Question

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 24% and a standard deviation of return of 31%. Stock B has an expected return of 17% and a standard deviation of return of 26%. The correlation coefficient between the returns of A and B is .5. The risk-free rate of return is 6%. The proportion of the optimal risky portfolio that should be invested in stock B is approximately _________.

a		30.01%
b		38.79%
c		58.76%
d		25.36%

Answer 1

Given the following information,

Stock A

Expected return E(Ra) = 24% = 0.24

Standard deviation σa = 31% = 0.31

Stock B

Expected return E(Rb) = 17% = 0.17

Standard deviation σb = 26% = 0.26

Corr(A,B) = 0.5

and

Cov(A,B) = Corr(A,B)*σa*σb = 0.5*0.31*0.26 = 0.0403

risk free rate Rf = 6% = 0.06

The proportion of the optimal risky portfolio invested in the stock A is given by

Wa = (E(Ra)-rf)*σb^2 - (E(Rb)-rf)*Cov(A,B)/ ((E(Ra)-rf)*σb^2) + ((E(Rb)-rf)*σa^2) - (E(Ra)-rf + E(Rb)-rf)*Cov(A,B))

Wa = (0.24-0.06)*0.26^2 - (0.17-0.06)*0.0403/ ((0.24-0.06)*0.26^2) + ((0.17-0.06)*0.31^2) - (0.24-0.06 + 0.17-0.06)*0.0403)

Wa = (0.18)*0.26^2 - (0.11)*0.0403/ ((0.18)*0.26^2) + ((0.11)*0.31^2) - (0.18 + 0.11)*0.0403)

Wa = (0.18)*0.0676 - (0.11)*0.0403/ ((0.18)*0.0676 ) + ((0.11)*0.0961) - (0.18 + 0.11)*0.0403)

Wa = (0.012168 - 0.004433)/ ((0.012168) + (0.010571) - (0.29)*0.0403)

Wa = 0.007735/ ((0.012168) + (0.010571) - 0.011687)

Wa = 0.007735/ 0.011052

Wa = 0.6999 or 69.99%

and

Wb = 1- Wa

= 1 - 0.6999

= 0.3001 or 30.01%

Therefore the proportion of the optimal risky portfolio that should be invested in stock B is approximately is 30.01%

Ans is a.