180 Optimizing Optimization
present some results in Section 8.2. These are a consequence of two results we
present, Proposition 8.1 and a generalization, Proposition 8.2, which prove
that the set of minimum risk portfolios are essentially the same under elliptic-
ity for a wide class of risk measures. In addition to these results, we present
three extensions. The extensions we propose in this chapter are threefold. First,
we consider extensions for portfolio simulation of those advocated for value
at risk simulation by Bensalah (2002). Second, under normality we compute
explicit expressions for mean/value at risk, mean/expected loss, and mean/
semivariance frontiers in the two asset case and in the general N asset case,
complementing the results for mean/value at risk under normality provided by
Alexander and Bapista (2001). Finally, our framework allows us to consider
fairly arbitrary risk measures in the two asset case with arbitrary return dis-
tributions, and in particular some explorations under bivariate lognormality
are considered. In Section 8.6, we present issues to do with the simulation of
portfolios, pointing out some of the limitations of our proposed methodology.
These methodologies are applied to general downside risk frontiers for general
distributions. Conclusions follow in Section 8.7.
8.2 Main proposition
It is worth noting that although it is well known that normality implies mean –
variance analysis for an arbitrary utility function (see, for example, Sargent
(1979, p. 149) ), it is not clear what happens to the mean/downside risk fron-
tier under normality. What is known is that the ( μ (^) p , θ (^) p ) frontier should be
concave under appropriate assumptions for θ (^) p and that the set of minimum
downside risk portfolios should be the same as those that make up the set of
minimum variance portfolios. We note that Wang (2000) provides examples
that show that when returns are not elliptical, the two sets do not coincide. A
proof of this second assertion is provided below in Proposition 8.1. We shall
initially assume that the ( N 1) vector of returns is distributed as N N ( μ , Σ )
where μ is an ( N 1) vector of expected returns and Σ is an ( N N ) positive
definite covariance matrix. We define the scalars α μ Σ ^1 μ , β μ Σ ^1 e , and
γ e Σ ^1 e , where e is an ( N 1) vector of ones.
Proposition 8.1: For any risk measure φφμσppp (),^2 , where μ (^) p μ x ,
σ^2 pxx′Σ, φ 1 ∂∂φμ/ p , φφσ 2 (),∂∂/^2 p and φ 2 is assumed non-zero, the
mean minimum risk frontier ( μ (^) p , φ (^) p ) is spanned by the same set of vectors for
all φ, namely, x Σ^ 1 E ( E Σ ^1 E )^ ^1 Ψ (^) p , where Ψp ( μ (^) p ,1) and E ( μ , e ).
Proof : Our optimization problem is to minimize φ (^) p subject to E x Ψ (^) p. That is,
min ,
x pp p
φμ σ() ()^2 −λ Ex Ψ
(8.1)