Histogram-valued data on value at risk measures: a symbolic approach for risk attribution

In this article, we develop the concept of histogram-valued data on value at risk for the classification of hedge fund risk. By using recent developments in data mining, it is a question of the classification of heterogeneous data in order to sort hedge funds by risk class. In practical terms, risk levels relative to measures of histogram-valued data on VaR are calculated as an aid to decision-making. The empirical study was carried out on 1023 HFR-based hedge funds, where we had estimated monthly ARMA-GARCH or asymmetric GARCH VaR and CVaR measures between 01 January 2003 and 31 December 2008. We identify two sub-periods: from 2003 to 2005, and from 2006 to 2008 in order to identify a recovery period after the 2001-2002 crisis and the impact of the 2007-2008 crisis. First, the symbolic approach allows us to construct the measures of histogram-valued data on VaR by optimizing the definition of categories. A symbolic principal component analysis shows that the indices coming from the VaR of the GARCH and asymmetrical GARCH are the most pertinent. Second, we apply a criterion of inter-class inertia and retain a partitioning of hedge funds into three classes by dynamic <italic>K</italic>-means cluster analysis. For each of our sub-periods and for each class, a risk level is defined on the basis of the categories of the most discriminating variable.