In structor s Ma nual Chapter 12 Page 146

5. Suppose that you have the opportunity to buy stock in AT&T and Microsoft.

a. What is the minimum risk (variance) portfolio of AT&T and Microsoft if the correlation between the two stocks is 0? .5? 1? -1?

What do you notice about the change in the allocations between AT&T and Microsoft as their correlation moves from -1 to 0? to .5? to +1? Why might this be?

b. What is the variance of each of the minimum-variance portfolios in part a?

c. What is the optimal combination of these two securities in a portfolio for each value of the correlation, assuming the

existence of a money market fund that currently pays 4.5% (.045)? Do you notice any relation between these weights and the weights for the minimum variance portfolios?

d. What is the variance of each of the optimal portfolios?

e. What is the expected return of each of the optimal portfolios?

f. Derive the risk-reward trade-off line for the optimal portfolio when the correlation is .5. How much extra expected return can

you anticipate if you take on an extra unit of risk?

SOLUTION:

a. Minimum risk portfolios if correlation is:

-1: 62.5% AT&T, 37.5% Microsoft

0: 73.5% AT&T, 26.5% Microsoft

.5: 92.1% AT&T, 7.9% Microsoft

1: 250% AT&T, short sell 150% Microsoft

As the correlati on moves from -1 to +1, the allocati on to AT&T in creases. When two stocks have n egative correlati on, sta ndard deviati on can be reduced dramatically by mixing them in a portfolio. It is to the in vestors ben efit to weight more heavily the stock with the higher expected return since this will produce a high portfolio expected return while the standard deviation of the portfolio is decreased. This is why the highest allocation to Microsoft is observed for a correlation of -1, and the allocation to Microsoft decreases as the correlation becomes positive and moves to +1. With correlation of +1, the returns of the two stocks will move closely together, so you want to weight most heavily the stock with the lower in dividual sta ndard deviati on.

b. Varia nces of each of the mini mum varia nee portfolios: 62.5% AT&T, 37.5% Microsoft Var 0

73.5% AT&T, 26.5% Microsoft Var .016

5

92.1% AT&T, 7.9% Microsoft Var .0222 250% AT&T, short 150% Microsoft Var 0

c. Optimal portfolios if correlati on is:

-1: 62.5% AT&T, 37.5% Microsoft

0: 48.1% AT&T, 51.9% Microsoft

.5: 11.4% AT&T, 88.6% Microsoft 1:

250% AT&T, short 150% Microsoft d. Varia nces of the optimal portfolios:

62.5% AT&T, 37.5% Microsoft Var = 0

48.1% AT&T, 51.9% Microsoft Var = .0220

11.4% AT&T, 88.6% Microsoft Var = .0531

250% AT&T, short 150% Microsoft Var = 0

e. Expected returns of the optimal portfolios:

62.5% AT&T, 37.5% Microsoft E[r]= 14.13%

48.1% AT&T, 51.9% Microsoft E[r]= 15.71%

11.4% AT&T, 88.6% Microsoft E[r]= 19.75%

250% AT&T, short 150% Microsoft E[r]= -6.5%

f. Risk-reward trade-off li ne for optimal portfolio with correlati on = .5: E[r] = .045 + .66 二