This article considers an optimization problem for an insurance buyer in the context of proportional insurance and furnishing effort to reduce the size of a potential loss. The buyer's risk preference is given by a distortion risk measure, with risk probabilities being evaluated via a non-concave distortion function. This kind of distortion function reflects potential cognitive biases in the way in which individuals perceive risk probabilities. The buyer will select the optimal level of both insurance coverage and the prevention effort that he/she will furnish to reduce the amount that he/she stands to lose. The distribution of losses is given by a family of stochastically-ordered probability measures, indexed by the prevention effort. Contrary to what is found in the standard economic literature, introducing a non-concave distortion function leads to indeterminacy in the relationship between market insurance and self-insurance. Self-insurance and market insurance may be either substitutes, with a rise in one producing a fall in the other, or complements, so that the two rise or fall together, depending on the price elasticity.