Chapter 10 Arbitrage Pricing Theory and Multifactor Models of Risk and Return GVHD: TS Trần Thị Hải Lý Nhóm Lâm Bá Du Huỳnh Thái Huy Phan Tuyết Trinh Trần Thị Ngọc Hạnh Tô Thị Phương Thảo Nguyễn Hoàng Minh Huy OUTLINE Multifactor Models: An Overview Arbitrage Pricing Theory The APT, the CAPM, and the Index Model A Multifactor APT The Fama-French (FF) Three-Factor Model 10.1 Multifactor Models: An Overview  SECURITY RISK INDEX MODEL: Total risk = Systematic + firm-specific risk  SINGLE – INDEX MODEL 10.1 Multifactor Models: An Overview SINGLE-FACTOR MODEL Ri = E(Ri) + βiF + ei (1) If the macro factor (F) = in any particular period (i.e., no macro surprises), then the excess security return on the its previously expected value the effect of firm-specific events E(Ri) ei Ri nonsystematic components of returns, the e , are assumed to be uncorrelated across The i stocks and with the factor F 10.1 Multifactor Models: An Overview SINGLE-FACTOR MODEL Ri = E(Ri) + βiF + ei (1) F: the deviation of the common factor from its expected value βi: the sensitivity of firm i to that factor ei: the firm-specific disturbance The actual excess return on firm i will equal its initially expected value plus a (zero expected value) random amount attributable to unanticipated economywide events, plus another (zero expected value) random amount attributable to firm-specific events 10.1 Multifactor Models: An Overview SINGLE-FACTOR MODEL Ex: Ri = E(Ri) + βiF + ei (1) Suppose F is taken to be news about the state of the business cycle • • Consensus is GDP will increase by 4% this year stock’s β value is 1.2 If GDP increases by only 3% => F = -1%, then (1): Ri = E(Ri) -1.2% + ei F and ei determines the total departure of the stock’s return from its E(Ri) 10.1 Multifactor Models: An Overview • • Systematic risk is not confined to a single factor SINGLE-FACTOR MODEL Systematic risk is representated explicitly => different stocks to exhibit different sensitivities to its various components multifactor models can provide better descriptions of security returns 10.1 Multifactor Models: An Overview MULTIFACTOR MODELS Suppose: • • two-factor model macroeconomic sources of risk are measured by unanticipated growth in GDP and unexpected changes in interest rates IR The return on any stock will respond both to sources of macro risk and to its own firm-specific influences Then: Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2) 10.1 Multifactor Models: An Overview MULTIFACTOR MODELS two-factor model two-factor model Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2) • • Both macro factors have zero expectation • • An increase in interest rates is bad news for most firms βiGDP and βiIR measure the sensitivity of share returns to that factor loadings or factor betas ei reflects firmspecific influences βiIR < factor 10.1 Multifactor Models: An Overview MULTIFACTOR MODELS two-factor model Ri = E(Ri) + βiGDPGDP + βiIRIR +ei (2) • • • electric-power utility firm’s stock: βeGDP low and βeIR high airline firm’s stock: βeGDP high and βeIR low Economy will expand suggestion both GDP and Interest rates are expected increase “macro news” are the bad news for the utility but good ones for the airline 10.2 Arbitrage Pricing Theory (APT) The No-Arbitrage Equation of the APT • If you sell short $1 million of B and buy $1 million of A, a zero-net-investment strategy, you would have a riskless payoff of $20,000, as follows: (.10 + 1.0 X F) X $1 million From long position in A -(.08 + 1.0 X F) X $1 million From short position in B 02 X $1 million = $20,000 Net proceeds Your profit is risk-free because the factor risk cancels out across the long and short positions 10.2 Arbitrage Pricing Theory (APT) The No-Arbitrage Equation of the APT • What about portfolios with different betas? Their risk premiums must be proportional to beta 10.2 Arbitrage Pricing Theory (APT) The No-Arbitrage Equation of the APT • • We have a new portfolio, D, composed of half of portfolio A and half of the risk-free asset Portfolio D’s beta will be (0.5 X + 0.5 X 1.0) = 0.5, and its expected return will be (0.5 X + 0.5 X 10) = 7% • • Now portfolio D has an equal