CHAPTER 7

Optimal Risky Portfolios

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

Two-Security Portfolio: Return

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED E

P

D

D

E

E

r

r

w

r

w

r

w wr r

( ) ( ) ( )p D D E EE r w E r w E r

= Variance of Security D

= Variance of Security E

= Covariance of returns for Security D and Security E

Two-Security Portfolio: Risk

2 2 2 2 2 2 ( , )P D D E E D E D Ew w w Cov r r

2D

2E

( , )D ECov r r

Two-Security Portfolio: Risk Continued

• Another way to express variance of the portfolio:

2 ( , ) ( , ) 2 ( , )P D D D D E E E E D E D Ew w Cov r r w w Cov r r w w Cov r r

D,E = Correlation coefficient of returns

Cov(rD,rE) = DEDE

D = Standard deviation of returns for Security DE = Standard deviation of returns for Security E

Covariance

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

Table 7.1 Descriptive Statistics for Two Mutual Funds

2p = w1

212 + w2

212

+ 2w1w2 Cov(r1,r2)

+ w323

2

Cov(r1,r3)+ 2w1w3

Cov(r2,r3)+ 2w2w3

Three-Security Portfolio

1 1 2 2 3 3( ) ( ) ( ) ( )pE r w E r w E r w E r

Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

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

Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

• 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

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

The Sharpe Ratio

• Maximize the slope of the CAL for any possible portfolio, p

• The objective function is the slope:

( )P fP

P

E r rS

Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the

Optimal Risky Portfolio

Figure 7.8 Determination of the Optimal Overall Portfolio

Figure 7.9 The Proportions of the Optimal Overall Portfolio

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

Figure 7.10 The Minimum-Variance Frontier of Risky Assets

Markowitz Portfolio Selection Model Continued

• We now search for the CAL with the highest reward-to-variability ratio

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

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

2

1 1

( , )n n

P i j i ji j

ww Cov r r

Figure 7.12 The Efficient Portfolio Set

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

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

Diversification and Portfolio Risk

• Market risk– Systematic or nondiversifiable

• Firm-specific risk– Diversifiable or nonsystematic

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:

2

1 1

( , )n n

P i j i ji j

ww Cov r r

2 21 1P

nCov

n n

2 2

1

1 1

1

1( , )

( 1)

n

ii

n n

i jj ij i

n

Cov Cov r rn n

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

Universes

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