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Chapter 7 Introduction to Risk and Return 169estimate of required returns over the next year or so. However, changes in the dividend yield
tell companies next to nothing about the expected risk premium over the next 10 or 20 years.
It seems that, when estimating the discount rate for longer term investments, a firm can safely
ignore year-to-year fluctuations in the dividend yield.Out of this debate only one firm conclusion emerges: Do not trust anyone who claims to
know what returns investors expect. History contains some clues, but ultimately we have to
judge whether investors on average have received what they expected. Many financial econo-
mists rely on the evidence of history and therefore work with a risk premium of about 7%. The
remainder generally use a somewhat lower figure. Brealey, Myers, and Allen have no official
position on the issue, but we believe that a range of 5% to 8% is reasonable for the risk pre-
mium in the United States.◗ FIGURE 7.4
Dividend yields in the
U.S. 1900–2014.
Source: R. J. Shiller, “Long
Term Stock, Bond, Interest
Rate and Consumption Data
since 1871,” http://www.econ.yale.
edu/~shiller/data.htm. Used
with permission.012345678910190019051910191519201925193019351940194519501955196019651970197519801985199019952000200520102014
YearDividend yield, %7-2 Measuring Portfolio RiskYou now have a couple of benchmarks. You know the discount rate for safe projects, and you
have an estimate of the rate for average-risk projects. But you don’t know yet how to estimate
discount rates for assets that do not fit these simple cases. To do that, you have to learn (1) how
to measure risk and (2) the relationship between risks borne and risk premiums demanded.
Figure 7.5 shows the 115 annual rates of return for U.S. common stocks. The fluctua-
tions in year-to-year returns are remarkably wide. The highest annual return was 57.6% in
1933—a partial rebound from the stock market crash of 1929–1932. However, there were
losses exceeding 25% in six years, the worst being the –43.9% return in 1931.
Another way of presenting these data is by a histogram or frequency distribution. This is
done in Figure 7.6, where the variability of year-to-year returns shows up in the wide “spread”
of outcomes.Variance and Standard Deviation
The standard statistical measures of spread are variance and standard deviation. The vari-
ance of the market return is the expected squared deviation from the expected return. In other
words,Variance (r ̃ (^) m) = the expected value of ( ̃r (^) m − rm)^2