The equity risk premium is a long-term prediction of how much the stock market will outperform risk-free debt instruments. In this article, we take a deeper look at the assumptions and validity of the risk premium by looking at the calculation process in action with actual data. Recall the three steps of calculating the risk premium: (1) estimate the expected return on stocks, (2) estimate the expected return on risk-free bonds and (3) subtract the difference to get the equity risk premium. (For background reading, see *The Equity-Risk Premium: More Risk For Higher Returns.*)

## Step One: Estimate the Expected Total Return on Stocks

Estimating future stock returns is the most difficult (if not impossible) step. Here are the two methods of forecasting long-term stock returns:

*Plugging into the Earnings Model *

The earnings-based model says that the expected return is equal to the earnings yield. Consider the 15-year history of the S&P 500 Index, from Dec. 31, 1988 to Dec. 31, 2003:

In the above graph, we split the S&P 500 Index (violet line) into two pieces: earnings per share (green line) and the P/E multiple (blue solid line). At every point, you can multiply EPS by the P/E multiple to get the index value. For example, on the last day of December 2003, the S&P index reached 1112 (rounded up from 1111.92). At that time, the EPS of the combined companies was $45.20 and the P/E multiple therefore was 24.6 ($45.20 x 24.6 = 1112).

As the index finished the year with a P/E of almost 25, the earnings yield was 4% (1 ÷ 25 = .04). According to the earnings-based approach, the expected real return – before inflation – was therefore 4%. The underlying intuitive idea is 'mean reversion': the theory that P/E multiples cannot get too high or too low before they revert back to some natural middle ground. Consequently, a high P/E implies lower future returns and a low P/E implies higher future returns.

Graphically, we can also see why some academics warned that next decade's equity returns couldn't keep pace with the double-digit returns of the 1990s. Consider the 10-year period from 1988 to 1998, omitting the acute bubble at the end of the decade. EPS grew at an annualized rate of 6.4%, but the S&P index grew a whopping 16%! The difference arose from so-called "multiple expansion": an increase in the P/E multiple from about 12 to 28. Academic skeptics use simple logic. If you start from a base P/E multiple of about 25 at the end of 2003, you can only realize aggressive long-term returns that outpace earnings growth with further expansion of the P/E multiple.

*Plugging into the Dividend Model *

The dividend model says that expected return equals dividend yield plus growth in dividends (all in a percentage). Here is the dividend yield on the S&P 500 from 1988 to 2003:

The index ended 2003 with a dividend yield of 1.56%. We only need to add a long-term forecast of growth in the markets' dividends per share. One way to do this is to assume that dividend growth will track with economic growth. And we have several economic measures to choose from, including gross national product, per capita GDP and per capita gross national product.

Let's take real GDP which has been consistent over the long run at 3-4%. To use this measure for estimating future equity returns, we need to acknowledge a realistic relationship between it and dividend growth. It is a big leap to assume that 4% real GDP growth will translate into 4% growth in dividends per share. Dividend growth has rarely, if ever, kept pace with GDP growth and there are two good reasons why.

First, private entrepreneurs create a disproportionate share of economic growth; the public markets often do not participate in the economy's most rapid growth. Second, the dividend yield approach is concerned with per share growth, and there is "leakage" because companies dilute their share base by issuing stock options (while it is true that stock buybacks have an offsetting effect, they rarely compensate for stock option dilution. Publicly traded companies are remarkably consistent "net diluters").

History tells us that real GDP growth of 4% translates, at best, into roughly 2% growth in real dividends per share, or 3% if we are really optimistic. If we add our growth forecast to the dividend yield at the end of 2003, we get about 3.5% to 4.5% (1.56% + 2 to 3% = 3.5% to 4.5%). We happen to match the 4% predicted by the earnings model, and both numbers are expressed in real terms, before inflation.

## Step Two: Estimate the Expected "Risk-Free" Rate

The nearest thing to a safe long-term investment is the Treasury Inflation Protected Security (TIPS). Because the coupon payments and principal are adjusted semi-annually for inflation, the TIPS yield is already a real yield. TIPS are not truly risk-free; if interest rates move up or down, their price moves, respectively, down or up. However, if you hold a TIPS bond to maturity, you can lock in a real rate of return.

