Before revision:  This model allows us to obtain semi-explicit solutions for the insurance contract and is applied to the context of fuel poverty  .
After revision:  This model allows us to obtain semi-explicit solutions for the insurance contract and is applied to the context of fuel poverty . For this application, our model allows to conclude that, even in the least efficient scenario from the households point of view, i.e., when the insurance is provided by a monopoly, this mechanism improves significantly the living conditions of the riskiest households by ensuring them a sufficient consumption of energy .