Risk and Rates of Return
What does it mean to take risk when investing?
How are risk and return of an investment measured?
For what type of risk is an average investor
rewarded?
How can investors reduce risk?
What actions do investors take when the return they
require to purchase an investment is different from
the return the investment is expected to produce?
1
RISK AND RATES OF RETURN
Definitions and General Concepts
Probability Distributions
 Expected return
 Standard deviation, 
Risk Attitudes
Coefficient of Variation
Portfolio Risk and Return
 Diversification
 Relevant risk
 Beta coefficients
Determining Return—Capital Asset Pricing Model
Real (Physical) Assets Versus Financial Assets
2
What is Risk?



Dictionary definition—chance of loss
In finance we define risk as the chance that
something other than what is expected occurs—
that is, variability of returns
Risk can be considered “bad”—that is, when the
results are worse than expected (lower-thanexpected returns)—or “good”—that is, when the
results are better than expected (higher-thanexpected returns)
3
Risk
Stand-alone risk—risk of an investment if it was
held by itself, or alone
Portfolio risk—risk of an investment when it is
combined in a portfolio with other investments
4
Risk
We know that an investment is risky if more
than one future outcome is possible—that is,
there are two or more possible payoffs
associated with the investment
A probability distribution summarizes each
possible outcome along with the chance, or
probability, that the outcome will occur
5
Probability Distributions—Example
Economy
Booming
Normal
Recession
Probability
0.2
0.5
0.3
1.0
Risky
Payoff
Asset
18.0%
8.0
-2.0
Risk-Free
Asset
5.0%
5.0
5.0
6
Probability Distributions
Risk-Free Asset
Probability
Probability
0.5
0.5
0.4
0.4
0.3
0.3
0.2
Investment B
0.1
-5
-2 0
Investment A
0.2
0.1
5
8 10
15
18
Return (%)
Discrete Distributions
-5
-2 0
5
8 10
15
18
Return (%)
Continuous Distributions
7
Expected Return
Weighted average of the various possible outcomes
based on the probability that each outcome will
occur
Average of the outcomes if the action—for example,
an investment—was continued over and over again
and the probability for each outcome remained the
same—that is, the probability distribution does not
change
8
Expected Return
Expected rate = r̂ = Pr r + Pr r + L + Pr r
2 2
1 1
n n
of return
n
=  Pr i r
i =1
i
ri
= the result of outcome i
Pri = the probability that outcome i will occur
9
Expected Return
Economy Probability, Pri
Boom
Normal
Recession
0.2
0.5
0.3
Payoff, ri
Pri x r
18.0%
3.6%
8.0
4.0
-2.0
-0.6
r̂ = 7.0%
10
Measuring Stand-Alone Risk
Standard Deviation, 
Measures the tightness, or variability, of a set of
outcomes, or a probability distribution
The tighter the distribution, the less the
variability of the outcomes and the less risk
associated with the event
Measures risk for a single investment—that is an
investment held by itself (standing alone)
11
Measuring Stand-Alone Risk
Standard Deviation, 
Variance, 2—measures the variability of outcomes
2 = ( r1 - r̂ ) 2 Pr1 + ( r 2 - r̂ ) 2 Pr2 + L + ( r n - r̂ ) 2 Prn
n
=  (ri - r̂ ) 2 Pri
i =1
Standard deviation, 
n
2
)
σ = σ = ∑(r i - r̂ Pri
2
i =1
12
Standard Deviation, 
ri
–
r̂
= ri– r̂
(ri– r̂ )2 x Pri
18.0% – 7.0% = 11.0% 121.0 x 0.2
8.0
– 7.0 = 1.0
1.0 x 0.5
-2.0
– 7.0 = -9.0
81.0 x 0.3
2
(ri– r̂)2Pri
= 24.2
= 0.5
= 24.3
= 49.0
σ = 49.0 =
7.0%
13
Risk Attitudes
Risk Aversion—all else equal, risk averse
investors prefer higher returns to lower
returns as well as less risk to more risk
Risk averse investors demand higher returns
for investments with higher risk
14
Risk Aversion
Return
Risk Premium = RP
rRF
r = rRF + RP
Risk-Free Return = rRF
0
Risk
15
Coefficient of Variation
Measures the relationship between risk and
return
Allows for comparisons among various
investments that have different risks and