In the above chart, we compare the nominal 10-year Treasury yield (blue line) to its equivalent real yield (violet). The real yield simply deducts inflation. The short green line is important. It is the 10-year TIPS yield during the year 2002. We expect the inflation-adjusted yield on the regular 10-year Treasury (violet) to track closely with the 10-year TIPS (green). At the end of 2003, they were close enough. The 10-year TIPS yield was just shy of 2%, and the real yield on the Treasury was about 2.3%. Therefore, the 2% real yield becomes our best guess at future real returns on a safe bond investment.

## Step Three: Subtract the Estimated Bond Return from the Estimated Stock Return

When we subtract our forecast of bond returns from stock returns, we get an estimated equity risk premium of +1.5% to +2.5%:

For an easy method, see *How do I calculate the equity risk premium in Excel? *

## All Sorts of Assumptions

The model attempts a forecast and therefore requires assumptions – enough for some experts to reject the model entirely. However, some assumptions are safer than others. If you reject the model and its outcome, it is important to understand exactly where and why you disagree with it. There are three kinds of assumptions, ranging from safe to dubious.

First, the model does assume that the entire stock market will outperform risk-free securities over the long term. But we could say this is a safe assumption because it allows for the varying returns of different sectors and the short-term vagaries of the market. Take the calendar year 2003, during which the S&P 500 jumped 26% while experiencing a modest decline in the P/E multiple.

No equity risk premium model would have predicted such a jump, but this jump does not invalidate the model. It was caused largely by phenomena that cannot be sustained over the long haul: a 17% increase in the combined forward EPS (i.e. EPS estimates for four future quarters) and an almost unbelievable 60%-plus increase in trailing EPS (according to S&P, from $27.60 to $45.20).

Second, the model requires that real growth in dividends per share – or EPS, for that matter – be limited to very low single-digit growth rates in the long run. This assumption seems secure but is reasonably debated. On the one hand, any serious study of historical returns (like those by Robert Arnott, Peter Bernstein or Jeremy Siegel) proves the sad fact that such growth rarely gets above 2% for a sustained period.

On the other hand, optimists allow for the possibility that technology could unleash a discontinuous leap in productivity that could lead to higher growth rates; who knows, maybe the new economy is just around the bend. But even if this happens, the benefits will surely accrue to selected sectors of the market rather than all stocks. Also, it is plausible that publicly traded companies could reverse their historical conduct, executing more share buybacks, granting fewer stock options and reversing the eroding effects of dilution.

Finally, the model's dubious assumption is that current valuation levels are approximately correct. We've assumed that, at the end of 2003, the P/E multiple of 25 and the price-to-dividend yield of 65 (1 ÷ 1.5% dividend yield) are going to hold going forward. Clearly, this is just a guess! If we could predict valuation changes, the full form of the equity risk premium model would read as follows:

## The Bottom Line

The equity risk premium is calculated as the difference between the estimated real return on stocks and the estimated real return on safe bonds – that is, by subtracting the risk-free return from the expected asset return (the model makes a key assumption that current valuation multiples are roughly correct). The U.S. Treasury bill (T-bill) rate is most often used as the risk-free rate. The risk-free rate is merely hypothetical, as all investments have some risk of loss. However, the T-bill rate is a good measure since they are very liquid assets, easy to understand, and the U.S. government has never defaulted on its debt obligations.

When the dividend yield on stocks is close enough to the TIPS yield (like at the end of 2003, when they were less than 50 basis points apart), the subtraction conveniently reduces the premium to a single number: the long-term growth rate of dividends paid per share.

The equity risk premium can provide some guidance to investors in evaluating a stock, but it attempts to forecast the future return of a stock based upon its past performance. The assumptions about stock returns can be problematic because it is difficult to predict future returns. The equity risk premium assumes the market will always provide greater returns than the risk-free rate, which may not be a valid assumption. The equity risk premium can provide a guide for investors, but it is a tool with significant limitations.