Answer these questions. Address all questions effectively.

During a period of length 7 years, you observe a total of / lives between the ages of x and x+1. You do not necessarily observe each life for the entire year of age. The total time spent under observation by the N lives is I . d deaths are observed. (i) State the assumptions underlying the Poisson model for d, given that the force of mortality between the ages of x and x + 1 is a constant, /. [2] (ii) Show that the maximum likelihood estimator of / is D/I, under the assumptions in (i) above. [2] ili) Show that the maximum likelihood estimator has: (a) an expected value of uf (b) a variance of u/V . [2] [Total 6]An insurance company operates a no claims discount system with discount levels of 0%, 30%, 40%, 50% and 60%. The rules are as follows: At the end of a claim free year, a policyholder moves up one level (or remains on the maximum discount level). At the end of a year in which exactly one claim was made, a policyholder drops back two levels (or moves to zero discount). At the end of a year in which more than one claim was made, a policyholder drops back to zero discount. For a particular driver in any year, the probability of a claim free year is 0.7, the probability of exactly one claim is 0.2, and the probability of more than one claim is 0.1. (i) Write down the transition matrix for this time-homogeneous Markov chain. [2] (ii) If the policyholder starts with no discount, calculate the probability that he is at the maximum discount level 6 years later. [2] (iii) If a large number of people having the same claims distribution take out policies at the same time, calculate the proportion you would expect to be in each discount category in the long run. [6] [Total 10]14 A company wishes to raise additional finance but it has been prevented from borrowing by the conditions imposed by an existing loan agreement. Explain why a lender might impose such a restriction before granting a loan. [5] 15 A company's board of directors is revising the company's investment appraisal criteria. The directors have asked for an explanation of the concept of opportunity cost and the manner in which it might impact on the selection of projects. Outline the points you would make. [5] 16 A company is planning to raise additional funds by issuing shares. Describe the advantages of doing so by means of a rights issue. [5]