Lecture Topic 10: Return and Risk
Asset Pricing Model - CAPM
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Students
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to Cox
Cox Business
MBA Students
Financial
Management
FINAFINA
6214:3320:
International
Financial
Markets
Return and Risk
The Capital Asset Pricing Model
(CAPM)
What is Investment Risk
• Risk, in general, refers to the chance that
some unfavorable event will occur
• Investment risk pertains to the probability of
realized (actual) returns being less than
expected returns
– The greater the chance of low or negative returns, the
riskier the investment
Types of Risk
• Stand-alone risk
– Riskiness of an asset held in isolation
• Portfolio risk
– Riskiness of an asset held as one of a number of assets
in a portfolio
– In a portfolio context, risk can be divided into two
components
• Diversifiable (firm-specific) risk
• Market (non-diversifiable) risk
Individual Securities
• The characteristics of individual securities
that are of interest are the:
– Expected Return: Return on a risky asset expected in the future
E ( Ra ) 
S
 p(state)  R(state)
state1
a
– Variance and Standard Deviation: Measures of dispersion of
an asset’s returns around its expected, or mean, return
Var ( Ra )   a2 
SD( Ra )   a 
S
2
p
(
state
)

[
R
(
state
)

E
(
R
)]

a
a
state1
S
2
p
(
state
)

[
R
(
state
)

E
(
R
)]

a
a
state1
Individual Securities
• The characteristics of individual securities
that are of interest are the:
– Covariance and Correlation (to another security or
index): statistics measuring the interrelationship between two securities
Cov( Ra , Rb ) 
S
 p(state)  [ R(state)
state1
a
 E ( Ra )] [ R( state)b  E ( Rb )]
Cov( Ra , Rb )
Corr ( Ra , Rb ) 
 a  b
Expected Return, Variance, Standard
Deviation, and Covariance
• Consider the following two risky asset world
– There is a 1/3 chance of each state of the economy
– The only assets are a stock fund and a bond fund
Scenario
Recession
Normal
Boom
Rate of Return
Probability Stock Fund Bond Fund
33.3%
-7%
17%
33.3%
12%
7%
33.3%
28%
-3%
Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond
Rate of
Return
17%
7%
-3%
7.00%
0.0067
8.2%
Fund
Squared
Deviation
0.0100
0.0000
0.0100
Expected Return
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
E ( RS )  1  (7%)  1  (12%)  1  (28%)
3
3
3
E ( RS )  11%
Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
(7%  11%)  .0324
2
Variance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
1
.0205  (.0324  .0001  .0289)
3
Standard Deviation
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
14.3%  0.0205
Covariance
Scenario
Recession
Normal
Boom
Sum
Covariance
Stock
Bond
Deviation Deviation
-18%
10%
1%
0%
17%
-10%
Product
-0.0180
0.0000
-0.0170
Weighted
-0.0060
0.0000
-0.0057
-0.0117
-0.0117
– Deviation compares return in each state to the
expected return
– Weighted takes the product of the deviations
multiplied by the probability of that state
Correlation
Cov( Ra , Rb )

