Chapter 9Chapter 9Charles P. Jones, Investments: Analysis and Charles P. Jones, Investments: Analysis and

Management,Management,Ninth Edition, John Wiley & SonsNinth Edition, John Wiley & Sons

Prepared byPrepared byG.D. Koppenhaver, Iowa State UniversityG.D. Koppenhaver, Iowa State University

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Asset Pricing ModelsAsset Pricing Models

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Capital Asset Pricing ModelCapital Asset Pricing Model

Focus on the equilibrium relationship Focus on the equilibrium relationship between the risk and expected between the risk and expected return on risky assetsreturn on risky assets

Builds on Markowitz portfolio theoryBuilds on Markowitz portfolio theory Each investor is assumed to diversify Each investor is assumed to diversify

his or her portfolio according to the his or her portfolio according to the Markowitz modelMarkowitz model

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CAPM AssumptionsCAPM Assumptions

All investors:All investors:– Use the same Use the same

information to information to generate an generate an efficient frontier efficient frontier

– Have the same one-Have the same one-period time horizonperiod time horizon

– Can borrow or lend Can borrow or lend money at the risk-money at the risk-free rate of returnfree rate of return

No transaction No transaction costs, no personal costs, no personal income taxes, no income taxes, no inflationinflation

No single investor No single investor can affect the price can affect the price of a stockof a stock

Capital markets Capital markets are in equilibrium are in equilibrium

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Borrowing and Lending Borrowing and Lending PossibilitiesPossibilities

Risk free assets Risk free assets – Certain-to-be-earned expected return Certain-to-be-earned expected return

and a variance of return of zeroand a variance of return of zero– No correlation with risky assetsNo correlation with risky assets– Usually proxied by a Treasury securityUsually proxied by a Treasury security

Amount to be received at maturity is free of Amount to be received at maturity is free of default risk, known with certaintydefault risk, known with certainty

Adding a risk-free asset extends and Adding a risk-free asset extends and changes the efficient frontierchanges the efficient frontier

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Risk

B

A

TE(R)

RF

L

Z X

Risk-Free LendingRisk-Free Lending

Riskless assets can Riskless assets can be combined with be combined with any portfolio in the any portfolio in the efficient set ABefficient set AB– Z implies lendingZ implies lending

Set of portfolios on Set of portfolios on line RF to T line RF to T dominates all dominates all portfolios below itportfolios below it

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Impact of Risk-Free LendingImpact of Risk-Free Lending

If wIf wRFRF placed in a risk-free asset placed in a risk-free asset– Expected portfolio returnExpected portfolio return

– Risk of the portfolioRisk of the portfolio

Expected return and risk of the Expected return and risk of the portfolio with lending is a weighted portfolio with lending is a weighted averageaverage

))E(R-w (RF w) E(R XRFRFp 1

XRFp )σ-w ( σ 1

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Borrowing PossibilitiesBorrowing Possibilities

Investor no longer restricted to own Investor no longer restricted to own wealthwealth

Interest paid on borrowed moneyInterest paid on borrowed money– Higher returns sought to cover expenseHigher returns sought to cover expense– Assume borrowing at RFAssume borrowing at RF

Risk will increase as the amount of Risk will increase as the amount of borrowing increasesborrowing increases– Financial leverageFinancial leverage

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The New Efficient SetThe New Efficient Set

Risk-free investing and borrowing Risk-free investing and borrowing creates a new set of expected return-creates a new set of expected return-risk possibilitiesrisk possibilities

Addition of risk-free asset results inAddition of risk-free asset results in– A change in the efficient set from an arc A change in the efficient set from an arc

to a straight line tangent to the feasible to a straight line tangent to the feasible set without the riskless assetset without the riskless asset

– Chosen portfolio depends on investor’s Chosen portfolio depends on investor’s risk-return preferencesrisk-return preferences

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Portfolio ChoicePortfolio Choice

The more conservative the investor The more conservative the investor the more is placed in risk-free the more is placed in risk-free lending and the less borrowinglending and the less borrowing

The more aggressive the investor the The more aggressive the investor the less is placed in risk-free lending and less is placed in risk-free lending and the more borrowingthe more borrowing– Most aggressive investors would use Most aggressive investors would use

leverage to invest more in portfolio Tleverage to invest more in portfolio T

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Capital Market Line Capital Market Line

A risk averse investor makes investment A risk averse investor makes investment decisions based on Markowitz principles.decisions based on Markowitz principles.

Investors can borrow and lend freely at the Investors can borrow and lend freely at the RFR.RFR.

Each investor should construct an optimal Each investor should construct an optimal portfolio that matches his or her preferred portfolio that matches his or her preferred risk return combinations. risk return combinations.

All investors can construct an efficient All investors can construct an efficient portfolio by combining an efficient portfolio by combining an efficient portfolio M, with a risk free asset. portfolio M, with a risk free asset.

