Chapter 11

1. Chapter 11 Diversification and Risky Asset Allocation

2. Learning Objectives 1. How to calculate expected returns and variances for a security. 2. How to calculate expected returns and variances for a portfolio. 3. The importance of portfolio diversification. 4. The efficient frontier and the importance of asset allocation.

3. Expected Return of an Asset Expected return = the “weighted average” return on a risky asset expected in the future. Prs = probability of a state Rs = return if a state occurs s = number of states

4. Expected Return & Risk Premium Risk-free rate = 8% Risk Premium = 15% - 8% = 7%

5. Measuring Variance and Standard Deviation

6. Calculating Dispersion of Returns

7. Calculating Dispersion of Returns

8. Portfolios • Portfolios = groups of assets, such as stocks and bonds, that are held by an investor. • Portfolio Description = list the proportion of the total value of the portfolio that is invested into each asset. • Portfolio Weights = proportions • Sometimes expressed in percentages. • In calculations, make sure you use proportions (i.e., decimals).

9. Portfolio Return The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio proportions as weights. xi = Proportion of funds in Security i E(Ri) = Expected return on Security i

10. Portfolio Return 7.0% = .60(-5%) + .40(25%)

11. Portfolio Return Alternate Method – Step 1

12. Portfolio Return Alternate Method – Step 2 Assume 60% in Stock A; 40% in Stock B

13. Portfolio Variance & Standard Deviation

14. Portfolio Variance & Standard Deviation

15. Risk-Return Comparison

16. Stocks A & B vs. States

17. Portfolio Returns: 60% A – 40% B

18. Diversification and Risk

19. Diversification and Risk

20. Why Diversification Works • Correlation = The tendency of the returns on two assets to move together. • Positively correlated assets tend to move up and down together. • Negatively correlated assets tend to move in opposite directions. • Imperfect correlation, positive or negative, is why diversification reduces portfolio risk.

21. Correlation Coefficient Correlation Coefficient = ρ (rho) Scales covariance to [-1,+1] r = -1.0  Two stocks can be combined to form a riskless portfolio r = +1.0No risk reduction at all In general, stocks have r≈ 0.35 - 0.67 Risk is lowered but not eliminated Will Not Be on a Test

22. Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0) Stock W Stock M Portfolio WM 25 15 0 0 0 -10 -10 25 25 15 15 -10

23. Returns distribution for two perfectly positively correlated stocks (ρ = 1.0) Stock M’ Portfolio MM’ Stock M 25 25 25 15 15 15 0 0 0 -10 -10 -10

24. Why Diversification Works ρ = +1 ρ = -1 ρ = 0

25. Why Diversification Works

26. Why Diversification Works

27. Covariance of Returns Measures how much the returns on two risky assets move together Will Not Be on a Test

28. Covariance vs. Variance of Returns Will Not Be on a Test

29. Covariance Deviation = RA,S – E(RA) Covariance (A:B) = -0.009 Will Not Be on a Test

30. Covariance Will Not Be on a Test

31. Correlation Coefficient Will Not Be on a Test

32. Calculating Portfolio Risk • For a portfolio of two assets, A and B, the variance of the return on the portfolio is: Where: xA = portfolio weight of asset A xB = portfolio weight of asset B such that xA + xB = 1

33. Portfolio Risk Example Continuing our 2-stock example

34. s of n-Stock Portfolio • Subscripts denote stocks i and j • ri,j = Correlation between stocks i and j • σi and σj=Standard deviations of stocks i and j • σij = Covariance of stocks i and j Will Not Be on a Test

35. Portfolio Risk-2 Risky Assets

36. Diversification & the Minimum Variance Portfolio Assume the following statistics for two portfolios, one of stocks and one of bonds:

37. Correlation and Diversification

38. Correlation and Diversification

39. The Minimum Variance Portfolio 100% Stocks MVP 100% Bonds

40. The Minimum Variance Portfolio XA* = 28.8136% Putting 28.8136% in stocks and 71.1864% in bonds yields an E(R) = 7.73% and a standard deviation of 8.69% as the minimum variance portfolio. (Ex 11.7 p.366)

41. Correlation and Diversification • The various combinations of risk and return available all fall on a smooth curve. • This curve is called an investment opportunity set ,because it shows the possible combinations of risk and return available from portfolios of these two assets. • A portfolio that offers the highest return for its level of risk is said to be an efficient portfolio. • The undesirable portfolios are said to be dominated orinefficient.

42. The Markowitz Efficient Frontier • The Markowitz Efficient frontier = the set of portfolios with the maximum return for a given risk AND the minimum risk given a return. • For the plot, the upper left-hand boundary is the Markowitz efficient frontier. • All the other possible combinations are inefficient. That is, investors would not hold these portfolios because they could get either • More return for a given level of risk, or • Less risk for a given level of return.

43. Markowitz Efficient Frontier Efficient Frontier Inefficient Frontier

44. Efficient Frontier

45. The Importance of Asset Allocation • Suppose we invest in three mutual funds: • One that contains Foreign Stocks, F • One that contains U.S. Stocks, S • One that contains U.S. Bonds, B • Figure 11.6 shows the results of calculating various expected returns and portfolio standard deviations with these three assets.

46. Risk and Return with Multiple Assets

47. Useful Internet Sites • www.411stocks.com (to find expected earnings) • www.investopedia.com (for more on risk measures) • www.teachmefinance.com (also contains more on risk measure) • www.morningstar.com (measure diversification using “instant x-ray”) • www.moneychimp.com (review modern portfolio theory) • www.efficentfrontier.com (check out the reading list)