Business question. Address all the questions appropriately.

The premiums payable under a deferred annuity contract issued to women aged exactly 60 are limited to 5 years. The annuity commences at age 65, provided the policyholder is still alive at that age. The annuity provides payments of $3,500 payable annually in advance for 5 years certain (ie it continues to be paid for 5 years even if the annuitant dies before age 69) and for life thereafter. There is no benefit if the policyholder dies before age 65. (i) Calculate the annual premium. [6] (ii) Calculate the retrospective and prospective reserves after the policy has been in force for each of 5 and 10 years. [10] Basis: PFA92C20 mortality, 4% pa interest [Total 16]An author is about to start writing a book that will contain 20 chapters. The author plans to write a new chapter each week. However, when he reviews his work at the end of each week, there is a probability of 0.25 (which is independent of the current state of the book) that he will not be happy with one of the chapters he has written. In this case, he will spend the following week rewriting that particular chapter instead of embarking on a new one. He may decide to rewrite any one chapter, including a new one he has just finished or one that he has previously rewritten. Let X, denote the number of chapters that the author is happy with at the end of week & , and define Xo = 0. (i) Explain why Xx can be modelled as a Markov chain. [2] Calculate the probability that the author will complete the book in exactly 25 weeks. [2] (iii) Calculate the expected number of weeks it will take the author to complete the book. [2] [Total 6]