LECTURE 8 :LECTURE 8 :

FACTOR MODELSFACTOR MODELS

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

ContentsContents

The CAPMThe CAPM Single index modelSingle index model Arbitrage portfolioSArbitrage portfolioS Which factors explain asset prices Which factors explain asset prices

?? Empirical resultsEmpirical results

IntroductionIntroduction

CAPM : Equilibrium model CAPM : Equilibrium model – One factor, where the factor is the One factor, where the factor is the

excess return on the market. excess return on the market. – Based on mean-variance analysisBased on mean-variance analysis

Stephen Ross (1976) developed Stephen Ross (1976) developed alternative model alternative model Arbitrage Arbitrage Pricing Theory (APT)Pricing Theory (APT)

Single Index ModelSingle Index Model

Single Index Model Single Index Model

Alternative approach to portfolio theory. Alternative approach to portfolio theory.

Market return is the single index. Market return is the single index. Return on a stock can be written as : Return on a stock can be written as :

RRii = a = aii + + iiRRmm

aaii = = ii + e + eii

Hence RHence Rii = = ii + + iiRRmm + e + eii Equation (1)Equation (1)

Assume : Assume : Cov(eCov(eii, R, Rmm) = 0) = 0

E(eE(eiieejj) = 0 for all i and j (i ≠ j)) = 0 for all i and j (i ≠ j)

Single Index Model Single Index Model (Cont.)(Cont.)Obtain OLS estimates of Obtain OLS estimates of ii, , ii and and eiei (using (using

OLS)OLS)

Mean return : Mean return : ERERii = = ii + + iiERERmm

Variance of security return : Variance of security return : 22

ii = = 22ii22

mm + + 22eiei

Covariance of returns between securities : Covariance of returns between securities : ijij = = iijj22

mm

Portfolio Theory and Portfolio Theory and the Market Modelthe Market Model Suppose we have a 5 Stock PortfolioSuppose we have a 5 Stock Portfolio Estimates required Estimates required

– Traditional MV-approach Traditional MV-approach 5 Expected returns5 Expected returns 5 Variances of returns5 Variances of returns 10 Covariances10 Covariances

– Using the Single Index Model Using the Single Index Model 5 OLS regressions 5 OLS regressions

– 5 alphas and 5 betas5 alphas and 5 betas– 5 Variances of error term5 Variances of error term

1 Expected return of the market portfolio1 Expected return of the market portfolio 1 Variance of market return1 Variance of market return

Factor ModelsFactor Models

Single Factor Model Single Factor Model

ER

Factor

a

Slope = b

Factor Model : Factor Model : ExampleExample RRii = a = aii + b + biiFF11 + e + eii Example : Example :

Factor-1 is predicted rate of growth in Factor-1 is predicted rate of growth in industrial productionindustrial production

ii mean Rmean Rii bbii

Stock 1Stock 1 15%15% 0.90.9

Stock 2Stock 2 21%21% 3.03.0

Stock 3Stock 3 12%12% 1.81.8

The APT : Some The APT : Some ThoughtsThoughts The Arbitrage Pricing Theory The Arbitrage Pricing Theory

– New and different approach to determine New and different approach to determine asset prices.asset prices.

– Based on the law of one price : two items Based on the law of one price : two items that are the same cannot sell at different that are the same cannot sell at different prices. prices.

– Requires fewer assumptions than CAPMRequires fewer assumptions than CAPM– Assumption : each investor, when given the Assumption : each investor, when given the

opportunity to increase the return of his opportunity to increase the return of his portfolio without increasing risk, will do so. portfolio without increasing risk, will do so.

