1. CHAPTER 6 Risk and Rates of Return Stand-alone risk Portfolio risk Risk & return: CAPM/SML

2. What is investment risk? Investment risk pertains to the probability of actually earning a low or negative return. The greater the chance of low or negative returns, the riskier the investment.

3. Probability distribution Expected Rate of Return Rate of return (%) 100150-70 Firm X Firm Y

4. Annual Total Returns,1926-1998 AverageStandard ReturnDeviationDistribution Small-company stocks 17.4% 33.8% Large-company stocks 13.2 20.3 Long-term corporate bonds 6.1 8.6 Long-term government 5.7 9.2 Intermediate-term government 5.5 5.7 U.S. Treasury bills 3.8 3.2 Inflation 3.2 4.5

5. Investment Alternatives (Given in the problem) EconomyProb.T-BillHTCollUSRMP Recession0.18.0%-22.0%28.0%10.0%-13.0% Below avg.0.28.0-2.014.7-10.0 1.0 Average0.48.020.00.07.0 15.0 Above avg.0.28.035.0-10.045.0 29.0 Boom0.18.050.0-20.030.0 43.0 1.0

6. Why is the T-bill return independent of the economy? Will return the promised 8% regardless of the economy.

7. Do T-bills promise a completely risk-free return? No, T-bills are still exposed to the risk of inflation. However, not much unexpected inflation is likely to occur over a relatively short period.

8. Do the returns of HT and Coll. move with or counter to the conomy? HT: Moves with the economy, and has a positive correlation. This is typical. Coll: Is countercyclical of the economy, and has a negative correlation. This is unusual.

9. Calculate the expected rate of return on each alternative: k = expected rate of return. k HT = (-22%)0.1 + (-2%)0.20 + (20%)0.40 + (35%)0.20 + (50%)0.1 = 17.4%. ^ ^

10. k HT17.4% Market15.0 USR13.8 T-bill8.0 Coll.1.7 HT appears to be the best, but is it really? ^

11. What’s the standard deviation of returns for each alternative?  = Standard deviation.  = =  =

12.  T-bills = 0.0%.  HT = 20.0%.  Coll =13.4%.  USR =18.8%.  M =15.3%. 1/2  T-bills =           (8.0 – 8.0) 2 0.1 + (8.0 – 8.0) 2 0.2 + (8.0 – 8.0) 2 0.4 + (8.0 – 8.0) 2 0.2 + (8.0 – 8.0) 2 0.1

13. Prob. Rate of Return (%) T-bill USR HT 0813.817.4

14. Standard deviation (  i ) measures total, or stand-alone, risk. The larger the  i, the lower the probability that actual returns will be close to the expected return.

15. Expected Returns vs. Risk Security Expected return Risk,  HT 17.4% 20.0% Market 15.0 15.3 USR 13.8* 18.8* T-bills 8.0 0.0 Coll. 1.7* 13.4* *Seems misplaced.

16. Coefficient of Variation (CV) Standardized measure of dispersion about the expected value: Shows risk per unit of return. CV = =. Std dev  ^ k Mean

17. 0 A B  A =  B, but A is riskier because larger probability of losses. = CV A > CV B.  ^ k

18. Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections. Calculate k p and  p. ^

19. Portfolio Return, k p k p is a weighted average: k p = 0.5(17.4%) + 0.5(1.7%) = 9.6%. k p is between k HT and k COLL. ^ ^ ^ ^ ^^ ^^ k p =   w i k i  n i = 1

20. Alternative Method k p = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%. ^ Estimated Return EconomyProb.HTColl.Port. Recession 0.10-22.0% 28.0% 3.0% Below avg. 0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0 10.0 Above avg. 0.20 35.0 -10.0 12.5 Boom 0.10 50.0 -20.0 15.0

21. CV p = = 0.34. 3.3% 9.6%  p = = 3.3%. 12/                       (3.0 – 9.6) 2 0.10 + (6.4 – 9.6) 2 0.20 + (10.0 – 9.6) 2 0.40 + (12.5 – 9.6) 2 0.20 + (15.0 – 9.6) 2 0.10

22.  p = 3.3% is much lower than that of either stock (20% and 13.4%).  p = 3.3% is lower than average of HT and Coll = 16.7%.  Portfolio provides average k but lower risk. Reason: negative correlation. ^

23. General statements about risk Most stocks are positively correlated. r k,m  0.65.  35% for an average stock. Combining stocks generally lowers risk.

24. Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM 25 15 0 -10 0 0 15 25 Stock WStock MPortfolio WM...............

25. Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’ Stock M 0 15 25 -10 Stock M’ 0 15 25 -10 Portfolio MM’ 0 15 25 -10

26. What would happen to the riskiness of an average 1-stock portfolio as more randomly selected stocks were added?  p would decrease because the added stocks would not be perfectly correlated but k p would remain relatively constant. ^

27. Large 0 15 Prob. 2 1 Even with large N,  p  20%

28. # Stocks in Portfolio 102030 40 2,000+ Company Specific Risk Market Risk 20 0 Stand-Alone Risk,  p  p (%) 35

29. As more stocks are added, each new stock has a smaller risk-reducing impact.  p falls very slowly after about 10 stocks are included, and after 40 stocks, there is little, if any, effect. The lower limit for  p is about 20% =  M.

