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

182 Optimizing Optimization

satisfies Equation (8.9) for any φ 1 and φ 2. This is because the right-hand side
of Equation (8.9) can be written as:

φμ 1 ΣΣ Σ^1111 EE()′′E E  2 φ 2 ΣΣΨ^111 EE()′E p

(8.11)

But ,

()′′

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

EEΣΣE ⎟⎟

11 1^1

0

μ

(8.12)

so that

φμ^1 φμ

1111

1

ΣΣ Σ ΣEE()′′E E  ^1

(8.13)

and Equation (8.9) then simplifies to:

22 φφ 22 xEEE ΣΣΨ^111 ()′ p

(8.14)

or

xEEEΣΣΨ^111 ()′ p

(8.15)

Corollary 8.1: The above applies to a large family of risk measures for a range
of distributions that reduce to the “ mean – variance ” analysis, namely the ellip-
tical class as outlined in Ingersoll (1987).
Ingersoll (1987, p. 104) defines a vector of N random variables to be ellipti-
cally distributed if its density ( pdf ) can be written as:

pdf y()g y y

 

ΩΩ

(^121)
()()()μμ′
(8.16)
If means exist, then E [ y ]  μ and if variances exist, then the covariance
matrix cov( y ) is proportional to Ω. The characteristic function of y is:
φμN()tE ity⎣⎡exp( )′ ⎦⎤ exp(it tt′′) (ψΩ)
(8.17)
for some function ψ that does not depend on N. It is apparent from Equation
(8.17) that if ω y  z is a portfolio of elliptical variables, then
EiszEisy⎡⎣exp()⎤⎦⎢⎡⎣exp()ωω′ ⎤⎦⎥ exp()( )is s′′μψωω^2 Ω
(8.18)