Portfolio risk is in an important way driven by the ‘abnormal’ returns emanating
from heavy-tailed distributed asset returns. Therefore the …nancial industry
often employs so called downside risk measures to characterize the asset and
portfolio risk, since it is widely recognized that large losses are more frequent
than a normal distribution based statistic like the standard deviation suggests.
A formal portfolio selection criterion which incorporates the concern for
downside risk is the safety …rst criterion, see Roy (1952) and Arzac and Bawa
(1977). The paper by Gourieroux, Laurent and Scaillet (2000) (hereafter Gourieroux
et al.) analyzes the sensitivity of Value at Risk with respect to portfolio
allocation, which is essentially the same problem as portfolio selection with the safety …rst criterion. They discuss how to check the convexity of the estimated
VaR e¢cient portfolio set. They concerns the entire distribution instead of the
tails of the distribution. Jansen, Koedijk and de Vries (2000) (hereafter Jansen
et al.) apply the safety …rst criterion and exploit the fact that returns are fattailed.
They propose a semi-parametric method for modeling tail events and
use extreme Value at Risk VaR) as the measure for downside risk.
If one selects assets on the basis of the tail properties of the return distribution,
there is a tendency to end up in a corner solution whereby the asset the
highest tail is selected, see e.g. Jansen et al. (2000), Hartmann, Straetmans and
de Vries (2000) and Poon, Rockinger and Tawn (2003). This follows from Geluk
and de Haan (1987), which states that a convolution of two regularly varying
variables produces a random variable which has the same tail properties as the
fattest tail of the two convoluting variables, i.e. the fattest tail dominates.
In this paper, we show that a second order expansion of the downside risk
often includes a more balanced solution, whereas the …rst order expansion selects
the corner solution. The portfolio selection follows Jansen et al. (2000). We
calculate the failure probability by a convolution theory based on a second
order expansion of the downside risk. Lastly, we demonstrate the relevance of
our theory by re-examining the cases used in Jansen et al. (2000).
