Computing optimal mean/downside risk frontiers: the role of ellipticity 189
Finally , if r 1 and r 2 are any two mean – variance efficient portfolios, then they
span the set of minimum risk portfolios as described in Proposition 8.1, and
thus the result follows.
Corollary 8.3: If we wish to consider expected loss under normality, which we
denote by L where L E [ r p | r p t ], then
Lttt
pp
ppp
p
()μμ
σσφμ
σΦ⎛⎝⎜⎜
⎜⎜⎜⎞⎠⎟⎟
⎟⎟
⎟⎛⎝⎜⎜
⎜⎜⎜⎞⎠⎟⎟
⎟⎟
⎟(8.53)Proof : The same argument as before.
The above results can be generalized to any elliptical distribution with a
finite second moment since in all cases we know, at least in principle, the mar-
ginal distribution of any portfolio. Let f ( · ) and F ( · ) be the pdf and cdf of any
portfolio return r p , and let μ (^) p and σ (^) p be the relevant mean and scale param-
eters. Then, it is a consequence of ellipticity that a density (if it exists) f ( z ) has
the following property. For any ( μ (^) p , σ (^) p ) and r p μ (^) p σ (^) p z ,
f z pdf
r
p
pp
p
()
σ
μ
σ
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
(8.54)
Furthermore , there are incomplete moment distributions:
FxkkzfzdzFx Fx
x
()() () ()
∫∞
,^0
(8.55)
for k a positive integer (existence of the k th moment is required for F k ( x ) to
exist). So,
Lt F
t
F
t
p
p
p
p
p
p
()μ
μ
σ
σ
μ
σ
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
⎛
⎝
⎜⎜
⎜⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟
1
(8.56)
and letting ω ( t μ (^) p )/ σ (^) p ,
Semivariance ( ) ( ) ( ) ( )
()
sv t F t F
F
ppp
p
μω μσω
σω
21
22
2
(8.57)
To illustrate the problems that arise when the pdf of r p is not available in
closed form, we consider the case of bivariate lognormality; as we see below,
the previous simplifications no longer occur. Suppose that:
r
P
P
1
1
0
1
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
(8.58)