Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory 7-2

7.1 The Capital Asset Pricing Model 7-3

4 Capital Asset Pricing Model uIntroduction uSystematic and unsystematic risk uFundamental risk/return relationship revisited

5 Introduction The Capital Asset Pricing Model (CAPM) is a theoretical description of how the market must price individual securities in relation to their security risk class (Asset Pricing) CAPM tells us 1) what is the price of risk? (Market price of risk) 2) what is the risk of asset i? (quantity of risk asset i)

Systematic and unsystematic risk Specifically: Total risk = systematic risk + unsystematic risk CAPM says: (1)Unsystematic risk can be diversified away. It can be avoided by diversifying at NO cost, the market will not reward the holder of unsystematic risk at all. (2)Systematic risk cannot be diversified away without cost. investors need to be compensated by a certain risk premium for bearing systematic risk

7 Diversification and Beta Beta measures systematic risk –Investors differ in the extent to which they will take risk, so they choose securities with different betas E.g., an aggressive investor could choose a portfolio with a beta of 2.0 E.g., a conservative investor could choose a portfolio with a beta of 0.5 A measure of the sensitivity of a stock’s return to the returns on the market portfolio β i = Cov(R i, R m )/Var(R m )

Risk and return n investors need to be compensated by a certain risk premium for bearing systematic risk n the reward-to-risk ratio for any individual security in the market is equal to the market reward-to-risk ratio, thus:  The market reward-to-risk ratio is effectively the Market Risk Premium which is defined as n The market is defined as a portfolio of all wealth including a broad based stock index, such as the S&P 500 or the portfolio of all NYSE stocks, is generally used. n rearranging the above equation and solve for the expected return

The CAPM formula So E(R i )=R f + β i (E(R m ) – R f ) R f + Units × Price. Number of units of systematic risk (  ) Market Risk Premium or the price per unit risk or,

10 Security Market Line The graphical relationship between expected return and beta is the security market line (SML) –The slope of the SML is the market price of risk –The slope of the SML changes periodically as the risk-free rate and the market’s expected return change

Sample Calculations for SML E(r m ) - r f = r f =  x = 1.25 E(r x ) =  y =.6 E(r y ) = Equation of the SML E(r i ) = r f +  i [E(r M ) - r f ] (.08) =.13 or 13% (0.08) = or 7.8% Return per unit of systematic risk = 8% & the return due to the TVM = 3% 7-11

E(r) SML ß ßMßMßMßM 1.0 R M =11% 3% R x =13% ßxßxßxßx 1.25 R y =7.8% ßyßyßyßy.6.08 Graph of Sample Calculations If the CAPM is correct, only β risk matters in determining the risk premium for a given slope of the SML. 7-12

13 CAPM (cont’d) The CAPM deals with expectations about the future Excess returns on a particular stock are directly related to: –The beta of the stock –The expected excess return on the market

More on alpha and beta E(r M )= β S = r f = Required return= r f + β S [E(r M ) – r f ] = If you believe the stock will actually provide a return of ____, what is the implied alpha(abnormal return=actual- expected)?  = [14 – 5] = 18.5% 17% 14% 1.5 5% A stock with a negative alpha plots below the SML & gives the buyer a negative abnormal return % % = -1.5%

Portfolio Betas β P = If you put half your money in a stock with a beta of ___ and ____ of your money in a stock with a beta of ___and the rest in T-bills, what is the portfolio beta? β P = 0.50(1.5) (0.9) (0) = % 0.9  W i β i 7-15

7.3 The CAPM and the Real World 7-16

17 CAPM (cont’d) CAPM assumptions: –Variance of return and mean return are all investors care about –Investors are price takers They cannot influence the market individually –All investors have equal and costless access to information –There are no taxes or commission costs

18 CAPM (cont’d) CAPM assumptions (cont’d): –Investors look only one period ahead –Everyone is equally at analyzing securities

19 Note on the CAPM Assumptions Several assumptions are unrealistic: –People pay taxes and commissions –Many people look ahead more than one period Theory is useful to the extent that it helps us learn more about the way the world acts –Empirical testing shows that the CAPM works reasonably well

Evaluating the CAPM The CAPM could still be a useful predictor of expected returns. That is an empirical question. Huge measurability problems because the market portfolio is unobservable. Conclusion: As a theory the CAPM is untestable. 7-20

Evaluating the CAPM However, practically the CAPM is testable. Betas are ___________ at predicting returns as other measurable factors may be. More advanced versions of the CAPM that do a better job at ___________________________ are useful at predicting stock returns. not as useful estimating the market portfolio Still widely used and well understood. 7-21

