Capital Asset Pricing and Arbitrage Pricing Theory

Capital Asset Pricing and Arbitrage Pricing Theory

Bodie, Kane and MarcusEssentials of Investments 9th Global Edition

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7.1 THE CAPITAL ASSET PRICING MODEL

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Assumptions Markets are competitive, equally profitable

No investor is wealthy enough to individually affect prices All information publicly available; all securities public No taxes on returns, no transaction costs Unlimited borrowing/lending at risk-free rate

Investors are alike except for initial wealth, risk aversion Investors plan for single-period horizon; they are rational,

mean-variance optimizers Use same inputs, consider identical portfolio opportunity

sets


Hypothetical EquilibriumAll investors choose to hold market portfolioMarket portfolio is on efficient frontier, optimal

risky portfolioRisk premium on market portfolio is proportional

to variance of market portfolio and investor’s risk aversion

Risk premium on individual assets Proportional to risk premium on market portfolioProportional to beta coefficient of security on market portfolio

FIGURE 7.1 EFFICIENT FRONTIER AND CAPITAL MARKET LINE


Passive Strategy is Efficient Mutual fund theorem: All investors desire same

portfolio of risky assets, can be satisfied by single mutual fund composed of that portfolio

If passive strategy is costless and efficient, why follow active strategy?If no one does security analysis, what brings about efficiency of market portfolio?


Risk Premium of Market Portfolio Demand drives prices, lowers expected rate of

return/risk premiums When premiums fall, investors move funds into risk-

free asset Equilibrium risk premium of market portfolio

proportional to Risk of marketRisk aversion of average investor


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The Security Market Line (SML) Represents expected return-beta relationship of CAPM Graphs individual asset risk premiums as function of

asset risk Alpha

Abnormal rate of return on security in excess of that predicted by equilibrium model (CAPM)

FIGURE 7.2 THE SML AND A POSITIVE-ALPHA STOCK


Applications of CAPM Use SML as benchmark for fair return on risky asset SML provides “hurdle rate” for internal projects In setting pricing for government offerings and tenders


What is the expected rate of return for a stock that has a beta of 1 if the expected return on the market is 15%?

a. 15%. b. More than 15%. c. Cannot be determined without the risk-free rate.

7.2 CAPM AND INDEX MODELS

7.2 CAPM AND INDEX MODELS•

TABLE 7.1 MONTHLY RETURN STATISTICS 01/06 - 12/10

Statistic (%) T-Bills S&P 500 GoogleAverage rate of return 0.184 0.239 1.125

Average excess return - 0.055 0.941

Standard deviation* 0.177 5.11 10.40

Geometric average 0.180 0.107 0.600

Cumulative total 5-year return 11.65 6.60 43.17

Gain Jan 2006-Oct 2007 9.04 27.45 70.42

Gain Nov 2007-May 2009 2.29 -38.87 -40.99

Gain June 2009-Dec 2010 0.10 36.83 42.36

* The rate on T-bills is known in advance, SD does not reflect risk.

FIGURE 7.3A: MONTHLY RETURNS

FIGURE 7.3B MONTHLY CUMULATIVE RETURNS

FIGURE 7.4 SCATTER DIAGRAM/SCL: GOOGLE VS. S&P 500, 01/06-12/10

TABLE 7.2 SCL FOR GOOGLE (S&P 500), 01/06-12/10

Linear Regression

Regression StatisticsR 0.5914 R-square 0.3497 Adjusted R-square 0.3385 SE of regression 8.4585 Total number of observations 60

Regression equation: Google (excess return) = 0.8751 + 1.2031 × S&P 500 (excess return)ANOVA

df SS MS F p-level Regression 1 2231.50 2231.50 31.19 0.0000 Residual 58 4149.65 71.55

Total 59 6381.15

CoefficientsStandard

Error t-Statistic p-value LCL UCL Intercept 0.8751 1.0920 0.8013 0.4262 -1.7375 3.4877 S&P 500 1.2031 0.2154 5.5848 0.0000 0.6877 1.7185

t-Statistic (2%) 2.3924

LCL - Lower confidence interval (95%)

UCL - Upper confidence interval (95%)

7.2 CAPM AND INDEX MODELS• Estimation results

• Security Characteristic Line (SCL)• Plot of security’s expected excess return over risk-free rate as function of excess return on market

• Required rate = Risk-free rate + β x Expected excess return of index

7.2 CAPM AND INDEX MODELS Predicting Betas

Mean reversionBetas move towards mean over timeTo predict future betas, adjust estimates from historical data to account for regression towards 1.0

TRUE OR FALSE

Stocks with a beta of zero offer an expected rate of return of zero.

The CAPM implies that investors require a higher return to hold highly volatile securities.

You can construct a portfolio with beta of .75 by investing .75 of the investment budget in T-bills and the remainder in the market portfolio.

10) The market price of a security is $ 30. Its expected rate of return is 10%. The risk-free rate is 4%, and the market risk premium is 8%. What will the market price of the security be if its beta doubles (and all other variables remain unchanged)? Assume the stock is expected to pay a constant dividend in perpetuity.

