We analyze spectral risk measures (SRMs) including its most popular representative, Expected Shortfall (ES), with respect to Gollier and Pratt (1996)’s concept of risk vulnerability. We find that SRMs and risk vulnerability are mutually exclusive, owing to the property of subadditivity: while subadditivity is commonly regarded as the axiomatic cornerstone of SRMs, risk vulnerability, by contrast, prevails if and only if superadditivity holds. The lack of risk vulnerability yields questionable predictions in portfolio problems: SRM-decision makers who split their wealth between a risk free and a risky asset do constantly opt for an increase in the risky investment when their deterministic background wealth is complemented by some additional background risk. The more general setting where background wealth is already random and then becomes more risky is not as clear-cut: Any SRM-decision maker may both increase or decrease the risky investment, depending on the concrete instance of the portfolio problem. However, when random background wealth and the risky asset are jointly normally distributed, SRM-decision makers will again unambiguously increase their risky investment. We further conduct a data analysis and discuss possible implications of the findings for regulatory risk management.

We study portfolio selection under Conditional Value-at-Risk and, as its natural extension, spectral risk measures, and compare it with traditional mean–variance analysis. Unlike the previous literature that considers an investor’s mean-spectral risk preferences for the choice of optimal portfolios only implicitly, we explicitly model these preferences in the form of a so-called spectral utility function. Within this more general framework, spectral risk measures tend towards corner solutions. If a risk free asset exists, diversification is never optimal. Similarly, without a risk free asset, only limited diversification is obtained. The reason is that spectral risk measures are based on a regulatory concept of diversification that differs fundamentally from the reward-risk tradeoff underlying the mean–variance framework.

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.