 a b
 .0117

 0.998
(.143)(.082)
Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock Fund
Rate of
Squared
Return Deviation
-7%
0.0324
12%
0.0001
28%
0.0289
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
0.0100
7%
0.0000
-3%
0.0100
7.00%
0.0067
8.2%
– Note that stocks have a higher expected return than
bonds and higher risk
– Now turn to the risk-return tradeoff of a portfolio that
is 50% invested in bonds and 50% invested in stocks
Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
– The rate of return on the portfolio is a weighted
average of the returns on the stocks and bonds in the
portfolio
RP  wB RB  wS RS
5%  50%  (7%)  50%  (17%)
Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
– The expected rate of return on the portfolio is a
weighted average of the expected returns on the
securities in the portfolio
E ( RP )  wB E ( RB )  wS E ( RS )
9%  50%  (11%)  50%  (7%)
Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
– The variance of the rate of return on the two risky asset
portfolio is:
σ P2  wB2 σ B2  wS2σ S2  2wB wS σ B σ S ρB,S
σ P2  (50%) 2 (0.67%)  (50%) 2 (2.05%)  2(50%)(50%)(8.16%)(14.31%)  B,S
Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
– The variance of the rate of return on the two risky asset
portfolio is:
σ P2  wB2 σ B2  wS2σ S2  2wB wS CovB,S
σ P2  (50%) 2 (0.67%)  (50%) 2 (2.05%)  2(50%)(50%)CovB,S
Portfolios
• CovarianceCov( R , R )   p(state)  [ R(state)
S
B
S
state1
B
( 13 )  [17%  7.0%]  [7.0%  11.0%]  0.006
( 13 )  [7%  7.0%]  [12.0%  11.0%]  0.000
( 13 )  [3%  7.0%]  [28.0%  11.0%]  0.005667
Cov( RB , RS )  0.011667
• Correlation
Corr ( RB , RS ) 
Corr ( RB , RS ) 
Cov( RB , RS )
 B  S
 0.011667
 0.999
0.0816  0.1431
 E ( RB )] [ R( state) S  E ( RS )]
Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.0016
0.0000
0.0012
9.0%
0.0010
3.08%
– Observe the decrease risk that diversification offers
• Particularly when the two assets are almost perfectly
negatively correlated!!!
– An equally weighted portfolio (50% in stocks and
50% in bonds) has less risk than either stocks or
bonds held in isolation
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
100%
stocks
11.0%
10.0%
9.0%
8.0%
100%
bonds
7.0%
6.0%
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
10.0%
100%
stocks
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level of
risk or less.
return
Portfolios with Various Correlations
100%
stocks
 = -1.0
100%
bonds
 = 1.0
 = 0.2
P
• Relationship depends on correlation coefficient
 1.0    1.0
• If ρ = +1.0, no risk reduction is possible
• If ρ = -1.0, complete risk reduction is possible
return
The Efficient Set for Many Securities
Individual Assets
P
– Consider a world with many risky assets
– We can still identify the opportunity set of risk-return
combinations of various portfolios
return
The Efficient Set for Many Securities
minimum
variance
portfolio
Individual Assets
P
– The section of the opportunity set above the minimum
variance portfolio is the efficient frontier
Stand-alone vs. Portfolio Risk
• Stand-alone risk
– Measured by the dispersion of returns about the mean
(standard deviation) of an individual asset and is
relevant only for assets held in isolation
• Portfolio risk
– The risk of an asset when held in a portfolio
– In portfolio context, risk can be divided into:
• Diversifiable risk (also called firm-specific, unique, or
unsystematic)
• Non-diversifiable risk (also called market or systematic)
– As long as correlation coefficient < 1, portfolio risk is
lower than stand-alone risk
Diversification and Portfolio Risk
• Diversification can substantially reduce the
variability of returns without an equivalent
reduction in expected returns
– This reduction in risk arises because worse than
expected returns from one asset are offset by better
than expected returns from another asset
– However, there is a minimum level of risk that cannot
be diversified away, and that is the systematic portion
Portfolio Risk and Number of Stocks
– In a large portfolio, the variance terms are effectively
diversified away, but the covariance terms are not!

Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
What is Diversifiable Risk?
• Caused by company specific events (e.g.,
lawsuits, strikes, winning or losing major
contracts, etc.)
– Risk factors that affect a limited number of assets
– Also known as unique risk or unsystematic risk
– Risk that can be eliminated by combining assets into a
portfolio
• Effects of such events on a portfolio can be eliminated by
diversification
– If we hold only one asset, or assets in the same
industry, then we are exposing ourselves to risk that we
could diversify away
What is Market Risk?
• Stems from such external events as war,
inflation, recession, changes in GDP and/or
interest rates
– Risk factors that affect a large number of assets
– Also known as non-diversifiable risk or systematic risk
• Known as systematic risk since it shows the degree to which
a stock moves with other stocks
– Because all firms are effected simultaneously by these
factors, market risk cannot be eliminated by combining
assets into a portfolio
• Effects of such factors on a portfolio cannot be eliminated by
diversification
What is Total Risk?
• Total risk = systematic risk + unsystematic
risk
– The standard deviation of returns of an individual asset
is a measure of total risk
– For well-diversified portfolios, unsystematic risk is
very small
– Consequently, the total risk for a diversified portfolio
is essentially equivalent to the systematic risk
Stock Prices and Information
• Actual (realized) return = expected return +
unexpected return (surprise)
– Surprise is risk of investment (what we couldn’t
forecast prior to buying the asset)
• General diversification information
–
–
–
–
Most stocks are positively correlated: Corrx, y  0.65
Average stand-alone risk:  x  49.24%
Average portfolio risk:  P  20%
Combining stocks in a portfolio lowers risk
• Except when : Corr  1.0
What would happen to the
riskiness of a 1 stock portfolio as
more randomly selected stocks
were added
The standard deviation of the
portfolio would decrease because the
added stocks would not be perfectly
correlated
Standard Deviations of Annual
Portfolio Returns
Number of Stocks
in Portfolio
Average Standard
Deviation of Annual
Portfolio Returns
1
10
50
100
300
500
1,000
49.24%
23.93%
20.20%
19.69%
19.34%
19.27%
19.21%
Ratio of Portfolio
Standard Deviation to
Standard Deviation
of a Single Stock
1.00
0.49
0.41
0.40
0.39
0.39
0.39
These figures are from Table 1 in Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of
Financial and Quantitative Analysis 22 (September 1987), pp. 353–64. They were derived from E. J. Elton and M. J.
Gruber, “Risk Reduction and Portfolio Size: An Analytic Solution,” Journal of Business 50 (October 1977), pp. 415–37.
Risk and the Sensible Investor
If you chose to hold a one-stock
portfolio and thus are exposed to
more risk than diversified investors,
would you be compensated for all the
risk that you bear?
Risk and the Sensible Investor
– NO!
– Stand-alone risk (as measured by individual stock’s σ)
is not important to a well-diversified investor since it
can be reduced through diversification
– Since diversifiable risk can be easily eliminated,
rational investors will do so
• Rational risk averse investors are concerned with portfolio
risk σP, which is based on market risk, and the contribution
of a security to the risk of the entire portfolio
– In equilibrium, there can be only one price, or return,
for a given security
• If majority of investors are diversified and they determine
prices, then no compensation can be earned for the additional
risk of a one-stock portfolio
Risk Free Assets
• So far we have examined portfolios of risky
assets…
• What happens if one of the assets in our
portfolio is risk free, i.e., σf = 0?
E ( R p )  wx E ( Rx )  w f R f
SD( R p )  w   w f   2wx w f Covx , f  wx x
2
x
2
x
2
f
• Or
SD( R p )  wx2 x2  w f  2f  2wx w f  x f Corrx , f  wx x
Risk Free Assets
• Assume there is a risky asset, x, and a risk
free asset, f
E ( Rx )  0.16,  x  8%
• Risky Asset:
• Risk Free Asset: E ( R f )  0.06,  f  0%
– You have $100, you put $50 in x and $50 in f (i.e.,
lending $50 at the risk free rate)
– The weights are: wx  AmountInX  $50  0.50
InitialWea lth
wf 
$100
AmountInF
$50

 0.50
InitialWea lth $100
E ( R p )  wx E ( Rx )  w f R f  0.5(16%)  0.5(6%)  11%
SD( R p )  wx x  0.5(8%)  4%
return
Optimal Portfolio with a Risk-Free
Asset
100%
stocks
Balanced
fund
Rf
100%
bonds
σ
– In addition to stocks and bonds, consider a world that
also has risk-free securities, e.g., T-bills
– Now investors can allocate their money across the
Treasury securities and a balanced mutual fund
Riskless Borrowing and Lending
• The Capital Market Line (CML)
– Expected Portfolio Return:
E ( R p )  R f  [ Slope ] p
– Slope = Market Price of Risk:
Slope 
E ( RM )  R f
M
– Any portfolio on CML is a combination of M and f
E ( R p )  wM E ( RM )  (1  wM ) R f
 p  wM  M
– If we invest in the risk-free asset, f, and in M:
 p M
– If we borrow money at the risk-free rate and invest in M
 p M
Riskless Borrowing and Lending
return
• The Capital Market Line (CML)
Rf
P
– With a risk-free asset available and the efficient frontier
identified, we choose the capital allocation line with the
steepest slope
– This is the Capital Market Line (CML)
return
Market Equilibrium
M
Rf
P
– With the optimal capital allocation line (i.e., the CML)
identified, all investors choose a point along the line
• i.e., some combination of the risk-free asset and the market
portfolio, M
– In a world with homogeneous expectations, M is the
same for all investors
return
Market Equilibrium
100%
stocks
Balanced
fund
Rf
100%
bonds
P
– Where the investor chooses along the Capital Market
Line (CML) depends on her risk tolerance
– The important point is that all investors have the same
CML
The Systematic Risk Principal
• Risk when holding the Market Portfolio, M
– Researchers have shown that the best measure of the
risk of a security in a large portfolio is the beta (β) of
the security
– Beta measures the responsiveness of a security to
movements in the market portfolio, i.e., systematic
risk
• The reward for bearing risk depends only upon systematic
risk since unsystematic risk can be diversified away
i 
Cov( Ri , RM )
 2 ( RM )
Security Returns
Estimating β with Regression
Slope = 
bi
Return on
market %
Ri = a i + iRm + ei
Portfolio Betas (βp)
• Portfolio Betas
– While portfolio variance is not equal to a simple
weighted sum of individual security variances,
portfolio betas are equal to the weighted sum of
individual security betas
N
 p   wi  i
i 1
• Where w is the proportion of security i’s market value to
that of the entire portfolio, and N is the number of securities
in the portfolio
Relationship between Risk and
Expected Return (CAPM)
• Beta and the Risk Premium
• A risk-free asset has a beta of zero
• When a risky asset is combined with a risk-free asset, the
resulting portfolio expected return is a weighted sum of their
expected returns and the portfolio beta is the weighted sum
of their betas
• Reward-to-Risk Ratio
• We can vary the amount invested in each type of asset and
get an idea of the relationship between portfolio expected
return and portfolio beta
E(Rp )  R f
Re wardToRisk Ratio 
p
Relationship between Risk and
Expected Return (CAPM)
• What happens if two assets have different
reward-to-risk ratios?
• Since systematic risk is all that matters in determining
expected return, the reward-to-risk ratio must be the same
for all assets and portfolios in equilibrium
– If not, investors would only buy the assets (or portfolios) that offer a
higher reward-to-risk ratio
• Because the reward-to-risk ratio is the same for all assets, it
must hold for the risk-free asset as well as for the market
portfolio
E ( Ra )  R f E ( Rb )  R f
• Result:

