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Arbitrage Pricing Theory and Multifactor Models

This chapter explores the concepts of Single-Factor Models, Multifactor Models, and the application of Arbitrage Pricing Theory. It discusses the sources of returns on securities, factor sensitivities, and the relationship between macro-economic factors and firm-specific events. The chapter also covers the APT in relation to well-diversified portfolios, the CAPM, and the Index Model.

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Arbitrage Pricing Theory and Multifactor Models

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  1. Chapter Ten Arbitrage Pricing Theory and Multifactor Models of Risk and Return

  2. Single Factor Model • Returns on a security come from two sources: • Common macro-economic factor • Firm specific events • Possible common macro-economic factors • Gross Domestic Product Growth • Interest Rates

  3. Single-Factor Model ri = αi + βi f + ei , ri = stock i’s return f = a macro economic factor βi = sensitivity of stock i’s return to the macro economic factor ei = return component due to stock specific events

  4. Single-Factor Model in Alternate Form (1) ri = αi + βi f + ei , Taking expectation of (1), we have (2) E(ri)= E(αi + βi f + ei) = αi + βi E(f) Subtract (2) from (1) ri - E(ri) = βi f - βi E(f) + ei= βi [f - E(f)] + ei ri = E(ri) + βi [f - E(f)] + ei (10.1) ri = E(ri) + βi F + ei

  5. Single Factor Model Equation ri= Return for security i βi = Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (F could be positive, negative or zero) ei = Firm specific events

  6. Multifactor Models Use more than one factor in addition to market return Examples include gross domestic product, expected inflation, interest rates etc. Estimate a beta or factor loading for each factor using multiple regression.

  7. Multifactor Model Equation ri = E(ri) + GDP GDP + IR IR + ei ri= Return for security i GDP = Factor sensitivity for GDP IR= Factor sensitivity for Interest Rate ei = Firm specific events

  8. Multifactor SML Models E(r) = rf + GDP RPGDP + IR RPIR GDP= Factor sensitivity for GDP RPGDP = Risk premium for GDP IR = Factor sensitivity for Interest Rate RPIR= Risk premium for Interest Rate

  9. The risk-free rate The sensitivity to GDP times the risk premium for bearing GDP risk The sensitivity to interest rate risk times the risk premium for bearing interest rate risk Interpretation The expected return on a security is the sum of:

  10. Arbitrage Pricing Theory • Arbitrage occurs if there is a zero investment portfolio with a sure profit • No investment  investors create large positions to obtain large profits • All investors will want an infinite position in the risk-free arbitrage portfolio • In efficient markets, profitable arbitrage opportunities will quickly disappear

  11. APT and Well-Diversified Portfolios • For a well-diversified portfolio, • eP  0 as the number of securities increases • and their associated weights decrease

  12. Figure 10.1 Returns as a Function of the Systematic Factor

  13. Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

  14. Figure 10.3 An Arbitrage Opportunity

  15. Single-Index Model (From Chapters 8 &9) • Rit = αi+ βiRMt + eit where Rit = rit – rft and RMt = rMt – rft • For a well-diversified portfolio, Rpt = αp+ βpRMt(10.5) E(Rp) = αp + βp E(RM), where αp = 0 Hence, E(Rp) = βp E(RM) (10.6)

  16. No-Arbitrage Equation of APT

  17. APT CAPM the APT, the CAPM and the Index Model • Assumes a well-diversified portfolio, but residual risk is still a factor. • Does not assume investors are mean-variance optimizers. • Uses an observable, market index • Reveals arbitrage opportunities Model is based on an inherently unobservable “market” portfolio. Rests on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium.

  18. Multifactor APT • Use of more than a single systematic factor • Requires formation of factor portfolios • What factors? • Factors that are important to performance of the general economy • What about firm characteristics?

  19. Two-Factor Model • The multifactor APT is similar to the one-factor case.

  20. Two-Factor Model • Track with diversified factor portfolios: • beta=1 for one of the factors and 0 for all other factors. • The factor portfolios track a particular source of macroeconomic risk, but are uncorrelated with other sources of risk.

  21. Fama-French Three-Factor Model • SMB = Small Minus Big (firm size) • HML = High Minus Low (book-to-market ratio) • Are these firm characteristics correlated with actual (but currently unknown) systematic risk factors?

  22. The Multifactor CAPM and the APT • A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge • The APT is largely silent on where to look for priced sources of risk

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