This is an open book, open notes, take-home exam. You may use any written source but do not consult or work with any other person. Answer questions completely and show your work, including sending me your spreadsheets by email. Make sure the answers and explanations to all of the questions are specifically written down, not just sitting in a cell somewhere on the spreadsheet. You can either email the answers and explanations along with the spreadsheet, write them out on the spreadsheet, or put a separate page(s) with answers and explanations in my mailbox or under my office door. It must be returned to me no later than May 3, 10 AM.
Let u(t) be a collective risk surplus process defined by u(t)=u+ct-S(t) for a starting value u(0)=u of surplus where time t is measure in years, where S(t)=X_{ 1}+…+X_{N(t)} for N(t) a Poisson process with λ=100, i.e. for each t, N(t) is a Poisson random variable with mean 100t and the usual independence assumptions apply, and the X’s are identically and independently distributed, independent of N(t), where each separate X has the same distribution as the sum of 5 identical and independent exponential random variables, each exponential random variable having mean 1,000. The premium accumulation rate c=1,000,000 per year.
Let L_{ 1} = - u(T) be the random variable representing the absolute size of the deficit at the time T of ruin, conditional on ruin occurring, assuming u=u(0)=0. Let L be the random variable representing the largest value of the absolute size of the deficit (- u(t))_{ +} as 0<t<∞, assuming u=u(0)=0.

Is L_{ 1} a gamma random variable with α=6 and θ=1000? If so, explain why; if not, explain exactly how L _{1} and the gamma 6, 1000 are related. (Think of the surface interpretation.)

What is the probability of ruin ψ(0) for starting surplus of u=u(0)=0?

Using a discrete approximation for L_{ 1}, rounding and with discrete unit equal to one-tenth (10%) of the mean of X, calculate the approximate value of the smallest starting surplus u=u(0)>0 for which ψ(u)<0.05.

For that value of u (the value in question (3)) calculate an approximate value for the conditional tail expectation E[LIL>u]. (Such a conditional tail expectation at some confidence level such as ψ(u)<0.05 is an increasingly popular measure of the surplus required to manage a portfolio of insurance risk, although usually the confidence level would be set at a more stringent level than 0.05.)