different returns
Coefficient
= Risk = σ
of Variation Return rˆ
16
Coefficient of Variation
Economy Probability
Boom
0.2
Normal
0.5
Recession
0.3
A
18.0%
8.0
-2.0
Payoffs
B
-5.0%
7.0
15.0
Expected return, r̂
Standard deviation, 
Coeff of variation, CV
7.0%
7.0%
1.00
7.0%
6.9%
0.99
C
55.0%
14.0
-10.0
15.0%
22.5%
1.50
17
Portfolio Risk
By combining investments to form a portfolio,
or collection of investments, diversification can
be achieved
When evaluated in a portfolio, the performance
of a single investment is not very important,
because some investments will perform better
than expected while others will perform worse
than expected
The performance of the portfolio as a whole is
important
18
Portfolio Return
Expected return of a portfolio = weighted average of the
expected returns of the individual investments in the portfolio
r̂P = w1r̂1 + w 2 r̂2 + L + w N r̂N
N
= ∑w jr̂j
j=1
N
 w j = 1.0
j=1
rˆj = expected return for Investment j
wj = proportion of funds invested in Asset j
19
Portfolio Return
Payoffs
Probability
A
B
Portfolio
wA=0.6; wB=0.4
Boom 0.2 18.0% -5.0% 18(0.6) + (-5)(0.4) =
Norm 0.5 8.0
7.0
8(0.6) + 7(0.4) =
Recess 0.3 -2.0
15.0
(-2)(0.6) + 15(0.4) =
r̂
7.0% 7.0%
7(0.6) + 7(0.4) =

7.0% 6.9%
1.5
CV
1.00
0.99
0.22
8.8
7.6
4.8
7.0
r̂ = 0.2(8.8%) + 0.5(7.6%) + 0.3(4.8) = 7.0%
σ = 0.2(8.8 -7.0)2 + 0.5(7.6 -7.0)2 + 0.3(4.8 -7.0)2 =1.5%
20
Portfolio Risk—Diversification
When investments that are not perfectly correlated—that
is, do not mirror each others’ movements on a relative
basis—are combined to form a portfolio, the risk of the
portfolio can be reduced (diversification)
The amount of the risk reduction depends on how the
investments in a portfolio are related
The smaller (greater) the positive (negative) relationship
among the various investments included in a portfolio, the
greater the diversification
Diversification—investing in a combination of stocks
generally reduces risk overall
21
Risk
Stand-alone risk
=  = total risk
= firm-specific risk + market risk
= diversifiable
+ nondiversifiable
22
Firm-Specific Risk
Caused by actions that are specific to the firm—
management decisions, labor characteristics, etc.
The impact of this type of risk on the expected
return associated with an investment is generally
fairly random
This risk component is often called unsystematic
risk
This risk is also called diversifiable risk, because
this portion of total risk can be reduced in a
portfolio of investments
23
Market Risk
Results from movements in factors that affect the
economy as a whole—interest rates, employment, etc.
This risk affects all companies, thus all investments; it
is a system wide risk that cannot be diversified away
This risk is called systematic, or nondiversifiable, risk
Even though all investments are affected by systematic
risk, they are not all affected to the same degree
24
Relevant Risk
Risk that cannot be reduced or diversified away
Relevant risk = systematic, or market risk
“Irrelevant” risk = firm-specific, or unsystematic risk,
because this portion of total risk can be reduced
through diversification
Investors should not be rewarded for taking
“irrelevant” risk—that is, for not diversifying
Risk premiums are based on the amount of
systematic risk associated with an investment
25
Relevant Risk
Return
Risk Premium
based on
systematic risk
rRF
r = rRF + RP
Risk-Free Return
0
Risk (systematic)
26
Concept of Beta
Market, or systematic, risk is measured by
comparing the return on an investment with the
return on the market in general, or an average stock
The market is very well diversified so that any
movements should be the result on systematic risk
only
Beta coefficient, β—measures the relationship
between an individual investment’s returns and the
market’s returns, thus the systematic risk of the
investment
27
Concept of Beta
Return on..the
Stock, r j
.
