This work develops a partial ordering of risks in the presence of a correlated background risk, which is parallel to the structure of Ekern (1980)’s risk increase. We introduce the notion of truncation expectation dependence to measure the primary risk’s dependence on the background risk. We provide its integral characterization of distribution functions. Within a bivariate expected utility framework, we show an equivalence between the sign of cross-derivatives of the utility function with the preference toward truncation expectation dependence. This choice-theoretic foundation is helpful for studying some economic and financial decisions with background risks, e.g., optimal portfolio, precautionary effort.
Wang et al. (Fri,) studied this question.