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

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

Introduction 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)

Single Index Model

Single Index Model Alternative approach to portfolio theory. Market return is the single index. Return on a stock can be written as : Ri = ai + biRm ai = ai + ei Hence Ri = ai + biRm + ei Equation (1) Assume : Cov(ei, Rm) = 0 E(eiej) = 0 for all i and j (i ≠ j)

Single Index Model (Cont.) Obtain OLS estimates of ai, bi and sei (using OLS) Mean return : ERi = ai + biERm Variance of security return : s2i = b2is2m + s2ei Covariance of returns between securities : sij = bibjs2m

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

Factor Models

Single Factor Model ER Slope = b a Factor

Factor Model : Example Ri = ai + biF1 + ei Example : Factor-1 is predicted rate of growth in industrial production i mean Ri bi Stock 1 15% 0.9 Stock 2 21% 3.0 Stock 3 12% 1.8

The APT : Some Thoughts 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

An Arbitrage Portfolio

Arbitrage Portfolio Arbitrage portfolio requires no ‘own funds’ Assume there are 3 stocks : 1, 2 and 3 Xi denotes the change in the investors holding (proportion) of security i, then X1 + X2 + X3 = 0 No sensitivity to any factor, so that b1X1 + b2X2 + b3X3 = 0 Example : 0.9 X1 + 3.0 X2 + 1.8 X3 = 0 (assumes zero non factor risk)

Arbitrage Portfolio (Cont.) Let X1 be 0.1. Then 0.1 + X2 + X3 = 0 0.09 + 3.0 X2 + 1.8 X3 = 0 2 equations, 2 unknowns. Solving this system gives X2 = 0.075 X3 = -0.175

Arbitrage Portfolio (Cont.) Expected return X1 ER1 + X2 ER2 + X3 ER3 > 0 Here 15 X1 + 21 X2 + 12 X3 > 0 (= 0.975%) Arbitrage portfolio is attractive to investors who Wants higher expected returns Is not concerned with risk due to factors other than F1

Portfolio Stats / Portfolio Weights (Example) Old Portf. Arbitr. Portf. New Portf. X1 1/3 0.1 0.433 X2 0.075 0.408 X3 -0.175 0.158 Properties ERp 16% 0.975% 16.975% bp 1.9 0.00 sp 11% small approx 11%

Pricing Effects Stock 1 and 2 Stock 3 Buying stock 1 and 2 will push prices up Hence expected returns falls Stock 3 Selling stock 3 will push price down Hence expected return will increase Buying/selling stops if all arbitrage possibilities are eliminated. Linear relationship between expected return and sensitivities ERi = l0 + l1bi where bi is the security’s sensitivity to the factor.

Interpreting the APT ERi = l0 + l1bi l0 = rf l1 = pure factor portfolio, p* that has unit sensitivity to the factor For bi = 1 ERp* = rf + l1 or l1 = ERp* - rf (= factor risk premium)

Two Factor Model : Example Ri = ai + bi1F1 + bi2F2 + ei i ERi bi1 bi2 Stock 1 15% 0.9 2.0 Stock 2 21% 3.0 1.5 Stock 3 12% 1.8 0.7 Stock 4 8% 2.0 3.2

Multi Factor Models Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + ei ERi = l0 + l1 bi1 + l2 bi2 + … + lkbik

Identifying the Factors Unanswered questions : How many factors ? Identity of factors (i.e. values for lamba) Possible factors (literature suggests : 3 – 5) Chen, Roll and Ross (1986) Growth rate in industrial production Rate of inflation (both expected and unexpected) Spread between long-term and short-term interest rates Spread between low-grade and high-grade bonds

Testing the APT

Testing the Theory Proof of any economic theory is how well it describes reality. 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 Ri = ai + S bijFj + ei APT model can be written as ERi = rf + S bijlj

Testing the Theory (Cont.) bij : are unique to each security and represent an attribute of the security Fj : any I affects more than 1 security (if not all). lj : the extra return required because of a security’s sensitivity to the jth attribute of the security

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

Principal Component Analysis (PCA) 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 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

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

Summary 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.

References 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)

END OF LECTURE