 
Summary: SPECTRAL RISK MEASURES FOR CREDIT PORTFOLIOS
CLAUDIO ALBANESE AND STEPHAN LAWI
ABSTRACT. In this article, we experiment with several different risk measures such as standard deviation,
valueatrisk, expected shortfall and powerlaw spectral measures. We consider several families of test
portfolios, one with a typical market risk profitandloss profile, and the others containing defaultable bonds
of various credit ratings and various degree of diversification. We find that the risk measures are roughly
equivalent on the market risk portfolios but differ significantly on the credit ones. In fact, valueatrisk as well
as some coherent risk measures including expected shortfall have counterintuitive features as a function of
the degree of diversification for return distributions deviating substantially from the normal. Certain spectral
risk measures possess the most intuitive properties regarding diversification.
1. INTRODUCTION
The quest for good risk measures that capture the risk exposure of a portfolio in terms of a single
number, has led to the consideration of standard deviation [7] and valueatrisk [8]. More recently,
coherent risk measures have been introduced [1, 2] and examples such as the expected shortfall [3] and
spectral risk measures [4] have been considered. This article is a rewrite of a 1997 working paper [5] in
the light of the recent findings in [4].
Valueatrisk ( VaR ) is the most commonly used measure of market risk and is recommended by the
Basel committee [6] and broadly endorsed by regulatory agencies [8]. If is a random variable of zero
mean modeling portfolio returns within a given time horizon, the VaR corresponding to the percentile
level (0, 1) is defined as follows:
