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Homework answers / question archive / 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 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 0.16 0.06 -0.011 0.06 0.09 -0.02251 -0.01 -0.0225 0.0625 For the assets solve the following problems: Find the minimum variance portfolio. For the minimum variance portfolio find the following values: Expected portfolio return: ]%. (write the return percentage as decimal number) Portfolio variance (write it as decimal number with 4 digits after 0 (as 0.xyzt)) weight of asset A (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345)) weight of asset B: (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345)) weight of asset C (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345)) Now suppose that you want an expected return of 25%. Solve the Markowitz problem and find the following variables: Portfolio variance: (write it as decimal number with 4 digits after 0 (as 0.xyzt)) weight of asset A (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345)) weight of asset B: (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345) weight of asset (write it as decimal number with 2 digits after 0 (as 0.11 for a weight of 0.1074345))
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