beta but a greater expected return than portfolio C To preclude arbitrage opportunities, the expected return on all well-diversified portfolios must lie on the straight line from the risk-free asset in Figure 10.3 10.2 Arbitrage Pricing Theory (APT) The No-Arbitrage Equation of the APT •• • Risk premiums are indeed proportional to portfolio betas in Figure 10.3   As in the simple CAPM, the risk premium is zero for and rises in direct proportion to 10.3 The APT, CAPM, Index Model The APT, CAPM It should be noted, however, that when we replace the unobserved market portfolio of the CAPM with an observed, broad index portfolio that may not be efficient, we no longer can be sure that the CAPM predicts risk premiums of all assets with no bias Neither model therefore is free of limitations Compar-ing the APT arbitrage strategy to maximization of the Sharpe ratio in the context of an index model may well be the more useful framework for analysis 10.3 The APT, CAPM, Index Model 10.3 The APT, CAPM, Index Model APT, Index Model In effect, the APT shows how to take advantage of security mispricing when diversification opportunities are abundant When you lock in and scale up an arbitrage opportunity, you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio, but only if the arbitrage portfolio is truly risk-free! 10.4 A Multifactor APT •   too simplistic !!!!!!!!! single-factor  multifactor E() = E() =   Cov(,) = Cov(, ) = 10.4 AAMultifactor 10.4 Multifactor APT APT Suppose: factor portfolio, and risk-free rate: 4% Portfolio Porfolio risk premium = 6% risk premium = 8% Portfolio A: a well-diversified portfolio : β A1 = 0,5 ; βA2 = 0,75 The risk premium attributable to risk factor 1: βA1[E(r1) – rf] = 3% The risk premium attributable to risk factor 2: βA2[E(r2) – rf] = 6% The total risk premium: 3% + 6% = 9% The total return on the portfolio: 4% + 9% = 13% 10.4 AAMultifactor 10.4 Multifactor APT APT • Portfolio Q:   βP1 in the first factor portfolio, βP2 in the second factor portfolio, - βP1 - βP2 in T-bills Portfolio Q will have betas equal to those of portfolio A and expected return: 10.4 AAMultifactor 10.4 Multifactor APT APT that ● • Suppose    an arbitrage opportunity :D :D :D  form portfolio Q with the same betas  10.5 The Fama-French (FF) three-factor model ++   - SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks - HML = High Minus Low, i.e., the return of aportfolio of stock with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low high book-to-market ratio 10.5 The Fama-French (FF) three-factor model THE FAMA-FRENCH (FF) THREE-FACTOR MODEL High ratio of book- Small stocks to-market value Capture sensitive to risk factor in the macroeconomy THE FAMA-FRENCH (FF) THREE-FACTOR MODEL 10.5 The Fama-French (FF) three-factor model • • The Fama-French model use proxies for extramarket sources of risk, is that none of the factors in the proposed models can be clearly identified as hedging a significant source of uncertainty Fama and French have shown that size and book-to-market ratios have predicted average returns in various time periods and in markets all over the world THE FAMA-FRENCH (FF) THREE-FACTOR MODEL 10.5 The Fama-French (FF) three-factor model the FF model reflect a multi-index ICAPM based on extra-market •• Whether  hedging demands or just represent yet-unexplained anomalies? Firm characteristics are correlated with alpha values A deviation from rational equilibrium () Firm characteristics identified as empirically associated with average returns are correlated with other risk factors () ... linking expected returns to risk • Arbitrage: Creation of riskless profits made possible by relative mispricing among securities 10.2 Arbitrage Pricing Theory (APT) Arbitrage Pricing Theory (APT)... returns (10.4) (10.5) 10.2 Arbitrage Pricing TheoryArbitrage (APT )Pricing Theory Arbitrage: • An arbitrage opportunity arises when an investor can earn riskless profits without making a net investment... example of an arbitrage opportunity would arise if shares of a stock sold for different prices on two different exchanges 10.2 Arbitrage Pricing TheoryArbitrage (APT )Pricing Theory Arbitrage: