lecture 7 h08

*Portfolio Management3-228-07 Albert Lee ChunMultifactor Equity Pricing Models Lecture 7 6 Nov 2008

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*Todays LectureSingle Factor ModelMultifactor ModelsFama-FrenchAPT

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Alpha*

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*Suppose a security with a particular is offering expected return of 17% , yet according to the CAPM, it should be 14.8%.Its under-priced: offering too high of a rate of return for its level of riskIts alpha is 17-14.8 = 2.2%According to CAPM alpha should be equal to 0.

Alpha

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*Frequency Distribution of Alphas

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*The CAPM and RealityIs the condition of zero alphas for all stocks as implied by the CAPM met?Not perfect but one of the best availableIs the CAPM testable?Proxies must be used for the market portfolio CAPM is still considered the best available description of security pricing and is widely accepted.

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*Single Factor ModelReturns on a security come from two sourcesCommon macro-economic factorFirm specific eventsPossible common macro-economic factorsGross Domestic Product GrowthInterest Rates

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*i = index of a securitys particular return to the factorF= some macro factor; in this case F is unanticipated movement; F is commonly related to security returnsSingle Factor ModelAssumption: a broad market index like the S&P/TSX Composite is the common factor

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*Regression Equation: Single Index Modelai = alphabi(rM-ri) = the component of return due to market movements (systematic risk)ei = the component of return due to unexpected firm-specific events (non-systematic risk)

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*Risk Premium FormatThe above equation regression is the single-index model using excess returns.

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*i2 = total variancei2 m2 = systematic variance2(ei) = unsystematic variance

Measuring Components of Risk

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*Index Models and Diversification

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*The Variance of a Portfolio

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*Security Characteristic Line for X

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*Multi Factor Models

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*More than 1 factor?CAPM is a one factor model: The only determinant of expected returns is the systematic risk of the market. This is the only factor.What if there are multiple factors that determine returns?Multifactor Models: Allow for multiple sources of risk, that is multiple risk factors.

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*Multifactor ModelsUse other factors in addition to market returns:Examples include industrial production, expected inflation etc.Estimate a beta or factor loading for each factor using multiple regression

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*Example: Multifactor Model EquationRi = E(ri) + BetaGDP (GDP) + BetaIR (IR) + eiRi = Return for security iBetaGDP= Factor sensitivity for GDP BetaIR = Factor sensitivity for Interest Rateei = Firm specific events

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*Multifactor SMLE(r) = rf + BGDPRPGDP + BIRRPIR

BGDP = Factor sensitivity for GDPRPGDP = Risk premium for GDPBIR = Factor sensitivity for Interest RatesRPIR = Risk premium for GDP

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*Multifactor ModelsCAPM say that a single factor, Beta, determines the relative excess return between a portfolio and the market as a whole. Suppose however there are other factors that are important for determining portfolio returns.The inclusion of additional factors would allow the model to improve the model`s fit of the data. The best known approach is the three factor model developed by Gene Fama and Ken French.

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*Fama French 3-Factor Model

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*The Fama-French 3 Factor ModelFama and French observed that two classes of stocks tended to outperform the market as a whole:

(i) small caps

(ii) high book-to-market ratio

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*Small Value Stocks Outperform

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*

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*Fama-French 3-Factor ModelThey added these two factors to a standard CAPM:

SMB = small [market capitalization] minus big "Size" This is the return of small stocks minus that of large stocks. When small stocks do well relative to large stocks this will be positive, and when they do worse than large stocks, this will be negative.HML = high [book/price] minus low"Value" This is the return of value stocks minus growth stocks, which can likewise be positive or negative. The Fama-French Three Factor model explains over 90% of stock returns.

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*Arbitrage Pricing Theory (APT)

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*APTRoss (1976): intuitive model, only a few assumptions, considers many sources of risk

Assumptions:There are sufficient number of securities to diversify away idiosyncratic risk The return on securities is a function of K different risk factors.No arbitrage opportunities

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*APTAPT does not require the following CAPM assumptions:Investors are mean-variance optimizers in the sense of Markowitz.Returns are normally distributed.The market portfolio contains all the risky securities and it is efficient in the mean-variance sense.

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*APT & Well-Diversified PortfoliosF is some macroeconomic factorFor a well-diversified portfolio eP approaches zero

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*Returns as a Function of the Systematic FactorWell-diversified portfolioSingle Stocks

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*Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

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*Example: An Arbitrage OpportunityRisk premiums must be proportional to Betas!

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*Disequilibrium ExampleShort Portfolio C, with Beta = .5One can construct a portfolio with equivalent risk and higher return : Portfolio DD = .5x A + .5 x Risk-Free AssetD has Beta = .5 Arbitrage opportunity: riskless profit of 1%Risk premiums must be proportional to Betas!

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*APT Security Market LineRisk premiums must be proportional to Betas!This is CAPM!

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*APT applies to well diversified portfolios and not necessarily to individual stocksWith APT it is possible for some individual stocks to be mispriced that is to not lie on the SMLAPT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolioAPT can be extended to multifactor modelsAPT and CAPM Compared

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*A Multifactor APTA factor portfolio is a portfolio constructed so that it would have a beta equal to one on a given factor and zero on any other factorThese factor portfolios are the building blocks for a multifactor security market line for an economy with multiple sources of risk

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*Where Should we Look for Factors?The multifactor APT gives no guidance on where to look for factorsChen, Roll and RossReturns a function of several macroeconomic and bond market variables instead of market returnsFama and FrenchReturns a function of size and book-to-market value as well as market returns

9-*

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*In theory, the APT supposes a stochastic process that generates returns and that may be represented by a model of K factors, such that

where: Ri = One period realized return on security i, i= 1,2,3,n E(Ri) = expected return of security i

= Sensitivity of the reutrn of the ith stock to the jth risk factor

= j-th risk factor

=captures the unique risk associated with security i

Similar to CAPM, the APT assumes that the idiosyncratic effects can be diversified away in a large portfolio.Generalized Factor Model

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*Multifactor APTAPT Model

The expected return on a secutity depends on the product of the risk premiums and the factor betas (or factor loadings)

E(Ri) rf is the risk premium on the ith factor portfolio.