CHAPTER 6
The Meaning and Measurement
of Risk and Return
CHAPTER ORIENTATION
In this chapter, we examine the factors that determine rates of return (discount rates) in the
capital markets. We are particularly interested in the relationship between risk and rates of
return. We look at risk both in terms of the riskiness of an individual security and that of a
portfolio of securities.
CHAPTER OUTLINE
I. Expected Return Defined and Measured
A. The expected benefits or returns to be received from an investment come from the cash
flows the investment generates.
B. The rate of return earned from an investment can be calculated as the ratio of the dollar
gain divided by the amount of the investment at the beginning of the period. We can
formalize these calculations as follows:
Holding-period dollar gain
Holding-period
dollar gain
= Price end of +
period

cash distribution
(dividend)
 Price beginning
(6-1)
of period
Price end of period  Price beginning of period
Price beginning of period
Holding-period rate of return
Rate of
return, r


dollar gain
Price beginning of period
(6-2)
Price end of period +dividend-Price begin of period
Period begin of Period
C. In an uncertain world, we must use expected cash flow, CF , for computing expected
gains and rates of return. We compute an expected cash flow as follows:
6-1
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6-2  Keown/Martin/Petty
CF =
3)
Instructor’s Solutions Manual
CF1(Pb1) + CF2(Pb)2 + … + CFn(Pbn)
where
n
CFi
Pbi
=
=
=
(6-
the number of possible states of the economy
the cash flow in the ith state of the economy
the probability of the ith state of the economy
D. Similar to the calculation of expected cash flow, the expected rate of return can be
calculated as follows:
Expected rate
4)
of return, r
where
r1(Pb1) + r2(Pb2) + … + rn(Pbn)
=
n
ri
Pbi
=
=
=
(6-
the number of possible states of the economy
the rate of return in the ith state of the economy
the probability of the ith state of the economy
II. Risk Defined and Measured
A. Risk can be defined as the potential variability in expected future cash flows.
B. Statistically, risk may be measured by the standard deviation about the expected cash
flow.
1. The standard deviation of returns is the square root of the variance of returns, a
statistical measure of the dispersion of returns from their expected value.
Variance is calculated as follows:
Variance in = [(r1 – r)2(Pb1)] + [(r2 – r)2(Pb2)] + … [(rn – r)n(Pbn)]
5)
rates of return
where
n
=
the number of possible states of the economy
ri
=
the rate of return in the ith state of the economy
r
=
the expected rate of return
Pbi
=
the probability of the ith state of the economy
(6-
2. The standard deviation of returns is the square room of the variance of returns,
calculated as follows:
Standard deviation in =
rates of return
where
n
ri
r
Pbi
 r – r 2  Pb      r 2 – r 2  Pb      r – r   Pb  
1
2 
n 
n
 n
 1

=
=
=
=
the number of possible states of the economy
the rate of return in the ith state of the economy
the expected rate of return
the probability of the ith state of the economy
III. Rates of Return: The Investor's Experience
A. Data have been compiled by Ibbotson Associates on the actual returns for various
portfolios of securities since 1926.
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Foundations of Finance, Tenth Edition  6-3
B. The following portfolios, plus the inflation rate, have been studied:
1. Common stocks of large firms
2. Common stocks of small firms
3. Corporate bonds
4. Intermediate U.S. government bonds
5. U.S. Treasury bills
6. Inflation rate
C. From studying the average returns and standard deviations of the above portfolios, we
learn that investors historically have received greater returns for greater risk taking.
D. All of these portfolios earned returns exceeding the inflation rate. However, the
portfolio that consistently earned the highest rate of return, on average, has been a
portfolio made up of common stocks.
IV. Risk and Diversification
A. The market rewards diversification. We can lower risk without sacrificing expected
return, or we can increase expected return without having to assume more risk by
holding a diversified portfolio of investments.
B. Total variability of returns can be divided into the following:
1. The variability of returns unique to the security (diversifiable or unsystematic
risk)
2. The risk related to market movements (nondiversifiable or systematic risk)
C. By diversifying, the investor can eliminate the “unique” security risk. Systematic risk,
however, cannot be diversified away. A portfolio of 20–25 different securities can
essentially eliminate diversifiable risk from a portfolio of stocks.
D. For a sample of observed holding period returns, the average holding period return and
standard deviation of returns are calculated below.
Average holding = return in period 1 + return in period 2 + … + return in period n
8)
period return
number of periodic returns
 r1 – r  2     r2 – r  2     rn – r  n 




