We introduce the notion of cross-risk vulnerability to generalize the concept of risk vulnerability introduced by Gollier and Pratt [Gollier, C., Pratt, J.W. 1996. Risk vulnerability and the tempering effect of background risk. Econometrica 64, 1109-1124]. While risk vulnerability captures the idea that the presence of an unfair financial background risk should make risk-averse individuals behave in a more risk-averse way with respect to an independent financial risk, cross-risk vulnerability extends this idea to the impact of a non-financial background risk on the financial risk. It provides an answer to the question of the impact of a background risk on the optimal coinsurance rate and on the optimal deductible level. We derive necessary and sufficient conditions for a bivariate utility function to exhibit cross-risk vulnerability both toward an actuarially neutral background risk and toward an unfair background risk. We also analyze the question of the sub-additivity of risk premia and show to what extent cross-risk vulnerability provides an answer.
