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LECTURE 8 : FACTOR MODELS

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.

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LECTURE 8 : FACTOR MODELS

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

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

  3. 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)

  4. Single Index Model

  5. 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)

  6. 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

  7. 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

  8. Factor Models

  9. Single Factor Model ER Slope = b a Factor

  10. 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

  11. 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

  12. An Arbitrage Portfolio

  13. 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)

  14. 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

  15. 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

  16. Portfolio Stats / Portfolio Weights (Example)

  17. Pricing Effects • Stock 1 and 2 • 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.

  18. 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)

  19. 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

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

  21. 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

  22. Testing the APT

  23. 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

  24. 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

  25. 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.

  26. 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

  27. 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

  28. 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.

  29. 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)

  30. END OF LECTURE

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