Survey_LM_2006

Abstract

There are n identical, risk-averse individuals who maximize expected utility with respect to an increasing, concave utility function u (·). Each individual is endowed with initial wealth w0 and faces a loss of size Xi, i = 1, ..., n. We assume that losses are independent and identically distributed according to a continuous distribution function F1 with F1 (0) = 0 and density function f 1. The aggregate loss in the economy, Pni=1 Xi, is then distributed according to the n-fold convolutionF n = ¡F1¢?(n) with density function fn.Risk sharing is organized through an insurance company, which can either be a stock insurer or a mutual insurer. The insurer is run by a risk-neutral manager. We consider the following three stages of setting up a stock insurance company. At the first stage, the company raises risk capital C from shareholders. At the second stage, the manager sells insurance policies offering full coverage to the n individuals at a premium P per policy. At the third stage, losses are realized, and the total capital, nP + C, is distributed to policyholders and shareholders. A mutual insurance company is owned by its policyholders, who own the right to the insurer’s surplus. The company does not raise capital from shareholders, but through selling policies at the second stage, i.e. the mutual has no capital other than the collected insurance premiums. Suppose that each policyholder pays a premium P m for full coverage. At the third stage, losses are realized, and the company distributes the total capital, nP m, amongst policyholders.