.
.
.
.
.
. . .
.
.
.
b = slope
Return on
.. the
Market, r M
28
Portfolio Beta Coefficients
A portfolio’s beta, βp is a function of the betas of the individual
investments in the portfolio
A portfolio beta is the weighted average of the betas associated
with the individual investments contained in the portfolio
b P = w 1b1 + w 2 b 2 + L + w N b N
N
=  w jb j
j=1
wj = % of total funds invested in asset j
βj = asset j’s beta coefficient
29
Portfolio Beta Coefficients—Example
Investment
Amount
Invested
Stock A
Stock B
Stock C
Stock D
$ 30,000
20,000
10,000
40,000
100,000
Beta, bj Weight, wj
2.0
1.5
1.0
0.5
bj x wj
0.3
0.2
0.1
0.4
0.6
0.3
0.1
0.2
1.0
bp= 1.2
30
Relationship between Risk
and Rates of Return
Market risk premium = RPM = rM - rRF
RPM = return associated with the riskiness of a portfolio that
contains all the investments in the market
RPM is based on how risk averse investors are on average
Because an investment’s beta coefficient indicates volatility
relative to the market, we can use β to determine the risk
premium for an investment
Investment risk premium = RPInvest = RPM x βInvest
A more volatile investment—that is, an investment with a
higher β—will earn a higher return than a less volatile
investment
31
Relationship between Risk
and Rates of Return
Return = Risk-free rate + Risk Premium
rInvest
=
rRF
5.0
+
RPInvest
6.0
=
rRF
5.0
+
( RP
1.5
4.0M ) β
Invest
=
rRF
5.0
+
( 9.0
rM – 5.0
rRF ) β1.5
Invest
= 11.0
Capital Asset Pricing Model (CAPM)
rRF = 5.0%
rM = 9%
bj = 1.5
32
CAPM Graph—SML
Return, %
Security Market Line, SML
RPM
Risk Premium
based on b
rM
rRF
rj = rRF + RPMbj
Risk-Free
Return
0
1.0
Risk—Measured by b
33
CAPM—Inflation Effects
Return, %
RPM2 = 4
RPM1 = 4
r1= 6 + 4(1.5) = 12
r2= 8 + 4(1.5) = 14
rM2 = 12
rM1 = 10
rRF2 = 8
rRF1 = 6
0
Risk Premium
Risk
Premium
based
on b
based on b
Risk-Free
Risk-Free
Return
Return
1.0
Risk—Measured by b
34
CAPM—Changes in Risk Aversion
Return, %
RP
=5
RPM2
M1 = 4
r1= 6 + 4(1.5) = 12.0
r2= 6 + 5(1.5) = 13.5
rM2 = 11
rM1 = 10
Risk Premium
based on b
rRF = 6
Risk-Free
Return
0
1.0
Risk—Measured by b
35
CAPM—Changes in Beta
Return, %
rB2 = 11
RPM = 4
rB1 = 12
rM = 10
Risk Premium
based on b
rRF = 6
Risk-Free
Return
0
1.0
1.5
Risk—Measured by b
1.25
36
Changes in Equilibrium Stock Prices
Stock prices are not constant due to changes
in rRF, RPM, bx, and so forth.
If the required rate of return, rs, and the
expected rate of return, r̂s , are not equal,
then the price of the investment will change
until rˆs = rs .
37
Risk and Rates of Return
What does it mean to take risk when investing?
 More than one outcome is possible
How are risk and return of an investment
measured?
 Variability of its possible outcomes; greater
variability = greater risk
How can investors reduce risk?
 Risk can be reduced through diversification
38
Risk and Rates of Return
For what type of risk is an average investor
rewarded?
 Investors should only be rewarded for risks they must
take
What actions do investors take when the return they
require to purchase an investment is different from
the return the investment is expected to produce?
 Investors will purchase a security only when its
expected return is greater than or equal to its
required return
39