READING 12: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND RETURN

Multifactor Model

        The inputs to a multifactor model, for any stock, are:

• Expected return for the stock.
  
• Factor betas.
  
• Deviation of macroeconomic factors from their expected values.
  
• Firm-specific return.

        The factor beta equals the sensitivity of the stock return to a 1-unit change in the factor. The firm-specific return is that portion of the stock’s return that is unexplained by the macro
factors. The expected value of the firm-specific return equals zero, because, by definition, firm-specific events are random.

        The equation for a multifactor model for stock i can be expressed as follows:

R_i = E(R_i) + β_i1F₁ + β_i2F₂ + … + β_ikF_k + e_i

        where:

• R_i = return on stock i
  
• E(R_i) = expected return for stock i
  
• β_i = j_th factor beta for stock i
  
• F = deviation of macroeconomic factor j from its expected value
  
• e = firm-specific return for stock i

        The return equals its expected value if none of the macro factors deviate from their expected values and if the firm-specific return equals zero. If macro factor Fj deviates from its expected value, then Fj is nonzero. If the firm experiences a nonfactor related surprise, then the firm-specific component, e, will be nonzero. The multifactor model can be used to calculate the expected return after new macroeconomic and/or firm-specific information is released.

Law of One Price

        Identical assets selling in different locations should be priced identically in the different locations.

Well-Diversified Portfolios

        Risk reduction benefits achieved through diversification come from reducing nonsystematic risk. Therefore, the expected return on a well-diversified portfolio is determined by systematic risk as measured by beta.

The Single-Factor Security Market Line

        A single-factor security market line (SML) is analogous to the capital asset pricing model (CAPM). In the single-factor SML, systematic risk is measured as the exposure of the asset to a well-diversified market index portfolio. The index portfolio can be any well-diversified portfolio thought to be highly correlated with the systematic factor that affects the returns of assets. The equation for the single-factor SML for any well-diversified portfolio is: E(R_P) = R_F + β_P[E(R_M) − R_F] , where R_F is the risk-free rate, M is an observable well-diversified market index, and β_P is the beta of any portfolio, P, relative to the market index.

Difference Between CAPM & Single-factor Model

        The key difference is that the CAPM relies on the existence of the mean-variance efficient market portfolio (i.e., an unobservable portfolio that lies on the efficient frontier, and consists of all marketable assets). In contrast, the equation for the single-factor security market line merely relies on the assumptions that security returns can be explained by a single-factor model, that well-diversified portfolios can be created, and that no arbitrage opportunities exist.

Hedging Exposures to Multiple Factors

        A multifactor model can be used to hedge away multiple factor risks. To do so, the investor can create factor portfolios, which are well-diversified portfolios with beta equal to one for a single risk factor, and betas equal to zero on the remaining risk factors. Factor portfolios can be used to hedge multiple risk factors by combining the original portfolio with offsetting positions in the factor portfolios.

Arbitrage pricing theory (APT)

        Arbitrage pricing theory describes expected returns as a linear function of exposures to common (i.e., macroeconomic) risk factors: E(R_i) = R_F + β_i1R_P1 + β_i2R_P2 + … + β_ikR_Pk, where R_Pj is the risk premium associated with risk factor j. The CAPM is a special case of the APT where there is only one priced risk factor (market risk).

        The assumptions underlying the APT model are as follows:

• Returns follow a k-factor process: R_i = E(R_i) + β_i1F₁ + β_i2F₂ + … + β_ikF_k + ei.
  
• Well-diversified portfolios can be formed.
  
• No arbitrage opportunities exist.

The Fama-French Three-Factor Model

        The Fama-French three-factor model describes returns as a linear function of the market index return, firm size, and book-to-market factors. The firm size factor, SMB, equals the difference in returns between portfolios of small and big firms. The book-to-market factor, HML, equals the difference in returns between portfolios of high and low book-to-market firms.

        The equation for the Fama-French three-factor model is:

R_i − R_F = α_i + β_i,M(R_M − R_F) + β_i,SMBSMB + β_i,HMLHML + e_i

• SMB (small minus big) is the firm size factor equal to the difference in returns between portfolios of small and big firms (R_S − R_B).
  
• HML (high minus low) is the book-to-market (i.e., book value per share divided by stock price) factor equal to the difference in returns between portfolios of high and low book-to-market firms (R_H − R_L).