# 091114_APT - Empirical Validation of APT November 2009 APT...

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Empirical Validation of APT Lecture 11 14 November 2009

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APT: a solution to CAPM shortcomings z Factors other than β influence stock returns: Size of the firm (can be measured by market capitalization) Firm’s perspectives (can be measured by book-to-market ratio) Momentum (average stock’s return over the past six months) z Actual SML is flatted than predicted by CAPM
APT: a reminder z Factor model + absence of arbitrage = APT z Returns are determined by the sensitivity to a common set of factors z The expected return of a security is determined by the following equation: Where λ k = E(RP k ) – R f (risk premium of k th factor) () k ki i i f i R R E λ β + + + + = ... 2 2 1 1

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Estimating the factors z Factor analysis A statistical procedure aimed at finding factor- replicating portfolios z Using macroeconomic variables to generate factors Macro variables are proxies for the factors z Using characteristic-sorted portfolios to estimate factors Portfolios of securities with some common characteristic are proxies for the factors
Factor analysis: intuition z Consider 2 stocks whose returns follow a one- factor model: R 1 = a 1 + b 1 F + ε 1 R 2 = a 2 + b 2 F + ε 2 z From historical data, we can obtain their variances, and covariance between returns ( σ 1 ) 2 , ( σ 1 ) 2 , σ 12 z Therefore, we can find the factor properties, and the sensitivities of the two stock’s returns b 1 , b 2 , ( σ F ) 2

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Factor analysis: intuition z Once we’ve found the sensitivities of individual securities to the risk factor, we can construct a factor-replicating portfolio RP 1 RP 1 = A + F Factor risk-premium = λ 1 = E(RP 1 ) – R f = A – R f z The same logic can be applies to a sample of N securities whose returns are described by a (N-1) factor model
Factor analysis: pros and cons z Advantages: RPs found in the process of factor analysis provide the best possible explanation of the covariance between the stocks’ returns estimated from historical data z Disadvantages: Assumes covariances do not change over time Doesn’t name the economic variables to which factors are linked

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091114_APT - Empirical Validation of APT November 2009 APT...

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