Portfolio Management 3-228-07 Albert Lee Chun

1. Portfolio Management3-228-07Albert Lee Chun Multifactor Equity Pricing Models Lecture 7 6 Nov 2008

2. Today’s Lecture Single Factor Model Multifactor Models Fama-French APT

3. Alpha

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

5. Frequency Distribution of Alphas

6. The CAPM and Reality • Is the condition of zero alphas for all stocks as implied by the CAPM met? • Not perfect but one of the best available • Is 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.

7. 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

8. Single Factor Model ßi = index of a security’s particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P/TSX Composite is the common factor

9. Regression Equation: Single Index Model ai = alpha bi(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)

10. Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Ri = i + ßiRm + ei Risk Premium Format The above equation regression is the single-index model using excess returns.

11. Measuring Components of Risk i2 = total variance i2m2= systematic variance 2(ei)= unsystematic variance

12. Index Models and Diversification

13. The Variance of a Portfolio

14. Excess Returns (i) SCL . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . . Ri = i + ßiRm + ei Security Characteristic Line for X

15. Multi Factor Models

16. 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.

17. Multifactor Models • Use 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

18. Example: Multifactor Model Equation • Ri= E(ri) + BetaGDP (GDP) + BetaIR (IR) + ei Ri= Return for security i BetaGDP= Factor sensitivity for GDP BetaIR= Factor sensitivity for Interest Rate ei= Firm specific events

19. Multifactor SML E(r) = rf + BGDPRPGDP + BIRRPIR BGDP = Factor sensitivity for GDP RPGDP = Risk premium for GDP BIR = Factor sensitivity for Interest Rates RPIR = Risk premium for GDP

20. Multifactor Models • CAPM 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.

21. Fama French 3-Factor Model

22. The Fama-French 3 Factor Model • Fama 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

23. Small Value Stocks Outperform

25. Fama-French 3-Factor Model • They 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.

26. Arbitrage Pricing Theory (APT)

27. APT • Ross (1976): intuitive model, only a few assumptions, considersmany sources of risk Assumptions: • There are sufficientnumber of securities to diversifyawayidiosyncraticrisk • The return on securitiesis a function of K differentriskfactors. • No arbitrage opportunities

28. APT • APT 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.

29. APT & Well-Diversified Portfolios • F is some macroeconomic factor • For a well-diversified portfolio eP approaches zero

30. Returns as a Function of the Systematic Factor Well-diversified portfolio Single Stocks

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

32. E(r)% 10 A D 7 6 C Risk Free = 4 .5 1.0 Beta for F Example: An Arbitrage Opportunity Risk premiums must be proportional to Betas!

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

34. APT Security Market Line This is CAPM! Risk premiums must be proportional to Betas!

35. APT and CAPM Compared • APT applies to well diversified portfolios and not necessarily to individual stocks • With APT it is possible for some individual stocks to be mispriced – that is to not lie on the SML • APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio • APT can be extended to multifactor models

36. A Multifactor APT • A factor portfoliois a portfolio constructed so that it would have a beta equal to one on a given factor and zero on any other factor • These factor portfolios are the building blocks for a multifactor security market line for an economy with multiple sources of risk

37. Where Should we Look for Factors? • The multifactor APT gives no guidance on where to look for factors • Chen, Roll and Ross • Returns a function of several macroeconomic and bond market variables instead of market returns • Fama and French • Returns a function of size and book-to-market value as well as market returns 9-36

38. Generalized Factor Model • 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.

39. Multifactor APT APT Model The expected return on a secutitydepends on the product of the risk premiums and the factor betas (or factor loadings) E(Ri) – rfis the risk premium on the ithfactor portfolio.

40. Sample APT Problem • Suppose that the equity market in a large economy can be described by 3 sources of risk: A, B and C. Factor Risk Premium A .06 B .04 C .02

41. Example APT Problem Suppose that the return on Maggie’s Mushroom Factory is given by the following equation, with an expected return of 17%. r(t) = .17 + 1.0 x A + .75 x B + .05 x C + error(t)

42. Sample APT problem • The risk free rate is given by 6% • 1. Find the expected rate of return of the mushroom factory under the APT model. • 2. Is the stock-under or over-valued? Why?

43. Sample APT Problem Factor Risk Premium A .06 B .04 C .02 Risk-Free Rate = 6% Return(t) = .17 + 1.0*A + 0.75*B + .05*C + e(t) The factor loadings are in green.

44. Sample APT Problem Factor Risk Premium A .06 B .04 C .02 Risk-Free Rate = 6% Return = .17 + 1.0*A + 0.75*B + .05*C + e So plug in risk-premia into the APT formula E[Ri] = .06 + 1.0*0.06+0.75*0.04+0.5*0.02 = .16 16% < 17% => Undervalued!

45. Quick Review of Underpricing • Undervalued = Underpriced = Return Too High • Overvalued = Overpriced = Return Too Low P(t) = P(t+1)/ 1+ r r = P(t+1)/P(t) – 1 where r is the return for a risky payoff P(t+1). This is easy to remember if you think about the inverse relationship between price (value) today and return.

46. Examples 9.3 and 9.4 Factor portfolio 1: E(R1) = 10% Factor Portfolio 2: E(R2) = 12% Rf = 4% Portfolio A with B1 = .5 and B2 = .75 Construct aPortfolio Q using weights of B1 = .5 on factor portfolio 1 and a weight of B2 = .75 on factor portfolio 2 and a weight of 1- B1 – B2 = -.25 on the risk free rate. E(Rq) = B1E(R1) + B2 E(R2) + (1-B1-B2) Rf = rf + B1(E(R1) –rf )+ B2(E(R2) – rf) =13%

47. Example 9.4 Suppose that: E(RA) = 12% < 13% Portfolio Q Ponderation B1 = .5: facteur portefeuille 1 Ponderation B2 = .75: facteur portefeuille 2 Ponderation 1- B1 – B2 = -.25 : rf E(Rq ) = 12% $1 x E(Rq) - $1x E(RA)=1% There is a riskless arbitrage opportunity of 1%!

48. Next Week • We will continue our lecture with Chapter 12 • Market Efficiency (Chapter 10; Section 11.1)