1. A failure time T is uniformly distributed on the interval [0, 20]. The force of interest is a constant 0.05.
(a) A deferred insurance contract provides for a payment of 1 at the moment of failure provided that this occurs after time 5. If Z‾ is the present value of the benefits, find the 80th percentile of Z‾. That is, find the point z such that P(Z‾ ≤ z) = 0.80.
(b) Find the 80th percentile of Z‾ when the contract is a term insurance policy that pays 1 unit if failure occurs in the first 5 years.
2. Consider a deferred term insurance contract. It provides for 1 unit payable at the moment of failure provided that this occurs between N and 2N years from now. Nothing is paid if failure occurs in the first N years or after 2N years. You are given that the force of failure μ and the force of interest δ are constants such that μ = δ/2. Moreover, you are given that e-Nδ = 0.36. If Z is the present value of the benefits, calculate FZ (z) for all nonnegative values of z.