Computing optimal mean/downside risk frontiers: the role of ellipticity 183
and all portfolios from the joint pdf given by Equation (8.16) will have the
same marginal distribution, which can be obtained by inverting Equation
(8.17). Furthermore, the distribution is location scale, in the sense that all
portfolios differ only in terms of ω μ and ω Ω ω and it is for this reason that
mean – variance analysis holds for these families.
Corollary 8.2: Our result includes as a special case the value at risk calcula-
tions of Alexander and Bapista (2001) since φμ σ σ μpp, t p p
2
() for t 0.
We note that Alexander and Bapista (2001) are more concerned with the
efficient set than the minimum risk set. The distinction between the efficient
set and the minimum risk set is addressed by finding the minimum point on
the mean minimum risk frontier. Nevertheless, their Proposition 2 (page 1,168)
that implicitly relates normality goes to some length to show that the mean –
value at risk efficient set of portfolios is the same as the mean minimum vari-
ance portfolios. This follows as a consequence of our Proposition 8.2 and its
corollaries.
We now turn to the question as to whether the mean minimum risk frontier
is concave (i.e., ( ∂ 2 v / ∂ μ 2 ) 0). Suppose that returns are elliptical so that our
risk measure v can be expressed as:
νφμσ μμ(), pp^2 and ,
(8.19)where
σμγ βμ α
p2(^22)
Δ
(8.20)
and
Δαγ β
(^2) ,
(8.21)
so that
∂
∂
ν
μ
φ
φ
σ
μγ β
1
2
p
()
Δ
(8.22)
so that ∂ v/ ∂ μ 0 when
φ
φ
σ
μγ β
1
^2 0
p
()
Δ
(8.23)