ch 08 mgmt 1362002

8/14/2019 Ch 08 Mgmt 1362002

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The Capital Asset Pricing Model (Chapter 8)

Premise of the CAPM

Assumptions of the CAPMUtility FunctionsThe CAPM With Unlimited Borrowing and Lendingat a Risk-Free Rate of Return

Capital Market Line Versus Security Market LineRelationship Between the SML and the CharacteristicLineThe CAPM With No Risk-Free Asset

The CAPM With Lending at the Risk-Free Rate, butNo BorrowingThe CAPM With Lending at the Risk-Free Rate, andBorrowing at a Higher RateMarket Efficiency

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Premise of the CAPM

The Capital Asset Pricing Model (CAPM) is a model toexplain why capital assets are priced the way they are.

The CAPM was based on the supposition that allinvestors employ Markowitz Portfolio Theory to find the

portfolios in the efficient set. Then, based on individualrisk aversion, each of them invests in one of theportfolios in the efficient set.Note, that if this supposition is correct, the Market

Portfolio would be efficient because it is the aggregate ofall portfolios. Recall Property I - If we combine two ormore portfolios on the minimum variance set, we getanother portfolio on the minimum variance set.

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One Major Assumption of the CAPM

Investors can choose between portfolios on the basis ofexpected return and variance. This assumption is validif either:

1. The probability distributions for portfolio

returns are all normally distributed, or2. Investors’ utility functions are all in quadraticform.

If data is normally distributed, only two parameters

are relevant: expected return and variance. There isnothing else to look at even if you wanted to.

If utility functions are quadratic, you only want tolook at expected return and variance, even if other

parameters exist.

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Evidence Concerning Normal Distributions

Returns on individual stocks may be “fairly”normally distributed using monthly returns.For yearly returns, however, distributions of

returns tend to be skewed to the right. (-100%is the largest possible loss; upside gains aretheoretically unlimited, however.Returns on portfolios may be normallydistributed even if returns on individual stocksare skewed.

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Utility Functions

Utility is a measure of well-being.A utility function shows the relationship betweenutility and return (or wealth) when the returns arerisk-free.

Risk-Neutral Utility Functions : Investors areindifferent to risk. They only analyze return whenmaking investment decisions.

Risk-Loving Utility Functions : For any given rateof return, investors prefer more risk.

Risk-Averse Utility Functions : For any given rateof return, investors prefer less risk.

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Utility Functions (Continued)To illustrate the different types of utilityfunctions, we will analyze the following riskyinvestment for three different investors:

Possible Return (%)(r i)

_________10%

50%

Probability(pi)

_________.5

.5

20%30%).5(50%30%).5(10%)σ(r

30%.5(50%).5(10%))E(r

22i

i

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Risk-Neutral Investor

Assume the following linear utility function:

u i = 10r i

Return (%)

(r i) __________

01020304050

Total Utility

(u i) __________

0100200300400500

Constant

Marginal Utility __________

100100100100100

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Risk-Neutral Investor (Continued)

Expected Utility of the Risky Investment:

Note: The expected utility of the riskyinvestment with an expected return of 30%

(300) is equal to the utility associated withreceiving 30% risk-free (300).

300.5(500).5(100)E(u)

u(50%)*.5u(10%)*.5E(u)

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Risk-Neutral Utility Function

u i = 10r i

0

100

200

300

400

500

600

0 10 20 30 40 50 60

Total Utility

Percent Return

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Risk-Loving Investor

Assume the following quadratic utility function:u i = 0 + 5r i + .1r i

2

Return (%)(r

i)

__________ 0

1020

304050

Total Utility(u

i)

__________ 0

60140

240360500

IncreasingMarginal Utility

__________

6080

100120140

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Risk-Loving Investor (Continued)


Note: The expected utility of the risky investment withan expected return of 30% (280) is greater than theutility associated with receiving 30% risk-free (240).

That is, the investor would be indifferent betweenreceiving 33.5% risk-free and investing in a risky assetthat has E(r) = 30% and (r) = 20%

280.5(500).5(60)E(u)

u(50%)*.5u(10%)*.5E(u)

33.5%2(.1)

)4(.1)(-280-25+5- :EquivalentCertainty

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Risk-Loving Utility Function

u i = 0 + 5r i + .1r i2

0

600

0 60

Total Utility

Percent Return

500

280240

60

10 30 33.5 50

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Risk-Averse Investor

Assume the following quadratic utility function:

u i = 0 + 20r i - .2r i2

Return (%)

(r i) __________ 0

1020304050

Total Utility

(u i) __________ 0

180320420480500

Diminishing

Marginal Utility __________

1801401006020

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Risk-Averse Investor (Continued)


Note: The expected utility of the risky investment withan expected return of 30% (340) is less than the utilityassociated with receiving 30% risk-free (420).

