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1. An insurer charges a single premium of E(X) + 0.1√Var(X) for a contract with present value of benefits equal to X. Use a normal approximation.
(a) How many contracts must be sold so that the probability is 95% that premiums cover benefits?
(b) What is the probability that premiums cover benefits if 225 contracts are sold?
2. A failure time has a hazard rate of μ(t) = 2/20 - t.
A contract provides for a benefit of 1 at time 10 provided that failure occurs before time 10 (so, for example, if failure occurs at time 6, the benefit is not paid until 4 years later). You are given that v(10) = 0.6. Find the expectation and variance of the present value of the benefits.
For each mutual fund, find an interval in which you would expect 95.44 percent of all yearly returns to fall. Assume returns are normally distributed.
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