Higher-order risk vulnerability

We add an independent unfair background risk to higher-order risk-taking models in the current literature and examine its interaction with the main risk under consideration. Parallel to the well-known concept of risk vulnerability, which is defined by Gollier and Pratt (Econometrica 64: 1109-1123, 1996), an agent is said to have a type of higher-order risk vulnerability if adding an independent unfair background risk to wealth raises his level of this type of higher-order risk aversion. We derive necessary and sufficient conditions for all types of higher-order risk vulnerabilities and explain their behavioral implications. We find that as in the case of risk vulnerability, all familiar HARA utility functions have all types of higher-order risk vulnerabilities except for a type of third-order risk vulnerability corresponding to a downside risk aversion measure called the Schwarzian derivative.