a
b
Relationship between Risk and
Expected Return (CAPM)
• The Security Market Line (SML)
– The security market line is the line which gives the
expected return – systematic risk (beta) combinations
of assets in a well functioning, active financial market
• In an active, competitive market in which only systematic
risk affects expected return, the reward-to-risk ratio must be
the same for all assets in the market
• The slope of the SML is the difference between the expected
return on the market portfolio and the risk-free rate (i.e., the
market risk premium)
E ( Ri )  R f
• Result:
 E ( RM )  R f
i
Relationship between Risk and
Expected Return (CAPM)
• Expected Return on the Market
E ( RM )  R f  Market Risk Premium
• Expected Return on an individual security
E ( Ri )  R f
– Or
i
 E ( RM )  R f
E ( Ri )  R f   i [ E ( RM )  R f ]
Market Risk Premium
• This applies to individual securities held within welldiversified portfolios
Relationship between Risk and
Expected Return (CAPM)
Expected return
• Capital Asset Pricing Model (CAPM)
E ( Ri )  R f   i [ E ( RM )  R f ]
E ( RM )
Rf
βM =1.0

The Capital Asset Pricing Model
• The Capital Asset Pricing Model (CAPM)
– An equilibrium model of the relationship between risk
and required return on assets in a diversified portfolio
– What determines an asset’s expected return?
• The risk-free rate: the pure time value of money
• The market risk premium: the reward for bearing systematic
risk
• The beta coefficient: a measure of the amount of systematic
risk present in a particular asset
– The CAPM: E ( Ri )  R f   i [ E ( RM )  R f ]
The Capital Asset Pricing Model
• Example: Capital Asset Pricing Model
– Suppose a stock has 1.5 times the systematic risk as the
market portfolio
– The risk-free rate as measured by the T-bill rate is 3%
and the expected risk premium on the market portfolio
is 7%
– What is the stock’s expected return according to the
CAPM?
E ( Ri )  R f   i [ E ( RM )  R f ]
E ( Ri )  3%  1.5[7%]  13.5%
The Capital Asset Pricing Model
Expected
return
• Example: Capital Asset Pricing Model
13.5%
3%
1.5
E ( Ri )  R f   i [ E ( RM )  R f ]
E ( Ri )  3%  1.5[7%]  13.5%

Summary of Risk and Return
I.
II.
III.
IV.
V.
VI.
VII.
Total risk: the variance (or the standard deviation) of an individual,
stand-alone asset’s return
Total return: the expected return + the unexpected return (surprise)
Systematic and unsystematic risks
Systematic risks are unanticipated events that affect almost all
assets to some degree
Unsystematic risks are unanticipated events that affect single assets
or small groups of assets
The effect of diversification: the elimination of unsystematic risk
via the combination of assets into a portfolio
The systematic risk principle and beta: the reward for bearing risk
depends only on its level of systematic risk
The reward-to-risk ratio: the ratio of an asset’s risk premium to its
beta
The capital asset pricing model: E(Ri) = Rf + [E(RM) - Rf] i
Thank You!
Charles B. (Chip) Ruscher, PhD
Department of Finance and Business Economics