0 Portfolio Management Albert Lee Chun Multifactor Equity Pricing Models Lecture 7 6 Nov 2008

Albert Lee Chun Portfolio Management 1 Today’s Lecture Single Factor Model Multifactor Models Fama-FrenchAPT

Albert Lee Chun Portfolio Management Alpha 2

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

Albert Lee Chun Portfolio Management 4 Frequency Distribution of Alphas

Albert Lee Chun Portfolio Management 5 The CAPM and Reality Is the condition of zero alphas for all stocks as implied by the CAPM met? 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? 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. CAPM is still considered the best available description of security pricing and is widely accepted.

Albert Lee Chun Portfolio Management 6 Single Factor Model Returns on a security come from two sources Returns on a security come from two sources – Common macro-economic factor – Firm specific events Possible common macro-economic factors Possible common macro-economic factors – Gross Domestic Product Growth – Interest Rates

Albert Lee Chun Portfolio Management 7 ß i = 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 Single Factor Model Assumption: a broad market index like the S&P/TSX Composite is the common factor

Albert Lee Chun Portfolio Management 8 Regression Equation: Single Index Model a i = alpha b i (r M -r i ) = the component of return due to market movements (systematic risk) e i = the component of return due to unexpected firm- specific events (non-systematic risk)

Albert Lee Chun Portfolio Management 9 Let: R i = (r i - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i R m + e i Risk Premium Format The above equation regression is the single-index model using excess returns.

Albert Lee Chun Portfolio Management 10  i 2 = total variance  i 2  m 2 = systematic variance  2 (e i ) = unsystematic variance Measuring Components of Risk

Albert Lee Chun Portfolio Management 11 Index Models and Diversification

Albert Lee Chun Portfolio Management 12 The Variance of a Portfolio

Albert Lee Chun Portfolio Management 13 Security Characteristic Line for X Excess Returns (i) SCL Excess returns on market index R i =  i + ß i R m + e i

Albert Lee Chun Portfolio Management 14 Multi Factor Models

Albert Lee Chun Portfolio Management 15 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. 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? What if there are multiple factors that determine returns? Multifactor Models: Allow for multiple sources of risk, that is multiple risk factors. Multifactor Models: Allow for multiple sources of risk, that is multiple risk factors.

Albert Lee Chun Portfolio Management 16 Multifactor Models Use other factors in addition to market returns: 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

Albert Lee Chun Portfolio Management 17 Example: Multifactor Model Equation R i = E(r i ) + Beta GDP (GDP) + Beta IR (IR) + e i R i = E(r i ) + Beta GDP (GDP) + Beta IR (IR) + e i R i = Return for security i Beta GDP = Factor sensitivity for GDP Beta IR = Factor sensitivity for Interest Rate e i = Firm specific events

Albert Lee Chun Portfolio Management 18 Multifactor SML E(r) = r f +  GDP RP GDP +  IR RP IR  GDP = Factor sensitivity for GDP RP GDP = Risk premium for GDP  IR = Factor sensitivity for Interest Rates RP IR = Risk premium for GDP

Albert Lee Chun Portfolio Management 19 Multifactor Models CAPM say that a single factor, Beta, determines the relative excess return between a portfolio and the market as a whole. 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. 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 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. The best known approach is the three factor model developed by Gene Fama and Ken French.

Albert Lee Chun Portfolio Management 20 Fama French 3-Factor Model

Albert Lee Chun Portfolio Management 21 The Fama-French 3 Factor Model Fama and French observed that two classes of stocks tended to outperform the market as a whole: Fama and French observed that two classes of stocks tended to outperform the market as a whole: (i) small caps (i) small caps (ii) high book-to-market ratio (ii) high book-to-market ratio

Albert Lee Chun Portfolio Management 22 Small Value Stocks Outperform

Albert Lee Chun Portfolio Management 23

Albert Lee Chun Portfolio Management 24 Fama-French 3-Factor Model They added these two factors to a standard CAPM: They added these two factors to a standard CAPM: SMB = “small [market capitalization] minus big” 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.

Albert Lee Chun Portfolio Management 25 Arbitrage Pricing Theory (APT)

Albert Lee Chun Portfolio Management 26 APT Ross (1976): intuitive model, only a few assumptions, considers many sources of risk Ross (1976): intuitive model, only a few assumptions, considers many sources of riskAssumptions: 1. There are sufficient number of securities to diversify away idiosyncratic risk 2. The return on securities is a function of K different risk factors. 3. No arbitrage opportunities

Albert Lee Chun Portfolio Management 27 APT APT does not require the following CAPM assumptions: APT does not require the following CAPM assumptions: 1. Investors are mean-variance optimizers in the sense of Markowitz. 2. Returns are normally distributed. 3. The market portfolio contains all the risky securities and it is efficient in the mean-variance sense.