7.4 Multifactor Models and the CAPM 7-22

Fama-French (FF) 3 factor Model Fama and French noted that stocks of ____________ and stocks of firms with a _________________ have had higher stock returns than predicted by single factor models. – Problem: Empirical model without a theory high book to market smaller firms 7-23

Fama-French (FF) 3 factor Model FF proposed a 3 factor model of stock returns as follows: r M – rf = Market index excess return Ratio of ______________________________________ measured with a variable called ____: –HML: High minus low or difference in returns between firms with a high versus a low book to market ratio. _______________ measured by the ____ variable –SMB: Small minus big or the difference in returns between small and large firms. book value of equity to market value of equity HML Firm size variable SMB 7-24

Fama-French (FF) 3 factor Model r GM – rf =α GM + β M (r M – rf ) + β HML r HML + β SMB r SMB + e GM 7-25

7.5Factor Models and the Arbitrage Pricing Theory 7-26

Arbitrage Pricing Theory (APT) Arbitrage: Zero investment: Efficient markets: Arises if an investor can construct a zero investment portfolio with a sure profit Since no net investment outlay is required, an Arbitrageurs can create arbitrarily large positions to secure large levels of profit With efficient markets, profitable arbitrage opportunities will quickly disappear 7-27

Selected Problems 7-28

Problem 1 5% + 0.8(14% – 5%) = 12.2% 14% – 12.2% = 1.8% 5% + 1.5(14% – 5%) = 18.5% 17% – 18.5% = –1.5% a. CAPM: E(r i ) = 5% + β(14% -5%) CAPM: E(r i ) = r f + β(E(r M )-r f ) 7-29  E(r X )=   X =  E(r Y ) =   Y =

Problem 1 b.Which stock? i.Well diversified: Relevant Risk Measure? Best Choice? b.Which stock? ii.Held alone: Relevant Risk Measure? Best Choice? β: CAPM Model Stock X with the positive alpha  Calculate Sharpe ratios  X = 1.8%  Y = -1.5% 7-30

Problem 1 b.(continued) Sharpe Ratios ii.Held Alone: Sharpe Ratio X = Sharpe Ratio Y = Sharpe Ratio Index = (0.14 – 0.05)/0.36 = 0.25 (0.17 – 0.05)/0.25 = 0.48 (0.14 – 0.05)/0.15 = 0.60 Better 7-31

Problem 2 E(r P ) = r f +  [E(r M ) – r f ] 20% = 5% +  (15% – 5%)  = 15/10 =

Problems 5 & 6 6. Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower. Possible. Portfolio A's lower expected rate of return can be paired with a higher standard deviation, as long as Portfolio A's beta is lower than that of Portfolio B

Problem 7 7. Calculate Sharpe ratios for both portfolios: Not possible. The reward-to-variability ratio for Portfolio A is better than that of the market, which is not possible according to the CAPM, since the CAPM predicts that the market portfolio is the portfolio with the highest return per unit of risk

Problem 9 9. Given the data, the SML is: E(r) = 10% +  (18% – 10%) A portfolio with beta of 1.5 should have an expected return of: E(r) = 10% + 1.5(18% – 10%) = 22% Not Possible: The expected return for Portfolio A is 16% so that Portfolio A plots below the SML (i.e., has an  = –6%), and hence is an overpriced portfolio. This is inconsistent with the CAPM

Problem Sharpe A = Sharpe M = Possible: Portfolio A's ratio of risk premium to standard deviation is less attractive than the market's. This situation is consistent with the CAPM. The market portfolio should provide the highest reward-to-variability ratio. (16% - 10%) / 22% =.27 (18% - 10%) / 24% =

Problem 13 b. r 1 = 19%; r 2 = 16%;  1 = 1.5;  2 = 1.0 We can’t tell which adviser did the better job selecting stocks because we can’t calculate either the alpha or the return per unit of risk. r 1 = 19%; r 2 = 16%;  1 = 1.5;  2 = 1.0, rf = 6%; r M = 14%  1 =  2 = The second adviser did the better job selecting stocks (bigger + alpha) 19% – 16% – 19% – 18% = 1% 16% – 14% = 2% CAPM: r i = 6% + β(14%-6%) Part c? [6% + 1.5(14% – 6%)] = [6% + 1.0(14% – 6%)] = 7-37

Problem 13 c. r 1 = 19%; r 2 = 16%;  1 = 1.5;  2 = 1.0, rf = 3%; r M = 15%  1 =  2 = Here, not only does the second investment adviser appear to be a better stock selector, but the first adviser's selections appear valueless (or worse). 19% – [3% + 1.5(15% – 3%)] = 16% – [3%+ 1.0(15% – 3%)] = 19% – 21% = –2% 16% – 15% = 1% CAPM: r i = 3% + β(15%-3%) 7-38