Capital Market Line
 Line from RF to L is
L

M
E(RM)

x
RF
y

M
Risk

capital market line (CML)
x = risk premium
= E(RM) - RF
y = risk = M
Slope = x/y
= [E(RM) - RF]/M
y-intercept = RF
Capital Market Line
 Slope of the CML is the market price of risk for
efficient portfolios, or the equilibrium price of risk
in the market
 Relationship between risk and expected return for
portfolio P (Equation for CML):
E(R p )  RF 
E(R M )  RF
M
p
Security Market Line
 CML Equation only applies to markets in
equilibrium and efficient portfolios
 The Security Market Line depicts the tradeoff
between risk and expected return for individual
securities
 Under CAPM, all investors hold the market
portfolio

How does an individual security contribute to the risk of
the market portfolio?
Security Market Line
 Equation for expected return for an individual
stock similar to CML Equation
E(R i )  RF 
E(R M )  RF  i,M
M
M
 RF   i E(R M )  RF
Security Market Line
 Beta = 1.0 implies as
risky as market
 Securities A and B are
more risky than the
market
SML
E(R)
A
E(RM)
RF
B

C
 Security C is less risky
than the market

0
Beta > 1.0
0.5
1.0 1.5
BetaM
2.0
Beta < 1.0
Security Market Line
 Beta measures systematic risk
 Measures relative risk compared to the market portfolio of
all stocks
 Volatility different than market
 All securities should lie on the SML
 The expected return on the security should be only that
return needed to compensate for systematic risk
SML and Asset Values
Er
Underpriced
SML: Er = rf +  (Erm – rf)
Overpriced
rf
β
 expected return > required return according to CAPM
 lie “above” SML
Overpriced
 expected return < required return according to CAPM
 lie “below” SML
Correctly priced  expected return = required return according to CAPM
 lie along SML
Underpriced
CAPM’s Expected Return-Beta
Relationship
 Required rate of return on an asset (ki) is
composed of


risk-free rate (RF)
risk premium (i [ E(RM) - RF ])

Market risk premium adjusted for specific security
ki = RF +i [ E(RM) - RF ]

The greater the systematic risk, the greater the required
return
Estimating the SML
 Treasury Bill rate used to estimate RF
 Expected market return unobservable
 Estimated using past market returns and taking an expected
value
 Estimating individual security betas difficult
 Only company-specific factor in CAPM
 Requires asset-specific forecast
Estimating Beta
 Market model
 Relates the return on each stock to the return on the
market, assuming a linear relationship
Rit =i +i RMt + eit
Test of CAPM
 Empirical SML is “flatter” than predicted SML
 Fama and French (1992)
 Market
 Size
 Book-to-market ratio
 Roll’s Critique
 True market portfolio is unobservable
 Tests of CAPM are merely tests of the mean-variance
efficiency of the chosen market proxy
Arbitrage Pricing Theory
 Based on the Law of One Price
 Two otherwise identical assets cannot sell at different
prices
 Equilibrium prices adjust to eliminate all arbitrage
opportunities
 Unlike CAPM, APT does not assume
 single-period investment horizon, absence of personal
taxes, riskless borrowing or lending, mean-variance
decisions
Factors
 APT assumes returns generated by a factor model
 Factor Characteristics
 Each risk must have a pervasive influence on stock returns
 Risk factors must influence expected return and have
nonzero prices
 Risk factors must be unpredictable to the market
APT Model
 Most important are the deviations of the factors
from their expected values
 The expected return-risk relationship for the
APT can be described as:
E(Rit) =a0+bi1 (risk premium for factor 1) +bi2
(risk premium for factor 2) +… +bin (risk
premium for factor n)
APT Model
 Reduces to CAPM if there is only one factor and
that factor is market risk
 Roll and Ross (1980) Factors:





Changes in expected inflation
Unanticipated changes in inflation
Unanticipated changes in industrial production
Unanticipated changes in the default risk premium
Unanticipated changes in the term structure of interest
rates
Problems with APT
 Factors are not well specified ex ante
 To implement the APT model, the factors that account
for the differences among security returns are required

