Markowitz Model

1
Markowitz Model
For 3-stock portfolio, short selling allowed
Eg. RA 20 RB 10 RC 8
?2A100 ?2B25 ?2C16 ?AB 15 ?AC
20 ?BC 4
2
Markowitz Model
Form the Lagrangian
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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
4
Single Index Model
Rt A BRmt et E(Rt) A BE(Rmt) Cov(ei,ek)
0 E(ei)0 Cov(ei,Rm)0 ?2p?PiRi - E(R)2 by
substitution ?2?PiA BRmt ei -A -
BE(Rm)2 ?2?PiBRmi-E(Rm) ei2 ?2?PiB2Rmi-
E(Rm)2 ei22BRmi-E(Rm) ei ?2p B2p ?2m
?2e
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Single Index Model (Contd)
?2p B2p?2m ?2ep We also know that Bp
Cov(Rp, Rm)/ ?2m Bp?wjBj ?2ep ?wj2
?2ej Hence, the portfolio variance is ?2p
(?xjBj )2?2m ?xj2 ?2ej
6
Single Index Model (Contd)
For 3-stock portfolio, short selling allowed
Eg. RA 20 RB 10 RC 8
?2A?20 ?2B?15 ?2C?9 ?A 1.5 ?B
0.75 ?C 0.50
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Single Index Model (Contd)
Form the Lagrangian
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Single Index Model Solution
When R 15 wA 0.57 wB 0.08 wC
0.35 Standard Deviation