15) If the CAPM is valid, is the below situation possible?

16) If the CAPM is valid, is the below situation possible?

24) Two investment advisers are comparing performance. One averaged a 19% return and the other a 16% return. However, the beta of the first adviser was 1.5, while that of the second was 1.a) Can you tell which adviser was a better selector

of individual stocks (aside from the issue of general movements in the market)?

b) If the T-bill rate were 6% and the market return during the period were 14%, which adviser would be the superior stock selector? c. What if the T-bill rate were 3% and the market return 15%?

22) Assume the risk-free rate is 4% and the expected rate of return on the market is 15%. I am buying a firm with an expected perpetual cash flow of $ 600 but am unsure of its risk. If I think the beta of the firm is zero, when the beta is really 1, how much more will I offer for the firm than it is truly worth?

7.3 CAPM AND THE REAL WORLD CAPM is false based on validity of its assumptions

Useful predictor of expected returns Untestable as a theory Principles still valid

Investors should diversifySystematic risk is the risk that mattersWell-diversified risky portfolio can be suitable for wide range of investors

7.3 CAPM AND THE REAL WORLD

Consider the statement: “If we can identify a portfolio that beats the S& P 500 Index portfolio, then we should reject the single-index CAPM.” Do you agree or disagree? Explain.

7.4 MULTIFACTOR MODELS AND CAPM•

TABLE 7.3 MONTHLY RATES OF RETURN, 01/06-12/10

Monthly Excess Return % * Total Return

Security AverageStandard Deviation

Geometric Average

Cumulative Return

T-bill 0 0 0.18 11.65Market index ** 0.26 5.44 0.30 19.51SMB 0.34 2.46 0.31 20.70HML 0.01 2.97 -0.03 -2.06Google 0.94 10.40 0.60 43.17 *Total return for SMB and HML ** Includes all NYSE, NASDAQ, and AMEX stocks.

TABLE 7.4 REGRESSION STATISTICS: ALTERNATIVE SPECIFICATIONS Regression statistics for: 1.A Single index with S&P 500 as market proxy

1.B Single index with broad market index (NYSE+NASDAQ+AMEX)

2. Fama French three-factor model (Broad Market+SMB+HML)

Monthly returns January 2006 - December 2010 Single Index Specification FF 3-Factor Specification

Estimate S&P 500Broad Market

Index with Broad Market Index Correlation coefficient 0.59 0.61 0.70Adjusted R-Square 0.34 0.36 0.47Residual SD = Regression SE (%) 8.46 8.33 7.61Alpha = Intercept (%) 0.88 (1.09) 0.64 (1.08) 0.62 (0.99)Market beta 1.20 (0.21) 1.16 (0.20) 1.51 (0.21)SMB (size) beta - - -0.20 (0.44)HML (book to market) beta - - -1.33 (0.37) Standard errors in parenthesis

7.5 ARBITRAGE PRICING THEORY Arbitrage

Relative mispricing creates riskless profit Arbitrage Pricing Theory (APT)

Risk-return relationships from no-arbitrage considerations in large capital markets

Well-diversified portfolio Nonsystematic risk is negligible Arbitrage portfolio Positive return, zero-net-investment, risk-free portfolio

7.5 ARBITRAGE PRICING THEORY Calculating APT

Returns on well-diversified portfolio

TABLE 7.5 PORTFOLIO CONVERSION

*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

FIGURE 7.5 SECURITY CHARACTERISTIC LINES

7.5 ARBITRAGE PRICING THEORY Multifactor Generalization of APT and CAPM

Factor portfolioWell-diversified portfolio constructed to have beta of 1.0 on one factor and beta of zero on any other factor

Two-Factor Model for APT

7.5 ARBITRAGE PRICING THEORY29. Assume a market index represents the common factor and all stocks in the economy have a beta of 1. Firm-specific returns all have a standard deviation of 45%. Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 3.5%, and one-half have an alpha of –3%. The analyst then buys $ 1 million of an equally weighted portfolio of the positive-alpha stocks and sells short $ 1.7 million of an equally weighted portfolio of the negative-alpha stocks.a) What is the expected profit (in dollars), and what is

the standard deviation of the analyst’s profitb) How does your answer change if the analyst

examines 50 stocks instead of 20? 100 stocks?

7.5 ARBITRAGE PRICING THEORY

31) The APT itself does not provide information on the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Is industrial production a reasonable factor to test for a risk premium? Why or why not?

7.5 ARBITRAGE PRICING THEORY31) As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh insights concerning the firm’s cost of capital. She decides that this would be a good opportunity to try out the new material on the APT that she learned last semester. As such, she decides that three promising factors would be (i) the return on a broad-based index such as the S& P 500; (ii) the level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and (iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta of Pork Products against each of these factors and to estimate the risk premium associated with exposure to each factor. Comment on Jennifer’s choice of factors. Which are most promising with respect to the likely impact on her firm’s cost of capital? Can you suggest improvements to her specification?

7.5 ARBITRAGE PRICING THEORY

Capital Asset Pricing and Arbitrage Pricing Theory

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