3.17 Insurance company losses of a particular type are to reasonable approximati

Question

3.17 Insurance company losses of a particular type are to reasonable approximation normally distributed with a mean of $150 million and standard deviation of $50 million. (Assume no different between loses in a risk neutral world and losses in the real world). The one year risk rate is 5%. Estimate the cost of the following
A) a contract will pays $100 million in one year time 60% of the insurance company%u2019s cost on a pro rata basis
B) A contract that pays $100 million in one year time if losses exceed $200 million

Explanation / Answer

Answer:

A) a contract will pays $100 million in one year time 60% of the insurance company%u2019s cost on a pro rata basis:

The losses are in the millions of dollars are approx. %u2C77 (150,50). The re-insurance contract would pay out 60% of the losses. The payout from the reinsurance contract is therefore %u2C77(90,30). The cost of the re-insurance is the expected payout in a risk neutral world, discounted at the risk free rate. In this case the expected payout is the same in a risk neutral world as it is in the real world. The value of re-insurance contract is:

90%u212E-0.5x1= 85.61

B) A contract that pays $100 million in one year time if losses exceed $200 million

The probability that losses will be greater than $200 million is the probability that a normally distributed variable is greater than one standard deviation above the mean. This equals 0.1587. The expected payoff in millions of dollars is:

0.1587 x 100 = 15.87 and the value of the contract is 15.87%u212E-0.5x1= 15.10

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