1. 6-1 Risk and Return: Modern Portfolio Theory

2. 6-2 Returns and Return Distribution Dollar return Dollar return dividend income + capital gains dividend income + capital gains Percentage return Percentage return R t+1 = [D t+1 + (P t+1 – P t )] / P t R t+1 = [D t+1 + (P t+1 – P t )] / P t Holding period return Holding period return R T=3 = (1 + R 1 ) × (1 + R 2 ) × (1 + R 3 ) R T=3 = (1 + R 1 ) × (1 + R 2 ) × (1 + R 3 ) Average return (Arithmetic) Average return (Arithmetic) R = (R 1 + ··· + R T ) / T R = (R 1 + ··· + R T ) / T Variance and Standard Deviation Variance and Standard Deviation SD =  (Var) =  [(1 / T –1) × ((R 1 –R) 2 + ··· + (R T –R) 2 )] SD =  (Var) =  [(1 / T –1) × ((R 1 –R) 2 + ··· + (R T –R) 2 )]

3. 6-3 The Historical Record What is the average year-to-year return on the following financial investments: What is the average year-to-year return on the following financial investments: Large-company stocks Large-company stocks Small-company stocks Small-company stocks Long-term corporate bonds Long-term corporate bonds Long-term government bonds Long-term government bonds U.S. Treasury bills U.S. Treasury bills Consumer Price Index (CPI) Consumer Price Index (CPI)

4. A $1 Investment in Different Types of Portfolios: 1926-1996 Index ($) $4,495.99 $33.73 $13.54 $8.85 $1,370.95 Small Company Stocks Large Company Stocks Long-Term Government Bonds Treasury BillsInflation Year-End 6-4

5. 6-5 What does capital market history tell us about risk and return? Historical Average Returns and Standard Deviations, 1926 - 1996

6. 6-6 Risk Premium: Risk Premium: the excess return required from an investment in a risky asset over that required from a risk-free asset. Treasury bill rate is used as the risk-free rate. Risk Premium

7. Historical Returns, Standard Deviations, and Frequency Distributions: 1926-1996 Source: © Stocks, Bonds, Bills, and Inflation 1997 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. – 90% + 90% 0% Average Standard Series Annual Return DeviationDistribution Large Company Stocks12.7%20.3% Small Company Stocks17.734.1 Long-Term Corporate Bonds6.08.7 Long-Term Government Bonds5.49.2 U.S. Treasury Bills3.83.3 Inflation3.24.5

8. Frequency Distribution of Returns on Common Stocks, 1926-1996 2 16 1 8 11 6 10 13 1 3 Number of Years Return (%)

9. The Normal Distribution Probability Return on large company stocks 68% 95% > 99% – 3  – 48.2% – 2  – 27.9% – 1  – 7.6% 0 12.7% + 1  33.0% + 2  53.3% + 3  73.6%

10. 6-10 Investment risk pertains to the probability of realized returns being less than expected return. The greater the chance of low or negative returns, the riskier the investment. Normal distribution is assumed States of Nature  Future scenarios What is investment risk?

11. 6-11 Expected Return and Variance Expected Return: Return on a risky asset expected in the future Expected Return: Return on a risky asset expected in the future Variance: Measures the dispersion of an asset's returns around its expected return. Variance: Measures the dispersion of an asset's returns around its expected return. Standard deviation: The square root of the variance. Standard deviation: The square root of the variance.

12. State of the Weather Very Cold ColdAverageHotProbability  100% 0.10.30.40.2 Estimated Rate of Return Estimated Rate of Return Investment Alternatives: 2 risky investments 9-5 Amusement ParkSki Resort -15.0% 35.0% -15.0% 35.0% -5.0 15.0 -5.0 15.0 10.0 5.0 10.0 5.0 30.0 -5.0 30.0 -5.0 Adds up to 100%

13. 6-13 Calculate the expected rate of return on each investment alternative. E(R A ) = (-15%)  0.1 + (-5%)  0.3 + (10%)  0.4 + (30%)  0.2 = 7.0% E(R S ) = (35%)  0.1 + (15%)  0.3 + (5%)  0.4 + (-5%)  0.2 = 9.0% Mean or expected value: E(X) = p(1)  X 1 + p(2)  X 2 + … + p(n)  X n wherei = possible outcome p(i)= probability of outcome i X i = return if outcome I happens n= total number of possible outcomes

