lecture 7 h08

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Portfolio ManagementPortfolio Management3-228-073-228-07

Albert Lee ChunAlbert Lee Chun

Multifactor Equity Pricing Multifactor Equity Pricing Models Models

Lecture 7Lecture 7

6 Nov 2008

Albert Lee Chun Portfolio Management 2

Today’s LectureToday’s Lecture

Single Factor ModelSingle Factor ModelMultifactor ModelsMultifactor ModelsFama-FrenchFama-FrenchAPTAPT

Albert Lee Chun Portfolio Management

AlphaAlpha

3


• Suppose a security with a particular Suppose a security with a particular is offering is offering expected return of 17% , yet according to the expected return of 17% , yet according to the CAPM, it should be 14.8%.CAPM, it should be 14.8%.

• It’s under-priced: offering too high of a rate of It’s under-priced: offering too high of a rate of return for its level of riskreturn for its level of risk

• Its Its alphaalpha is 17-14.8 = 2.2% is 17-14.8 = 2.2%• According to CAPM alpha should be equal to 0.According to CAPM alpha should be equal to 0.

AlphaAlpha


Frequency Distribution of AlphasFrequency Distribution of Alphas


The CAPM and RealityThe CAPM and Reality

• Is the condition of zero alphas for all stocks as Is the condition of zero alphas for all stocks as implied by the CAPM met?implied by the CAPM met?• Not perfect but one of the best available

• Is the CAPM testable?Is the CAPM testable?• Proxies must be used for the market portfolio

• CAPM is still considered the best available CAPM is still considered the best available description of security pricing and is widely description of security pricing and is widely accepted.accepted.


Single Factor ModelSingle Factor Model

• Returns on a security come from two sourcesReturns on a security come from two sources– Common macro-economic factor– Firm specific events

• Possible common macro-economic factorsPossible common macro-economic factors– Gross Domestic Product Growth– Interest Rates

iiii eFrEr )(


ßßii = index of a security’s particular return to the factor = index of a security’s particular return to the factor

F= some macro factor; in this case F is unanticipated F= some macro factor; in this case F is unanticipated movement; F is commonly related to security movement; F is commonly related to security returnsreturns

Single Factor ModelSingle Factor Model

iiii eFrEr )(

Assumption: a broad market index like the S&P/TSX Composite is the common factor


Regression Equation: Regression Equation: Single Index ModelSingle Index Model

ifMiifi e)rr()rr(

ai = alpha

bi(rM-ri) = the component of return due to market movements (systematic risk)

ei = the component of return due to unexpected firm-specific events (non-systematic risk)


Let: Ri = (ri - rf)

Rm = (rm - rf)Risk premiumformat

Ri = i + ßiRm + ei

Risk Premium FormatRisk Premium Format

The above equation regression is the single-index model using excess returns.


ii2 2 = total variance= total variance

ii22 mm

22 = systematic variance= systematic variance

22(e(eii)) = unsystematic variance= unsystematic variance

Measuring Components of RiskMeasuring Components of Risk

)e( i22M

2i

2i


Index Models and DiversificationIndex Models and Diversification

)(

1

1

1

2222

1

1

1

PMP

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iPP

N

iPP

N

iPP

PPPP

e

eNe

N

N

eR

p


The Variance of a PortfolioThe Variance of a Portfolio


Security Characteristic Line for XSecurity Characteristic Line for X

Excess Returns (i)SCL

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.

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.

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. .. .Excess returnson market index

Ri = i + ßiRm + ei


Multi Factor ModelsMulti Factor Models


More than 1 factor?More than 1 factor?

CAPM is a one factor model: The only determinant CAPM is a one factor model: The only determinant of expected returns is the systematic risk of the of expected returns is the systematic risk of the market. This is the only factor.market. This is the only factor.

What if there are multiple factors that determine What if there are multiple factors that determine returns?returns?

Multifactor Models: Allow for multiple sources of Multifactor Models: Allow for multiple sources of risk, that is risk, that is multiple risk factorsmultiple risk factors..


Multifactor ModelsMultifactor Models

• Use other factors in addition to market returns:Use other factors in addition to market returns:– Examples include industrial production,

expected inflation etc.– Estimate a beta or factor loading for each

factor using multiple regression


Example: Multifactor Model EquationExample: Multifactor Model Equation

RRi i = E(r= E(rii) + Beta) + BetaGDPGDP (GDP) + Beta (GDP) + BetaIRIR (IR) + e (IR) + eii

RRi i = Return for security i= Return for security i

BetaBetaGDPGDP= Factor sensitivity for GDP = Factor sensitivity for GDP

BetaBetaIR IR = Factor sensitivity for Interest Rate= Factor sensitivity for Interest Rate

eei i = Firm specific events= Firm specific events


Multifactor SMLMultifactor SML

E(r) = rf + GDPRPGDP + IRRPIR

GDP = Factor sensitivity for GDP

RPGDP = Risk premium for GDP

IR = Factor sensitivity for Interest Rates

RPIR = Risk premium for GDP


Multifactor ModelsMultifactor Models

CAPM say that a single factor, CAPM say that a single factor, BBeta, determines the eta, determines the relative excess return between a portfolio and the relative excess return between a portfolio and the market as a whole. market as a whole.

Suppose however there are other factors that are Suppose however there are other factors that are important for determining portfolio returns.important for determining portfolio returns.

The inclusion of additional factors would allow the The inclusion of additional factors would allow the model to improve the model`s fit of the data.model to improve the model`s fit of the data.

The best known approach is the three factor model The best known approach is the three factor model developed by Gene Fama and Ken French. developed by Gene Fama and Ken French.


Fama French 3-Factor ModelFama French 3-Factor Model


The Fama-French 3 Factor ModelThe Fama-French 3 Factor Model

Fama and French observed that two classes of stocks Fama and French observed that two classes of stocks tended to outperform the market as a whole:tended to outperform the market as a whole:

(i) small caps (i) small caps

(ii) high book-to-market ratio (ii) high book-to-market ratio


Small Value Stocks OutperformSmall Value Stocks Outperform


Fama-French 3-Factor ModelFama-French 3-Factor Model They added these two factors to a standard CAPM: They added these two factors to a standard CAPM:

SMB = “small [market capitalization] minus big” SMB = “small [market capitalization] minus big” ""SizeSize" This is the return of small stocks minus that of large stocks. " This is the return of small stocks minus that of large stocks.

When small stocks do well relative to large stocks this will be When small stocks do well relative to large stocks this will be positive, and when they do worse than large stocks, this will be positive, and when they do worse than large stocks, this will be negative.negative.

HML = “high [book/price] minus low”HML = “high [book/price] minus low”""ValueValue" This is the return of value stocks minus growth stocks, " This is the return of value stocks minus growth stocks,

which can likewise be positive or negative. which can likewise be positive or negative.

titititftmiitf ti HMLbSMBbrRb = rR ,32,,1,, )(

The Fama-French Three Factor model explains over 90% of stock returns.


Arbitrage Pricing Theory (APT)Arbitrage Pricing Theory (APT)


APTAPT

Ross (1976): intuitive model, only a few Ross (1976): intuitive model, only a few assumptions, considers many sources of risk assumptions, considers many sources of risk

Assumptions:Assumptions:1.1. There are sufficient number of securities to diversify There are sufficient number of securities to diversify

away idiosyncratic risk away idiosyncratic risk 2.2. The return on securities is a function of K different The return on securities is a function of K different

risk factors.risk factors.3.3. No arbitrage opportunitiesNo arbitrage opportunities


APTAPT

APT does not require the following CAPM APT does not require the following CAPM assumptions:assumptions:

1.1. Investors are mean-variance optimizers in the sense Investors are mean-variance optimizers in the sense of Markowitz.of Markowitz.

2.2. Returns are normally distributed.Returns are normally distributed.3.3. The market portfolio contains all the risky securities The market portfolio contains all the risky securities

and it is efficient in the mean-variance sense.and it is efficient in the mean-variance sense.


