This note determines a sufficient condition on (von Neumann-Morgenstern) utility functions to preserve (reserve) comparative risk aversion under general background risks. Our condition is weaker than the one determined by Nachman (1982, Journal of Economic Theory). Nachmanfs condition requires...

This note gives sufficient conditions of cross risk vulnerability introduced by Malevergne and Rey (2005), which is the equivalent condition to guarantee that an unfair non-monetary background risk makes decision makers more risk averse. The sufficient conditions determined by this note expand...