1. Optimal Risky Portfolios
CHAPTER 7

2. 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

3. Two-Security Portfolio: Return

4. Two-Security Portfolio: Risk
= Variance of Security D
= Variance of Security E
= Covariance of returns for
Security D and Security E

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

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

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

8. Table 7.1 Descriptive Statistics for Two Mutual Funds

9. Three-Security Portfolio
2p = w1212
+ w2212
+ w3232
+ 2w1w2
Cov(r1,r2)
Cov(r1,r3)
+ 2w1w3
+ 2w2w3
Cov(r2,r3)

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

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

12. 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

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

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

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

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

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

18. Figure 7.8 Determination of the Optimal Overall Portfolio

19. Figure 7.9 The Proportions of the Optimal Overall Portfolio

20. 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

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

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

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

24. 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

25. Figure 7.12 The Efficient Portfolio Set

26. 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

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

28. Diversification and Portfolio Risk
Market risk
Systematic or nondiversifiable
Firm-specific risk
Diversifiable or nonsystematic

29. 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:

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

31. Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio