1. RISK MANAGEMENT PROBLEM SET 1 - RM 302 Lou's Cleaners is a family owned dry cleaner. Lou can use the following probability distribution to estimate his expected worker's compensation loss for the year at his cleaning plant: Probability Loss amount 0.4 0.3 4000 0.2 8000 0.1 12,000 Lou is considering how to protect his business from this loss exposure. (a) Based on the above experience, estimate Lou's expected loss and standard deviation of the loss. the (b) Lou is considering buying insurance from the Self Insurance Company (SIC), a new non-profit dry-cleaning industry self-insurer that is able to pool Lou's loss experience with other dry cleaning firms. Assuming that SIC is able to pool Lou's losses with other firms that are independent of each other, estimate the expected value and the standard deviation of the mean loss distribution when SIC insures 100, 400, and 1600 independent and homogeneous firms. (c) Calculate the premiums that SIC will charge to the three pools (i.e., 100, 400, and 1600 exposure units) if it wants to be 95% confident that the premium will be sufficient to pay for the underlying losses. To keep things simple, do not try to include insurer administrative expenses in your premium calculations. 2. (a) Lou is also considering buying insurance from Huge Insurance Company (HIC), a big commercial insurer. Assume HIC can pool an unlimited number of independent and homogeneous exposure units, but it must add an additional $400 to its premium (i.e., in addition to the mean loss plus the risk charge) to cover its overhead expense and to cover profit. What premium would HIC charge to cover its underlying losses? Assume that HIC wants to be 95% confident that the premium charge covers its underlying losses. (b) If SIC grows large enough to insure 400 dry cleaners (including Lou), will Lou buy insurance from HIC or SIC? In your answer, assume that the non-profit nature of SIC means that it does not add a charge to its premium to cover overhead or profits. 3. Al's Toy Store faces the following probability distribution of fire losses in its store over the next year: Probability Loss .85 $0 .10 .05 $20000 $40000 (a) (b) Calculate the mean and standard deviation of Al's losses for the year. Assume Al pools his losses with Ed's store, which has an identical loss distribution. Ed's losses are independent of Al's. Al and Ed agree to split the total losses in the pool equally. Show the revised probability distribution for the mean loss from the pool, and calculate the mean and standard deviation of the pooled mean losses. 4. Sid's Subs is a family owned sandwich shop. Based on his loss experience over the years, Sid estimates that his average annual cost to pay for medical expenses incurred by workers injured at his store is $5000 with a standard deviation of $8000. A year later, Sid decides to open three additional stores. Since each store is built identically to the first store, Sid decides that he can pool the cost of his worker medical costs as if they are independent and homogeneous across all four stores. If Sid assumes the mean loss is normally distributed, how much money would each store need to keep on hand as a reserve to be 95% certain that they had sufficient funds to cover their losses? 5. The Sweaty Tool Company makes industrial equipment. It is trying to decide which safety features to include in a particular piece of equipment. Find the optimal level of safety (probability of an accident) from a societal perspective using the data on the costs of safety given in the table below. Assume that an accident imposes costs on a victim equal to $500,000.