1. McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Capital Asset Pricing and Arbitrage Pricing Theory 7

2. 7-2 The Capital Asset Pricing Model (CAPM) What is the capital market line (CML)? Given the capital market line (CML), what is the separation theorem? Assumptions of t he Capital Asset Pricing Model Given the CML, what is the relevant risk measure for an individual risky asset? What is the security market line (SML) and how does it differ from the CML? What is beta and why is it referred to as a standardized measure of systematic risk?

3. 7-3 Separation Property The property that implies portfolio choice can be separated into two independent tasks: 1. Determination of the optimal risky portfolio, which a purely technical problem, and 2. The personal choice of the best mix of the risky portfolio and risk- free asset.

4. 7-4 7.1 The Capital Asset Pricing Model 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

5. 7-5 7.1 The Capital Asset Pricing Model Hypothetical Equilibrium Given above assumptions, we can state that: All investors choose to hold market portfolio Market portfolio is on efficient frontier, optimal risky portfolio Risk premium on market portfolio is proportional to variance of market portfolio and investor’s risk aversion, A, E(r M –r F ) = A    Risk premium on individual assets will be proportional to risk premium on market portfolio and proportional to beta coefficient of security on market portfolio Beta measures the extent to which returns respond to the market portfolio.

6. 7-6 Figure 7.1 Efficient Frontier and Capital Market Line

7. 7-7 7.1 The Capital Asset Pricing Model The Passive Strategy is Efficient The CAPM implies that a passive strategy using the CML as the optimal CAL is a powerful alternative to an active strategy. 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?

8. 7-8 7.1 The Capital Asset Pricing Model Risk Premium of Market Portfolio When investors purchase stocks, their demand drives up prices and thereby lowers expected rate of return/risk premiums When premiums fall, investors move funds into risk-free asset In equilibrium, the risk premium on the market portfolio must be high enough to induce investors to hold available supply of stocks. If the risk premium is too high, there will be excess demand for securities, and prices rise; if it is too low, investors will not hold enough stock to absorb the supply and prices will fall. Therefore: the equilibrium risk premium of market portfolio proportional to 1.Risk of market 2. Risk aversion of average investor E(r M –r F ) = A   

9. 7-9 Example Suppose- risk-free =5%, the average investor’s risk aversion coefficient =2 and standard deviation of th market portfolio =20%, then using equation E(r M –r F ) = A     the equilibrium value of the market risk premium E(r M –r F ) = 2 x (20%) 2 =8% The expected rate of return on the market portfolio is: E(r M )=r F + Equilibrium risk premium E(r M )= 5% + 8% = 13% If the investor were more risk averse, it would take a higher risk premium to induce them to hold shares. For example, if the average degree of risk aversion were 3, the market risk premium would be: E(r M –r F ) = 3 x (20%) 2 =12% E(r M )= 5% + 12% = 17%

10. 7-10 7.1 The Capital Asset Pricing Model Expected Returns on Individual Securities Expected return-beta relationship Implication of CAPM that security risk premiums (expected excess return) will be proportional to beta E(r i ) = R f +  i ( (E(r M – R f )) - SML 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) For example- Dell beta =1.2 and risk-free rate = 6% and market expected rate of return =14% E(r i ) = R f +  i ( (E(r M – R f )) = 6% +1.2( 14%-6%) =15.6% If one believes the stock will provide instead a return of 17%, its implied alpha would be 1.4% as show in the next graph.

11. 7-11 Figure 7.2 The SML and a Positive-Alpha Stock

12. 7-12 7.1 The Capital Asset Pricing Model Applications of CAPM Use SML as benchmark for fair return on risky asset SML provides “hurdle rate” for internal projects

13. 7-13 7.2 CAPM and Index Models

14. 7-14 7.2 CAPM and Index Models

15. 7-15 Table 7.1 Monthly Return Statistics 01/06 - 12/10 Statistic (%)T-BillsS&P 500Google Average rate of return0.1840.2391.125 Average excess return-0.0550.941 Standard deviation*0.1775.1110.40 Geometric average0.1800.1070.600 Cumulative total 5-year return11.656.6043.17 Gain Jan 2006-Oct 20079.0427.4570.42 Gain Nov 2007-May 20092.29-38.87-40.99 Gain June 2009-Dec 20100.1036.8342.36 * The rate on T-bills is known in advance, SD does not reflect risk.