14. 6-14 measure of total or “stand-alone” riskmeasure of total or “stand-alone” risk the larger the  the lower the probability that actual returns will be close to the expected returns.the larger the  the lower the probability that actual returns will be close to the expected returns. What is the standard deviation of returns for each alternative? = Variance =  p(i) × [X i - E(X) ] 2 n i= 1 

15. 6-15 =  p(i) × [X i - E(X)] 2 For the amusement park and ski resort we have: n i= 1 

16. 6-16 Stand-alone risk consists of: diversifiable risk diversifiable risk company specific, unique, or unsystematic company specific, unique, or unsystematic non-diversifiable risk non-diversifiable risk market or systematic market or systematic It is measured by dispersion of returns about the mean and is relevant only for assets held in isolation. It is measured by dispersion of returns about the mean and is relevant only for assets held in isolation. What is stand-alone risk?

17. 6-17 Probability Distributions Expected Rate of Return Probability Firm X Firm Y 015 100-70 Rate of Return%

18. 6-18 AB 0 Probability Expected Return  A =  B but A is riskier because it has a larger probability of losses.

19. 6-19 Caused by company or industry specific events like lawsuits, strikes, winning or losing major contracts, etc. Caused by company or industry specific events like lawsuits, strikes, winning or losing major contracts, etc. Effects of such events on a portfolio can be eliminated by diversification and should therefore not be rewarded. Effects of such events on a portfolio can be eliminated by diversification and should therefore not be rewarded. What is diversifiable risk?

20. 6-20 Stems from such external events as war, inflation, recession, and interest rates. Because all firms are affected simultaneously by these factors, market risk cannot be eliminated by diversification. Because all firms are affected simultaneously by these factors, market risk cannot be eliminated by diversification. Market risk is also known as systematic risk since it shows the degree to which a stock moves systematically with other stocks. Market risk is also known as systematic risk since it shows the degree to which a stock moves systematically with other stocks. What is market risk?

21. 6-21 E(x) CV = = Define coefficient of variation (CV) Standardized measure of dispersion about the expected value:  Shows risk per unit of return. Std. Dev. Mean

22. 6-22 measure of how 2 (X and Y) investments move together, or how an investment moves with the entire marketmeasure of how 2 (X and Y) investments move together, or how an investment moves with the entire market For our amusement park and ski resort:For our amusement park and ski resort:  AS = 0.1(-.15 -.07)(.35 -.09) + 0.3(-.05 -.07)(.15 -.09) + 0.4(.10 -.07)(.05 -.09) + 0.2(.30 -.07)(-.05 -.09)  AS = 0.1(-.15 -.07)(.35 -.09) + 0.3(-.05 -.07)(.15 -.09) + 0.4(.10 -.07)(.05 -.09) + 0.2(.30 -.07)(-.05 -.09) = - 0.0148 What is the covariance of returns for each alternative? Covariance =  p(i) × (X i - E(X) ) × (Y i - E(Y)) n i= 1

23. 6-23 What is the correlation coefficient? CORR XY = COV XY /  X  Y =  XY CORR XY = COV XY /  X  Y =  XY The correlation coefficient is standardized to always be between -1.0 and +1.0 The correlation coefficient is standardized to always be between -1.0 and +1.0 For the amusement park and ski resort we have: For the amusement park and ski resort we have:  AS =  AS /  A  S = – 0.0148 / (0.1418)(0.1114) = – 0.0148 / (0.1418)(0.1114) = – 0.9375 = – 0.9375

24. 6-24 Stock M     Stock W Portfolio WM Returns for Two Perfectly Negatively Correlated (  = -1.0) Stocks and Portfolio WM

25. 6-25 Returns for Two Perfectly Positively Correlated (  = +1.0) Stocks and Portfolio MM’:     Stocks M and M ’ (Identical returns) Portfolio MM’

26. 6-26 Portfolio Theory Portfolio: A group of securities, such as stocks and bonds, held by an investor. Portfolio: A group of securities, such as stocks and bonds, held by an investor. Portfolio weights: Percentages of the portfolio's total value invested in each security. Portfolio weights: Percentages of the portfolio's total value invested in each security. Example: Your portfolio consists of IBM stock and GM stock. You have $2,500 invested in IBM and $7,500 invested in GM. What are the portfolio weights? Example: Your portfolio consists of IBM stock and GM stock. You have $2,500 invested in IBM and $7,500 invested in GM. What are the portfolio weights?

