1. Optimal portfolios and index model

2.  Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio? ◦ Uncertainty at the market level ◦ Uncertainty at the firm level  Market risk ◦ Systematic or Nondiversifiable  Firm-specific risk ◦ Diversifiable or nonsystematic  If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk  If it is diversified, the total risk has only market risk

6.  Why the std (total risk) decreases when more stocks are added to the portfolio?  The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them  Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond state Prob.stock (%)Bond (%)Portfolio Recession 0.3-1116 Normal 0.4136 Boom 0.327-4

7.  What is the E(r s ) and σ s ?  What is the E(r b ) and σ b ?  What is the E(r p ) and σ p ?  E(r) σ Bond67.75 Stock1014.92 Portfolio8.45.92

8.  When combining the stocks into the portfolio, you get the average return but the std is less than the average of the std of the 2 stocks in the portfolio  Why?  The risk of a portfolio also depends on the correlation between 2 stocks  How to measure the correlation between the 2 stocks  Covariance and correlation

9.  Prob r s E( r s ) r b E( r b ) P( r s - E( r s ))( r b - E( r b )) 0.3-1110 16 6 -63 0.41310 6 6 0 0.32710 -4 6 -51 -114  Cov ( r s, r b ) = -114  The covariance tells the direction of the relationship between the 2 assets, but it does not tell the whether the relationship is weak or strong  Corr( r s, r b ) = Cov ( r s, r b )/ σ s σ b = -114/(14.92*7.75) = -0.99

10. Covariance and Correlation  Portfolio risk depends on the correlation between the returns of the assets in the portfolio  Covariance and the correlation coefficient provide a measure of the way returns two assets vary

11. Two-Security Portfolio: Return

13. Two-Security Portfolio: Risk Continued  Another way to express variance of the portfolio:

14.  D,E = Correlation coefficient of returns Cov(r D, r E ) =  DE  D  E  D = Standard deviation of returns for Security D  E = Standard deviation of returns for Security E Covariance

15. Range of values for  1,2 + 1.0 >  >-1.0 If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Correlation Coefficients: Possible Values

16.  2 p = w 1 2  1 2 + w 2 2  2 2 + 2w 1 w 2 Cov(r 1, r 2 ) + w 3 2  3 2 Cov(r 1, r 3 ) + 2w 1 w 3 Cov(r 2, r 3 )+ 2w 2 w 3 Three-Security Portfolio

17. Table 7.1 Descriptive Statistics for Two Mutual Funds

18. Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

19. Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

20. Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

21. Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

22. Minimum Variance Portfolio as Depicted in Figure 7.4  Standard deviation is smaller than that of either of the individual component assets  Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

23. Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

24.  The relationship depends on the correlation coefficient  -1.0 <  < +1.0  The smaller the correlation, the greater the risk reduction potential  If  = +1.0, no risk reduction is possible Correlation Effects

25. Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

26. The Sharpe Ratio  Maximize the slope of the CAL for any possible portfolio, p  The objective function is the slope:

27. 27 The solution of the optimal portfolio is as follows

28. Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

29. Figure 7.8 Determination of the Optimal Overall Portfolio

30. Figure 7.9 The Proportions of the Optimal Overall Portfolio An investor with risk-aversion coefficient A = 4 would take a position in a portfolio P The investor will invest 74.39% of wealth in portfolio P, 25.61% in T- bill. Portfolio P consists of 40% in bonds and 60% in stock, therefore, the percentage of wealth in stock =0.7349*0.6=44.63%, in bond = 0.7349*0.4=29.76%

31. Markowitz Portfolio Selection Model  Security Selection ◦ First step is to determine the risk- return opportunities available ◦ All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk- return combinations

32. Figure 7.10 The Minimum-Variance Frontier of Risky Assets

33. Markowitz Portfolio Selection Model Continued  We now search for the CAL with the highest reward-to-variability ratio

34. Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

35. Markowitz Portfolio Selection Model Continued  Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

36. Figure 7.12 The Efficient Portfolio Set

37. Capital Allocation and the Separation Property  The separation property tells us that the portfolio choice problem may be separated into two independent tasks ◦ Determination of the optimal risky portfolio is purely technical ◦ Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference

38. Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

39. The Power of Diversification  Remember:  If we define the average variance and average covariance of the securities as:  We can then express portfolio variance as:

40. Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

41.  The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990.  However, the approach involved too many inputs, calculations ◦ If a portfolio includes only 2 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? ◦ If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need? ◦ If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need?  n variances  n(n-1)/2 covariances

43. Single-Index Model Continued  Risk and covariance: ◦ Total risk = Systematic risk + Firm-specific risk: ◦ Covariance = product of betas x market index risk: ◦ Correlation = product of correlations with the market index

44. Index Model and Diversification  Portfolio’s variance:  Variance of the equally weighted portfolio of firm-specific components:  When n gets large, becomes negligible

45. Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient β p in the Single-Factor Economy

46. When we diversify, all the specific risk will go away, the only risk left is systematic risk component Now, all we need is to estimate beta1, beta2,...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before

47.  Run a linear regression according to the index model, the slope is the beta  For simplicity, we assume beta is the measure for market risk  Beta = 0  Beta = 1  Beta > 1  Beta < 1

48. Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006

49. Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP

50. Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard

51. Figure 8.4 Excess Returns on Portfolio Assets

52. Figure 8.5 Efficient Frontiers with the Index Model and Full-Covariance Matrix

53. Table 8.2 Comparison of Portfolios from the Single-Index and Full-Covariance Models

54.  Reduces the number of inputs for diversification  Easier for security analysts to specialize