optimization_lagrange
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Quantitative Method and Lagrange Optimization
Transcript of optimization_lagrange
FIN4814 [1/2012]
Minimum variance portfolio (two assets)
Min 0.0225WA
2+ 0.0289W
B
2+ 0.0255W
AW
B
S.t. WA
+WB
=1
WA
+WB
=1 →1−WA
−WB
= 0
L = 0.0225WA
2+ 0.0289W
B
2+ 0.0255W
AW
B+ λ(1−W
A−W
B)
Take partial derivatives w.r.t. each choice variable and each Lagrange multiplier
∂L
∂WA
= 0.045WA
+ 0.0255WB
− λ = 0
∂L
∂WB
= 0.0578WB
+ 0.0255WA
− λ = 0
∂L
∂λ=1−W
A−W
B= 0
Rearrange in Matrix form
0.045 0.0255 −1
0.0255 0.0578 −1
−1 −1 0
×
WA
WB
λ
=
0
0
−1
WA
WB
λ
=
0.6236
0.3764
0.0377
FIN4814 [1/2012]
Minimum variance portfolio (three assets)
Min 0.0225WA
2+ 0.0289W
B
2+ 0.04W
C
2+ 0.0255W
AW
B+ 0.0204W
BW
C+ 0.024W
AW
C
S.t. WA
+WB
+WC
=1
WA
+WB
+WC
=1 →1−WA
−WB
−WC
= 0
L = 0.0225WA
2+ 0.0289W
B
2+ 0.04W
C
2+ 0.0255W
AW
B+ 0.0204W
BW
C+ 0.024W
AW
C+ λ(1−W
A−W
B−W
C)
Take partial derivatives w.r.t. each choice variable and Lagrange multiplier
∂L
∂WA
= 0.045WA
+ 0.0255WB
+ 0.024WC
− λ = 0
∂L
∂WB
= 0.0578WB
+ 0.0255WA
+ 0.0204WC
− λ = 0
∂L
∂WC
= 0.08WC
+ 0.0204WB
+ 0.024WA
− λ = 0
∂L
∂λ=1−W
A−W
B−W
C= 0
Rearrange in Matrix form
0.045 0.0255 0.024 −1
0.0255 0.0578 0.0204 −1
0.024 0.0204 0.08 −1
−1 −1 −1 0
×
WA
WB
WC
λ
=
0
0
0
−1
WA
WB
WC
λ
=
0.4793
0.3125
0.2082
0.0345
FIN4814 [1/2012]
Minimum variance portfolio (two assets with additional constraints)
Min 0.0225WA
2+ 0.0289W
B
2+ 0.0255W
AW
B
S.t. WA
+WB
=1
11.014.01.0 =+BA
WW
WA
+WB
=1 →1−WA
−WB
= 0
014.01.011.011.014.01.0 =−−→=+BABA
WWWW
L = 0.0225WA
2+ 0.0289W
B
2+ 0.0255W
AW
B+ λ
1(1−W
A−W
B) + λ
2(0.11− 0.1W
A− 0.14W
B)
Take partial derivatives w.r.t. each choice variable and each Lagrange multiplier
01.00255.0045.0 21 =−−+= λλ∂
∂BA
A
WWW
L
∂L
∂WB
= 0.0578WB
+ 0.0255WA
− λ1
− 0.14λ2
= 0
∂L
∂λ1
=1−WA
−WB
= 0
∂L
∂λ2
= 0.11− 0.1WA
− 0.014WB
= 0
Rearrange in Matrix form
0.045 0.0255 −1 −0.1
0.0255 0.0578 −1 −0.14
−1 −1 0 0
−0.1 −0.14 0 0
×
WA
WB
λ1
λ2
=
0
0
−1
−0.11
WA
WB
λ1
λ2
=
0.75
0.25
0.0565
−0.16375
FIN4814 [1/2012]
Maximize portfolio return (three assets with additional constraints)
cBAWWWMax 08.015.011.0 ++
1.19.02.1.. ≤++CBA
WWWtS
10 ≤≤A
W
10 ≤≤B
W
10 ≤≤C
W
1=++CBA
WWW
Reduce the function to two variables by using the last constraint
1=++CBA
WWW � BACWWW −−= 1
)1(08.015.011.0BABA
WWWWMax −−++
1.1)1(9.02.1... ≤−−++BABA
WWWWtrw
10 ≤≤A
W
10 ≤≤B
W
1)1(0 ≤−−≤BA
WW
Or
08.007.003.0 ++BA
WWMax
2.03.01.0... ≤+BA
WWtrw
10 ≤≤A
W
10 ≤≤B
W
1)(0 ≤+≤BA
WW
Using the graphical method
Draw graph and locate the corner points
Find the optimum solution
Vesarach Aumeboonsuke
