Question

answer them in typed please.

1. The (annual) expected return and standard deviation of returns for 2 assets are as follows: Asset A Asset B E[r] 10% 20% SD[r] 30% 50% The correlation between the returns is 0.15. a. Calculate the expected returns and standard deviations of the following portfolios: (i) 80% in A, 20% in B (ii) 50% in A, 50% in B (iii) 20% in A, 80% in B

b. Find the weights for a portfolio with an expected return of 25%? What is the standard deviation of this portfolio?

Answer 1

ANSWER

a. part

(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 %

b. part

let Weight(A) be x, and Weight(B) be (1-x)

Now we can Solve this by 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 StockA}

Weight (B) = 1 - (-0.5) = 1.5

PROOF

ER (portfolio) = { ER(A) * Weight(A) } + { ER(B) * Weight(B) }

= { 10 * -0.5 } + { 20 * 1.5 }

= { - 5 } + { 30 }

= 25%

HENCE our Weights are Correct

NOW, lets Calculate 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

= 74.25 %