Question

Assume a portfolio of assets is made up of two securities, security A and security B. An amount of investment of GHS12,000 is made in security A and GHS8000 is made in security B. The return for security A is 30% and the return for security B is 20%. The standard deviation for security A is 15% whilst the standard deviation of security B is 10%. The correlation of return of security A and return of security B is given as 0.70.

Required:

i. Calculate the return on this portfolio 2 marks

ii. Calculate the weighted standard deviation of this portfolio 2 marks

iii. Calculate the standard deviation for this portfolio as proposed by Markowitz and

explain your observation. 5 marks

iv. Indicate what will happen to the standard deviation of this portfolio if the correlation

between these two assets is 0.40.

v. Indicate what will happen to this portfolio if the correlation is 1.

vi. Indicate what will happen if the correlation coefficient is -1.

Answer 1

Weight of A = investment in A/total investment = 12,000/(12,000+8,000) = 0.6

Weight of B = 1 - weight of A = 1-0.6 = 0.4

i). Portfolio return = sum of weighted returns = (0.6*30%) + (0.4*20%) = 26.00%

ii). Variance = Sum of weight*(return -expected return)^2/[((N-1)/N)*sum of weights] where N = number of non-zero weights = 2 (in this case)

= [0.6*(30%-26%)^2 + 0.4*(20%-26%)^2]/[((2-1)/2)*1] = 0.0048

Portfolio weighted standard deviation = 0.0048^0.5 = 6.93%

iii). Portfolio variance = [(wA*SDA)^2 + (wB*SDB)^2 + (2*wA*wB*SDA*SDB*Correlation)]

= [(0.6*15%)^2 + (0.4*10%)^2 + (2*0.6*0.4*15%*10%*0.7)] = 0.01474

Portfolio standard deviation = variance^0.5 = 0.01474^0.5 = 12.14&

iv). If correlation is 0.40 then variance becomes = [(0.6*15%)^2 + (0.4*10%)^2 + (2*0.6*0.4*15%*10%*0.4)] = 0.01258

Portfolio standard deviation = 0.01258^0.5 = 11.22%

v). If correlation is 1 then variance becomes = [(0.6*15%)^2 + (0.4*10%)^2 + (2*0.6*0.4*15%*10%*1)] = 0.0169

Portfolio standard deviation = 0.0169^0.5 = 13.00%

vi). If correlation is -1 then variance becomes = [(0.6*15%)^2 + (0.4*10%)^2 + (2*0.6*0.4*15%*10%*-1)] = 0.0025

Portfolio standard deviation = 0.0025^0.5 = 5.00%