The aim of this paper is to go further in the understanding of the determinants of individual insurance choices. We propose a two-step model: the choice of insurance is primarily determined by a need of protection, and then by an assessment of the willingness to pay for insurance coverage. In standard models it is assumed that individuals make a unique evaluation of their subjective insurance value that determines both the willingness to insure, and the willingness to pay for insurance coverage. However, empirical evidence exists in favor of our model: variables explaining the decision to buy insurance or not are different from those explaining the amount of insurance coverage. Our hypothesis is that the willingness to insure is triggered by an emotional mechanism (emotions anticipated in case of loss and emotional sensitivity induce a need for protection), the assessment of the willingness to pay being then done according to the objective financial risk characteristics (probabilities of loss and amounts of loss). To verify this hypothesis we performed a laboratory experiment in which subjects had to make hypothetical insurance choices in different contextualized scenarios. We elicited simultaneously the anticipated individual discontent (or dissatisfaction), the willingness to insure, and the willingness to pay for insurance. We measured also some psychological individual characteristics (personality traits, risk attitudes measured from lottery choices with financial incentives etc.). Our results confirm the dissociation of the factors driving willingness to take insurance and willingness to pay for insurance. We show that the willingness to insure depends on the anticipated dissatisfaction but the direction of the relationship depends on the individuals' emotional sensitivity: a high anticipated discontent in case of loss increases the willingness to insure for subjects with low emotional sensitivity, but the inverse relationship is observed for subjects with high emotional sensitivity. Concerning the willingness to pay for insurance, it strongly depends on the willingness to insure, but also on common factors as loss probabilities, loss amounts and insurance domain.