Central Limit Theorem

Consider risk pools of independent and identically distributed risks. Each individual risk has an expected loss of $15,000 and a standard deviation of $45,000. Consider three safety levels, = 0.90, = 0.95 and = 0.99. Create a column where you vary the size of the risk pool from 5 to 1,000 in increments of 5 (i.e., 5, 10, 15, ..., 1,000). For each pool size, use the CLT to determine the total amount of capital needed to ensure a shortfall probability less than (1 ) for the entire risk pool. For each pool size, also determine the amount of capital per risk and the proportional loading relative to the expected loss that is necessary to raise this amount of capital. Find the total amount of capital, the amount of capital per risk, and the proportional loading to raise this amount of capital for each of the three safety levels. Assume that the owners of the individual risks are willing to transfer the risk into the pool as long as the loading is below 61%. This number corresponds roughly to a level of risk aversion of about 3. (Ask me for details if you are curious. ) Use your spreadsheet to find the minimum pool size necessary to ensure a loading below 61%. Now assume that the owners of the individual risks are willing to transfer the risk into the pool as long as the loading is below 26%. This number corresponds roughly to a level of risk aversion of about 1.5. (Ask me for details if you are curious. ) Use your spreadsheet to find the minimum pool size necessary to ensure a loading below 26%. Summarize your main observations in a few sentences in your spreadsheet.