This paper introduces the concept of proper risk aversion to expected utility theory. A decision-maker displays proper risk aversion if, given two independent undesirable monetary lotteries and being required to take one of them he continues to find the other undesirable. This condition implies decreasing risk-aversion. The first main result is an equivalence theorem between properness and some conditions on certainty equivalents. A large class of widely-used utility functions are shown to be proper, and analytical necessary and sufficient conditions are derived.