was almost 25. This theory says that the expected return is equal to the earnings yield of 4% (1 ÷ 25 = 4%). If that seems low, remember it's a real return. Add a rate of inflation to get a nominal return.

Here is the math that gets you the earnings-based approach:

Whereas a growth factor is explicitly added to the dividend-based approach, growth is implicit to the earnings model. It assumes the P/E multiple already impounds future growth. For example, if a company has a 4% earnings yield but doesn't pay dividends, then the model assumes the earnings are profitably reinvested at 4%.

Even experts disagree here. Some "rev up" the earnings model on the idea that, at higher P/E multiples, companies can use high-priced equity to make progressively more profitable investments. Arnott and Bernstein - authors of perhaps the definitive study - prefer the dividend approach precisely for the opposite reason. They show that, as companies grow, the retained earnings they often opt to reinvest result in only sub-par returns - in other words, the retained earnings should have instead been distributed as dividends.

Handle with Care
Let's remember that the equity premium refers to a long-term estimate for the entire market of publicly-traded stocks. Lately several studies have cautioned that we should expect a fairly conservative premium in the future.

There are two reasons why academic studies, regardless of when they are conducted, are certain to produce low equity risk premiums.

The first is that the studies make an assumption that the market is correctly valued. In both the dividend-based approach and earnings-based approach, the dividend yield and earnings yield have reciprocal valuation multiples:

Both models assume that the valuation multiples - the price-to-dividend and P/E ratio - are correct in the present and will not change going forward. This is understandable, for what else can these models do? It is notoriously difficult to predict an expansion or contraction of the market's valuation multiple. The earnings model might forecast 4% based on a P/E ratio of 25. And earnings may grow at 4%, but if the P/E multiple expands to 30 in the next year, then the total market return will be 25%, where multiple expansion alone contributes 20%! (30/25 -1 = +20%)

The second reason low equity premiums tend to characterize academic estimates is that the total market growth is limited over the long-term. You'll recall that we have a factor for dividend growth in the dividend-based approach. Academic studies assume that dividend growth for the overall market - and, for that matter, earnings or EPS growth - cannot exceed the total economy's growth over the long term. If the economy grows at 4% as measured by gross domestic product (GDP) or national income, then studies assume that markets cannot collectively outpace this growth rate. Therefore, if you start with an assumption that the market's current valuation is approximately correct and you set the economy's growth as a limit on long-term dividend growth (or earnings or earnings per share growth), a real equity premium of 4 or 5% is pretty much impossible to exceed.

Now that we have explored the risk premium models and their challenges, it is time to look at them with actual data. The first step is to find a reasonable range of expected equity returns; step two is to deduct a risk-free rate of return and step three is to try to arrive at a reasonable equity risk premium.

Calculating The Equity Risk Premium

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