Chapter 13. Risk & Return in Asset Pricing Models

1. Chapter 13. Risk & Return inAsset Pricing Models • Portfolio Theory • Managing Risk • Asset Pricing Models

2. I. Portfolio Theory • how does investor decide among group of assets? • assume: investors are risk averse • additional compensation for risk • tradeoff between risk and expected return

3. goal • efficient or optimal portfolio • for a given risk, maximize exp. return • OR • for a given exp. return, minimize the risk

4. tools • measure risk, return • quantify risk/return tradeoff

5. Measuring Return • R is ex post • based on past data, and is known • R is typically annualized change in asset value + income return = R = initial value

6. example 1 • Tbill, 1 month holding period • buy for $9488, sell for $9528 • 1 month R: 9528 - 9488 = .0042 = .42% 9488

7. annualized R: (1.0042)12 - 1 = .052 = 5.2%

8. example 2 • 100 shares IBM, 9 months • buy for $62, sell for $101.50 • $.80 dividends • 9 month R: 101.50 - 62 + .80 = .65 =65% 62

9. annualized R: (1.65)12/9 - 1 = .95 = 95%

10. Expected Return • measuring likely future return • based on probability distribution • random variable E(R) = SUM(Ri x Prob(Ri))

11. example 1 R Prob(R) 10% .2 5% .4 -5% .4 E(R) = (.2)10% + (.4)5% + (.4)(-5%) = 2%

12. example 2 R Prob(R) 1% .3 2% .4 3% .3 E(R) = (.3)1% + (.4)2% + (.3)(3%) = 2%

13. examples 1 & 2 • same expected return • but not same return structure • returns in example 1 are more variable

14. Risk • measure likely fluctuation in return • how much will R vary from E(R) • how likely is actual R to vary from E(R) • measured by • variance (s2) • standard deviation (s)

15. s2 = SUM[(Ri - E(R))2 x Prob(Ri)] s = SQRT(s2)

16. example 1 s2 = (.2)(10%-2%)2 + (.4)(5%-2%)2 + (.4)(-5%-2%)2 = .0039 s = 6.24%

17. example 2 s2 = (.3)(1%-2%)2 + (.4)(2%-2%)2 + (.3)(3%-2%)2 = .00006 s = .77%

18. same expected return • but example 2 has a lower risk • preferred by risk averse investors • variance works best with symmetric distributions

19. prob(R) prob(R) R R E(R) E(R) symmetric asymmetric

20. II. Managing risk • Diversification • holding a group of assets • lower risk w/out lowering E(R)

21. Why? • individual assets do not have same return pattern • combining assets reduces overall return variation

22. two types of risk • unsystematic risk • specific to a firm • can be eliminated through diversification • examples: -- Safeway and a strike -- Microsoft and antitrust cases

23. systematic risk • market risk • cannot be eliminated through diversification • due to factors affecting all assets -- energy prices, interest rates, inflation, business cycles

24. example • choose stocks from NYSE listings • go from 1 stock to 20 stocks • reduce risk by 40-50%

25. s unsystematic risk total risk systematic risk # assets

26. measuring relative risk • if some risk is diversifiable, • then sis not the best measure of risk • σ is an absolute measure of risk • need a measure just for the systematic component

27. Beta, b • variation in asset/portfolio return relative to return of market portfolio • mkt. portfolio = mkt. index -- S&P 500 or NYSE index % change in asset return b = % change in market return

28. interpreting b • if b = 0 • asset is risk free • if b = 1 • asset return = market return • if b > 1 • asset is riskier than market index • b < 1 • asset is less risky than market index

29. Sample betas (monthly returns, 5 years back)

30. measuring b • estimated by regression • data on returns of assets • data on returns of market index • estimate

31. problems • what length for return interval? • weekly? monthly? annually? • choice of market index? • NYSE, S&P 500 • survivor bias

32. # of observations (how far back?) • 5 years? • 50 years? • time period? • 1970-1980? • 1990-2000?

33. III. Asset Pricing Models • CAPM • Capital Asset Pricing Model • 1964, Sharpe, Linter • quantifies the risk/return tradeoff

34. assume • investors choose risky and risk-free asset • no transactions costs, taxes • same expectations, time horizon • risk averse investors

35. implication • expected return is a function of • beta • risk free return • market return

36. or where is the portfolio risk premium is the market risk premium

37. so if b >1, > • portfolio exp. return is larger than exp. market return • riskier portfolio has larger exp. return >

38. so if b <1, < • portfolio exp. return is smaller than exp. market return • less risky portfolio has smaller exp. return <

39. so if b =1, = • portfolio exp. return is same than exp. market return • equal risk portfolio means equal exp. return =

40. so if b = 0, = 0 • portfolio exp. return is equal to risk free return =

41. example • Rm = 10%, Rf = 3%, b = 2.5

42. CAPM tells us size of risk/return tradeoff • CAPM tells use the price of risk

43. Testing the CAPM • CAPM overpredicts returns • return under CAPM > actual return • relationship between β and return? • some studies it is positive • some recent studies argue no relationship (1992 Fama & French)

45. other factors important in determining returns • January effect • firm size effect • day-of-the-week effect • ratio of book value to market value

46. problems w/ testing CAPM • Roll critique (1977) • CAPM not testable • do not observe E(R), only R • do not observe true Rm • do not observe true Rf • results are sensitive to the sample period

47. APT • Arbitrage Pricing Theory • 1976, Ross • assume: • several factors affect E(R) • does not specify factors

48. implications • E(R) is a function of several factors, F each with its own b

49. APT vs. CAPM • APT is more general • many factors • unspecified factors • CAPM is a special case of the APT • 1 factor • factor is market risk premium

50. testing the APT • how many factors? • what are the factors? • 1980 Chen, Roll, and Ross • industrial production • inflation • yield curve slope • other yield spreads