Markowitz Model

For 3-stock portfolio, short selling allowed

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Eg. RA= 20% RB= 10% RC= 8% 2

A=100 2B=25 2

C=16 AB= 15 AC= 20 BC= 4

Markowitz Model

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Markowitz Model: Solution

For R* = 15%wA = 50.214%wB = 48.715%wC = 1.075%

Standard Dev = 6.22%

For R* = 20%wA = 91%wB = 56%wC = -47% (short sale)

Standard Dev = 9.47%

Single Index ModelRt= A + BRmt + et

E(Rt) = A + BE(Rmt)

Cov(ei,ek)=0E(ei)=0Cov(ei,Rm)=0

2p=Pi[Ri - E(R)]2

by substitution2=Pi[A + BRmt +ei -A - BE(Rm)]2

2=Pi{B[Rmi-E(Rm)] + ei}2

2=Pi{B2[Rmi-E(Rm)]2 +ei2+2B[Rmi-E(Rm)] ei}

2p= B2

p 2m + 2

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Single Index Model (Cont’d)

2p= B2

p2m + 2

ep

We also know that

Bp= Cov(Rp, Rm)/ 2m

Bp=wjBj2

ep= wj2 2

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Hence, the portfolio variance is:

2p= (xjBj )22

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Single Index Model (Cont’d)

For 3-stock portfolio, short selling allowed

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Eg. RA= 20% RB= 10% RC= 8% 2

A=20 2B=15 2

C=9 A= 1.5 B= 0.75 C= 0.50

Single Index Model (Cont’d)

Form the Lagrangian

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Single Index Model Solution

When R* = 15%

wA = 0.57wB = 0.08wC = 0.35

Standard Deviation =