A (full) insurance contract consists in a premium paid by the insuree in order to fully eliminate a risk, which is then entirely borne by the insurer. Insurance relies on risk-pooling: as insurees face risks that are statistically independent (or at least do not share a common risk), the risk the insurer faces is predictable and can be redistributed across its shareholders. If an insurance company has M identical shareholders and N insurees with independent and identically distributed risks , i = 1,..., N, then each shareholder bears a share 1/M of the total risk = 1 M N xi. i=1 1. Suppose that each ; is zero with probability (1-p) and equals -L < 0 with probability p. What is the distribution of ? More precisely, give the formula for Pr(y = -nL/M) for n=0,..., N. 2. Suppose that M and N become very large, with a fixed ratio N/M = K. What is the limit distribution of y? 3. In this limit, what is the minimum price P at which the insurer should sell a contract that takes all the risk from an insuree? why does the answer not depend on the risk aversion of the shareholders? 4. Suppose that insuree i has a concave Bernoulli index u. What is the maximum price P which she is willing to pay for a contract that fully eliminates her risk ? 5. Compare P and P, and conclude.