Answer:
Part A
(i) Weight(A) = 0.80 , Weight(B) = 0.20
ER(portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }
= { 10 * 0.80 } + { 20 * 0.20 }
= 12%
SD(portfolio) = { SD(A)^2 * W(A)^2 + SD(B)^2 * W(B)^2 + 2*SD(A) * SD(B) * W(A) * W(B) * CORR }^1/2
= { 900*0.64 + 2500*0.04 + 2*30*50*0.8*0.2*0.15}^1/2
= {748}^1/2
= 27.35%
(ii) Weight(A) = 0.50 , Weight(B) = 0.50
ER(portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }
= { 10 * 0.50 } + { 20 * 0.50 }
= 15%
SD(portfolio) = { SD(A)^2 * W(A)^2 + SD(B)^2 * W(B)^2 + 2*SD(A) * SD(B) * W(A) * W(B) * CORR }^1/2
= { 900*0.25 + 2500*0.25 + 2*30*50*0.5*0.5*0.15}^1/2
= {917.5}^1/2
= 30.29 %
(iii) Weight(A) = 0.20 , Weight(B) = 0.80
ER(portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }
= { 10 * 0.20 } + { 20 * 0.80 }
= 18 %
SD(portfolio) = { SD(A)^2 * W(A)^2 + SD(B)^2 * W(B)^2 + 2*SD(A) * SD(B) * W(A) * W(B) * CORR }^1/2
= { 900*0.04 + 2500*0.64 + 2*30*50*0.2*0.8*0.15}^1/2
= {1708}^1/2
= 41.33 %
Part B
Let Weight(A) be x, and Weight(B) be (1-x)
Solving the ER(portfolio) Equation :
ER(portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }
25 = {10 * x } + {20 * (1 - x) }
25 = 10x + 20 - 20x
25 - 20 = -10x
x = - 0.5
Weight (A) = - 0.5 {its Negative which means Short Selling of Stock A}
Weight (B) = 1 - (-0.5) = 1.5
<u><em>Cross-Proof</em></u>
ER (portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }
= { 10 * -0.5 } + { 20 * 1.5 }
= { - 5 } + { 30 }
= 25%
. Therefore, our Weights are Correct
Calculation of SD (portfolio)
SD(portfolio) = { SD(A)^2 * W(A)^2 + SD(B)^2 * W(B)^2 + 2*SD(A) * SD(B) * W(A) * W(B) * CORR }^1/2
= { 900*0.25 + 2500*2.25 + 2*30*50*-0.5*1.5*0.15}^1/2
= { 225 + 5625 - 337.5 }^1/2
= {5512.5}1/2
= 74.2 %
