Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

234 Optimizing Optimization

The corresponding results for^ IR^ and^ TE are given by:

EIR

n

nh n

()

()

→

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

1 ⎟⎟

1

1

12







/

and

ETE

n

nh n

()

()

→

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

1

1

1

1

12

λ 





/

We now examine portfolio optimization without a benchmark. Here, we maxi-
mize ω μ  λ /2 ω Ω ω subject to ω i  1. The associated Lagrangian is given by:

Wi  ωμ ω ω θωλ 2 Ω () 1

(10.11)

with

∂

∂

W

i

ω

μλωθΩ   0

implying

ωμθλ^1 ()ΩΩ^11 i

since i ω  1, we have immediately that

θμλ ^

()/iiiΩΩ^11

i.e.,

θβλγ()/

Consequently,

ˆ ˆ ˆ

ˆ

ˆ

ωμβλˆ

λ γ



1 ^11 

ΩΩ

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟⎟

⎛

⎝

⎜⎜

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

⎟

i