Introduction to Corporate Finance

6: The Trade-Off Between Risk and Return

If we use the complete 111-year history of US returns given, rather than just the last 18 years, we

arrive at the following estimates of the average and standard deviation of historical returns:

Average return = 11.4 % Standard deviation=20.0%

These figures indicate that the US experience from 1993–2010, described in Table 6.4, is roughly

similar to the entire twentieth century.

Let’s return to our assumption that the underlying probability distribution governing share returns is

approximately normal. From the last 111 years, we have estimates of the mean (11.4%) and the standard deviation

(20.0%) of that distribution. Those estimates allow us to make a few other interesting descriptive statements

about the behaviour of ordinary shares. First, for any normal distribution, 68% of all observations fall between

one standard deviation above and one standard deviation below the mean, and 95% of the observations should

be within two standard deviations of the mean. In the present context, this implies that share returns should fall

between –8.6% and 31.4% (11.4%, plus or minus 20.0%) in a little more than two-thirds of the years. How does

that prediction compare with the historical evidence? From 1900–2010, returns fell within the range of –8.6 to

31.4% in 75 years, or 67.6% of the time. All in all, the distribution of historical returns seems close to a bell curve.

Table 6.5 shows the average annual return and the standard deviation of returns for equities (shares),

Treasury bonds and Treasury bills during the last 111 years. We saw the average returns previously in

Table 6.1a, but now we have a specific measure of risk to couple with the mean returns. Once again,

we see evidence that risk and return are positively linked, at least if we define risk to mean volatility (as

captured by the standard deviation). The average return on shares is more than double the average bond

return, but shares are almost 2.5 times more volatile than bonds. Bonds offer a premium over bills, but

the standard deviation of bond returns is roughly three times the standard deviation for bills. Switching

from nominal to real returns lowers the average returns, but it does not change the basic story. Asset

classes that display greater volatility pay higher returns on average.

TABlE 6.5 AVERAGE RETURNS AND STANDARD DEVIATION FOR EQUITIES, BONDS AND BILLS, 1900–2010

Nominal returns Real returns

Asset Average (%) Std. dev. (%) Average (%) Std. dev. (%)

Equities 11.4 20.0 8.3 20.3

Bonds 5.6 8.3 2.4 10.1

Bills 3.9 2.8 1.0 4.7

Source: Elroy Dimson, Paul Marsh and Mike Staunton, ‘Triumph of the Optimists,’ in Global Investment Returns Yearbook 2010. Published by

ABN AMRO, London. Updates provided by Dimson et al. to 2009. Author’s calculations for 2010. Reprinted with permission.

CONCEPT REVIEW QUESTIONS 6-3

6 Use Figure 6.5 to estimate the probability that a portfolio of ordinary shares will earn a return of at

least 20% in a given year.

7 Suppose nominal bond returns approximately follow a normal distribution. Using the data in Table 6.5,

construct a range that should contain 95% of historical bond returns. (Hint: Use the mean and standard

deviation of bond returns to calculate the endpoints of this range.) Next, refer to Figure 6.4. Is the number

of years with bond returns outside the range you just calculated approximately what you expected?

8 Suppose there is an asset class with a standard deviation that lies about halfway between the

standard deviations of shares and bonds. Based on Figure 6.6, what would you expect the average

return on this asset class to be?