27. 6-27 Portfolio Return and Variance Expected Return on a portfolio: Weighted average of the expected returns on the individual securities in the portfolio. Portfolio Variance: Unlike the expected return, the variance of a portfolio is not a simple weighted average of the individual security variances.

28. 6-28 Expected Portfolio Return In our earlier example, there were two stocks: the amusement park and the ski resort. In our earlier example, there were two stocks: the amusement park and the ski resort. E(R A ) = 7%  A = 14.177%Cov(A,S) = – 0.0148 E(R A ) = 7%  A = 14.177%Cov(A,S) = – 0.0148 E(R S ) = 9%  S = 11.136% E(R S ) = 9%  S = 11.136% Assume we have $100 and invest $50 in A and $50 in S. What is our expected portfolio return? Assume we have $100 and invest $50 in A and $50 in S. What is our expected portfolio return? E(R P ) = 0.5 (7%) + 0.5 (9%) = 8% E(R P ) = 0.5 (7%) + 0.5 (9%) = 8% w A = 50/100 = 0.5 w A = 50/100 = 0.5 w S = 50/100 = 0.5 w S = 50/100 = 0.5

29. 6-29 Expected Portfolio Risk To measure the risk of the portfolio, we must account for the risk of the individual stocks and how they move together. To measure the risk of the portfolio, we must account for the risk of the individual stocks and how they move together. Var(R p ) = (0.5) 2 (0.1418) 2 + (0.5) 2 (0.1114) 2 Var(R p ) = (0.5) 2 (0.1418) 2 + (0.5) 2 (0.1114) 2 + 2 (0.5)(0.5)(-0.0148) + 2 (0.5)(0.5)(-0.0148) =0.000725 =0.000725 Standard Deviation (R p ) = (0.000725) 1/2 Standard Deviation (R p ) = (0.000725) 1/2 = 0.0269 = 2.69%

30. State of the WeatherProb Very Cold0.1 Cold0.3 Average0.4 Hot0.2 1.0 Estimated Rate of Return Investment Alternatives 9-22 Amusement Ski50-50 ParkResortPortfolio -15.0%35.0%10.0%(a) -5.015.0 5.0% -5.015.0 5.0% 10.0 5.0 7.5% 10.0 5.0 7.5% 30.0 -5.012.5% 30.0 -5.012.5% 8.0% (b) 8.0% (b) (a) R P (very cold) = 0.5 (-15%) + 0.5 (35%) = 10% (b) R P = 0.1(10) + 0.3(5) + 0.4(7.5) + 0.2(12.5) = 8.0% Var(R P ) = 0.1(10 - 8) 2 + 0.3(5 - 8) 2 + 0.4(7.5 - 8) 2 + 0.2(12.5 - 8) 2 = 7.25 Std. Dev. (R P ) = (7.25) 0.5 = 2.69%

31. 6-31 If we alter the weights on the two stocks:

32. 6-32 Plotting the risk-return combinations gives: 100% S 100% A

33. 6-33 Expected Portfolio Return, R P Risk,  P Efficient Set Feasible Set A B C DE

34. 6-34 Risk Free Assets What happens if one of the assets in our portfolio is risk free, i.e.  Y = 0? What happens if one of the assets in our portfolio is risk free, i.e.  Y = 0?

35. 6-35 Risk Free Assets Assume there is a risky asset (X) and a risk free asset (F): Risky Asset: E(R X ) = 0.16,  X = 8% Risk Free Asset: E(R F ) = 0.06,  F = 0% You have $100, you put $50 in X and $50 in F (i.e., lending $50 at the risk free rate). The weights are:

36. 6-36 Risk Free Assets What happens if you have $100 and you borrow $50 from F and put $150 in X? What happens if you have $100 and you borrow $50 from F and put $150 in X? Note: The weights always add up to one!

