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5 - 1CHAPTER 5

Risk and Return: Portfolio Theory and Asset Pricing

Models

Portfolio Theory

Capital Asset Pricing Model (CAPM)Efficient frontier Capital Market Line (CML)Security Market Line (SML)

Beta calculationArbitrage pricing theoryFama-French 3-factor model

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Portfolio Theory

Suppose Asset A has an expected returnof 10 percent and a standard deviation of

20 percent. Asset B has an expectedreturn of 16 percent and a standarddeviation of 40 percent. If the correlationbetween A and B is 0.6, what are theexpected return and standard deviation for a portfolio comprised of 30 percent AssetA and 70 percent Asset B?

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Portfolio Expected Return

%.2.14142.0)16.0(7.0)1.0(3.0

r)w1(rwr BAAAP

!!

!

!

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Portfolio Standard Deviation

309.0)4.0)(2.0)(4.0)(7.0)(3.0(2)4.0(7.0)2.0(3.0

)W1(W2)W1(W

2222

BAABAA2B

2A

2A

2Ap

!

!

! W W VW W W

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Attainable Portfolios: VAB = 0.4

VAB = +0.4 A i li / i i

0%

5%

0%

5%

0%

0% 0% 0% 0% 40%

i , W

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Attainable Portfolios: VAB = 1 V AB = 1 A

1

1

1

W

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Attainable Portfolios: VAB

= -1

V AB = -1 A

1

1

1

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Attainable Portfolios with Risk-Free

Asset (Expected risk-free return = )

Attainable et of Risk Returno binations with Risk-Free Asset

Risk W p

E x p e c

t e d r e

t u r n

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5 - 9ExpectedPortfolioReturn, r p

Risk, W p

Efficient Set

Feasible Set

Feasible and Efficient Portfolios

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The feasible set of portfolios representsall portfolios that can be constructedfrom a given set of stocks.

An efficient portfolio is one that offers:the most return for a given amount of risk,or

the least risk for a give amount of return.

The collection of efficient portfolios iscalled the efficient set or efficientfrontier .

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IB2 IB1

IA2IA1

Optimal Portfolio

Investor A

Optimal PortfolioInvestor B

Risk W p

ExpectedReturn, r p

Optimal Portfolios

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Indifference curves reflect aninvestors attitude toward risk asreflected in his or her risk/return

tradeoff function. They differ among investors because of differences in risk aversion.

An investors optimal portfolio isdefined by the tangency pointbetween the efficient set and theinvestors indifference curve.

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W hat is the CAPM?

The CAPM is an equilibrium modelthat specifies the relationshipbetween risk and required rate of return for assets held in well-diversified portfolios .

It is based on the premise that onlyone factor affects risk.W hat is that factor?

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Investors all think in terms of a single holding period .

All investors have identical expectations .

Investors can borrow or lend unlimitedamounts at the risk-free rate .

W hat are the assumptionsof the CAPM?

(More...)

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All assets are perfectly divisible .

There are no taxes and no transactionscosts .

All investors are price takers , that is,investors buying and selling wontinfluence stock prices.

Quantities of all assets are given andfixed.

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W hen a risk-free asset is added to thefeasible set, investors can createportfolios that combine this asset with aportfolio of risky assets.

The straight line connecting r RF with M, the

tangency point between the line and theold efficient set, becomes the new efficientfrontier .

W hat impact does r RF have onthe efficient frontier?

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M

Z

.A

r RF

W M Risk, W p

Efficient Set with a Risk-Free Asset

The Capital MarketLine (CML):

New Efficient Set

..B

r M^

ExpectedReturn, r p

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The Capital Market Line (CML) is alllinear combinations of the risk-freeasset and Portfolio M.Portfolios below the CML are inferior.

The CML defines the new efficient set.All investors will choose a portfolio onthe CML.

W hat is the Capital Market Line?

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r p = r RF

SlopeIntercept

^ W p .

The CML Equation

r M - r RF^

W M

Riskmeasure

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The expected rate of return on anyefficient portfolio is equal to therisk-free rate plus a risk premium .The optimal portfolio for anyinvestor is the point of tangencybetween the CML and theinvestors indifference curves.

W hat does the CML tell us?

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r RF

W M Risk, W p

I1I2

CML

R = OptimalPortfolio

.R .M

r Rr M

W R

^^

ExpectedReturn, r p

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The CML gives the risk/return

relationship for efficient portfolios .The Security Market Line (SML), alsopart of the CAPM, gives the risk/returnrelationship for individual stocks .

W hat is the Security Market Line (SML)?

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The measure of risk used in the SMLis the beta coefficient of company i, b i.