16. 7-16 Figure 7.3A: Monthly Returns

17. 7-17 Figure 7.3B Monthly Cumulative Returns

18. 7-18 Figure 7.4 Scatter Diagram/SCL: Google vs. S&P 500, 01/06- 12/10

19. 7-19 Table 7.2 SCL for Google (S&P 500), 01/06-12/10 Linear Regression Regression Statistics R 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 dfSSMSFp-level Regression 12231.50 31.190.0000 Residual 584149.6571.55 Total 596381.15 CoefficientsStandard Errort-Statisticp-valueLCLUCL Intercept 0.87511.09200.80130.4262-1.73753.4877 S&P 500 1.20310.21545.58480.00000.68771.7185 t-Statistic (2%) 2.3924 LCL - Lower confidence interval (95%) UCL - Upper confidence interval (95%)

20. 7-20 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

21. 7-21 7.2 CAPM and Index Models Predicting Betas Mean reversion Betas move towards mean over time To predict future betas, adjust estimates from historical data to account for regression towards 1.0

22. 7-22 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 diversify Systematic risk is the risk that matters Well-diversified risky portfolio can be suitable for wide range of investors

23. 7-23 7.4 Multifactor Models and CAPM

24. 7-24 7.4 Multifactor Models and CAPM

25. 7-25 Table 7.3 Monthly Rates of Return, 01/06-12/10 Monthly Excess Return % *Total Return SecurityAverageStandard DeviationGeometric AverageCumulative Return T-bill000.1811.65 Market index **0.265.440.3019.51 SMB0.342.460.3120.70 HML0.012.97-0.03-2.06 Google0.9410.400.6043.17 *Total return for SMB and HML ** Includes all NYSE, NASDAQ, and AMEX stocks.

26. 7-26 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 SpecificationFF 3-Factor Specification EstimateS&P 500Broad Market Indexwith Broad Market Index Correlation coefficient0.590.610.70 Adjusted R-Square0.340.360.47 Residual SD = Regression SE (%)8.468.337.61 Alpha = Intercept (%)0.88 (1.09)0.64 (1.08) 0.62 (0.99) Market beta1.20 (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

27. 7-27 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

28. 7-28 7.5 Arbitrage Pricing Theory Calculating APT Returns on well-diversified portfolio

29. 7-29 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

30. 7-30 Table 7.6 Largest Capitalization Stocks in S&P 500 Stock Weight

31. 7-31 Table 7.7 Regression Statistics of S&P 500 Portfolio on Benchmark Portfolio, 01/06-12/10 Linear Regression Regression Statistics R0.9933 R-square0.9866 Adjusted R-square0.9864 Annualized Regression SE0.59682.067 Total number of observations60 S&P 500 = - 0.1909 + 0.9337 × Benchmark Coefficients Standard Errort-statp-level Intercept-0.19090.0771-2.47520.0163 Benchmark0.93370.014365.34340.0000

32. 7-32 Table 7.8 Annual Standard Deviation PeriodReal RateInflation RateNominal Rate 1/1 /06 - 12/31/101.46 0.61 1/1/96 - 12/31/000.570.540.17 1/1/86 - 12/31/900.860.830.37

33. 7-33 Figure 7.5 Security Characteristic Lines

34. 7-34 7.5 Arbitrage Pricing Theory Multifactor Generalization of APT and CAPM Factor portfolio Well-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

35. 7-35 Table 7.9 Constructing an Arbitrage Portfolio Constructing an arbitrage portfolio with two systemic factors