Albert Lee Chun Portfolio Management 28 APT & Well-Diversified Portfolios F is some macroeconomic factor F is some macroeconomic factor For a well-diversified portfolio e P approaches zero For a well-diversified portfolio e P approaches zero

Albert Lee Chun Portfolio Management 29 Returns as a Function of the Systematic Factor Well-diversified portfolioSingle Stocks

Albert Lee Chun Portfolio Management 30 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

Albert Lee Chun Portfolio Management 31 E(r)% Beta for F Risk Free = 4 A D C.51.0 Example: An Arbitrage Opportunity Risk premiums must be proportional to Betas!

Albert Lee Chun Portfolio Management 32 Disequilibrium Example Short Portfolio C, with Beta =.5 Short Portfolio C, with Beta =.5 One can construct a portfolio with equivalent risk and higher return : Portfolio D 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% Arbitrage opportunity: riskless profit of 1% Risk premiums must be proportional to Betas!

Albert Lee Chun Portfolio Management 33 APT Security Market Line Risk premiums must be proportional to Betas! This is CAPM!

Albert Lee Chun Portfolio Management 34 APT applies to well diversified portfolios and not necessarily to individual stocks 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 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 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 APT can be extended to multifactor models APT and CAPM Compared

Albert Lee Chun Portfolio Management 35 A Multifactor APT A 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 factor A 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 factor These factor portfolios are the building blocks for a multifactor security market line for an economy with multiple sources of risk These factor portfolios are the building blocks for a multifactor security market line for an economy with multiple sources of risk

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

Albert Lee Chun Portfolio Management 37 In theory, the APT supposes a stochastic process that generates returns and that may be represented by a model of K factors, such that In theory, the APT supposes a stochastic process that generates returns and that may be represented by a model of K factors, such thatwhere: R i = One period realized return on security i, i= 1,2,3…,n R i = One period realized return on security i, i= 1,2,3…,n E(Ri) = expected return of security i E(Ri) = expected return of security i = Sensitivity of the reutrn of the ith stock to the jth risk factor = 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 =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. Similar to CAPM, the APT assumes that the idiosyncratic effects can be diversified away in a large portfolio. Generalized Factor Model

Albert Lee Chun Portfolio Management 38 Multifactor APT APT 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.

Albert Lee Chun Portfolio Management 39 Sample APT Problem Suppose that the equity market in a large economy can be described by 3 sources of risk: A, B and C. Suppose that the equity market in a large economy can be described by 3 sources of risk: A, B and C. FactorRisk Premium A.06 A.06 B.04 B.04 C.02 C.02

Albert Lee Chun Portfolio Management 40 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) = x A +.75 x B +.05 x C + error(t) r(t) = x A +.75 x B +.05 x C + error(t)

Albert Lee Chun Portfolio Management 41 Sample APT problem The risk free rate is given by 6% The risk free rate is given by 6% 1. Find the expected rate of return of the mushroom factory under the APT model. 1. Find the expected rate of return of the mushroom factory under the APT model. 2. Is the stock-under or over-valued? Why? 2. Is the stock-under or over-valued? Why?

Albert Lee Chun Portfolio Management 42 Sample APT Problem FactorRisk Premium A.06 A.06 B.04 B.04 C.02 C.02 Risk-Free Rate = 6% Risk-Free Rate = 6% Return(t) = *A *B +.05*C + e(t) Return(t) = *A *B +.05*C + e(t) The factor loadings are in green. The factor loadings are in green.

Albert Lee Chun Portfolio Management 43 Sample APT Problem FactorRisk Premium A.06 A.06 B.04 B.04 C.02 C.02 Risk-Free Rate = 6% Risk-Free Rate = 6% Return = *A *B +.05*C + e Return = *A *B +.05*C + e So plug in risk-premia into the APT formula So plug in risk-premia into the APT formula E[Ri] = * * *0.02 =.16 E[Ri] = * * *0.02 =.16 16% Undervalued! 16% Undervalued!

Albert Lee Chun Portfolio Management 44 Quick Review of Underpricing Undervalued = Underpriced = Return Too High Undervalued = Underpriced = Return Too High Overvalued = Overpriced = Return Too Low Overvalued = Overpriced = Return Too Low P(t) = P(t+1)/ 1+ r P(t) = P(t+1)/ 1+ r r = P(t+1)/P(t) – 1 r = P(t+1)/P(t) – 1 where r is the return for a risky payoff 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.

Albert Lee Chun Portfolio Management Examples 9.3 and 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%

Albert Lee Chun Portfolio Management Example Suppose that: E(R A ) = 12% < 13% Portfolio Q Ponderation B1 =.5: facteur portefeuille 1 Ponderation B2 =.75: facteur portefeuille 2 Ponderation 1- B1 – B2 = -.25 : rf E(R q ) = 12% $1 x E(Rq) - $1x E(R A )=1% There is a riskless arbitrage opportunity of 1%!

Albert Lee Chun Portfolio Management 47 Next Week We will continue our lecture with Chapter 12 We will continue our lecture with Chapter 12 Market Efficiency (Chapter 10; Section 11.1) Market Efficiency (Chapter 10; Section 11.1)