Mechanism for doing so : arbitrage portfolioMechanism for doing so : arbitrage portfolio

An Arbitrage PortfolioAn Arbitrage Portfolio

Arbitrage PortfolioArbitrage Portfolio

Arbitrage portfolio requires no ‘own funds’ Arbitrage portfolio requires no ‘own funds’ – Assume there are 3 stocks : 1, 2 and 3Assume there are 3 stocks : 1, 2 and 3

– XXii denotes the denotes the changechange in the investors holding in the investors holding (proportion) of security i, then X(proportion) of security i, then X11 + X + X22 + X + X33 = 0 = 0

– No sensitivity to any factor, so that bNo sensitivity to any factor, so that b11XX11 + b + b22XX22 + b+ b33XX33 = 0 = 0

– Example : 0.9 XExample : 0.9 X11 + 3.0 X + 3.0 X22 + 1.8 X + 1.8 X33 = 0 = 0

– (assumes zero non factor risk) (assumes zero non factor risk)

Arbitrage Portfolio Arbitrage Portfolio (Cont.) (Cont.) Let XLet X11 be 0.1. be 0.1. Then Then

0.1 + X0.1 + X22 + X + X33 = 0 = 0

0.09 + 3.0 X0.09 + 3.0 X22 + 1.8 X + 1.8 X33 = 0 = 0

– 2 equations, 2 unknowns. 2 equations, 2 unknowns. – Solving this system gives Solving this system gives

XX22 = 0.075 = 0.075

XX33 = -0.175 = -0.175

Arbitrage Portfolio Arbitrage Portfolio (Cont.)(Cont.) Expected return Expected return

XX11 ER ER11 + X + X22 ER ER22 + X + X33 ER ER33 > 0 > 0

Here 15 XHere 15 X11 + 21 X + 21 X22 + 12 X + 12 X33 > 0 (= 0.975%) > 0 (= 0.975%)

Arbitrage portfolio is attractive to Arbitrage portfolio is attractive to investors who investors who – Wants higher expected returnsWants higher expected returns– Is not concerned with risk due to factors Is not concerned with risk due to factors

other than Fother than F11

Portfolio Stats / Portfolio Stats / Portfolio Weights Portfolio Weights (Example)(Example)

WeightsWeights Old Portf.Old Portf. Arbitr. Arbitr. Portf.Portf.

New Portf.New Portf.

XX11 1/31/3 0.10.1 0.4330.433

XX22 1/31/3 0.0750.075 0.4080.408

XX33 1/31/3 -0.175-0.175 0.1580.158

PropertiesProperties

ERERpp 16%16% 0.975%0.975% 16.975%16.975%

bbpp 1.91.9 0.000.00 1.91.9

pp 11%11% smallsmall approx approx 11%11%

Pricing EffectsPricing Effects

Stock 1 and 2 Stock 1 and 2 – Buying stock 1 and 2 will push prices upBuying stock 1 and 2 will push prices up– Hence expected returns fallsHence expected returns falls

Stock 3Stock 3– Selling stock 3 will push price downSelling stock 3 will push price down– Hence expected return will increaseHence expected return will increase

Buying/selling stops if all arbitrage possibilities Buying/selling stops if all arbitrage possibilities are eliminated. are eliminated.

Linear relationship between expected return and Linear relationship between expected return and sensitivities sensitivities

ERERii = = 00 + + 11bbii

where bwhere bii is the security’s sensitivity to the is the security’s sensitivity to the factor. factor.

Interpreting the APTInterpreting the APT

ERERii = = 00 + + 11bbii

00 = r = rff

11 = pure factor portfolio, p* that has unit = pure factor portfolio, p* that has unit sensitivity to the factorsensitivity to the factor

For bFor bii = 1 = 1

ERERp*p* = r = rff + + 11

or or 11 = ER = ERp*p* - r - rff (= factor risk premium) (= factor risk premium)

Two Factor Model : Two Factor Model : ExampleExample RRii = a = aii + b + bi1i1FF11 + b + bi2i2FF22 + e + eii

ii ERERii bbi1i1 bbi2i2

Stock 1Stock 1 15%15% 0.90.9 2.02.0

Stock 2Stock 2 21%21% 3.03.0 1.51.5

Stock 3Stock 3 12%12% 1.81.8 0.70.7

Stock 4Stock 4 8%8% 2.02.0 3.23.2

Multi Factor Models Multi Factor Models

RRii = a = aii + b + bi1i1 F F11 + b + bi2i2 F F22 + … + b + … + bikik F Fkk + + eeii

