Questions: answer the following questions appropriately. Explain your answers in detail.

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](i) In the context of the graduation of mortality data discuss the concepts of "smoothness" and "fidelity to data" and their relationship. [6] (ii) Comment on the graduation in the following table, mentioning briefly the limitations of any tests you apply. Exposed Actual Graduated mortality Expected Age to risk deaths rate qx deaths 65 60,000 1,370 0.0225 1,350 66 50,000 1,200 0.0250 1,250 67 43,000 1,200 0.0276 1,187 68 37,000 1,090 0.0304 1,125 60 30,000 1,010 0.0334 1,002 70 26,000 950 0.0368 957 71 23,000 980 0.0406 934 72 21,000 950 0.0449 943 73 20,000 1,000 0.0497 994 74 18,000 960 0.0551 992 328,000 10,710 10,734 The chi square value of the graduation is 7.29. You may assume that there are 7 degrees of freedom. [20] [Total 26]11 A company's directors are unable to decide whether to invest in a particular project. Discuss the extent to which a simulation exercise might help them to reach a conclusion. [5] 12 A company's income statement shows that it has generated substantial profits but its cash flow statement indicates that it has suffered a large outflow of cash during the same period. The figures are reliable and free from distortion. Explain whether this set of circumstances warrants any major concern. [5] 13 Explain why the interpretation of a company's accounting ratios requires some understanding of the nature of the business and the industry in which it operates. [5] 14 Explain the implications to the reader of a qualified independent auditor's report. [5] 15 Explain the implications to the shareholders of issuing company directors with stock options as part of their remuneration package. [5]