CAPM identifies market portfolio as single factor
 Neither CAPM or APT has been proven superior
 Both rely on unobservable expectations
Two-Security Case
 For a two-security portfolio containing Stock A and
Stock B, the variance is:
  x   x   2 xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
17
Two Security Case (cont’d)
Example
Assume the following statistics for Stock A and Stock
B:
Stock A
Stock B
Expected return
Variance
Standard deviation
.015
.050
.224
.020
.060
.245
Weight
Correlation coefficient
40%
60%
.50
18
Two Security Case (cont’d)
19
Example (cont’d)
What is the expected return and variance of this twosecurity portfolio?
Two Security Case (cont’d)
20
Example (cont’d)
Solution: The expected return of this two-security
portfolio is:
n
E ( R p )    xi E ( Ri ) 
i 1
  x A E ( RA )    xB E ( RB ) 
  0.4(0.015)    0.6(0.020) 
 0.018  1.80%
Two Security Case (cont’d)
21
Example (cont’d)
Solution (cont’d): The variance of this two-security
portfolio is:
 2p  xA2 A2  xB2 B2  2 xA xB  AB A B
 (.4) (.05)  (.6) (.06)  2(.4)(.6)(.5)(.224)(.245)
2
2
 .0080  .0216  .0132
 .0428
Minimum Variance Portfolio
22
 The minimum variance portfolio is the particular
combination of securities that will result in the least
possible variance
 Solving for the minimum variance portfolio requires
basic calculus
Minimum Variance
Portfolio (cont’d)
23
 For a two-security minimum variance portfolio, the
proportions invested in stocks A and B are:
   A B  AB
xA  2
2
 A   B  2 A B  AB
2
B
xB  1  x A
Minimum Variance
Portfolio (cont’d)
24
Example (cont’d)
Assume the same statistics for Stocks A and B as in
the previous example. What are the weights of the
minimum variance portfolio in this case?
Minimum Variance
Portfolio (cont’d)
25
Example (cont’d)
Solution: The weights of the minimum variance
portfolios in this case are:
 B2   A B  AB
.06  (.224)(.245)(.5)
xA  2

 59.07%
2
 A   B  2 A B  AB .05  .06  2(.224)(.245)(.5)
xB  1  xA  1  .5907  40.93%
Minimum Variance
Portfolio (cont’d)
26
Example (cont’d)
1.2
Weight A
1
0.8
0.6
0.4
0.2
0
0
0.01
0.02
0.03
0.04
Portfolio Variance
0.05
0.06
Correlation and
Risk Reduction
27
 Portfolio risk decreases as the correlation coefficient
in the returns of two securities decreases
 Risk reduction is greatest when the securities are
perfectly negatively correlated
 If the securities are perfectly positively correlated,
there is no risk reduction
The n-Security Case
28
 For an n-security portfolio, the variance is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
The n-Security Case (cont’d)
29
 A covariance matrix is a tabular presentation of
the pairwise combinations of all portfolio
components

The required number of covariances to compute a portfolio
variance is (n2 – n)/2

Any portfolio construction technique using the full covariance
matrix is called a Markowitz model
Computational Advantages
30
 The single-index model compares all securities to
a single benchmark

An alternative to comparing a security to each of the others

By observing how two independent securities behave relative
to a third value, we learn something about how the securities
are likely to behave relative to each other
Computational
Advantages (cont’d)
31
 A single index drastically reduces the number of
computations needed to determine portfolio variance

A security’s beta is an example:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
Portfolio Statistics With the Single-Index
Model
32
 Beta of a portfolio:
n


x


p
i
i
 Variance of a portfolio:
i 1
 2p   p2 m2   ep2
  p2 m2
Portfolio Statistics With the Single-Index
Model (cont’d)
33
 Variance of a portfolio component:
    
2
i
2
i
2
m
2
ei
 Covariance of two portfolio components:
 AB   A  B
2
m