The reduction of realised variance in deductible insurance

We derive a series of mathematical identities that connect insurance purchasers with insurance companies. In particular, we focus on the way in which variance is shared between the parties. We argue that, from the perspective of governmental oversight, a desirable property of insurance is that the total amount of variance experienced by the involved parties is smaller under an insurance contract than in the uninsured case. It is shown that this always holds in the case of a single insurer and a single insured, while for the case of a single insurer and multiple insured, we derive a condition which guarantees the relationship.