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.