1. Arbitrage Pricing Theorem Chapter 7 1

2. Learning Objectives Develop an understanding of multi-factor pricing models Use the APT to identify mispriced securities Compare and Contrast the APT and CAPM 2

3. Single Index v Multi Index Models Both break total risk into systematic and unique  Single Index models assumes that systematic risk comes from a single source  Multi Index models allow for systematic risk to come from several sources 3

4. Two Index Model Risk comes from 1) Business Cycle (S&P 500) 2) Interest Rate Changes (Treasury bond portfolio) Two Index Realized Returns r it = α i + β iM r Mt + β iTB r TBt + e it Two Factor SML E(r i ) = r f +β iM [E(r M )–r f ]+β iTB [E(r TB )–r f ] 4

5. Interpretation The expected return on a security is the sum of: 1. The risk-free rate 2. The sensitivity to the business cycle times the business cycle risk premium (S&P 500) 3. The sensitivity to interest rate risk times the interest rate risk premium (Treasury Portfolio) 5

6. Two Factor Example What is the expected return of a portfolio with a beta of 1.5 on the market and a beta of 0.75 on oil? The market risk premium is 8%, the oil risk premium is 7%, and the risk free rate is 3%.

7. Two Factor Example: Fun with Algebra Risk Comes from: Market and Oil Risk-free rate = 6% The following are well diversified portfolios: What are the expected returns for Market and Oil? PortfolioMarket BetaOil BetaExpected Return A1.52.031% B2.2-0.227%

8. 10-8 Fama-French Three-Factor Model Market Factor  Same as CAPM SML: Small minus Big (Market Cap)  Return on the averages small firm minus the average large firm HML: High minus Low (Book to Market)  Return on the average value firm minus the average growth firm Argues these firm characteristics are correlated with actual (but currently unknown) systematic risk factors

9. Example What is a stock’s expected return if its betas are: SML: 0.5; HML: 3.0; Mkt: 2.0 The expected returns are  16% small firms, 8% large firms  14% value firms, 9% growth firms  7% market, and the risk free rate is 3% 9

10. Measuring Model Success A model is successful if it can accurately explain a stock’s return  The return actually earned equals what we would expect given the actual movements in the market Measure success with:  Higher Adjusted R-Square  Lower residual standard deviation  Smaller α’s 10

11. What Drives These Models? Arbitrage ≡ Exploiting the mispricing of two (or more) securities to earn risk free profit  EX: Imagine that Google stock is selling for $10 on the NYSE and $12 on NASDAQ, how would you make money?  In truest form no investment is required so they can be scaled up easily How do risk averse investors feel about these? Technological advancements has made it extremely difficult to find simple arbitrage opportunities 11

12. Arbitrage Pricing Theory Starts with the idea that arbitrage opportunities cannot exist in an efficient market  In a well functioning economy no one should be able to earn money for nothing Uses the absence of arbitrage to derive the risk-return relation  Avoid the CAPM assumptions 12

13. APT and CAPM Assumes a well-diversified portfolio.  Residual still important Arbitrage Opportunities Equilibrium is quickly restored  Only takes a few arbitragers Uses an observable, market index APT CAPM Model is based on an inherently unobservable “market” portfolio. Rests on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium. 13

14. Arbitrage in a 1 Factor Economy Portfolio (P): has a positive alpha  We need to buy P to earn alpha However, P has systematic risk How do we remove the systematic risk? What if we had an investment that moved in the exact opposite way of the market?  We short the market (M) (β M of 1), but how much?  We short so that our total portfolio β T = 0 β T = w P * β p + w M * (-β M ) 0 = 1 * β p + w M * (-1) Use the risk free asset to ensure weights balance  Just used to make up cash difference, borrow/lend 14

15. Explanation Math *When alpha is negative, you would reverse the signs of each portfolio weight to achieve a portfolio A with positive alpha and no net investment. Steps to convert a well-diversified portfolio into an arbitrage portfolio Arbitrage Portfolio: Generates risk free profits with 0 net investment -This is a money making machine 15

16. APT: 2 Factors Now need two benchmark portfolios  P1: benchmark portfolio with a beta of 1 on factor 1 and a beta of 0 on factor 2  P2: benchmark portfolio with a beta of 0 on factor 1 and a beta of 1 on factor 2 Then follow the same basic methodology as in the single factor example

17. APT: 2 Factors Math Constructing an arbitrage portfolio with two systemic factors

18. Question Consider a one-factor economy. All portfolios are well diversified. Suppose that another portfolio, E, is well diversified with a beta of.6 and an expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy? PortfolioExpected ReturnBeta A12%1.2 F6%0.0 18

19. 10-19 Actual v Expected Returns E(r P ) rPrP

20. Two Factor Example Risk Comes from: Market and Oil Risk-free rate = 6% The follow is a well diversified portfolio: What is this portfolio’s expected return? If the Mkt earned 8% and Oil earned 13% what did our portfolio actually return? PortfoliosMkt BetaE(Mkt)Oil BetaE(Oil) A1.510%0.758%