That is, the investor would be indifferent betweenreceiving 21.7% risk-free and investing in a risky assetthat has E(r) = 30% and (r) = 20%.

340.5(500).5(180)E(u)

u(50%)*.5u(10%)*.5E(u)

21.7%.2)2(

0)4(-.2)(-34-400+20-

:EquivalentCertainty

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Risk-Averse Utility Function

u i = 0 + 20r i - .2r i2

0

600

0 60

Total Utility

Percent Return

500

420

340

180

10 21.7 30 50

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Indifference Curve

Given the total utility function, an indifference curvecan be generated for any given level of utility. First,for quadratic utility functions, the following equation

for expected utility is derived in the text:

2

2

1

2

0

2

22

2210

E(r)a

E(r)a

a

a

a

E(u)=σ(r)

:σ(r)forSolving

(r)σaE(r)aE(r)aaE(u)

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Indifference Curve (Continued)

Using the previous utility function for the risk-averseinvestor, (u i = 0 + 20r i - .2r i

2), and a given level ofutility of 180:

Therefore, the indifference curve would be:

2E(r).2

E(r)20.2

180σ(r)

E(r)

1020304050

(r)

026.534.638.740.0

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Risk-Averse Indifference CurveWhen E(u) = 180, and u

i = 0 + 20r

i - .2r

i

2

0

10

20

30

40

50

60

0 10 20 30 40 50

Expected Return

Standard Deviation of Returns

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Maximizing UtilityGiven the efficient set of investment possibilities and a“mass” of indifference curves, an investor would

maximize his/her utility by finding the point oftangency between an indifference curve and theefficient set.

0

10

20

30

40

50

60

0 10 20 30 40 50

Expected Return


Portfolio ThatMaximizesUtility

E(u) = 380 E(u) = 280

E(u) = 180

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Problems With Quadratic Utility Functions

Quadratic utility functions turn down after they reach

a certain level of return (or wealth). This aspect isobviously unrealistic:

0

100

200

300

400

500

600

0 20 40 60 80

Total Utility

Percent Return

Unrealistic

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Problems With Quadratic UtilityFunctions (Continued)

As discussed in the Appendix on utilityfunctions, with a quadratic utility function, asyour wealth level increases, your willingness totake on risk decreases (i.e., both absolute risk

aversion [dollars you are willing to commit torisky investments] and relative risk aversion[% of wealth you are willing to commit torisky investments] increase with wealth levels).In general, however, rich people are morewilling to take on risk than poor people.Therefore, other mathematical functions (e.g.,logarithmic) may be more appropriate.

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Two Additional Assumptions of the CAPM

Assumption II - All investors are in agreementregarding the planning horizon (i.e., all have the sameholding period), and the distributions of securityreturns (i.e., perfect knowledge exists).Assumption III - There are no frictions in the capital

market (i.e., no taxes, no transaction costs, norestrictions on short-selling).

Note: Many of the assumptions are obviouslyunrealistic. Later, we will evaluate the consequences of

relaxing some of these assumptions. The assumptionsare made in order to generate a model that examinesthe relationship between risk and expected returnholding many other factors constant.

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The CAPM With Unlimited Borrowing &Lending at a Risk-Free Rate of Return

First, using the Markowitz full covariance model weneed to generate an efficient set based on all riskyassets in the universe:

0

5

10

15

20

25

0 20 40

Expected Return


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Capital Market Line (CML)

Next, the risk-free asset is introduced. TheCapital Market Line (CML) is thendetermined by plotting a line that goesthrough the risk-free rate of return, and is

tangent to the Markowitz efficient set. Thispoint of tangency identifies the MarketPortfolio (M). The CML equation is:

)σ(r)σ(rr)E(rr)E(r p

M

FMFp

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Portfolio Risk and the CMLNote that all points on the CML except the MarketPortfolio dominate all points on the Markowitz

efficient set (i.e., provide a higher expected return forany given level of risk). Therefore, all investors shouldinvest in the same risky portfolio (M), and then lend orborrow at the risk-free rate depending on their risk

preferences.That is, all portfolios on the CML are somecombination of two assets: (1) the risk-free asset, and(2) the Market Portfolio. Therefore, for portfolios onthe CML:

