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 analysisStephen Ross (1976) developed alternative model Arbitrage Pricing Theory (APT)
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 + biRmai = ai + eiHence Ri = ai + biRm + ei Equation (1)Assume : Cov(ei, Rm) = 0E(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 + biERmVariance of security return :s2i = b2is2m + s2eiCovariance of returns between securities :sij = bibjs2m
7 Portfolio Theory and the Market Model Suppose we have a 5 Stock PortfolioEstimates requiredTraditional MV-approach5 Expected returns5 Variances of returns10 CovariancesUsing the Single Index Model5 OLS regressions5 alphas and 5 betas5 Variances of error term1 Expected return of the market portfolio1 Variance of market return
10 Factor Model : Example Ri = ai + biF1 + ei Example : Factor-1 is predicted rate of growth in industrial productioni mean Ri biStock 1 15% 0.9Stock 2 21% 3.0Stock 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 CAPMAssumption : 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
13 Arbitrage Portfolio Arbitrage portfolio requires no ‘own funds’ Assume there are 3 stocks : 1, 2 and 3Xi denotes the change in the investors holding (proportion) of security i, then X1 + X2 + X3 = 0No sensitivity to any factor, so that b1X1 + b2X2 + b3X3 = 0Example : 0.9 X X X3 = 0(assumes zero non factor risk)
14 Arbitrage Portfolio (Cont.) Let X1 be 0.1.Then0.1 + X2 + X3 = 0X X3 = 02 equations, 2 unknowns.Solving this system givesX2 = 0.075X3 =
15 Arbitrage Portfolio (Cont.) Expected returnX1 ER1 + X2 ER2 + X3 ER3 > 0Here 15 X X X3 > 0 (= 0.975%)Arbitrage portfolio is attractive to investors whoWants higher expected returnsIs not concerned with risk due to factors other than F1
17 Pricing Effects Stock 1 and 2 Stock 3 Buying stock 1 and 2 will push prices upHence expected returns fallsStock 3Selling stock 3 will push price downHence expected return will increaseBuying/selling stops if all arbitrage possibilities are eliminated.Linear relationship between expected return and sensitivitiesERi = l0 + l1biwhere 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 factorFor bi = 1ERp* = rf + l1or l1 = ERp* - rf (= factor risk premium)
19 Two Factor Model : Example Ri = ai + bi1F1 + bi2F2 + eii ERi bi1 bi2Stock 1 15%Stock 2 21%Stock 3 12%Stock 4 8%
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 productionRate of inflation (both expected and unexpected)Spread between long-term and short-term interest ratesSpread between low-grade and high-grade bonds
23 Testing the TheoryProof of any economic theory is how well it describes reality.Testing the APT is not straight forwardtheory specifies a structure for asset pricingtheory does not say anything about the economic or firm characteristics that should affect returns.Multifactor return-generating processRi = ai + S bijFj + eiAPT model can be written asERi = rf + S bijlj
24 Testing the Theory (Cont.) bij : are unique to each security and represent an attribute of the securityFj : 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’sFirst method is to specify a set of attributes (firm characteristics) : bij are directly specifiedSecond 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 factorsPrincipal components (PC) serve as factorsFirst PC : (normalised) linear combination of asset returns with maximum varianceSecond 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 premiumEasy to computeDisadvantage :interpretation of the principal components, statistical approach
28 Summary APT alternative approach to explain asset pricing Factor model requiring fewer assumptions than CAPMBased on concept of arbitrage portfolioInterpretation : lamba’s are difficult to interpret, no economics about the factors and factor weightings.
29 ReferencesCuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 7Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.5 (The Arbitrage Pricing Theory)