30. Stand-alone Market Firm-specific Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification, and is measured by beta. Firm-specific risk is that part of a security’s stand-alone risk that can be eliminated by proper diversification. risk risk risk = +

31. By forming portfolios, we can eliminate about half the riskiness of individual stocks (35% vs. 20%).

32. If you chose to hold a one-stock portfolio and thus are exposed to more risk than diversified investors, would you be compensated for all the risk you bear?

33. NO! Stand-alone risk as measured by a stock’s  or CV is not important to a well-diversified investor. Rational, risk averse investors are concerned with  p, which is based on market risk.

34. There can only be one price, hence market return, for a given security. Therefore, no compensation can be earned for the additional risk of a one- stock portfolio.

35. Beta measures a stock’s market risk. It shows a stock’s volatility relative to the market. Beta shows how risky a stock is if the stock is held in a well-diversified portfolio.

36. How are betas calculated? Run a regression of past returns on Stock i versus returns on the market. Returns = D/P + g. The slope of the regression line is defined as the beta coefficient.

37. Yeark M k i 115% 18% 2 -5-10 312 16... kiki _ kMkM _ - 505101520 20 15 10 5 -5 -10 Illustration of beta calculation: Regression line: k i = -2.59 + 1.44 k M ^^

38. If beta = 1.0, average stock. If beta > 1.0, stock riskier than average. If beta < 1.0, stock less risky than average. Most stocks have betas in the range of 0.5 to 1.5.

39. List of Beta Coefficients Stock Beta Merrill Lynch 2.00 America Online 1.70 General Electric 1.20 Microsoft Corp. 1.10 Coca-Cola 1.05 IBM 1.05 Procter & Gamble 0.85 Heinz 0.80 Energen Corp. 0.80 Empire District Electric 0.45

40. Can a beta be negative? Answer: Yes, if r i, m is negative. Then in a “beta graph” the regression line will slope downward. Though, a negative beta is highly unlikely.

41. HT T-Bills b = 0 kiki _ kMkM _ - 2002040 40 20 -20 b = 1.29 Coll. b = -0.86

42. Riskier securities have higher returns, so the rank order is OK. HT 17.4% 1.29 Market 15.0 1.00 USR 13.8 0.68 T-bills 8.0 0.00 Coll. 1.7 -0.86 Expected Risk Security Return (Beta)

43. Use the SML to calculate the required returns. Assume k RF = 8%. Note that k M = k M is 15%. (Equil.) RP M = k M – k RF = 15% – 8% = 7%. SML: k i = k RF + (k M – k RF )b i. ^

44. Required Rates of Return k HT = 8.0% + (15.0% – 8.0%)(1.29) = 8.0% + (7%)(1.29) = 8.0% + 9.0%= 17.0%. k M = 8.0% + (7%)(1.00)= 15.0%. k USR = 8.0% + (7%)(0.68)= 12.8%. k T-bill = 8.0% + (7%)(0.00)= 8.0%. k Coll = 8.0% + (7%)(-0.86)= 2.0%.

45. HT 17.4% 17.0% Undervalued: k > k Market 15.0 15.0 Fairly valued USR 13.8 12.8 Undervalued: k > k T-bills 8.0 8.0 Fairly valued Coll. 1.7 2.0 Overvalued: k < k Expected vs. Required Returns ^ ^ ^ ^ k k

46. .. Coll.. HT T-bills. USR SML k M = 15 k RF = 8 -1 0 1 2. SML: k i = 8% + (15% – 8%) b i. k i (%) Risk, b i

47. Calculate beta for a portfolio with 50% HT and 50% Collections b p = Weighted average = 0.5(b HT ) + 0.5(b Coll ) = 0.5(1.29) + 0.5(-0.86) = 0.22.

48. The required return on the HT/Coll. portfolio is: k p = Weighted average k = 0.5(17%) + 0.5(2%) = 9.5%. Or use SML: k p = k RF + (k M – k RF ) b p = 8.0% + (15.0% – 8.0%)(0.22) = 8.0% + 7%(0.22) = 9.5%.

49. If investors raise inflation expectations by 3%, what would happen to the SML?

50. SML 1 Original situation Required Rate of Return k (%) SML 2 00.5 1.01.5 Risk, b i 18 15 11 8 New SML  I = 3%

51. If inflation did not change but risk aversion increased enough to cause the market risk premium to increase by 3 percentage points, what would happen to the SML?

52. k M = 18% k M = 15% SML 1 Original situation Required Rate of Return (%) SML 2 After increase in risk aversion Risk, b i 18 15 8 1.0  RP M = 3%

53. Has the CAPM been verified through empirical tests? Not completely. Those statistical tests have problems that make verification almost impossible.

54. Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of k i : k i = k RF + (k M – k RF )b + ?

55. Also, CAPM/SML concepts are based on expectations, yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.