Question

In: Finance

Consider a portfolio with three assets E[rA]=10% E[rB]=12% E[rC]=8%; σA2 =0.008 σB2 =0.010 σC2 =0.005; ρA,B...

Consider a portfolio with three assets E[rA]=10% E[rB]=12% E[rC]=8%; σA2 =0.008 σB2 =0.010 σC2 =0.005; ρA,B =0.2 ρB,C = 0.0 ρA,C = −0.2

a) Consider the portfolio weights xA = 0.3 and xB = 0.3. Calculate the portfolio weight xC , the expected portfolio return, and the variance of the portfolio returns.

b) Consider the portfolio weights xA = 0.3. Calculate the expected portfolio return as a function of xB

c) Consider the portfolio weights xA = 0.3. Calculate the portfolio return variance as a function of xB

d) Calculate the portfolio which has the smallest variance, for which xA = 0.3.

Expert Solution

a)

Total Weight =1

XA+XB+XC=1

0.3+0.3+Xc=1

Xc=1-0.3-0.3=0.4

W

R

W*R

Weight

Return(%)

Weight*Return

StockA

0.3

10

3

StockB

0.3

12

3.6

StockC

0.4

8

3.2

SUM

9.8

Expected PortfolioReturn

9.80%

Expected Portfolio variance:

(XA^2)*(VarianceA)+(XB^2)*(VarianceB)+(XC^2)*(VarianceC)+2XA*XB*CovarianceA,B+2XA*XC*Covaraince,A,C+2XB*XC*Covariane,B,C

XA=0.3, XB=0.3, XC=0.4

Variance A=0.008

Variance B=0.01

Variance C=0.005

Covariance A,B=0.2*SQRT (0.008*0.01)

0.001789

Covariance A,C=-0.2*SQRT (0.008*0.005)

-0.00126

Covariance B,C=

0

Expected Portfolio variance:

(0.3^2)*(0.008)+(0.3^2)*(0.01)+(0.4^2)*(0.005)+2*0.3*0.3*0.001789+2*0.3*0.4*(-0.00126)

Expected Variance of the portfolio Return

0.00244

Expected Standard Deviation of the portfolio

0.049393

(SQRT (0.00244)

b.

XA=0.3

XC=1-XA-XB=1-0.3-XB=0.7-XB

RP=Expected Portfolio Return (Percentage)

XA*10+XB*12+XC*8

0.3*10+XB*12+(0.7-XB)*8

RP=3+12XB+5.6-8XB

RP=4XB+8.6

RP=Expected Portfolio Return (Percentage)

XB=Weight of StockB in the portfolio

VP=Portfolio Variance:

(XA^2)*(VarianceA)+(XB^2)*(VarianceB)+(XC^2)*(VarianceC)+2XA*XB*CovarianceA,B+2XA*XC*Covaraince,A,C+2XB*XC*Covariane,B,C

XA=0.3,XC=0.7-XB

Variance A=0.008

Variance B=0.01

Variance C=0.005

Covariance A,B=0.2*SQRT (0.008*0.01)

0.001789

Covariance A,C=-0.2*SQRT (0.008*0.005)

-0.00126

Covariance B,C=

0

VP=Portfolio Variance:

(0.3^2)*(0.008)+(XB^2)*(0.01)+((0.7-XB)^2)*(0.005)+2*0.3*XB*0.001789+2*0.3*(0.7-XB)*(-0.00126)

Vp=Portfolio Variance=0.00072+0.01*(XB^2)+(0.49+(XB^2)-1.4XB)*0.005+0.001073XB-0.00053-0.00076XB

Vp=Portfolio Variance=0.00072+0.01*(XB^2)+0.00245+(XB^2)*0.005-0.007XB+0.001073XB-0.00053-0.00076XB

Vp=Portfolio Variance=0.015(XB^2)-0.00669XB+0.00264

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