)σ(rx)σ(rand)(rσx)(rσ

Free)(Risk-0)σ(rsinceHowever,

)σ(r)σ(rρxx2)(rσx)(rσx)(rσ

MMpM22

Mp2

F

MFM,rMrM22

MF22

rp2

FFF

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Portfolio Risk and the CML (Continued)

By definition, since (r p) = x M (r M ), all portfolios thatlie on the CML are perfectly positively correlated withthe Market Portfolio (i.e., 100% of the variance in theportfolio’s returns is explained by the variance in themarket’s returns, when the portfolio lies on the CML).

Recall the Single- Factor Model’s Measure of Variance

)σ(rβ)σ(r

)(rσβ)(rσ

:CMLth eonportfoliosforTherefore,

0)(εσ1.00,ρ :When

)(εσ)(rσβ)(rσ

Mpp

M22

pp2

p2

Mp,

p2

M22

pp2

Note, since (r M) is thesame for all portfolios,all of the risk of aportfolio on the CML isreflected in its beta.

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Capital Market Line (CMLVersus

Security Market Line (SML)

Recall Property II:Given a population of securities, there will be asimple linear relationship between the betafactors of different securities and theirexpected (or average) returns if and only if thebetas are computed using a minimum variancemarket index portfolio.Therefore:Given the CML, we can determine the SML(relationship between beta & expected return)

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CML Versus SML

0

0.1

0.2

0.3

0 0.5 1 1.5

0

0.1

0.2

0.3

0 0.48

E(r) E(r)

(r)

CML

SMLM C

BA

CM

BA

r F r F

E(r M) E(r M)

(r M)

P tf li Th t Li th CML

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Portfolios That Lie on the CMLWill Also Lie on the SML

CML Equation:

Can be restated as:

And, since for portfolios on the CML:

We can state that for portfolios on the CML:

)σ(r)σ(rr)E(rr)E(r p

M

FMFp

)σ(r

)σ(r]r)[E(rr)E(r

M

pFMFp

)σ(rβ)σ(r Mpp

)σ(r

)σ(rβ

M

pp

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Therefore, for portfolios on the CML:

Individual Securities Will Lie on the SML,But Off the CMLRecall:

However:

in well diversified portfolios (i.e., can be doneaway with)

EquationS ML

β]r)[E(rr)E(r

)σ(r)σ(r]r)[E(rr)E(r

pFMFp

MpFMFp

)(εσ)(rσβ)(rσ p2

M22

pp2

0)(εσ p2

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Therefore, Relevant Risk may be def ined as:

And since:

We can state that:

That is, a security’s contribution to the risk of a portfoliocan be measured by its beta. Since an individual security’sresidual variance can be diversified away in a portfolio,the market place will not reward this “unnecessary” risk.Since only beta is relevant, individual securities will bepriced to lie on the SML.

)(rσβ)(rσ M22

pp2

m

1 j

j jp βxβ

Risk Relevant

)(rσβx)(rσ M2

2m

1 j

j jp2

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Individual Security on the SML and Off the CML(Continued)

0

30

0 50

0

30

0 1 2

E(r) E(r)

(r)

CML

M MOff the CML

22

18

10

22.5 33.75

22

18

10

On the SML

SML

1.5

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Ch i i Li V SML

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Characteristic Line Versus SML(In Equilibrium)

-10

-5

0

5

10

15

20

25

30

0 10 20

0

5

10

15

20

25

30

0 0.5 1 1.5

2 = 1.5

1 =.5

r j

r M

E(r 2)

E(r M)

E(r 1)

r F

A1

A2Characteristic Line

E(r M) =

A1 = 10(1 - .5) = 5A2 = 10(1 - 1.5) = -5

E(r)

Security Market Line

E(r 2)

E(r M)

E(r 1)

r F

Ch i i Li V SML

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Characteristic Line Versus SML(In Disequilibrium: Undervalued Security)

0

5

10

15

20

25

30

0 0.5 1 1.5-10

-5

0

5

10

15

20

25

30

0 10 20

E(r 2)

E(r E)

E(r M)

r F

AEE(r M) =

2 = 1.5

r j

r M

Characteristic Line


E(r)

