CHAPTER 9&10

1. CHAPTER 9&10 Capital Asset Pricing Models

2. Risk and diversification • Market risk is the only risk left after diversification • Return that investors get in the market is rewarded for market risk only, not total risk • Hence market risk is relevant risk, and specific risk is irrelevant risk • In the market, higher beta gets higher return, not higher std gets higher return.

3. Example: std beta amount invested IBM 40% 0.95 2000 AT&T 20% 1.10 4000 • What is the beta of market portfolio • Does IBM have more or less risk than the market • Which stock has more total risk Which stock has more systematic risk Which stock is expected to have higher return in the market 4. In the boom market, which stock do you choose 5. In the recession market, which stock do you choose 6. How do investors know whether the return they get in the market is high enough to reward for the level of risk taken

4. Capital Asset Pricing Model (CAPM) • It is the equilibrium model that underlies all modern financial theory. • Derived using principles of diversification with simplified assumptions. • Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

5. CAPM Assumptions • Investors care only about the mean-variance trade-off of their portfolios in the next period • All investors are price-takers. i.e., no investor is dominant such that her action alone will change prices – perfect competition assumption • Investors have homogeneous beliefs and equal investment opportunities • There is a risk-free asset and investors can borrow and lend at the same risk-free rate • Markets are frictionless, i.e., with no taxes and transaction costs. No limitation on the size of trading and short sales • All of investors’ wealth is in market traded assets

6. CAPM (Equilibrium) Results • All investors will hold the same portfolio for risky assets – market portfolio. • Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value. • Risk premium on the the market depends on the average risk aversion of all market participants. • Risk premium on an individual security is a function of its covariance with the market.

7. Capital Market Line (CML) CML Borrowing Lending M Rf

8. The CML and the Separation Theorem • The CML leads all investors to invest in the M portfolio. The only difference is the location on the CML depending on risk preferences • Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio • Investors preferring more risk might borrow funds at Rf and invest everything in the market portfolio • Two-fund separation theorem or “mutual fund theorem”

9. The Market Portfolio • Because Portfolio M lies at the point of tangency, it has the highest portfolio possibility line • Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML • Therefore this portfolio must include ALL RISKY ASSETS • Because the market is in equilibrium, all assets are included in this portfolio in proportion to their market value • Therefore, Portfolio M must be the market portfolio

10. The Market Portfolio • The tangency portfolio M is the market portfolio • All assets included in this portfolio are weighted in proportion to their market value • Because portfolio M contains all risky assets, it is a completely diversified portfolio. Only systematic risk remains in the market portfolio. • Systematic risk may be measured by the standard deviation of returns on the market portfolio.

11. Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return Standard Deviation of Return Unsystematic (diversifiable) Risk Total Risk Standard Deviation of the Market Portfolio (systematic risk) Systematic Risk Number of Stocks in the Portfolio

12. Figure 9.1 The Efficient Frontier and the Capital Market Line

13. Market Risk Premium • The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:

14. The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio Return and Risk For Individual Securities

15. Using GE Text Example Covariance of GE return with the market portfolio: Therefore, the reward-to-risk ratio for investments in GE would be:

16. Using GE Text Example Continued Reward-to-risk ratio for investment in market portfolio: Reward-to-risk ratios of GE and the market portfolio: And the risk premium for GE:

17. Expected Return-Beta Relationship CAPM holds for the overall portfolio because: This also holds for the market portfolio:

18. Figure 9.2 The Security Market Line

19. Determining the Expected Rate of Return for Risky Assets Assume: RFR = 6% RM = 12% Implied market risk premium = 6% E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2% E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0% E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9% E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4% E(RE) = 0.06 + (-0.30) (0.12-0.06) = 0.042 = 4.2%

20. Capital Asset Pricing Model (CAPM) • CAPM gives the relationship between risk and return. • It gives the minimum return required by investors in order for them to buy stock • What is the meaning of E(R) calculated in the previous slide?

21. Capital Asset Pricing Model (CAPM) • Remember earlier, we have • In CAPM, we have • What is the difference in meaning between the two expected return?

22. Capital Asset Pricing Model (CAPM) • When forecasted E(R) > required E(R), stock is undervalued or the price is too low • When forecasted E(R) < required E(R), stock is undervalued or the price is too high • In equilibrium, forecasted E(R) = required E(R)

23. Capital Asset Pricing Model (CAPM) • Example: E(Rm) = 14%, Rf = 6% Stocks Beta E(R) (forecasted) IBM 1.2 17% ATT 1.5 14% • According to CAPM, what is the required E(R) for IBM and ATT • Which stock is undervalued, which stock is overvalued

24. Determining the Expected Rate of Return for a Risky Asset • Let alpha (α) be the difference between the actual (forecasted) E(R) and the required E(R) • In equilibrium, all assets and all portfolios of assets should plot on the SML ( i.e., α = 0) • Any security with an estimated return that plots above the SML is underpriced (α > 0 ) • Any security with an estimated return that plots below the SML is overpriced ( α < 0 ) • Previous example: • αIBM= 17 – 15.6 = 1.4 > 0. Actual E(R) is above the SML • αATT= 14-18 = -4 > 0. Actual E(R) is below the SML

25. Figure 7-2 The Security Market Line and Positive Alpha Stock

26. b for Portfolios • Expected return of a portfolio

27. Portfolio Beta (p289) Q3 Q: Suppose that the risk premium on the market portfolio is estimated at 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested 25% in GM and 75% in Ford, if they have betas of 1.10 and 1.25, respectively?