The SML equation:

r i = r RF (RP M) b i

The SML Equation

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Run a regression line of pastreturns on Stock i versus returnson the market.The regression line is called thecharacteristic line .

The slope coefficient of thecharacteristic line is defined as thebeta coefficient .

How are betas calculated?

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Illustration of beta calculation

Year r M r i1 1 182 - -103 12 16

r i _

r M _

- 0 10 1 20

20

1

10

-

-10

.

.

.

r i = -2. 9 1.44 k M^ ^

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(More...)

Method of Calculation

Analysts use a computer withstatistical or spreadsheet software toperform the regression.

At least 3 years of monthly returns or 1years of weekly returns are used.

Many analysts use years of monthlyreturns.

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If beta = 1.0, stock is average risk.

If beta > 1.0, stock is riskier thanaverage.

If beta < 1.0, stock is less risky thanaverage.

Most stocks have betas in the rangeof 0. to 1. .

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Interpreting Regression Results

The R 2 measures the percent of astocks variance that is explained bythe market. The typical R 2 is:

0.3 for an individual stockover 0.9 for a well diversified portfolio

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Interpreting Regression Results

(Continued)

The 9 confidence interval showsthe range in which we are 9 surethat the true value of beta lies. Thetypical range is:

from about 0. to 1. for an individual

stockfrom about .92 to 1.08 for a welldiversified portfolio

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W 2 = b 2 W 2 W e 2.

W 2 = variance= stand-alone risk of Stock j.

b2

W2

= market risk of Stock j. W e

2 = variance of error term= diversifiable risk of Stock j.

W hat is the relationship between stand-alone, market, and diversifiable risk.

j j M j

j

j

j M

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Beta stability testsTests based on the slopeof the SML

W hat are two potential tests that can beconducted to verify the CAPM?

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Tests of the SML indicate:

A more-or-less linear relationship

between realized returns and marketrisk.

Slope is less than predicted.

Irrelevance of diversifiable riskspecified in the CAPM model can bequestioned.

(More...)

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Betas of individual securities are notgood estimators of future risk.

Betas of portfolios of 10 or morerandomly selected stocks arereasonably stable.

Past portfolio betas are goodestimates of future portfolio volatility.

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Yes .Richard Roll questioned whether it waseven conceptually possible to test theCAPM.

Roll showed that it is virtuallyimpossible to prove investors behavein accordance with CAPM theory.

Are there problems with theCAPM tests?

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CAPM/SML concepts are based onexpectations , yet betas arecalculated using historical data. Acompanys historical data may notreflect investors expectations aboutfuture riskiness .

Other models are being developed

that will one day replace the CAPM,but it still provides a good frameworkfor thinking about risk and return.

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The CAPM is a single factor model.The APT proposes that therelationship between risk and return

is more complex and may be due tomultiple factors such as GDPgrowth, expected inflation, tax ratechanges, and dividend yield.

W hat is the difference between theCAPM and the Arbitrage

Pricing Theory (APT)?

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r i = r RF (r 1 - r RF )b 1 (r 2 - r RF )b 2... (r j - r RF )b j.

b j = sensitivity of Stock i to economicFactor j.

r j = required rate of return on a portfolio

sensitive only to economic Factor j.

Required Return for Stock iunder the APT

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The APT is being used for some realworld applications.

Its acceptance has been slow becausethe model does not specify whatfactors influence stock returns.

More research on risk and returnmodels is needed to find a model thatis theoretically sound, empiricallyverified, and easy to use.

W hat is the status of the APT?

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Fama-French 3-Factor Model

Fama and French propose threefactors:

The excess market return, r M-r RF .the return on, S, a portfolio of smallfirms (where size is based on the market

value of equity) minus the return on B, aportfolio of big firms. This return iscalled r SMB , for S minus B.

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Fama-French 3-Factor Model

(Continued)the return on, H, a portfolio of firms withhigh book-to-market ratios (using

market equity and book equity) minusthe return on L, a portfolio of firms withlow book-to-market ratios. This returnis called r HML, for H minus L.

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r i = r RF (r M - r RF )b i (r SMB )c i (r HMB)d i

b i = sensitivity of Stock i to the marketreturn.

c j = sensitivity of Stock i to the sizefactor.

d j = sensitivity of Stock i to the book-to-market factor.

Required Return for Stock iunder the Fama-French 3-Factor Model

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r i = r RF (r M - r RF )b i (r SMB )c i (r HMB)d i

r i = 6.8 (6.3 )(0.9) (4 )(-0. )( )(-0.3)

= 8.97

Required Return for Stock i: b i=0.9,

r RF =6.8 , the market risk premium is6.3 , c i=-0. , the expected value for thesize factor is 4 , d i=-0.3, and the

expected value for the book-to-market

factor is .

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