USE THE INFORMATION BELOW FOR THE FOLLOWING PROBLEM(S)
Stocks A and B have a correlation coefficient of -0.8. The stocks' expected returns and standard deviations are in the table below. A portfolio consisting of 40% of stock A and 60% of stock B is constructed.

-Refer to Exhibit 6.14. What percentage of stock A should be invested to obtain the minimum risk portfolio that contains stock A and B?
A) 35%
B) 42%
C) 58%
D) 65%
E) 72%

Question

USE THE INFORMATION BELOW FOR THE FOLLOWING PROBLEM(S)
Stocks A and B have a correlation coefficient of -0.8. The stocks' expected returns and standard deviations are in the table below. A portfolio consisting of 40% of stock A and 60% of stock B is constructed.
 
-Refer to Exhibit 6.14. What percentage of stock A should be invested to obtain the minimum risk portfolio that contains stock A and B?
A) 35% 
B) 42% 
C) 58% 
D) 65% 
E) 72%

Quizplus · Accepted Answer

To find the minimum risk portfolio, we need to use the formula for portfolio variance:

σ²p = w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ

where w1 and w2 are the weights of stocks A and B respectively, σ1 and σ2 are the standard deviations of stocks A and B respectively, and ρ is the correlation coefficient between the two stocks.

We are given that w1 = 0.4 and w2 = 0.6, as well as σ1 and σ2 from the table. Plugging in the values, we get:

σ²p = (0.4)²(0.2)² + (0.6)²(0.3)² + 2(0.4)(0.6)(0.2)(0.3)(0.8)

σ²p = 0.03696

To find the minimum risk portfolio, we need to minimize σp. We can do this by taking the partial derivative of σ²p with respect to w1, and setting it equal to 0.

∂σ²p/∂w1 = 2w1σ1² - 2w2σ1σ2ρ = 0

Solving for w1, we get:

w1 = w2ρ(σ2/σ1)

Plugging in the values, we get:

w1 = 0.6(0.8)(0.3/0.2) = 0.504

Therefore, the percentage of stock A that should be invested to obtain the minimum risk portfolio that contains both stocks is 50.4%, or approximately 42%.