Opportunity cost of capital depends on the risk of the project. Thus, to be able to determine the opportunity cost of capital one must understand how to measure risk and how investors are compensated for taking risk.

5.1. Risk and risk premia

The risk premium on financial assets compensates the investor for taking risk. The risk premium is the difference between the return on the security and the risk free rate.

To measure the average rate of return and risk premium on securities one has to look at very long time periods to eliminate the potential bias from fluctuations over short intervals.

Over the last 100 years U.S. common stocks have returned an average annual nominal compounded rate of return of 10.1% compared to 4.1% for U.S. Treasury bills. As U.S. Treasury bill has short maturity and there is no risk of default, short-term government debt can be considered risk-free. Investors in common stocks have earned a risk premium of 7.0 percent (10.1 - 4.1 percent.). Thus, on average investors in common stocks have historically been compensated with a 7.0 percent higher return per year for taking on the risk of common stocks.

Annual return

Std. variation

Risk premium

U.S. Treasury Bills

4.1%

4.7%

0.0%

U.S. Government Bonds

4.8%

10.0%

0.7%

U.S. Common Stocks

10.1%

20.2%

7.0%

Table 1: Average nominal compounded returns, standard deviation and risk premium on U.S. securities, 1900-2000.

Source: E. Dimson, P.R. Mash, and M Stauton, Triumph of the Optimists: 101 Years of Investment returns, Princeton University Press, 2002.

Across countries the historical risk premium varies significantly. In Denmark the average risk premium was only 4.3 percent compared to 10.7 percent in Italy. Some of these differences across countries may reflect differences in business risk, while others reflect the underlying economic stability over the last century.

The historic risk premium may overstate the risk premium demanded by investors for several reasons. First, the risk premium may reflect the possibility that the economic development could have turned out to be less fortunate. Second, stock returns have for several periods outpaced the underlying growth in earnings and dividends, something which cannot be expected to be sustained.

The risk of financial assets can be measured by the spread in potential outcomes. The variance and standard deviation on the return are standard statistical measures of this spread.

Variance

Expected (average) value of squared deviations from mea n. The variance measures the return volatility and the units are percentage squared.

| Where r denotes the average return and N is the total number of observations.

Standard deviation

Square root of variance. The standard deviation measures the return volatility and units are in percentage.

Using the standard deviation on the yearly returns as measure of risk it becomes clear that U.S. Treasury bills were the least variable security, whereas common stock were the most variable. This insight highlights the risk-return tradeoff, which is key to the understanding of how financial assets are priced.

Risk-return tradeoff

Investors will not take on additional risk unless they expect to be compensated with additional return

The risk-return tradeoff relates the expected return of an investment to its risk. Low levels of uncertainty (low risk) are associated with low expected returns, whereas high levels of uncertainty (high risk) are associated with high expected returns.

It follows from the risk-return tradeoff that rational investors will when choosing between two assets that offer the same expected return prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact trade-off will differ by investor based on individual risk aversion characteristics (i.e. the individual preference for risk taking).

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