Decision making with Conditional Value-at-Risk and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) on the one hand, and Ross (1981) on the other hand. The implications for two standard financial decision problems, namely the willingness to pay for insurance and portfolio selection, are studied. Within the framework of Arrow and Pratt, we show that the widely-applied spectral Arrow-Pratt-measure is not a consistent measure of Arrow-Pratt-risk aversion. A decision maker with a greater spectral Arrow-Pratt-measure may only be willing to pay less for insurance or to invest more in the risky asset than a decision maker with a smaller spectral Arrow-Pratt-measure. We further show how a proper measure of Arrow-Pratt-risk aversion should look like instead. Within the framework of Ross, we show that the popular subclasses of Conditional Value-at-Risk, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. As a consequence, these subclasses also exhibit counter-intuitive comparative static results. In the insurance problem, the willingness to pay for insurance may be decreasing with increasing risk parameter. In the portfolio selection problem, the investment in the risky asset may be increasing with increasing risk parameter. These shortcomings have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.