E(r 2)

E(r E)

E(r M)

r F

Ch t i ti Li V SML

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Characteristic Line Versus SML(In Disequilibrium: Overvalued Security)

0

5

10

15

20

25

0 0.5 1 1.5-15

-10

-5

0

5

10

15

20

25

0 10 20

E(r E)E(r 2)

r F

AEE(r M) =

2 = 1.5

r j

r M

Characteristic Line


E(r)

E(r 2)

E(r E)

E(r M)

r F

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CAPM With No Risk-Free Asset

0

0.25

0 0.5 1 1.5

0

0.25

0.5

0 0.48

E(r) E(r)

(r)

E(r M)

E(r M)

E(r Z)

E(r Z)M

X

SML

MVP

CAPM With No Risk Free Asset

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CAPM With No Risk-Free Asset(Continued)

Assumption: All investors take positions on theefficient set (Between MVP and X)In this case, the Markowitz efficient set (MVP to X) isthe Capital Market Line (CML).

M is the efficient Market Portfolio (the aggregateof all portfolios held by investors)

E(r Z) is the intercept of a line drawn tangent to(M)

From Property II, since (M) is efficient, a linearrelationship exists between expected return and beta.All assets (efficient and inefficient) will be priced to lieon the SML.

jZMZ j β)]E(r)[E(r)E(r)E(r

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Can Lend, but Cannot Borrow at theRisk-Free Rate (Continued)

Capital Market Line (CML): (r F - L - M - X)

Between r F and L:

Combinations of the risk-free asset and the risky(efficient) portfolio L.Between L and X:

Risky portfolios of assets.

Security Market Line (SML): All assets (efficient and inefficient) will be priced tolie on the SML.

jZMZ j β)]E(r)[E(r)E(r)E(r

Can Lend at the Risk Free Rate:

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Can Lend at the Risk-Free Rate:Borrowing is at a Higher Rate

0

0.25

0 0.5 1 1.5

0

0.25

0 0.48

E(r) E(r)

(r)

E(r M) E(r M)

r B

E(r Z) E(r Z)

SML

r F

(r M)

L

MB

X

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Can Lend at the Risk-Free Rate, and Borrow ata Higher Rate (Continued)

Capital Market Line (CML):(r

F - L - M - B - X)

Between r F and L:Combinations of the risk-free asset and the risky(efficient) portfolio L.

Between L and B:Risky portfolios of assets.

Between B and X:Combinations of the risky (efficient) portfolio B

and a loan with an interest rate of r B Security Market Line (SML):

All assets (efficient and inefficient) will be pricedto lie on the SML jZMZ j β)]E(r)[E(r)E(r)E(r

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Conditions Required for Market Efficiency

In order for the Market Portfolio to lie on theefficient set, the following assumptions musthold:

All investors must agree about the risk and

expected return for all securities.All investors can short-sell all securitieswithout restriction.No investor’s return is exposed to federal

or state income tax liability now in effect.The investment opportunity set ofsecurities is the same for all investors.

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When the Market Portfolio is InefficientInvestors Disagree About Risk and ExpectedReturn

In this case there will be no unique perceivedefficient set for the Market Portfolio to lie on (i.e.,different investors would have different perceived

efficient sets).Some Investors Cannot Sell Short

In this case, Property I no longer holds. If a“constrained” efficient set were constructed with

no short-selling, and each investor selected aportfolio lying on the “constrained” efficient set,the combination of these portfolios would not lieon the “constrained” efficient set.

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When the Market Portfolio is Inefficient(Continued)

Taxes Differ Among InvestorsWhen tax exposure differs among investors (e.g.,state, local, foreign, corporate versus personal), theafter-tax efficient set for one investor will be

different from that of others. There would be nounique efficient set for the Market Portfolio to lieon.

Alternative Investments Differ AmongInvestors

Efficient sets will differ among investors when thepopulations of securities used to construct theefficient sets differ (e.g., some may excludepolluters, others may include foreign assets, etc.).

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Summary of Market Portfolio Efficiency

In reality, assumptions underlying theefficiency of the Market Portfolio arefrequently violated. Therefore, the MarketPortfolio may well lie inside the efficient set

even if the efficient set is constructed using thepopulation of securities making up the market.In other words, perhaps the market can bebeaten. That is, there may be portfolios thatoffer higher risk-adjusted returns than theoverall Market Portfolio.

ch 08 mgmt 1362002

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