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

Computing optimal mean/downside risk frontiers: the role of ellipticity 191

we see that

()

()( )

()

VaR

t

pp

pp







μ

μγ βμ α

αγ β

2

22

2

2

(8.67)

Equation (8.67) is a general quadratic (conic) in μ (^) p and VaR p and we can
apply the methods of analytical geometry to understand what locus it describes.
The conic for v and u is:
vvuu u
(^22)   21 2()γθ βθ αθ 0
(8.68)^
where θ  t 2 /( α γ  β 2 ), u  μ (^) p , v  VaR p , and α , γ , and θ are always positive.
Following standard arguments, we can show that this conic must always be a
hyperbola since γ θ 0 (see Brown and Manson (1959, p. 284) ). Furthermore,
the center of the conic in ( u , v ) space is ( β / γ ,  β / γ ) so that the center corre-
sponds to the same expected return as the global minimum variance portfolio.
The center of the hyperbola divides the ( u , v ) space into two regions.
We now consider implicit differentiation of Equation (8.68) for the region
where μ  β / γ , which corresponds to the relevant region for computing our
frontier.
22 221 20v
v
u
v
u
uvu
∂
∂
∂
∂
()γθ βθ
(8.69)
So,
∂
∂
v
u
uv
uv



()γθ 1 βθ
(8.70)
Thus ,
vu
v
u
() whenγθ βθ 10 
∂
∂
(8.71)
or,
u
v



()
()
βθ
γθ (^1)
(8.72)
Substituting into Equation (8.68),
ν
βθ
γθ
βθ
γθ
γθ βθ
βθ
γθ
(^2) αθ
2
 (^2112121)








v 
()vv v
()
()
()
()
()
()
00
(8.73)