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LECTURE 8 : FACTOR MODELS (Asset Pricing and Portfolio Theory)

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Contents The CAPM The CAPM Single index model Single index model Arbitrage portfolioS Arbitrage portfolioS Which factors explain asset prices ? Which factors explain asset prices ? Empirical results Empirical results

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Introduction CAPM : Equilibrium model CAPM : Equilibrium model –One factor, where the factor is the excess return on the market. –Based on mean-variance analysis Stephen Ross (1976) developed alternative model Arbitrage Pricing Theory (APT) Stephen Ross (1976) developed alternative model Arbitrage Pricing Theory (APT)

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Single Index Model

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Alternative approach to portfolio theory. Market return is the single index. Return on a stock can be written as : R i = a i + i R m a i = i + e i Hence R i = i + i R m + e i Equation (1) Assume : Cov(e i, R m ) = 0 E(e i e j ) = 0 for all i and j (i ≠ j)

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Single Index Model (Cont.) Obtain OLS estimates of i, i and ei (using OLS) Mean return : Mean return : ER i = i + i ER m Variance of security return : Variance of security return : 2 i = 2 i 2 m + 2 ei Covariance of returns between securities : Covariance of returns between securities : ij = i j 2 m

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Portfolio Theory and the Market Model Suppose we have a 5 Stock Portfolio Suppose we have a 5 Stock Portfolio Estimates required Estimates required –Traditional MV-approach 5 Expected returns 5 Expected returns 5 Variances of returns 5 Variances of returns 10 Covariances 10 Covariances –Using the Single Index Model 5 OLS regressions 5 OLS regressions –5 alphas and 5 betas –5 Variances of error term 1 Expected return of the market portfolio 1 Expected return of the market portfolio 1 Variance of market return 1 Variance of market return

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Factor Models

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Single Factor Model ER Factor a Slope = b

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Factor Model : Example R i = a i + b i F 1 + e i R i = a i + b i F 1 + e i Example : Example : Factor-1 is predicted rate of growth in industrial production imean R i b i Stock 115%0.9 Stock 221%3.0 Stock 312%1.8

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The APT : Some Thoughts The Arbitrage Pricing Theory The Arbitrage Pricing Theory –New and different approach to determine asset prices. –Based on the law of one price : two items that are the same cannot sell at different prices. –Requires fewer assumptions than CAPM –Assumption : each investor, when given the opportunity to increase the return of his portfolio without increasing risk, will do so. Mechanism for doing so : arbitrage portfolio Mechanism for doing so : arbitrage portfolio

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An Arbitrage Portfolio

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Arbitrage Portfolio Arbitrage portfolio requires no ‘own funds’ Arbitrage portfolio requires no ‘own funds’ –Assume there are 3 stocks : 1, 2 and 3 –X i denotes the change in the investors holding (proportion) of security i, then X 1 + X 2 + X 3 = 0 –No sensitivity to any factor, so that b 1 X 1 + b 2 X 2 + b 3 X 3 = 0 –Example : 0.9 X X X 3 = 0 –(assumes zero non factor risk)

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Arbitrage Portfolio (Cont.) Let X 1 be 0.1. Let X 1 be 0.1. Then Then X 2 + X 3 = X 2 + X 3 = X X 3 = X X 3 = 0 –2 equations, 2 unknowns. –Solving this system gives X 2 = X 2 = X 3 = X 3 =

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Arbitrage Portfolio (Cont.) Expected return Expected return X 1 ER 1 + X 2 ER 2 + X 3 ER 3 > 0 Here 15 X X X 3 > 0 (= 0.975%) Arbitrage portfolio is attractive to investors who Arbitrage portfolio is attractive to investors who –Wants higher expected returns –Is not concerned with risk due to factors other than F 1

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Portfolio Stats / Portfolio Weights (Example) Weights Old Portf. Arbitr. Portf. New Portf. X1X1X1X11/ X2X2X2X21/ X3X3X3X31/ Properties ER p 16%0.975%16.975% bp bp bp bp pppp11%small approx 11%

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Pricing Effects Stock 1 and 2 Stock 1 and 2 –Buying stock 1 and 2 will push prices up –Hence expected returns falls Stock 3 Stock 3 –Selling stock 3 will push price down –Hence expected return will increase Buying/selling stops if all arbitrage possibilities are eliminated. Buying/selling stops if all arbitrage possibilities are eliminated. Linear relationship between expected return and sensitivities Linear relationship between expected return and sensitivities ER i = b i where b i is the security’s sensitivity to the factor.

