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Capital Market Line
Line from RF to L iscapital market line (CML)
x = risk premium
= E(RM) - RF
y = risk = M
Slope = x/y
= [E(RM) - RF]/ M
y-intercept = RF
E(RM)
RF
Risk
M
L
M
y
x
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Capital Market Line
Slope of the CML is the market price of risk forefficient portfolios, or the equilibrium price of risk
in the marketRelationship between risk and expected return for
portfolio P (Equation for CML):
p
M
Mp
RF)R(ERF)R(E
+
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Security Market Line
CML Equation only applies to markets inequilibrium and efficient portfolios
The Security Market Line depicts the tradeoffbetween risk and expected return for individualsecurities
Under CAPM, all investors hold the market
portfolio How does an individual security contribute to the risk of
the market portfolio?
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Security Market Line
Equation for expected return for an individual
stock similar to CML Equation
[ ]RF)R(ERFRF)R(E
RF)R(E
Mi
M
M,i
M
Mi
+
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Security Market Line
Beta = 1.0 implies asrisky as market
Securities A and B aremore risky than themarket Beta > 1.0
Security C is less riskythan the market Beta < 1.0
AB
C
E(RM)
RF
0 1.0 2.00.5 1.5
SM
L
BetaM
E(R)
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Security Market Line
Beta measures systematic risk Measures relative risk compared to the market portfolio of
all stocks
Volatility different than marketAll securities should lie on the SML The expected return on the security should be only that
return needed to compensate for systematic risk
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Er
Underpriced SML: Er= r
f+ (E
rm r
f)
Overpriced
rf
Underpriced expected return > required return according to CAPM lie above SML
Overpriced expected return < required return according to CAPM lie below SML
Correctly pricedexpected return = required return according to CAPM lie along SML
SML and Asset Values
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CAPMs Expected Return-BetaRelationship
Required rate of return on an asset (ki) is
composed of risk-free rate (RF)
risk premium (i[ E(R
M) - RF ])
Market risk premium adjusted for specific securityk
i= RF +
i[ E(R
M) - RF ]
The greater the systematic risk, the greater the requiredreturn
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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
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Estimating Beta
Market model Relates the return on each stock to the return on the
market, assuming a linear relationshipR
it=
i+
iR
Mt+ e
it
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Test of CAPM
Empirical SML is flatter than predicted SML
Fama and French (1992) Market
Size Book-to-market ratio
Rolls Critique True market portfolio is unobservable
Tests of CAPM are merely tests of the mean-varianceefficiency of the chosen market proxy
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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 arbitrageopportunities
Unlike CAPM, APT does not assume single-period investment horizon, absence of personal
taxes, riskless borrowing or lending, mean-variance
decisions
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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 havenonzero prices
Risk factors must be unpredictable to the market
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APT Model
Most important are the deviations of the factorsfrom their expected values
The expected return-risk relationship for the
APT can be described as:
E(Rit) =a
0+b
i1(risk premium for factor 1) +b
i2(risk
premium for factor 2) + +bin
(risk premium
for factor n)
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APT Model
Reduces to CAPM if there is only one factor andthat 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 interestrates
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Two-Security Case
17
For a two-security portfolio containing Stock A andStock B, the variance is:
2 2 2 2 2 2 p A A B B A B AB A B
x x x x = + +
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Two Security Case (contd)
18
Example
Assume the following statistics for Stock A and Stock B:
Stock A Stock B
Expected return .015 .020
Variance .050 .060
Standard deviation .224 .245
Weight 40% 60%
Correlation coefficient .50
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19
Two Security Case (contd)
Example (contd)
What is the expected return and variance of this two-securityportfolio?
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20
Two Security Case (contd)
Example (contd)
Solution: The expected return of this two-security portfolio is:
[ ] [ ]
1
( ) ( )
( ) ( )
0.4(0.015) 0.6(0.020)
0.018 1.80%
n
p i i
i
A A B B
E R x E R
x E R x E R
=
=
= + = +
= =
% %
% %
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21
Two Security Case (contd)
Example (contd)
Solution (contd): The variance of this two-security portfolio is:
2 2 2 2 2
2 2
2
(.4) (.05) (.6) (.06) 2(.4)(.6)(.5)(.224)(.245).0080 .0216 .0132
.0428
p A A B B A B AB A B x x x x = + +
= + += + +
=
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22
Minimum Variance Portfolio
The minimum variance portfolio is the particularcombination of securities that will result in the leastpossible variance
Solving for the minimum variance portfolio requiresbasic calculus
Mi i V i
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23
Minimum VariancePortfolio (contd)
For a two-security minimum variance portfolio, theproportions invested in stocks A and B are:
2
2 2 2
1
B A B ABA
A B A B AB
B A
x
x x
=
+
=
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Mi i V i
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25
Minimum VariancePortfolio (contd)
Example (contd)
Solution: The weights of the minimum variance portfolios in this caseare:
2
2 2
.06 (.224)(.245)(.5)59.07%
2 .05 .06 2(.224)(.245)(.5)
1 1 .5907 40.93%
B A B ABA
A B A B AB
B A
x
x x
= = =
+ +
= = =
Mi i V i
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26
Minimum VariancePortfolio (contd)
Example (contd)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.01 0.02 0.03 0.04 0.05 0.06
W
eightA
Portfolio Variance
C l ti d
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27
Correlation andRisk Reduction
Portfolio risk decreases as the correlation coefficientin the returns of two securities decreases
Risk reduction is greatest when the securities are
perfectly negatively correlatedIf the securities are perfectly positively correlated,
there is no risk reduction
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28
The n-Security Case
For an n-security portfolio, the variance is:
2
1 1
where proportion of total investment in Security
correlation coefficient between
Security and Security
n n
p i j ij i j
i j
i
ij
x x
x i
i j
= =
=
=
=
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29
The n-Security Case (contd)
Acovariance matrixis a tabular presentation ofthe pairwise combinations of all portfoliocomponents The required number of covariances to compute a portfolio
variance is (n2
n)/2
Any portfolio construction technique using the full covariancematrix is called aMarkowitz model
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30
Computational Advantages
The single-index modelcompares all securities toa single benchmark An alternative to comparing a security to each of the others
By observing how two independent securities behave relativeto a third value, we learn something about how the securitiesare likely to behave relative to each other
Computational
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31
ComputationalAdvantages (contd)
A single index drastically reduces the number ofcomputations needed to determine portfolio variance A securitys beta is an example:
2
2
( , )
where return on the market index
variance of the market returns
return on Security
i mi
m
m
m
i
COV R R
R
R i
=
=
=
=
% %
%
%
Portfolio Statistics With the Single Index
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Portfolio Statistics With the Single-IndexModel
Beta of a portfolio:
Variance of a portfolio:1
n
p i ii x =
=
2 2 2 2
2 2
p p m ep
p m
= +
Portfolio Statistics With the Single Index
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Portfolio Statistics With the Single-IndexModel (contd)
Variance of a portfolio component:
Covariance of two portfolio components:
2 2 2 2
i i m ei = +
2
AB A B m =