37. 6-37 If we compute the expected return and standard deviation for a variety of weights, we can build a table as we did before:

38. 6-38

39. 6-39 Expected Portfolio Return, R P Risk,  P Q C D E M RfRf.. Lending Borrowing

40. 6-40 Most stocks are positively correlated. Most stocks are positively correlated. average correlation x,y = 0.65 average correlation x,y = 0.65 Average  i = 49.24% Average  i = 49.24% Average  p = 20% Average  p = 20% Combining stocks in a portfolio usually lowers the risk. Combining stocks in a portfolio usually lowers the risk. This is referred to as diversification. This is referred to as diversification. Exception: Correlation = +1 Exception: Correlation = +1 General statements about risk

41. 6-41 What would happen to the risk of a 1 stock portfolio as more randomly selected stocks were added? The standard deviation of the portfolio would decrease because the added stocks would not be perfectly correlated.

42. 6-42 Standard Deviations of Annual Portfolio Returns (Table 13.7) Ratio of Portfolio Average StandardStandard Deviation to Number of StocksDeviation of AnnualStandard Deviation in PortfolioPortfolio Returnsof a Single Stock Ratio of Portfolio Average StandardStandard Deviation to Number of StocksDeviation of AnnualStandard Deviation in PortfolioPortfolio Returnsof a Single Stock 149.24%1.00 149.24%1.00 1023.93 0.49 1023.93 0.49 5020.20 0.41 5020.20 0.41 10019.69 0.40 10019.69 0.40 30019.34 0.39 30019.34 0.39 50019.27 0.39 50019.27 0.39 1,00019.21 0.39 1,00019.21 0.39 These figures are from Table 1 in Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22 (September 1987), pp. 353–64. They were derived from E. J. Elton and M. J. Gruber, “Risk Reduction and Portfolio Size: An Analytic Solution,” Journal of Business 50 (October 1977), pp. 415–37.

43. 6-43 Portfolio Diversification (Figure 13.1) Average annual standard deviation (%) Number of stocks in portfolio Diversifiable (firm-specific)risk Nondiversifiable (Market) risk 49.2 23.9 19.2 1102030401000

44. 6-44 If you hold a one-stock portfolio and thus are exposed to more risk than diversified investors, would you be compensated for all the risk that you bear?

45. 6-45 No! If you hold only one stock, you will not be compensated for the additional risk you bear. No! If you hold only one stock, you will not be compensated for the additional risk you bear. Stand-alone risk is not as important to a well- diversified investor, and most of it can be eliminated at virtually no cost through diversification. Stand-alone risk is not as important to a well- diversified investor, and most of it can be eliminated at virtually no cost through diversification. Rational risk averse investors are concerned with  P, which is based on market risk. Rational risk averse investors are concerned with  P, which is based on market risk.

46. 6-46 The Systematic Risk Principal The reward for bearing risk depends only upon systematic risk since unsystematic risk can be diversified away. The reward for bearing risk depends only upon systematic risk since unsystematic risk can be diversified away. This implies that the expected return on any asset depends only on that asset's systematic risk This implies that the expected return on any asset depends only on that asset's systematic risk Hence, the discount rate will depend only on the systematic risk of the project Hence, the discount rate will depend only on the systematic risk of the project

47. 6-47 Beta (  ) measures a stock’s market (or systematic) risk. It shows the relative volatility of a given stock compared to the average stock. An average stock (or the market portfolio) has a beta = 1.0. Beta (  ) measures a stock’s market (or systematic) risk. It shows the relative volatility of a given stock compared to the average stock. An average stock (or the market portfolio) has a beta = 1.0. Beta shows how risky a stock is if the stock is held in a well-diversified portfolio. Beta shows how risky a stock is if the stock is held in a well-diversified portfolio. Measuring Systematic Risk

48. 6-48 Beta Coefficients for Selected Companies (Table 13.8) Beta Company Coefficient (  i ) Beta Company Coefficient (  i ) Exxon0.65 AT&T0.90 IBM0.95 Wal-Mart1.10 General Motors1.15 Microsoft1.30 Harley-Davidson1.65 America Online2.40 Source: From Value Line Investment Survey, April 19, 1996.