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Interpreting the APT ER i = b i ER i = b i 0 = r f 0 = r f 1 = pure factor portfolio, p* that has unit sensitivity to the factor 1 = pure factor portfolio, p* that has unit sensitivity to the factor For b i = 1 For b i = 1 ER p* = r f + 1 or 1 = ER p* - r f (= factor risk premium)

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Two Factor Model : Example R i = a i + b i1 F 1 + b i2 F 2 + e i R i = a i + b i1 F 1 + b i2 F 2 + e i iER i b i1 b i2 Stock 115% Stock 221% Stock 312% Stock 48%2.03.2

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Multi Factor Models R i = a i + b i1 F 1 + b i2 F 2 + … + b ik F k + e i R i = a i + b i1 F 1 + b i2 F 2 + … + b ik F k + e i ER i = b i1 + 2 b i2 + … + k b ik ER i = b i1 + 2 b i2 + … + k b ik

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Identifying the Factors Unanswered questions : Unanswered questions : –How many factors ? –Identity of factors (i.e. values for lamba) Possible factors (literature suggests : 3 – 5) Possible factors (literature suggests : 3 – 5) 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 rates Spread between long-term and short-term interest rates Spread between low-grade and high-grade bonds Spread between low-grade and high-grade bonds

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Testing the APT

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Testing the Theory Proof of any economic theory is how well it describes reality. Proof of any economic theory is how well it describes reality. Testing the APT is not straight forward Testing the APT is not straight forward –theory specifies a structure for asset pricing –theory does not say anything about the economic or firm characteristics that should affect returns. Multifactor return-generating process Multifactor return-generating process R i = a i + b ij F j + e i APT model can be written as APT model can be written as ER i = r f + b ij j

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Testing the Theory (Cont.) b ij : are unique to each security and represent an attribute of the security F j : any I affects more than 1 security (if not all). j : the extra return required because of a security’s sensitivity to the j th attribute of the security j : the extra return required because of a security’s sensitivity to the j th attribute of the security

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Testing the Theory (Cont.) Obtaining the b ij ’s Obtaining the b ij ’s –First method is to specify a set of attributes (firm characteristics) : b ij are directly specified –Second method is to estimate the b ij ’s and then the j using the equation shown earlier.

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Principal Component Analysis (PCA) Technique to reduce the number of variables being studied without losing too much information in the covariance matrix. Technique to reduce the number of variables being studied without losing too much information in the covariance matrix. Objective : to reduce the dimension from N assets to k factors Objective : to reduce the dimension from N assets to k factors Principal components (PC) serve as factors Principal components (PC) serve as factors –First PC : (normalised) linear combination of asset returns with maximum variance –Second PC : (normalised) linear combination of asset returns with maximum variance of all combinations orthogonal to the first component

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Pro and Cons of Principal Component Analysis Advantage : Advantage : –Allows for time-varying factor risk premium –Easy to compute Disadvantage : Disadvantage : –interpretation of the principal components, statistical approach

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Summary APT alternative approach to explain asset pricing APT alternative approach to explain asset pricing –Factor model requiring fewer assumptions than CAPM –Based on concept of arbitrage portfolio Interpretation : lamba’s are difficult to interpret, no economics about the factors and factor weightings. Interpretation : lamba’s are difficult to interpret, no economics about the factors and factor weightings.

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References Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 7 Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 7 Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.5 (The Arbitrage Pricing Theory) Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.5 (The Arbitrage Pricing Theory)

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END OF LECTURE

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