The expected returns for three different assets are given as: Asset C Probability Asset A Asset B ET 3090 2096 1596 and the covariance matrix is given as 0

The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

 is the standard deviation of returns of security i.

and  is the correlation coefficient beltween returns of security i and security j

So for the given problem

THe minimum variance problem can be put as a linerar optimisation problem '

If wA,wB and wC are the weights of assets A, B and C , then the problem reduces to

Minimise :  

wA^2*0.16+wB^2*0.09+wC^2*0.0625+2*wA*wB*0.06+2*wA*wC*(-0.01)+2*wB*wC*(-0.0225)

subject to

wA,wB,wC >0

wA+wB+wC = 1

Solving the same using EXCEL,

wA =0.0479, wB =0.3928 and wC =0.5593

Expected Portfolio Return (Weighted average) = 0.0479*30%+0.3928*20%+0.5593*15% = 17.68%

Portfolio variance

=wA^2*0.16+wB^2*0.09+wC^2*0.0625+2*wA*wB*0.06+2*wA*wC*(-0.01)+2*wB*wC*(-0.0225)

=0.0256 (putting the values of wA,wB and wC as found by solving optimisation problem)

Weight of Asset A , wA = 0.05

Weight of Asset B , wB = 0.39

Weight of Asset C , wC = 0.56

If required return is 25%

the optimisation problem may be solved with an added constraint of

wA*0.3+wB*0.20+wC*0.15 = 0.25

Solving the same using EXCEL,

wA =0.6353, wB =0.0940 and wC =0.2706

Portfolio variance

=wA^2*0.16+wB^2*0.09+wC^2*0.0625+2*wA*wB*0.06+2*wA*wC*(-0.01)+2*wB*wC*(-0.0225)

=0.0725 (putting the values of wA,wB and wC as found by solving optimisation problem)

Weight of Asset A , wA = 0.64

Weight of Asset B , wB = 0.09

Weight of Asset C , wC = 0.27