Standard deviation in =
9)
rates of return
where
(6-
(6-
(n – 1)
n
ri
r
=
=
=
the number of possible states of the economy
the rate of return in the ith period
the expected rate of return
E. Measuring Market Risk
1. The characteristic line tells us the average movement in a firm’s stock price in
response to a movement in the general market, measured by a broad market
portfolio such as the S&P 500 Index. The slope of the characteristic line, which
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Instructor’s Solutions Manual
has come to be called beta, is a measure of a stock’s systematic or market risk.
The slope of the line is merely the ratio of the “rise” of the line relative to the
“run” of the line, that is, the change in a security's returns relative to the change in
returns in a market portfolio.
2. If a security’s beta equals one, a 10 percent increase (decrease) in market returns
will produce, on average, a 10 percent increase (decrease) in the security’s returns.
If a security’s beta is less than one, a 10 percent increase (decrease) in market
returns will produce, on average, a less than 10 percent increase (decrease) in the
security’s returns. If a security’s beta is greater than one, a 10 percent increase
(decrease) in market returns will produce, on average, a greater than 10 percent
increase (decrease) in the security’s returns.
3. A security with a higher beta is more volatile and thus more risky than a security
with a lower beta value.
F. A portfolio’s beta is equal to the average of the betas of the stocks in the portfolio, with
the weights being equal to the proportion of the portfolio invested in each security.
G. Diversifying among different kinds of assets is called asset allocation. Compared to
diversification within the different asset classes, the benefits received are far greater
through effective asset allocation.
H. Asset allocation between stocks and bonds clearly affects the expected returns of the
portfolios, as does how long you hold the portfolio. The more bonds in your portfolio
and the longer you hold the portfolio, the less the variability of returns will be.
V. The Investor’s Required Rate of Return
A. The required rate of return is the minimum rate necessary to compensate an investor
for accepting the risk he or she associates with the purchase and ownership of an asset.
B. Two factors determine the required rate of return for the investor:
1. The risk-free rate of return, which recognizes the time value of money
2. The risk premium, which considers the riskiness (variability of returns) of the
asset and the investor’s attitude toward risk
3. These factors can be expressed in the equation below.
Investor’s required
rate of return
=
risk-free rate
of return
+
risk
premium
(6-11)
(a) The risk-free rate of return is the rate of return on risk-free investments such
as short-term U. S. government securities.
(b) The risk premium is the additional return expected for assuming risk. It is
calculated as follows:
Risk
= investor’s required
premium
rate of return, r
− risk-free rate of
return, rf
(6-12)
C. Capital Asset Pricing Model—CAPM
1. The Capital Asset Pricing Model (CAPM) is a model that states that the
expected rate of return of an investment is a function of (1) the risk-free rate, (2)
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Foundations of Finance, Tenth Edition  6-5
the investment’s systematic risk, and (3) the expected risk premium for the market
portfolio comprised of all risky securities.
2. The required rate of return for a given security can be expressed as
Required rate  risk-free rate  beta   market return  risk-free rate 
(6-13)
3. Security market line
a. The security market line is a graphical representation of the CAPM.
b. The graph shows a security’s appropriate rate of return given its level of
systematic risk, beta, as defined according to the CAPM.
c. Traditionally, the security’s expected return is on the vertical axis, and the
level of systematic risk, beta, is on the horizontal axis.
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