238 Part 3 Financial Assets
(^13) See our tutorials on the text’s web site (http://academic.cengage.com/finance/brigham) or your calculator
manual for instructions on calculating historical standard deviations.
All " nancial calculators (and Excel) have easy-to-use functions for " nding #
based on historical data.^13 Simply enter the rates of return and press the key marked
S (or Sx) to obtain the standard deviation. However, neither calculators nor Excel
have a built-in formula for " nding # where probabilistic data are involved. In
those cases, you must go through the process outlined in Table 8-2.
8-2d Measuring Stand-Alone Risk:
The Coefficient of Variation
If a choice has to be made between two investments that have the same expected
returns but different standard deviations, most people would choose the one with
the lower standard deviation and, therefore, the lower risk. Similarly, given a
choice between two investments with the same risk (standard deviation) but dif-
ferent expected returns, investors would generally prefer the investment with the
higher expected return. To most people, this is common sense—return is “good”
and risk is “bad”; consequently, investors want as much return and as little risk as
possible. But how do we choose between two investments if one has the higher
expected return but the other has the lower standard deviation? To help answer
that question, we use another measure of risk, the coef! cient of variation (CV),
which is the standard deviation divided by the expected return:
8-3 Coe$ cient of variation " CV " !__
r&
The coef! cient of variation shows the risk per unit of return, and it provides a more mean-
ingful risk measure when the expected returns on two alternatives are not the same. Since
U.S. Water and Martin Products have the same expected return, the coef" cient of
variation is not necessary in this case. Here the " rm with the larger standard
deviation, Martin, must have the larger coef" cient of variation. In fact, the coef" -
cient of variation for Martin is 54.22/10 $ 5.42 and the coef" cient of variation for
U.S. Water is 3.87/10 $ 0.39. Thus, Martin is about 14 times riskier than U.S. Water
on the basis of this criterion.
8-2e Risk Aversion and Required Returns
Suppose you inherited $1 million, which you plan to invest and then retire on the
income. You can buy a 5% U.S. Treasury bill, and you will be sure of earning $50,000
of interest. Alternatively, you can buy stock in R&D Enterprises. If R&D’s research
programs are successful, your stock will increase to $2.1 million. However, if the
research is a failure, the value of your stock will be zero and you will be penniless.
You regard R&D’s chances of success or failure as 50-50, so the expected value of
the stock a year from now is 0.5($0) " 0.5($2,100,000) $ $1,050,000. Subtracting the
$1 million cost leaves an expected $50,000 pro" t and a 5% rate of return, the same
as for the T-bill:
Expected rate of return "
Expected ending value # Cost
___ (^) Cost
" $1,050,000 __# $1,000,000
$1,000,000
" __$50,000
$1,000,000
" 5%
Given the choice of the sure $50,000 pro" t (and 5% rate of return) and the risky
expected $50,000 pro" t and 5% return, which one would you choose? If you choose the
Coefficient of
Variation (CV)
The standardized measure
of the risk per unit of
return; calculated as the
standard deviation
divided by the expected
return.
Coefficient of
Variation (CV)
The standardized measure
of the risk per unit of
return; calculated as the
standard deviation
divided by the expected
return.