APT & Well-Diversified APT & Well-Diversified PortfoliosPortfolios

F is some macroeconomic factorF is some macroeconomic factor For a well-diversified portfolio eFor a well-diversified portfolio ePP approaches zero approaches zero

PPPP eF)r(Er


Returns as a Function of the Systematic FactorReturns as a Function of the Systematic Factor

Well-diversified portfolio Single Stocks


Returns as a Function of the Systematic Factor: An Returns as a Function of the Systematic Factor: An Arbitrage OpportunityArbitrage Opportunity


E(r)%

Beta for F

1076

Risk Free = 4

AD

C

.5 1.0

Example: An Arbitrage Opportunity

Risk premiums must be proportional to Betas!


Disequilibrium ExampleDisequilibrium Example

• Short Portfolio Short Portfolio CC, with Beta = .5, with Beta = .5• One can construct a portfolio with equivalent risk One can construct a portfolio with equivalent risk

and higher return : Portfolio and higher return : Portfolio DD• D = .5x A + .5 x Risk-Free Asset• D has Beta = .5

• Arbitrage opportunity: riskless profit of 1%Arbitrage opportunity: riskless profit of 1%



APT APT Security Market LineSecurity Market Line

)()( rfRmrfrE PP


This is CAPM!


• APT applies to well diversified portfolios and not APT applies to well diversified portfolios and not necessarily to individual stocksnecessarily to individual stocks

• With APT it is possible for some individual stocks to With APT it is possible for some individual stocks to be mispriced – that is to not lie on the SMLbe mispriced – that is to not lie on the SML

• APT is more general in that it gets to an expected APT is more general in that it gets to an expected return and beta relationship without the assumption of return and beta relationship without the assumption of the market portfoliothe market portfolio

• APT can be extended to multifactor modelsAPT can be extended to multifactor models

APT and CAPM ComparedAPT and CAPM Compared


A Multifactor APTA Multifactor APT

• A A factor portfoliofactor portfolio is a portfolio constructed so is a portfolio constructed so that it would have a beta equal to one on a given that it would have a beta equal to one on a given factor and zero on any other factorfactor and zero on any other factor

• These factor portfolios are the building blocks for These factor portfolios are the building blocks for a multifactor security market line for an economy a multifactor security market line for an economy with multiple sources of riskwith multiple sources of risk


Where Should we Look for Factors?Where Should we Look for Factors?

• The multifactor APT gives no guidance on where to The multifactor APT gives no guidance on where to look for factorslook for factors

• Chen, Roll and RossChen, Roll and Ross– Returns a function of several macroeconomic and

bond market variables instead of market returns• Fama and FrenchFama and French

• Returns a function of size and book-to-market value as well as market returns

9-37


In theory, the APT supposes a stochastic process that generates returns and that may be In theory, the APT supposes a stochastic process that generates returns and that may be represented by a model of K factors, such thatrepresented by a model of K factors, such that

where:where: RRii = One period realized return on security = One period realized return on security ii, , ii= 1,2,3…,n= 1,2,3…,n E(Ri) E(Ri) = expected return of security i = expected return of security i

= Sensitivity of the reutrn of the ith stock to the jth risk factor= Sensitivity of the reutrn of the ith stock to the jth risk factor

= j-th risk factor= j-th risk factor

=captures the unique risk associated with security i=captures the unique risk associated with security i

Similar to CAPM, the APT assumes that the idiosyncratic effects can be diversified Similar to CAPM, the APT assumes that the idiosyncratic effects can be diversified away in a large portfolio.away in a large portfolio.

Generalized Factor ModelGeneralized Factor Model

niFBFBFB RiE = Ri ikikii ...1 ...)( 2211

ijB

jF

i


Multifactor APTMultifactor APT

))((...).)2(())1(()( 21 rfRkE BrfREBrfREBRiE ikiirf

APT Model

The expected return on a secutity depends on The expected return on a secutity depends on the product of the risk premiums and the factor betas (or factor the product of the risk premiums and the factor betas (or factor loadings)loadings)

E(Ri) – rf is the risk premium on the ith E(Ri) – rf is the risk premium on the ith factor portfoliofactor portfolio..


