Need help in addressing the questions.

..1EI ..11 An author is about to start writing a book that will contain 2D 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 o.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 nished or one that he has previously rewdtten. Let It denote the number of chapters that the author is happy with at the end of week it . and define xi] = i] . [i] Explain why it} can be modelled as a Markov chain. [2] [ii] Calculate the probability that the author 1nrrill complete the book in exactly 25 weeks. [2] [iii] Calculate the expected number of weeks it will take the author to complete the book. [3] [Total 3'] The creditworthiness of debt issued by companies is assessed at the end of each year by a credit rating agency. The ratings are A {the most credit-worthy}, B and D [debt defaulted]. Historic evidence supports the view that the credit rating of a debt can be modelled as a Markov chain with one-year transition matrix: D32 CLUE [LIB P = EDS I135 Ill ID i] 1 [i] Determine the probability that a company Currently rated A will never be rated E in the future. [1] [ii] [a] Calculate the secondorder transition probabilities of the Markov chain. [b1 Hence calculate the expected number of defaults within the nest two years from a group of mo compa niesy all initially rated it. [2] The manager of a portfolio investing In company debt follows a 'downgrade trigger' strategy. Under this strategy, any debt in a company whose rating has fallen to B at the end of a year is sold and replaced with debt in an A-rated company. [iii] Calculate the expected number of defaults for this investment manager over the nest two years, given that the portfolio initially consists of 100 Fri-rated bonds. [2] [iv] Comment on the suggestion that the downgrade trigger strategy will improve the retum on the portfolio. [2] [Total 3]