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CMLCML

Depicts the equilibrium conditions Depicts the equilibrium conditions that prevail in the market for efficient that prevail in the market for efficient portfolio consisting of the optimal portfolio consisting of the optimal portfolio of risky assets and the risk portfolio of risky assets and the risk free asset. free asset.

All combinations of the risk free asset All combinations of the risk free asset and the risky portfolio M are on CML, and the risky portfolio M are on CML, and, in equilibrium, all investors will and, in equilibrium, all investors will end up with a portfolio somewhere end up with a portfolio somewhere on CML based on their risk tolerance. on CML based on their risk tolerance.

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E(RM)

RF

RiskM

L

M

y

x

Capital Market LineCapital Market Line

Line from RF to L Line from RF to L is capital market is capital market line (CML)line (CML)

x = risk premium x = risk premium =E(RM) - RF =E(RM) - RF

y =risk =y =risk =M

Slope =x/ySlope =x/y

=[E(RM) - RF]/=[E(RM) - RF]/M

y-intercept = RFy-intercept = RF

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Capital Market LineCapital Market Line

Slope of the CML is the market price Slope of the CML is the market price of risk for efficient portfolios, or the of risk for efficient portfolios, or the equilibrium price of risk in the marketequilibrium price of risk in the market

Relationship between risk and Relationship between risk and expected return for portfolio P expected return for portfolio P (Equation for CML):(Equation for CML):

pM

Mp σ

σRF)E(R

RF) E(R

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Important points about the Important points about the CMLCML

Only efficient portfolios consisting of the risk Only efficient portfolios consisting of the risk free asset and portfolio M lie on the CML. free asset and portfolio M lie on the CML.

The CML must always be upward sloping The CML must always be upward sloping because the price of risk must always be because the price of risk must always be positive. (risk & return relationship)positive. (risk & return relationship)

On a historical basis, for some particular time On a historical basis, for some particular time period, the CML can be downward sloping. period, the CML can be downward sloping. (RF>R(RF>RMM))

The CML can be used to determine the The CML can be used to determine the optimal expected returns associated with optimal expected returns associated with different portfolio risk levels. CML indicates different portfolio risk levels. CML indicates the required return for each portfolio risk the required return for each portfolio risk level. level.

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Market PortfolioMarket Portfolio

Most important implication of the Most important implication of the CAPM CAPM – All investors hold the same optimal All investors hold the same optimal

portfolio of risky assetsportfolio of risky assets– The optimal portfolio is at the highest The optimal portfolio is at the highest

point of tangency between RF and the point of tangency between RF and the efficient frontier efficient frontier

– The portfolio of all risky assets is the The portfolio of all risky assets is the optimal risky portfoliooptimal risky portfolio Called the market portfolioCalled the market portfolio

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Characteristics of the Characteristics of the Market PortfolioMarket Portfolio

All risky assets must be in portfolio, so All risky assets must be in portfolio, so it is completely diversifiedit is completely diversified– Includes only systematic riskIncludes only systematic risk

All securities included in proportion to All securities included in proportion to their market valuetheir market value

Unobservable but proxied by S&P 500Unobservable but proxied by S&P 500 Contains worldwide assetsContains worldwide assets

– Financial and real assetsFinancial and real assets

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The Separation TheoremThe Separation Theorem Investors use their preferences Investors use their preferences

(reflected in an indifference curve) to (reflected in an indifference curve) to determine their optimal portfoliodetermine their optimal portfolio

Separation Theorem:Separation Theorem:– The investment decision, which risky The investment decision, which risky

portfolio to hold, is separate from the portfolio to hold, is separate from the financing decisionfinancing decision

– Allocation between risk-free asset and Allocation between risk-free asset and risky portfolio separate from choice of risky portfolio separate from choice of risky portfolio, Trisky portfolio, T

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Separation TheoremSeparation Theorem

All investorsAll investors– Invest in the same portfolioInvest in the same portfolio– Attain any point on the straight line RF-Attain any point on the straight line RF-

T-L by by either borrowing or lending at T-L by by either borrowing or lending at the rate RF, depending on their the rate RF, depending on their preferencespreferences

Risky portfolios are not tailored to Risky portfolios are not tailored to each individual’s tasteeach individual’s taste

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Security Market LineSecurity Market Line

CML Equation only applies to markets CML Equation only applies to markets in equilibrium and efficient portfoliosin equilibrium and efficient portfolios

The Security Market Line depicts the The Security Market Line depicts the tradeoff between risk and expected tradeoff between risk and expected return for individual securitiesreturn for individual securities

Under CAPM, all investors hold the Under CAPM, all investors hold the market portfoliomarket portfolio– How does an individual security contribute How does an individual security contribute

to the risk of the market portfolio?to the risk of the market portfolio?