ERERii = = 00 + + 11 b bi1i1 + + 22 b bi2i2 + … + + … + kkbbikik

Identifying the FactorsIdentifying the Factors

Unanswered questions : Unanswered questions : – How many factors ? How many factors ? – Identity of factors (i.e. values for lamba)Identity of factors (i.e. values for lamba)

Possible factors (literature suggests : 3 – Possible factors (literature suggests : 3 – 5)5)Chen, Roll and Ross (1986)Chen, Roll and Ross (1986)

Growth rate in industrial production Growth rate in industrial production Rate of inflation (both expected and unexpected)Rate of inflation (both expected and unexpected) Spread between long-term and short-term interest Spread between long-term and short-term interest

ratesrates Spread between low-grade and high-grade bondsSpread between low-grade and high-grade bonds

Testing the APTTesting the APT

Testing the TheoryTesting the Theory

Proof of any economic theory is how well it Proof of any economic theory is how well it describes reality. describes reality.

Testing the APT is not straight forward Testing the APT is not straight forward – theory specifies a structure for asset pricingtheory specifies a structure for asset pricing– theory does not say anything about the economic or theory does not say anything about the economic or

firm characteristics that should affect returns. firm characteristics that should affect returns.

Multifactor return-generating process Multifactor return-generating process RRii = a = aii + + b bijijFFjj + e + eii

APT model can be written as APT model can be written as ERERii = r = rff + + bbijijjj

Testing the Theory Testing the Theory (Cont.)(Cont.)bbijij : are unique to each security and represent : are unique to each security and represent

an attribute of the securityan attribute of the security

FFjj : any I affects more than 1 security (if not : any I affects more than 1 security (if not all). all).

jj : the extra return required because of a : the extra return required because of a security’s sensitivity to the jsecurity’s sensitivity to the jthth attribute of attribute of the security the security

Testing the Theory Testing the Theory (Cont.)(Cont.) Obtaining the bObtaining the bijij’s’s

– First method is to specify a set of First method is to specify a set of attributes (firm characteristics) : battributes (firm characteristics) : bijij are directly specifiedare directly specified

– Second method is to estimate the Second method is to estimate the bbijij’s and then the ’s and then the jj using the using the equation shown earlier. equation shown earlier.

Principal Component Principal Component Analysis (PCA)Analysis (PCA) Technique to reduce the number of variables Technique to reduce the number of variables

being studied without losing too much being studied without losing too much information in the covariance matrix. information in the covariance matrix.

Objective : to reduce the dimension from N Objective : to reduce the dimension from N assets to k factorsassets to k factors

Principal components (PC) serve as factors Principal components (PC) serve as factors – First PC : (normalised) linear combination of asset First PC : (normalised) linear combination of asset

returns with maximum variance returns with maximum variance – Second PC : (normalised) linear combination of asset Second PC : (normalised) linear combination of asset

returns with maximum variance of all combinations returns with maximum variance of all combinations orthogonal to the first componentorthogonal to the first component

Pro and Cons of Pro and Cons of Principal Component Principal Component AnalysisAnalysis Advantage : Advantage :

– Allows for time-varying factor risk Allows for time-varying factor risk premium premium

– Easy to computeEasy to compute

Disadvantage : Disadvantage : – interpretation of the principal interpretation of the principal

components, statistical approachcomponents, statistical approach

SummarySummary

APT alternative approach to explain APT alternative approach to explain asset pricing asset pricing – Factor model requiring fewer Factor model requiring fewer

assumptions than CAPMassumptions than CAPM– Based on concept of arbitrage portfolioBased on concept of arbitrage portfolio

Interpretation : lamba’s are difficult Interpretation : lamba’s are difficult to interpret, no economics about to interpret, no economics about the factors and factor weightings. the factors and factor weightings.

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapters 7 Economics’, Chapters 7

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.5 Derivatives Markets’, Chapter 10.5 (The Arbitrage Pricing Theory)(The Arbitrage Pricing Theory)

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