49. 6-49 Portfolio Betas Portfolio Betas: While portfolio variance is not equal to a simple weighed sum of individual security variances, portfolio betas are equal to the weighed sum of individual security betas. Portfolio Betas: While portfolio variance is not equal to a simple weighed sum of individual security variances, portfolio betas are equal to the weighed sum of individual security betas. You have $6,000 invested in IBM, $4,000 in GM. The beta of IBM and GM is 0.95 and 1.15 respectively. What is the beta of the portfolio? You have $6,000 invested in IBM, $4,000 in GM. The beta of IBM and GM is 0.95 and 1.15 respectively. What is the beta of the portfolio?

50. 6-50 Run a regression line of past returns on Stock i versus returns on the market. Run a regression line of past returns on Stock i versus returns on the market. The regression line is called the characteristic line. The regression line is called the characteristic line. The slope coefficient of the characteristic line is defined as the beta coefficient. The slope coefficient of the characteristic line is defined as the beta coefficient. How are betas calculated?

51. 6-51 Illustration of beta calculation. YearR M R i 115%18% 2-5-10 31216 RiRi _ RMRM _ - 505101520 20 15 10 5 -5 -10... R i = -2.59 + 1.44 R M ^^ Regression line

52. 6-52 If beta = 1.0, stock is average risk. If beta = 1.0, stock is average risk. If beta > 1.0, stock is riskier than average. If beta > 1.0, stock is riskier than average. If beta < 1.0, stock is less risky than average. If beta < 1.0, stock is less risky than average. Most stocks have betas in the range of 0.5 to 1.5. Most stocks have betas in the range of 0.5 to 1.5.

53. 6-53 Answer: Yes, if the correlation between the market and the stock is negative. Then, in a “beta graph”, the characteristic (regression) line will slope downward. Answer: Yes, if the correlation between the market and the stock is negative. Then, in a “beta graph”, the characteristic (regression) line will slope downward. But, in the real world, we have never seen a negative beta stock. But, in the real world, we have never seen a negative beta stock. Can a beta be negative?

54. 6-54 Beta and Risk Premium A risk free asset has a beta of zero. When a risky asset is combined with a risk free asset, the resulting portfolio expected return is a weighted average of their expected returns and the portfolio beta is the weighted average of their betas. Consider various portfolios comprised of an investment in stock A with a beta (  ) of 1.2 and expected return of 18%, and a Treasury bill with a 7% return. Compute the expected return and beta for different portfolios of stock A and a Treasury bill.

55. 6-55 0*18% + 1*7% = 7% 0*1.2 + 1*0 = 0.25*18% +.75*7% = 9.75 0.25*1.2 + 0.75*0 = 0.3.50*18% +.50*7% = 12.5 0.50*1.2 + 0.50*0 = 0.6.75*18% +.25*7% = 15.25 0.75*1.2 + 0.25*0 = 0.9 1*18% + 0*7% = 18% 1*1.2 + 0*0 = 1.2 1.5*18% + (-.5)*7% = 23.5% 1.5*1.2 + (-.5)*0 = 1.8

56. 6-56 7% 18%

57. 6-57 Reward to Risk Ratio We can vary the amount invested in each type of asset and get an idea of the relation between portfolio expected return and portfolio beta.

58. 6-58 0*20% + 1*7% = 7 0*1.6 + 1*0 = 0 0.25*20% +.75*7% = 10.25 0.25*1.6 + 0.75*0 = 0.4 0.50*20% +.50*7% = 13.5 0.50*1.6 + 0.50*0 = 0.8 0.75*20% +.25*7% = 16.75 0.75*1.6 + 0.25*0 = 1.2 1*20% + 0*7% = 20 1*1.6 + 0*0 = 1.6 1.5*20% + (-.5)*7% = 26.5 1.5*1.6 + (-.5)*0 = 2.4 Consider asset B, with a beta of 1.6 and an expected return of 20%. Compute the expected return and beta for different portfolios of B and a T-bill with a 7% return.

