Consider again the data of the previous exercise and first express the pre mium and the distribution of final wealth when the risk manager selects a deductible of 20 for each building (assume the loading is zero to simplify the computation).

Now,his insurer suggests an “umbrella policy” where, for the same premium, you receive a coverage with a deductible of 22.85 on the aggregate loss.

(a) Check that indeed this umbrella policy induces an actuarially fair pre mium of 10.

(b) Again by drawing the cumulative distribution function of final wealth, show that—as compared to the separate deductibles—the umbrella pol icy reduces the risk without changing the expected final wealth.