28. Q16, Chapter 9 (p. 306) Q: Two investment advisors: A: return=19%; beta=1.5 B: return=16%; beta=1 a) Who was better? b) If T-bill rate were 6% and market return were 14%, who would be better? c) What if T-bill rate were 3% and market return were 15%?

29. The Index Model and Realized Returns To move from expected to realized returns—use the index model in excess return form: The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship

30. Using CAPM: Estimating b • where: • Ri,t = rate of return for asset i during period t • RM,t = rate of return for the market portfolio M during t • Adjustments to b • Merrill-Lynch: Adjusted Beta = 2/3*(unadjusted. b) + 1/3 • Time interval problems • Different holding periods produce different beta • More pronounced for small company and illiquid stocks • Weekly and monthly returns better for estimation, not daily data.

31. Other Estimation Issues • Risk-free rate • Most use short-term Treasury bill returns • Notice that bill returns are variable, not truly risk-free. • Market risk premium • Historical data of excess return on market index • But expected return on market index may change over time • Proxies for market portfolio • S&P (U.S. equity only) • World indices (ignore other assets, like real estates, etc.)

32. Estimating alpha and beta in practice (using index model) • alpha is the abnormal return = actual return – return predicted by CAPM • According to CAPM, alpha should be = 0 • Beta is the systematic risk

33. Cumulative Returns for T-bills, S&P 500 and GM Stock

34. Figure 7-4 Characteristic Line for GM

35. Characteristic Line for GM • All the points are actual values • Line is the predicted relationship • If there are a lot of specific risk, there will be a wide scatter of points around the line. Hence, using market risk only in this case does not produce a precise estimate of expected return • If the points are close to the line, there is only small specific risk. Using market risk can explain most of the company return.

36. Security Characteristic Line for GM: Summary Output

37. Security Characteristic Line for GM: Summary Output • R-square: 0.2866 • ANOVA table Total risk = systematic risk + unsystematic risk 7449.17 = 2224.696 + 5224.45 (100%) = 29.87% + 70.13% • Alpha = 0.8890 > 0 (positive alpha, undervalued or overvalued?) • During the period Jan 99-Dec03: the risk-adjusted or abnormal return of GM = 0.8990% or actual return is higher than CAPM predicted • Is this value statistically different from 0? Is this still consistent with CAPM • 95% confidence interval (-1.5690 to 3.3470) • Beta = 1.2384 • Is beta statistically different from 0?

38. Applications of CAPM • CAPM is a benchmark about the fair (required) expected return on a risk asset. Investors calculate the return they actually earn based on their input and compare with the return they get from the CAPM • Compare the performance of the mutual fund: we use alpha or risk-adjusted return rather than regular return • Compute the cost of equity for capital budgeting

39. Empirical Tests of the CAPM • Typical Tests • Time-series test (Black-Jensen-Scholes) • Cross-sectional tests (Fama-MacBeth) • Most tests are done in portfolios

40. Empirical Tests of the CAPM • High beta portfolios do not necessarily generate high returns • Controversial results: low betas have positive alpha, high betas have negative alpha. • Size and book-to-market value ratio seem to have explanatory power for returns • Fama and French • Momentum in returns • Relative strength

41. Why still the CAPM? • Is the CAPM wrong? • Problems with the proxy for market portfolio • Possible missing risk factors -> Multi-factor models • Relaxing assumptions • Important intuitions from the CAPM • Diversification • Only covariance with systematic risks matters

42. Current status of CAPM • CAPM is powerful at the conceptual level. It is a useful way to think about risk and return • Empirical data does not support CAPM fully but it is simple, logical, easy to use, so use CAPM with caution

43. Extensions of the CAPM • Zero-Beta Model • Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks • Consideration of labor income and non-traded assets • Merton’s Multiperiod Model and hedge portfolios • Incorporation of the effects of changes in the real rate of interest and inflation

44. Other Asset Pricing Models • CAPM is limited (true market portfolio is unobservable), nice idea though! • Other factors also matter, e.g., Fama-French book-to-market and size factors • Arbitrage Pricing Theory (APT): no free lunch (for diversified portfolio)! Chapter 9: Asset Pricing Theories

45. Arbitrary pricing theory (APT) • CAPM is a single factor model. The market risk premium is the only factor • In CAPM, all the news, uncertainties affect the market, then the market affect the stock individually • In APT, there are n factors that can influence stock return so there will be n-sources of risk or n-channels of uncertainties • Empirical evidence support APT (more than 1 factor affect stock return), but unable to identify these factors. • So if the purpose is to get cost of capital only, then APT is appropriate • If we want to know sources of risk then APT is not useful

46. Multifactor APT Use of more than a single factor Requires formation of factor portfolios What factors? Factors that are important to performance of the general economy Fama-French Three Factor Model

47. Example of the Multifactor Approach • Work of Chen, Roll, and Ross • Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy • IP = % change in industrial production • EI = % change in expected inflation • UI = % change in unanticipated inflation • CG = excess return of long-term corporate bond over long-term government bond • GB = excess return of long-term government bond over T-bills

48. Fama-French three-factor model • Fama and French propose three factors: • The excess market return, rM-rRF. • the return on, S, a portfolio of small firms (where size is based on the market value of equity) minus the return on B, a portfolio of big firms. This return is called rSMB, for S minus B. • the return on, H, a portfolio of firms with high book-to-market ratios (using market equity and book equity) minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called rHML, for H minus L.

49. Another Example:Fama-French Three-Factor Model • The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums • Where: • SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks • HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio

50. The Multifactor CAPM and the APM A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge The APT is largely silent on where to look for priced sources of risk