59. 6-59 7% 18% 16.75%

60. 6-60 What happens if two assets have different reward-to-risk ratios? Since systematic risk is all that matters in determining expected return, the reward-to-risk ratio must be the same for all assets and portfolios. If not, investors would only buy the assets (portfolios) that offer a higher reward-to- risk ratio. Since systematic risk is all that matters in determining expected return, the reward-to-risk ratio must be the same for all assets and portfolios. If not, investors would only buy the assets (portfolios) that offer a higher reward-to- risk ratio. Because the reward-to-risk ratio is the same for all assets, it must hold for the risk free asset as well as for the market portfolio. Because the reward-to-risk ratio is the same for all assets, it must hold for the risk free asset as well as for the market portfolio. Result: Result:

61. 6-61 The Security Market Line The security market line is the line which gives the expected return-systematic risk (beta) combinations of assets in a well functioning, active financial market. the reward-to- risk ratio must be the same for all assets in the market. In an active, competitive market in which only systematic risk affects expected return, the reward-to- risk ratio must be the same for all assets in the market. The slope of the SML is the difference between the expected return on the market portfolio and the risk-free rate, or, the market risk premium.

62. 6-62 The Security Market Line (SML) (Figure 13.4) Asset expected return (E (R i )) Asset beta (  i) = E (R M ) – R f E (R M ) RfRf  M = 1.0

63. 6-63 The Capital Asset Pricing Model The Capital Asset Pricing Model (CAPM) - an equilibrium model of the relationship between risk and required return on assets in a diversified portfolio. The Capital Asset Pricing Model (CAPM) - an equilibrium model of the relationship between risk and required return on assets in a diversified portfolio. What determines an asset’s expected return? What determines an asset’s expected return? The risk-free rate - the pure time value of money The risk-free rate - the pure time value of money The market risk premium - the reward for bearing systematic risk The market risk premium - the reward for bearing systematic risk The beta coefficient - a measure of the amount of systematic risk present in a particular asset The beta coefficient - a measure of the amount of systematic risk present in a particular asset The CAPM: E(R i ) = R f + [E(R M ) - R f ] x  i

64. 6-64 Example of CAPM Suppose a stock has 1.5 times the systematic risk as the market portfolio (average asset). If the risk-free rate as measured by the Treasury bill rate is 5% and the expected risk premium on the market portfolio is 8%, what is the stock's expected return according to the CAPM? Suppose a stock has 1.5 times the systematic risk as the market portfolio (average asset). If the risk-free rate as measured by the Treasury bill rate is 5% and the expected risk premium on the market portfolio is 8%, what is the stock's expected return according to the CAPM? E(R) = 0.05 + 1.5  0.08 = 0.17 = 17%

65. 6-65 Summary of Risk and Return I.Total risk - the variance (or the standard deviation) of an asset’s return. II.Total return - the expected return + the unexpected return. III.Systematic and unsystematic risks Systematic risks are unanticipated events that affect almost all assets to some degree. Unsystematic risks are unanticipated events that affect single assets or small groups of assets. IV.The effect of diversification - the elimination of unsystematic risk via the combination of assets into a portfolio. V.The systematic risk principle and beta - the reward for bearing risk depends only on its level of systematic risk. VI.The reward-to-risk ratio - the ratio of an asset’s risk premium to its beta. VII. The capital asset pricing model: E(R i ) = R f + [E(R M ) - R f ]  i.

66. 6-66 Cost of Capital Cost of Equity (r E ) Cost of Equity (r E ) CAPM: E(R i ) = R f +  i × [E(R m ) – R f ] CAPM: E(R i ) = R f +  i × [E(R m ) – R f ] E(R i ) = r E if firm is 100% equity financed E(R i ) = r E if firm is 100% equity financed  i =  I,M / (  M ) 2  i =  I,M / (  M ) 2 Financial Leverage Financial Leverage  A = D/A ×  D + E/A×  E  A = D/A ×  D + E/A×  E  E = (1 + D/E) ×  A if  D = 0  E = (1 + D/E) ×  A if  D = 0  E = (1 + (1 – T c ) × D/E) ×  A if  D = 0  E = (1 + (1 – T c ) × D/E) ×  A if  D = 0 T c = corporate tax rate T c = corporate tax rate

67. 6-67 Weighted Average Cost of Capital r WACC = D/A × (1–T c ) × r D + E/A × r E D=market value of debt E=market value of equity A=market value of assets T c =corporate tax rate r D =expected return on debt r E =expected return on equity Weights based on expected or target values prevailing over the life of the project