Question

Stock A has an expected return of 16% and a standard deviation of 30%. Stock B has an expected return of 14% and a standard deviation of 13%. The risk-free rate is 4.7% and the correlation between Stock A and Stock B is 0.9. Build the optimal risky portfolio of Stock A and Stock B. What is the expected return on this portfolio?

Answer 1

Minimum Variance Portfolio or Optimal Risky Portfolio:

A minimum variance portfolio is a collection of securities that combine to minimize the price volatility of the overall portfolio. with the given weights to securities/ Assets in portfolio, portfolio risk will be minimal.

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]

Particulars	Amount
SD of A	30.0%
SD of B	13.0%
r(A,B)	0.9000

Weight in A = [ [ (SD of B)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.13)^2 ] - [ 0.3 * 0.13 * 0.9 ] ] / [ [ (0.3)^2 ] + [ ( 0.13 )^2 ] - [ 2 * 0.3 * 0.13 * 0.9 ] ]
= [ [ 0.0169 ] - [ 0.0351 ] ] / [ [ 0.09 ] + [ 0.0169 ] - [ 2 * 0.0351 ] ]
= [ -0.0182 ] / [ 0.0367 ]
= -0.495913

Weight in B = [ [ (SD of A)^2] - [ SD of A * SD of B * r(A,B) ] ] / [ [ (SD of A)^2 ]+ [ (SD of B)^2 ] - [ 2* SD of A * SD of B * r (A, B) ] ]
= [ [ (0.3)^2 ] - [ 0.3 * 0.13 * 0.9 ] ] / [ [ (0.3)^2 ] + [ ( 0.13 )^2 ] - [ 2 * 0.3 * 0.13 * 0.9 ] ]
= [ [ 0.09 ] - [ 0.0351 ] ] / [ [ 0.09 ] + [ 0.0169 ] - [ 2 * 0.0351 ] ]
= [ 0.0549 ] / [ 0.0367 ]
= 1.495913

Portfolio ret:

Portfolio Return is the weighted avg return of securities in that portfolio

Stock	Weight	Ret	WTd Ret
A	-0.4959	16.00%	-7.93%
B	1.4959	14.00%	20.94%
Portfolio Ret Return			13.01%