Capital Market Line (Rohit)

8/9/2019 Capital Market Line (Rohit)

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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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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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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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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


Example (contd)

What is the expected return and variance of this two-securityportfolio?


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20


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


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


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


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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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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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 =

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Capital Market Line (Rohit)