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

184 Optimizing Optimization

We shall assume the existence of a unique minimum risk portfolio; if it
exists, the minimum risk portfolio occurs when ( ∂ v / ∂ μ  0) or when

μφ

β

γ

σφ

γφ

∗

1

1

2

p Δ

(8.24)

We note that β / γ is the expected return of the global minimum variance port-
folio, so the global minimum “ v ” portfolio is to the right or to the left depend-
ing on the signs of φ 1 and φ 2.

For value at risk, φ 2  t 0 and φ 1   1, whereas for variance, φ 2  2 σ (^) p
and φ 1  0. In general, one may wish to impose the restriction that φ 2 0 and
φ 1  0, then μ *  ( β / γ ) if σ (^) p Δ  φ 2 γ. Other cases can be elaborated. Multiple
solutions may be possible, but we shall ignore these. Considering now the
second derivatives,
∂
∂
2
2 11
12 22
2
2
2
ν (^22)
μ
φ
φ
σ
μγ β φ
σ
μγ β φ
σ
γ





p p p
()()
Δ Δ Δ
(8.25)
For v^2 v / ∂ μ 2 0 as required, we need the matrix of second derivatives
φφ
φφ
11 12
21 22
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟ to be positive definite and φ
2 0. This condition is satisfied for
variance and for value at risk as in this case φ 12  0 whilst φ 2  t.

8.3 The case of two assets

We now consider the nature of the mean/semivariance frontier. When N  2,
some general expressions can be calculated for the frontier. In particular, for
any distribution, r p  ω r 1  (1  ω ) r 2 , and

μωμ ωμp 12 () 1

(8.26)

Two special cases arise: if μ 1  μ 2  μ , then μ (^) p always equals μ , and the
(^) ()μθpp,^2 frontier is degenerate consisting of a single point. Otherwise, assume
that μ 1  μ 2 when Equation (8.26) can be solved for ω * , so that
ω
μμ
μμ
∗ 

p 2
(^12)
(8.27)