## What is 'Equity Risk Premium'

Equity risk premium, also referred to as simply equity premium, is the excess return that investing in the stock market provides over a risk-free rate, such as the return from government treasury bonds. This excess return compensates investors for taking on the relatively higher risk of equity investing. The size of the premium will vary depending on the level of risk in a particular portfolio and will also change over time as market risk fluctuates. As a rule, high-risk investments are compensated with a higher premium.

## BREAKING DOWN 'Equity Risk Premium'

The equity risk premium is based on the idea of the risk-reward tradeoff. As a forward-looking quantity, the equity-risk premium is theoretical and cannot be known precisely, since no one knows how a particular stock, basket of stocks, or the stock market as a whole will perform in the future. It can be estimated as a backward-looking quantity by observing stock market and government bond performance over a defined period of time, for example from 1950 to the present. Estimates, however, vary wildly depending on the time frame and method of calculation.

Some economists argue that, although certain markets in certain time periods may display a considerable equity risk premium, it is not in fact a generalizable concept. They argue that too much focus on specific cases – e.g. the U.S. stock market in the last century – has made a statistical peculiarity seem like an economic law. Several stock exchanges have gone bust over the years, for example, so a focus on the historically exceptional U.S. market may distort the picture. This focus is known as survivorship bias.

## Estimates of the Equity Risk Premium

The majority of economists, however, agree that the concept of an equity risk premium is valid: over the long term, markets will compensate investors more for taking on the greater risk of investing in stocks. How exactly to calculate this premium is disputed. A survey of academic economists gives an average range of 3-3.5% for a 1-year horizon, and 5-5.5% for a 30-year horizon. CFOs, meanwhile, estimate the premium to be 5.6% over T-bills (U.S. government debt obligations with maturities of less than one year) and 3.8% over T-bonds (maturities of greater than ten years).

Earlier in the century, estimates were more muted: a 1938 book by John Burr Williams puts the premium at 1.5%. In fact, the second half of the 20th century saw a relatively high equity risk premium, over 8% by some calculations, versus just under 5% for the first half of the century. Given that the century ended at the height of the dot-com bubble, however, this arbitrary window may not be ideal.

## Calculating the Equity Risk Premium

To calculate the equity risk premium, we can begin with the capital asset pricing model (CAPM), which is usually written:

**R _{a} = R_{f} + β_{a} (R_{m} - R_{f})**

where:

R_{a} = expected return on investment in "a"

R_{f} = risk-free rate of return

β_{a} = beta of "a"

R_{m} = expected return of market

In the context of the equity risk premium, *a* is an equity investment of some kind, such as 100 shares of a blue-chip stock, or a diversified stock portfolio. If we are simply talking about the stock market (a = m), then R_{a} = R_{m}. The beta coefficient is a measure of a stock's volatility, or risk, versus that of the market; the market's volatility is conventionally set to 1, so if a = m, then β_{a} = β_{m} = 1. R_{m} - R_{f} is known as the market premium; R_{a} - R_{f} is the risk premium. If *a* is an equity investment, then R_{a} - R_{f} is the equity risk premium; if *a *=* m*, then the market premium and the equity risk premium are the same.

The equation for the equity risk premium, then, is a simple reworking of the CAPM:

**Equity Risk Premium = R _{a} - R_{f} = β_{a} (R_{m} - R_{f})**

This summarizes the theory behind the equity risk premium, but questions arise in practice. If, instead of calculating expected rates of return, we want to plug in historical rates of return and use those to estimate future rates, the calculation is fairly straightforward. If, however, we are attempting a forward-looking calculation, the question is: how do you estimate the expected rate of return?

One method is to use dividends to estimate long-term growth, using a reworking of the Gordon Growth Model:

**k = D / P + g**

where:

k = expected return, expressed as a percentage (this value could be calculated for R_{a} or R_{m})

D = dividends per share

P = price per share

g = annual growth in dividends, expressed as a percentage

Another is to use growth in earnings, rather than growth in dividends. In this model, expected return is equal to the earnings yield, the reciprocal of the P/E ratio.

**k = E / P**

where:

k = expected return

E = trailing twelve month earnings per share

P = price per share

The drawback for both of these models is that they do not account for valuation, that is, they assume the stocks' prices will never correct. Given that we can observe stock market booms and busts in the recent past, this drawback is not insignificant.

Finally, the risk-free rate of return is usually calculated using U.S. government bonds, since they have a negligible chance of default. This can mean T-bills or T-bonds. To arrive at a real rate of return, that is, adjusted for inflation, it is easiest to use Treasury inflation-protected securities (TIPS), as these already account for inflation. It is also important to note that none of these equations account for tax rates, which can dramatically alter returns.