1. 7-1 Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

2. 7-2 7.1 THE CAPITAL ASSET PRICING MODEL

3. 7-3 Capital Asset Pricing Model (CAPM) The CAPM is a centerpiece of modern financial economics, which was proposed by William Sharpe, who was awarded the 1990 Nobel Prize for economics 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)

4. 7-4 Capital Asset Pricing Model (CAPM) It is an “equilibrium” model derived using principles of diversification and some simplified assumptions for the behavior of investors and the market condition – The market equilibrium refers to a condition in which for all securities, market prices are established to balance the demand of buyers and the supply of sellers. These prices are called equilibrium prices

5. 7-5 Capital Asset Pricing Model (CAPM) The CAPM is a model that relates the expected required rate of return for any security to its risk as measured by beta 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

6. 7-6 Capital Asset Pricing Model (CAPM 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 ) 6

7. 7-7 Capital Asset Pricing Model (CAPM So E(R i )=R f + β i (E(R m ) – R f ) R f + Units × Price. If we know the expected rate of return of a security, the theoretical price of this security can be derived by discounting the cash flows generated from this security at this expected rate of return So, this expected return-beta relationship is viewed as a kind of asset pricing model Number of units of systematic risk (  ) Market Risk Premium or the price per unit risk

8. 7-8 Assumptions for CAPM Single-period investment horizon Investors can invest in the universal set of publicly traded financial assets Investors can borrow or lend at the risk-free rate unlimitedly No taxes and transaction costs Information is costless and available to all investors Assumptions associated with investors – Investors are price takers (there is no sufficiently wealthy investor such that his will or behavior can influence the whole market and thus security prices)

9. 7-9 Assumptions for CAPM – All investors have the homogeneous expectations about the expected values, variances, and correlations of security returns – All investors attempt to construct efficient frontier portfolios, i.e., they are rational mean-variance optimizers (Investors are all very similar except their initial wealth and their degree of risk aversion) Several assumptions are unrealistic

10. 7-10 Resulting Equilibrium Conditions Identical efficient frontier – All investors are mean-variance optimizers and face the same universal set of securities, so they all derive the identical efficient frontier and the same tangent portfolio (O) and the corresponding CAL given the current risk-free rate The market portfolio is the tangent portfolio O – All investors will put part of their wealth on the same risky portfolio O and the rest on the risk-free asset – The market portfolio is defined as the aggregation of the risky portfolios held by all investors – Hence, the composition of the market portfolio must be identical to that of the tangent portfolio O, and thus E(r M ) = E(r O ) and σ M = σ O

11. 7-11 Resulting Equilibrium Conditions (cont.) The capital market line (CML) As a result, all investors will hold the same portfolio of risky assets–market portfolio, which contains all publicly traded risky assets in the economy – The market portfolio is of course on the efficient frontier, and the line from the risk-free rate through the market portfolio is called the capital market line (CML) We call this result the mutual fund theorem :Only one mutual fund of risky assets–the market portfolio–is sufficient to satisfy the investment demands of all investors

12. 7-12 The Efficient Frontier and the Capital Market Line M = Market portfolio r f = Risk free rate E(r M ) - r f = Market risk premium [E(r M ) – r f ] / σ M = Slope of the CML = Sharpe ratio for the market portfolio or for all combined portfolios on the CML ※ Note that the CML is on the E(r)-σ plane

13. 7-13 Evaluating the CAPM theoretically the CAPM is untestable because Huge measurability problems because the market portfolio is unobservable. However, practically the CAPM is testable and could still be a useful predictor of expected returns. Empirical testing shows that the CAPM works reasonably well Betas are not as useful at predicting returns as other measurable factors may be. More advanced versions of the CAPM that do a better job at estimating the market portfolio are useful at predicting stock returns. Still widely used and well understood. 7-13

14. 7-14 CAPM and the Real World the principles we learn from the CAPM are still entirely valid – Investors should diversify (invest in the market portfolio) – Differences in risk tolerances can be handled by changing the asset allocation decisions in the complete portfolio – Systematic risk is the only risk that matters (thus we have the relationship between the expected return and the beta of each security)

15. 7-15 Expected Returns On Individual Securities The risk premium, defined as the expected return in excess of r f, reflect the compensation for securities holders In the equilibrium, the ratio of risk premium to beta should be the same for any two securities or portfolios (including the market portfolio)

16. 7-16 Expected Returns On Individual Securities – Therefore, for all securities, ※ The competition among investors for pursuing the securities with higher risk premiums and smaller betas will result in the above equality – Rearranging gives us the CAPM’s expected return-beta relationship or

