Question

There are three assets A, B and C. The returns for shares A, B and C over the next year have expected values 9%, 6%, 5% respectively. The standard deviations for A, B and C are 30%, 25% and 20% respectively. The covariance between these A and B is 0.0375, the covariance between A and C is -0.018, the covariance between B and C is -0.03. Ben currently has invested 40%, 40%, 20% in shares A, B and C respectively.

a) Calculate the expected return on Ben's portfolio. Show full working.

b) Calculate the correlations ?AB,?AC,?BC. Show full working.

c) Calculate the variance and standard deviation of the return on Ben's portfolio. Show full working.

d) Alice also wants to invest in shares A, B and C, but her risk appetite is higher than Ben. Suggest a weight allocation such that Alice's portfolio will have a higher risk than Ben's portfolio, and explain why. You are NOT required to provide calculations.

Answer 1

Part a : Expected return of Portfolio

Stock	Weights	Expected returns	Weight x Expected returns
A	40%	9%	3.6%
B	40%	6%	2.4%
C	20%	5%	1.0%
Sum of (Weight x Expected returns) ---> Portfolio expected return	7.0%

Part b : Correlation of AB, AC, and BC

St. Dev. Of A	30%
St. Dev. Of B	25%
St. Dev. Of C	20%

Covariance of A and B	0.0375
Covariance of A and C	(0.0180)
Covariance of B and C	(0.0300)

Correlation of A and B = Covariance of A and B / (St. Dev. Of A x St. Dev. Of B)

Correlation of A and B = 0.0375/ (30% x 25%)

Correlation of A and B = 0.0375/ (7.5%)

Correlation of A and B = 0.5

Correlation of A and C = Covariance of A and B / (St. Dev. Of A x St. Dev. Of C)

Correlation of A and B = -0.018 / (30% x 20%)

Correlation of A and B = -0.018/ (6%)

Correlation of A and B = -0.30

Correlation of B and C = Covariance of A and B / (St. Dev. Of B x St. Dev. Of C)

Correlation of A and B = -0.03 / (25% x 20%)

Correlation of A and B = -0.03 / (5%)

Correlation of A and B = -0.60

Part C : Variance and standard Deviation of portfolio

Stock	Weight	Standard Deviation	Proportion x Standard Deviation	Square of Weight x Standard Deviation
A	40%	30%	12.00%	1.44%
B	40%	25%	10.00%	1.00%
C	20%	20%	4.00%	0.16%
Step 1 : Sum of (Square of Weight x Standard Deviation)	2.60%
Step 2 : 2 x Weight of stock A x St. Dev. Of stock A x Weight of Stock B x St. Dev. Of stock B x Correlation Coef.of stock A and stock B	1.20%
Step 3 : 2 x Weight of stock A x St. Dev. Of stock A x Weight of Stock C x St. Dev. Of stock C x Correlation Coef.of stock A and stock C	-0.29%
Step 4 : 2 x Weight of stock B x St. Dev. Of stock B x Weight of Stock C x St. Dev. Of stock C x Correlation Coef.of stock B and stock C	-0.48%
Step 5 : Step 1+Step 2+Step 3+Step4 ---> Variance of Portfolio	3.03%
Step 6 : Square root of Step 5 ---> Standard Deviation of portfolio	17.41%

Part d : In case Alice prefers higher risk portfolio, weights assigned to A and B can be increased as they carry higher Standard Deviation implying higher risk.

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