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Hi starting a new thread for discussing various important concepts related to the exam,

C1. Betai= Cov(Ri,Rm)/stdDev(m)^2= correlation*stdDev(i)*stdDev(m)/stdDev(m)^2 =correlation*stdDev(i)/stdDev(m)

stdDev of portfolio= Beta(p)*stdDev(m) ; m stands for market

how to get the above formula see the below derivation-----

We know that CML are most efficient portfolios with maximum sharpe ratio and thus for CAL portfolio to be most efficient,

Sharpe ratio of CAL= Sharpe ratio of CML

E(Rp)-Rf/stdDev(p)= E(Rm)-Rf/stdDev(m)

E(Rp)-Rf=stdDev(p)*[E(Rm)-Rf/stdDev(m)]

E(Rp)=Rf+[stdDev(p)/stdDev(m)]*[E(Rm)-Rf]

compare it with CAPM E(Rp)=Rf+Beta(p)*[E(Rm)-Rf]

we get Beta(p)=[stdDev(p)/stdDev(m)] =>

C2. stdDev(p) of two Assets A and B

stdDev(p)= sqrt[wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)]

for minimum variance portfolio,

wA=[stdDev(B)^2-corr(A,B)*stdDev(A)*stdDev(B)]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B)]

for above formula see the listed below derivation;

wA+wB=1 =>wB=1-wA

stdDev(p)^2=wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)

substitute wB for 1-wA =>

stdDev(p)^2= wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B)

for portfolio variance stdDev(p)^2 to be minimum,

first derivative of portfolio variance should be 0,

d(stdDev(p)^2)/dwA=0

=>d/dwA(wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B))=0

=>2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0

=>wA*stdDev(A)^2+(wA)*stdDev(B)^2-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0 cancelling out 2 from both sides

=>wA*[stdDev(A)^2+stdDev(B)^2]-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0

=>wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]-stdDev(B)^2+Corr(A,B)*stdDev(A)*stdDev(B))=0

taking wA common

wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]=stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))

wA=[stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]

to be certain that the portfolio variance is minimal,

second derivative of variance should be greater than zero,

d^2(stdDev(p)^2)/dwA^2=d/dwA[2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))]

=2*stdDev(A)^2+2*stdDev(B)^2+2*Corr(A,B)*(-2)*stdDev(A)*stdDev(B))

=2[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))

=2[stdDev(A)^2+stdDev(B)^2-2*stdDev(A)*stdDev(B)+2*stdDev(A)*stdDev(B)-2*Corr(A,B)*stdDev(A)*stdDev(B))]

=2[(stdDev(A)-stdDev(B))^2+2*stdDev(A)*stdDev(B)(1-Corr(A,B))]>=0

as Corr(A,B)<=1

thus proved that variance is minimum for this value of

C1. Betai= Cov(Ri,Rm)/stdDev(m)^2= correlation*stdDev(i)*stdDev(m)/stdDev(m)^2 =correlation*stdDev(i)/stdDev(m)

stdDev of portfolio= Beta(p)*stdDev(m) ; m stands for market

how to get the above formula see the below derivation-----

We know that CML are most efficient portfolios with maximum sharpe ratio and thus for CAL portfolio to be most efficient,

Sharpe ratio of CAL= Sharpe ratio of CML

E(Rp)-Rf/stdDev(p)= E(Rm)-Rf/stdDev(m)

E(Rp)-Rf=stdDev(p)*[E(Rm)-Rf/stdDev(m)]

E(Rp)=Rf+[stdDev(p)/stdDev(m)]*[E(Rm)-Rf]

compare it with CAPM E(Rp)=Rf+Beta(p)*[E(Rm)-Rf]

we get Beta(p)=[stdDev(p)/stdDev(m)] =>

**stdDev(p)=Beta(p)*stdDev(m)**C2. stdDev(p) of two Assets A and B

stdDev(p)= sqrt[wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)]

for minimum variance portfolio,

wA=[stdDev(B)^2-corr(A,B)*stdDev(A)*stdDev(B)]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B)]

for above formula see the listed below derivation;

wA+wB=1 =>wB=1-wA

stdDev(p)^2=wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)

substitute wB for 1-wA =>

stdDev(p)^2= wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B)

for portfolio variance stdDev(p)^2 to be minimum,

first derivative of portfolio variance should be 0,

d(stdDev(p)^2)/dwA=0

=>d/dwA(wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B))=0

=>2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0

=>wA*stdDev(A)^2+(wA)*stdDev(B)^2-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0 cancelling out 2 from both sides

=>wA*[stdDev(A)^2+stdDev(B)^2]-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0

=>wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]-stdDev(B)^2+Corr(A,B)*stdDev(A)*stdDev(B))=0

taking wA common

wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]=stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))

wA=[stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]

to be certain that the portfolio variance is minimal,

second derivative of variance should be greater than zero,

d^2(stdDev(p)^2)/dwA^2=d/dwA[2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))]

=2*stdDev(A)^2+2*stdDev(B)^2+2*Corr(A,B)*(-2)*stdDev(A)*stdDev(B))

=2[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))

=2[stdDev(A)^2+stdDev(B)^2-2*stdDev(A)*stdDev(B)+2*stdDev(A)*stdDev(B)-2*Corr(A,B)*stdDev(A)*stdDev(B))]

=2[(stdDev(A)-stdDev(B))^2+2*stdDev(A)*stdDev(B)(1-Corr(A,B))]>=0

as Corr(A,B)<=1

thus proved that variance is minimum for this value of

**wA=[stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]**
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