Sample APT ProblemSample APT Problem

Suppose that the equity market in a large economy Suppose that the equity market in a large economy can be described by 3 sources of risk: A, B and C.can be described by 3 sources of risk: A, B and C.

FactorFactor Risk Premium Risk Premium AA .06.06 BB .04.04 CC .02.02


Example APT ProblemExample APT Problem

Suppose that the return on Maggie’s Mushroom Factory Suppose that the return on Maggie’s Mushroom Factory is given by the following equation, with an expected is given by the following equation, with an expected return of return of 17%.17%.

r(t) = r(t) = .17.17 + 1.0 x A + .75 x B + .05 x C + error(t) + 1.0 x A + .75 x B + .05 x C + error(t)


Sample APT problemSample APT problem

The risk free rate is given by The risk free rate is given by 6%6%

1. Find the expected rate of return of the mushroom 1. Find the expected rate of return of the mushroom factory under the APT model.factory under the APT model.

2. Is the stock-under or over-valued? Why?2. Is the stock-under or over-valued? Why?



FactorFactor Risk Premium Risk Premium AA .06.06 BB .04.04 CC .02.02 Risk-Free Rate = Risk-Free Rate = 6%6% Return(t) = Return(t) = .17.17 + + 1.01.0*A + *A + 0.750.75*B + .*B + .0505*C + e(t)*C + e(t)

The factor loadings are in The factor loadings are in green.green.



FactorFactor Risk Premium Risk Premium AA .06.06 BB .04.04 CC .02.02 Risk-Free Rate = Risk-Free Rate = 6%6% Return = Return = .17.17 + + 1.01.0*A + *A + 0.750.75*B + .05*C + e*B + .05*C + e So plug in risk-premia into the APT formulaSo plug in risk-premia into the APT formula

E[Ri] = E[Ri] = .06.06 + + 1.01.0**0.060.06++0.750.75**0.040.04++0.50.5**0.020.02 = .16 = .16 16% < 16% < 17%17% => Undervalued! => Undervalued!


Quick Review of Underpricing Quick Review of Underpricing Undervalued = Underpriced = Return Too HighUndervalued = Underpriced = Return Too High Overvalued = Overpriced = Return Too LowOvervalued = Overpriced = Return Too Low

P(t) = P(t+1)/ 1+ rP(t) = P(t+1)/ 1+ r

r = P(t+1)/P(t) – 1 r = P(t+1)/P(t) – 1

where r is the return for a risky payoff P(t+1).where r is the return for a risky payoff P(t+1).

This is easy to remember if you think about the inverse This is easy to remember if you think about the inverse relationship between price (value) today and return.relationship between price (value) today and return.


Examples 9.3 and 9.4Examples 9.3 and 9.4

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Factor portfolio 1: E(R1) = 10%Factor Portfolio 2: E(R2) = 12%

Rf = 4%

Portfolio A with B1 = .5 and B2 = .75

Construct aPortfolio Q using weights ofB1 = .5 on factor portfolio 1 and a weight of

B2 = .75 on factor portfolio 2 and a weight of 1- B1 – B2 = -.25 on the risk free rate.

E(Rq) = B1E(R1) + B2 E(R2) + (1-B1-B2) Rf= rf + B1(E(R1) –rf )+ B2(E(R2) – rf) =13%


Example 9.4Example 9.4

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Suppose that: E(RA) = 12% < 13%

Portfolio QPonderation B1 = .5: facteur portefeuille 1

Ponderation B2 = .75: facteur portefeuille 2Ponderation 1- B1 – B2 = -.25 : rf

E(Rq ) = 12%

$1 x E(Rq) - $1x E(RA)=1%

There is a riskless arbitrage

opportunity of 1%!


Next WeekNext Week

We will continue our lecture with Chapter 12We will continue our lecture with Chapter 12 Market Efficiency (Chapter 10; Section 11.1)Market Efficiency (Chapter 10; Section 11.1)

lecture 7 h08

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