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**LECTURE 8 : FACTOR MODELS**

(Asset Pricing and Portfolio Theory)

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**Contents The CAPM Single index model Arbitrage portfolioS**

Which factors explain asset prices ? Empirical results

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**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)

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Single Index Model

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**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)

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**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

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**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

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Factor Models

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Single Factor Model ER Slope = b a Factor

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**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

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**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

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**An Arbitrage Portfolio**

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**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 X X X3 = 0 (assumes zero non factor risk)

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**Arbitrage Portfolio (Cont.)**

Let X1 be 0.1. Then 0.1 + X2 + X3 = 0 X X3 = 0 2 equations, 2 unknowns. Solving this system gives X2 = 0.075 X3 =

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**Arbitrage Portfolio (Cont.)**

Expected return X1 ER1 + X2 ER2 + X3 ER3 > 0 Here 15 X X 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

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**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%

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**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.

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**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)

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**Two Factor Model : Example**

Ri = ai + bi1F1 + bi2F2 + ei i ERi bi1 bi2 Stock 1 15% Stock 2 21% Stock 3 12% Stock 4 8%

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**Multi Factor Models Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + ei**

ERi = l0 + l1 bi1 + l2 bi2 + … + lkbik

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**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

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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. 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

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**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

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**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.

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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. 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

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**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

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**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.

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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)

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END OF LECTURE

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