17. 7-17 Expected Returns On Portfolios Since the expected return-beta relationship according to the CAPM is linear and holds not only for ALL INDIVIDUAL ASSETS but also for ANY PORTFOLIO, the beta of a portfolio is simply the weighted average of the betas of the assets in the portfolio β P =  W i β i If you put half your money in a stock with a beta of 1.5 and 30% of your money in a stock with a beta of 0.9 and the rest in T-bills, what is the portfolio beta? β P = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02

18. 7-18 Security Market Line (SML) Relationships E(r i ) = r f + β i [E(r M ) – r f ] β i = cov(R i,R M ) / var(R M ) E(r M ) – r f = market risk premium For example: E(r M ) – r f = 8% and r f = 3% β x = 1.25  E(r x ) = 3% + 1.25 × (8%) = 13% β y = 0.6  E(r y ) = 3% + 0.6 × (8%) = 7.8% ※ SML: graphical representation of the expected return-beta relationship of the CAPM (on the E(r)-beta plane) ※ For the stock with a higher beta, since it is with higher systematic risk, it needs to offer a higher expected return to attract investors

19. 7-19 E(r)E(r)E(r)E(r) E(r x )=13% SML β βMβMβMβM 1 E(r M )=11% E(r y )=7.8% 3% βXβXβXβX 1.25 βyβyβyβy 0.6 slope is 0.08, which is the market risk premium Graph of Security Market Line ※ The CAPM implies that all securities or portfolios should lie on this SML β ※ Note that the SML is on the E(r)-β plane, and CML is on E(r)-σ plane E(r)-σ plane

20. 7-20 Applications of the CAPM In reality, not all securities lie on the SML in the economy Underpriced (overpriced) stocks plot above (below) the SML: Given their betas, their expected rates of return are higher (lower) than the predication by the CAPM and thus the securities are underpriced (overpriced) The difference between actually rate of return on a security and the expected and is the abnormal rate of return on this security, which is often denoted as alpha (α)

21. 7-21 Disequilibrium Example E(r) 15% SML  1.0 R m =11% r f =3% 1.25

22. 7-22 Disequilibrium Example Suppose a security with a  of 1.25 is offering expected return of 15%. According to SML, it should be 13%. Under-priced: offering too high of a rate of return for its level of risk. This kind of security is a more attractive investment target This kind of security is a more attractive investment target

23. 7-23 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 17%, what is the implied alpha(abnormal return=actual- expected)? 5 + 1.5 [14 – 5] = 18.5% 14% 1.5 5% A stock with a negative alpha plots below the SML & gives the buyer a negative abnormal return 7-23  = 17% - 18.5% = -1.5%

24. 7-24 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 ) 7-24  E(r X )=   X =  E(r Y ) =   Y =

25. 7-25 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% 7-25

26. 7-26 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 7-26

27. 7-27 Problem 2 E(r P ) = r f +  [E(r M ) – r f ] 20% = 5% +  (15% – 5%)  = 15/10 = 1.5 7-27

28. 7-28 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. 5. 6. 7-28

29. 7-29 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. 7. 7-29

30. 7-30 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. 9. 7-30

31. 7-31 Problem 11 11. 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% =.33 11. 7-31

32. 7-32 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%)] = 7-32

33. 7-33 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%) 7-33

34. 7-34 7.4 MULTIFACTOR MODELS AND THE CAPM

35. 7-35 Fama French Three-Factor Model In reality, the systematic risk is not from one source It is obvious that developing models that allow for several systematic risks can provide better descriptions of security returns In addition to the market risk premium, Fama and French propose the size premium and the book-to- market premium – The size premium is constructed as the difference in returns between small and large firms and is denoted by SMB (“small minus big”) – The book-to-market premium is calculated as the difference in returns between firms with a high versus low B/M ratio, and is denoted by HML (“high minus low”)

36. 7-36 Fama French Three-Factor Model The Fama and French three-factor model is – r SMB is the return of a portfolio consisting of a long position of $1 in a small-size-firm portfolio and a short position of $1 in a large-size-firm portfolio – r HML is the return of a portfolio consisting of a long position of $1 in a higher-B/M (value stock) portfolio and a short position of $1 in a lower-B/M (growth stock) portfolio – The roles of r SMB and r HML are to identify the average reward compensating holders of the security i exposed to the sources of risk for which they proxy – Note that it is not necessary to calculate the excess return for r SMB and r HML

37. 7-37 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 7-37

38. 7-38 Arbitrage Pricing Theory Arbitrage –Creation of riskless profits by trading relative mispricing among securities 1.Constructing a zero-investment portfolio today and earn a profit for certain in the future 2.2. Or if there is a security priced differently in two markets, a long position in the cheaper market financed by a short position in the more expensive market will lead to a profit as long as the position can be offset each other in the future Since there is no risk for arbitrage, an investor will create arbitrarily large positions to obtain large levels of profit – No arbitrage argument: in efficient markets, profitable arbitrage opportunities will quickly disappear