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Security Market LineSecurity Market Line

A security’s contribution to the risk of A security’s contribution to the risk of the market portfolio is based on betathe market portfolio is based on beta

Equation for expected return for an Equation for expected return for an individual stockindividual stock

RF)E(RβRF) E(R Mii

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AB

C

kM

kRF

0 1.0 2.00.5 1.5

SML

BetaM

E(R)

Security Market LineSecurity Market Line

Beta = 1.0 implies Beta = 1.0 implies as risky as marketas risky as market

Securities A and B Securities A and B are more risky than are more risky than the marketthe market– Beta >1.0Beta >1.0

Security C is less Security C is less risky than the risky than the marketmarket– Beta <1.0Beta <1.0

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Security Market LineSecurity Market Line

Beta measures systematic riskBeta measures systematic risk– Measures relative risk compared to the Measures relative risk compared to the

market portfolio of all stocksmarket portfolio of all stocks– Volatility different than marketVolatility different than market

All securities should lie on the SMLAll securities should lie on the SML– The expected return on the security The expected return on the security

should be only that return needed to should be only that return needed to compensate for systematic riskcompensate for systematic risk

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CAPM’s Expected CAPM’s Expected Return-Beta RelationshipReturn-Beta Relationship

Required rate of return on an asset Required rate of return on an asset (k(kii) is composed of) is composed of– risk-free rate (risk-free rate (RF))– risk premium (risk premium (i [ E(RM) - RF ]))

Market risk premium adjusted for specific Market risk premium adjusted for specific securitysecurity

ki = RF +i [ E(RM) - RF ]– The greater the systematic risk, the The greater the systematic risk, the

greater the required returngreater the required return

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Estimating the SMLEstimating the SML

Treasury Bill rate used to estimate RFTreasury Bill rate used to estimate RF Expected market return unobservableExpected market return unobservable

– Estimated using past market returns and Estimated using past market returns and taking an expected valuetaking an expected value

Estimating individual security betas Estimating individual security betas difficultdifficult– Only company-specific factor in CAPMOnly company-specific factor in CAPM– Requires asset-specific forecastRequires asset-specific forecast

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Estimating BetaEstimating Beta

Market modelMarket model– Relates the return on each stock to the Relates the return on each stock to the

return on the market, assuming a linear return on the market, assuming a linear relationshiprelationship

Ri =i +i RM +ei

Characteristic lineCharacteristic line– Line fit to total returns for a security Line fit to total returns for a security

relative to total returns for the market relative to total returns for the market indexindex

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How Accurate Are Beta How Accurate Are Beta Estimates?Estimates?

Betas change with a company’s situationBetas change with a company’s situation– Not stationary over time Not stationary over time

Estimating a Estimating a futurefuture beta beta– May differ from the historical betaMay differ from the historical beta

RRM represents the total of all marketable represents the total of all marketable assets in the economyassets in the economy– Approximated with a stock market indexApproximated with a stock market index– Approximates return on all common stocksApproximates return on all common stocks

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How Accurate Are Beta How Accurate Are Beta Estimates?Estimates?

No one correct number of No one correct number of observations and time periods for observations and time periods for calculating betacalculating beta

The regression calculations of the The regression calculations of the true true and and from the characteristic from the characteristic line are subject to estimation errorline are subject to estimation error

Portfolio betas more reliable than Portfolio betas more reliable than individual security betasindividual security betas

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Arbitrage Pricing TheoryArbitrage Pricing Theory

Based on the Law of One PriceBased on the Law of One Price– Two otherwise identical assets cannot Two otherwise identical assets cannot

sell at different pricessell at different prices– Equilibrium prices adjust to eliminate all Equilibrium prices adjust to eliminate all

arbitrage opportunitiesarbitrage opportunities Unlike CAPM, APT does Unlike CAPM, APT does notnot assume assume

– single-period investment horizon, single-period investment horizon, absence of personal taxes, riskless absence of personal taxes, riskless borrowing or lending, mean-variance borrowing or lending, mean-variance decisionsdecisions

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FactorsFactors

APT assumes returns generated by a APT assumes returns generated by a factor modelfactor model

Factor CharacteristicsFactor Characteristics– Each risk must have a pervasive Each risk must have a pervasive

influence on stock returnsinfluence on stock returns– Risk factors must influence expected Risk factors must influence expected

return and have nonzero pricesreturn and have nonzero prices– Risk factors must be unpredictable to Risk factors must be unpredictable to

the marketthe market

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APT ModelAPT Model

Most important are the deviations of Most important are the deviations of the factors from their expected valuesthe factors from their expected values

The expected return-risk relationship The expected return-risk relationship for the APT can be described as:for the APT can be described as:E(Ri) =RF +bi1 (risk premium for factor 1) +bi2 (risk premium for factor 2) +… +bin (risk premium

for factor n)

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Problems with APTProblems with APT

Factors are not well specified ex anteFactors are not well specified ex ante– To implement the APT model, need the To implement the APT model, need the

factors that account for the differences factors that account for the differences among security returnsamong security returns CAPM identifies market portfolio as single CAPM identifies market portfolio as single

factorfactor

Neither CAPM or APT has been Neither CAPM or APT has been proven superiorproven superior– Both rely on unobservable expectationsBoth rely on unobservable expectations