Question

Given the following:

• Stock A Expected return= 0.28, standard deviation = 0.40
• Stock B Expected return= 0.16, standard deviation = 0.25

If stock A and stock B have a positive correlation of 0.48, which portfolio represent the minimum variance portfolio?

1. Weight of Stock A in the minimum variance portfolio: _____
2. Weight of Stock B in the minimum variance portfolio: _____
3. The expected return and standard deviation of this minimum variance portfolio: ______ and ________

show all formulas and calculations please

Answer 1

SOLUTION:

W_A + W_B = 1
W_B = 1 - W_A

Variance of a Portfolio = W_A².σ_A² + W_B².σ_B² + 2.W_A.W_B.σ_A.σ_B.Cor_(A,B)
σ² = W_A².(0.4)². + (1 - W_A)².(0.25)² + 2.W_A.(1 - W_A).(0.4).(0.25).(0.48)
σ² = 0.16W_A² + (1 + W_A² - 2W_A)*0.0625 + 0.096W_A - 0.096W_A²
σ² = 0.16W_A² + 0.0625 + 0.0625W_A² - 0.125W_A + 0.096W_A - 0.096W_A²
σ² = 0.1275W_A² - 0.029W_A + 0.0625
Differentiate w.r.t. W_A
dσ²/dW_A = 0.1275W_A - 0.029
0 + 0.029 = 0.1275W_A
W_A = 0.029/0.1275 = 0.2275

1. Weight of Stock A in Minimum Variance Portfolio = 0.2275 or 22.75%

2. Weight of Stock B in Minimum Variance Portfolio = 1 - 0.2275 = 0.7725 or 77.25%

3. Expected Return = (28*0.2275) + (16*0.7725) = 18.73%
Standard Deviation = (0.1275W_A² - 0.029W_A + 0.0625)^1/2 = ((0.1275*0.2275² - 0.029*0.2275 + 0.0625)^1/2 =0.25 or 25%