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

Some properties of averaging simulated optimization methods 229

It is straightforward to compute the information ratio defined as ασˆ/.ˆ Notice
that in this problem all terms depend essentially on a single term ( a γ  β 2 / γ ) or
functions of it.

10.3 Remark 1

A related formulation of the above problem is the following: min1/2
( ω  b ) Ω ( ω  b ) subject to ( ω  b ) i  0 and ( ω  b ) μ  π. Here, the
Lagrangian is given by:

Lbb bi b ^12 ()()() (() )ωωθω θωμπΩ 12

resulting in the first-order conditions:

∂ ∂ ∂ ∂ ∂ ∂

L

bi

L

bi

L

b

ω

ωθθμ

θ

ω

θ

ωμπ







Ω()

()

()

12

1

2

0

0

0

Solving, we have ω  b  θ 1 Ω ^1 i  θ 2 Ω ^1 μ with θ 1  ( β π / a γ  β 2 ) and
θ 2  ( γ π / a γ  β 2 ).

Thus ,

ˆ

ˆˆˆ

ω

πγ

γβ

μ

β

γ

ωπω









b 

a

i

b

2

⎛ΩΩ 11

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

and consequently,

ˆˆˆ ˆˆ

ˆ

σπωω

σ

πγ

γβ

22

2

2

2







Ω

a

Comparing with Equation (10.2), we see immediately that π  1/ λ ( a  β 2 / γ ).
This second problem is simply the computation of the minimum variance
frontier. It differs from the earlier version in that it explicitly specifies π , the
expected rate of return, rather than λ , the risk aversion coefficient.