Chapter 7 Capital Asset Pricing and Arbitrage Pricing
Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory Mc. Graw-Hill/Irwin Copyright © 2010 by The Mc. Graw-Hill Companies, Inc. All rights reserved.
Chapter 7 Capital Asset Pricing and Arbitrage Pricing Theory 7 -2
7. 1 The Capital Asset Pricing Model 7 -3
Capital Asset Pricing Model u Introduction u Systematic and unsystematic risk u Fundamental risk/return relationship revisited 4
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) 5
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
Diversification and 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 βi = Cov(Ri, Rm)/Var(Rm) 7
Risk and return n n § n n investors need to be compensated by a certain risk premium for bearing systematic risk 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 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. rearranging the above equation and solve for the expected return
The CAPM formula or, Number of units of systematic risk (b) So E(Ri)=Rf + βi(E(Rm) – Rf) Rf + Units × Price. Market Risk Premium or the price per unit risk
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 10
Sample Calculations for SML Equation of the SML E(ri) = rf + bi[E(r. M) - rf] E(rm) - rf =. 08 rf =. 03 Return per unit of systematic risk = 8% & the return due to the TVM = 3% bx = 1. 25 E(rx) = 0. 03 + 1. 25(. 08) =. 13 or 13% by =. 6 E(ry) = 0. 03 + 0. 6(0. 08) = 0. 078 or 7. 8% 7 -11
Graph of Sample Calculations E(r) SML Rx=13% RM=11% Ry=7. 8% 3% . 08 If the CAPM is correct, only β risk matters in determining the risk premium for a given slope of the SML. . 6 1. 0 1. 25 ßy ßM ßx ß 7 -12
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 13
More on alpha and beta E(r. M) = 14% βS = 1. 5 rf = 5% Required return = rf + β S [E(r. M) – rf] = 5 + 1. 5 [14 – 5] = 18. 5% If you believe the stock will actually provide a return of 17% what is the implied alpha(abnormal return=actual____, expected)? 17% - 18. 5% = -1. 5% =A stock with a negative alpha plots below the SML & gives the buyer a negative abnormal return 7 -14
Portfolio Betas βP = Wi βi If you put half your money in a stock with a beta of 1. 5 ___ and 0. 9 30% of your money in a stock with a beta of ___and ____ the rest in T-bills, what is the portfolio beta? βP = 0. 50(1. 5) + 0. 30(0. 9) + 0. 20(0) = 1. 02 7 -15
7. 3 The CAPM and the Real World 7 -16
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 17
CAPM (cont’d) • CAPM assumptions (cont’d): – Investors look only one period ahead – Everyone is equally at analyzing securities 18
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 19
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 ______ not as useful at predicting returns as other measurable factors may be. • More advanced versions of the CAPM that do a estimating the market portfolio are better job at ______________ useful at predicting stock returns. 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 ______ smaller firms and stocks of firms with a high _________ book to market have had higher stock returns than predicted by single factor models. – Problem: Empirical model without a theory 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 ___________________ book value of equity to market value of equity measured with a variable called HML ____: – HML: High minus low or difference in returns between firms with a high versus a low book to market ratio. SMB variable size variable measured by the ____ • Firm ________ – SMB: Small minus big or the difference in returns between small and large firms. 7 -24
Fama-French (FF) 3 factor Model r. GM – rf =αGM + βM(r. M – rf ) + βHMLr. HML + βSMBr. SMB + e. GM 7 -25
7. 5 Factor Models and the Arbitrage Pricing Theory 7 -26
Arbitrage Pricing Theory (APT) • Arbitrage: Arises if an investor can construct a zero investment portfolio with a sure profit • Zero investment: Since no net investment outlay is required, an Arbitrageurs can create arbitrarily large positions to secure large levels of profit • Efficient markets: With efficient markets, profitable arbitrage opportunities will quickly disappear 7 -27
Selected Problems 7 -28
Problem 1 CAPM: E(ri) = rf + β(E(r. M)-rf) a. CAPM: E(ri) = 5% + β(14% -5%) · E(r. X) = 5% + 0. 8(14% – 5%) = 12. 2% · X = 14% – 12. 2% = 1. 8% · E(r. Y) = 5% + 1. 5(14% – 5%) = 18. 5% · Y = 17% – 18. 5% = – 1. 5% 7 -29
Problem 1 X = 1. 8% Y = -1. 5% b. Which stock? i. Well diversified: Relevant Risk Measure? β: CAPM Model Best Choice? Stock X with the positive alpha b. Which stock? ii. Held alone: Relevant Risk Measure? Best Choice? Calculate Sharpe ratios 7 -30
Problem 1 b. (continued) Sharpe Ratios ii. Held Alone: Sharpe Ratio X = (0. 14 – 0. 05)/0. 36 = 0. 25 Better Sharpe Ratio Y = (0. 17 – 0. 05)/0. 25 = 0. 48 Sharpe Ratio Index = (0. 14 – 0. 05)/0. 15 = 0. 60 7 -31
Problem 2 E(r. P) = rf + b[E(r. M) – rf] 20% = 5% + b(15% – 5%) b = 15/10 = 1. 5 7 -32
Problems 5 & 6 5. 6. 5. Not possible. Portfolio A has a higher beta than Portfolio B, but the expected return for Portfolio A is lower. 6. 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. 7 -33
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 -34
Problem 9 9. Given the data, the SML is: E(r) = 10% + b(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. 7 -35
Problem 11 11. Sharpe A = (16% - 10%) / 22% =. 27 Sharpe M = (18% - 10%) / 24% =. 33 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. 7 -36
Problem 13 a. r 1 = 19%; r 2 = 16%; b 1 = 1. 5; b 2 = 1. 0 b. 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. CAPM: ri = 6% + β(14%-6%) r 1 = 19%; r 2 = 16%; b 1 = 1. 5; b 2 = 1. 0, rf = 6%; r. M = 14% 1 = 19% – [6% + 1. 5(14% – 6%)] = 19% – 18% = 1% 2 = 16% – [6% + 1. 0(14% – 6%)] = 16% – 14% = 2% The second adviser did the better job selecting stocks (bigger + alpha) Part c? 7 -37
Problem 13 c. CAPM: ri = 3% + β(15%-3%) r 1 = 19%; r 2 = 16%; b 1 = 1. 5; b 2 = 1. 0, rf = 3%; r. M = 15% 1 = 19% – [3% + 1. 5(15% – 3%)] = 19% – 21% = – 2% 2 = 16% – [3%+ 1. 0(15% – 3%)] = 16% – 15% = 1% 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). 7 -38
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