Respond to the three questions in the attachments below. Elaborate your answers fully.

A new actuarial student is planning to sit one exam each session. He expects that his performance in any exam will only be dependent on whether he passed or failed the last exam he sat. If he passes a given exam, the probability of passing the next will be or, regardless of the nature of the exam. If he fails an exam, the probability of passing the next will be B . (i) Calculate the probability that: (a) the first exam he fails is the seventh, given that he passes the first (b) he passes the fifth exam, given that he fails the first three. [4] (ii) Explain the results above in terms of a Markov chain, specifying the state space and transition matrix. (For the purposes of this model, assume that we are only interested in predicting passing or failing, not in the number of exams passed so far.) [3] [Total 7]In a Markov jump process model of sickness and death there are three states: healthy, sick and dead, { H.S,D . The transition rates are given by the matrix: -o (1) - u(1 ) o (1 ) u (1 ) A(1 ) = p(1) - p(1) - v (t ) v (t ) 0 0 0 Explain the following equation by general reasoning: I-5 PSH (s, 1) = Pss (s,t - w) p(t - we li-w((u)+ u(u)du dw [7] 0The finance director is also concerned that the sales director has ignored tax when preparing the budgeted income statement. (i) Calculate the net present value of the project using a discount rate of 7% p.a. and ignoring tax. [8] (ii) Explain why the required rate of return might be higher than the 7% rate charged by the bank on the funding for the project. [6] (iii) Explain how adjusting for tax will affect the analysis of this project. You are not required to calculate the effects of tax. [6] [Total 20]The directors of Merchant ple are considering a five year project that has been proposed by the company's sales director. A customer is about to sell a very specialised milling machine that has become surplus to requirements. This type of machine would cost tens of millions of pounds to buy new. Used machines are rarely sold on the open market. The machine has an expected remaining useful life of five years and has been checked and certified by an independent engineer. The sales director proposes buying the used machine for f4 million and using it to manufacture a new product for which Merchant plc owns some patent rights and which cannot be manufactured in an economic manner without such equipment. The sales director proposes taking out a five year lease on a factory building and using an employment agency to provide labour for the five year period. The sales director estimates that it would be necessary to invest a further f1 million at the start of the first year in addition to buying the machine: a returnable deposit of $100,000 on the factory, f200,000 for the first year's lease, f400,000 for the opening inventory of raw materials and $300,000 for the initial payment for labour. The sales director has drafted the following budgeted annual income statement for the project: Em Revenue 2.6 Raw materials (0.4) Factory lease (0.2) Labour (0.3) Depreciation of machine (0.8) Other running costs (0.5) Profit 0.4 It is anticipated that materials, recurring lease payments and labour will be paid for annually at the start of each year. Other running costs comprise the cost of electricity and other operating costs and they will be paid for annually at the end of each year. Revenues will be received at the end of the year in which they are earned. At the conclusion of the project, the sales director anticipates that it will cost (300,000 to dismantle the equipment and at that time it will be possible to reclaim 100% of the deposit paid to the factory owner. The sales director has shown this proposal to a potential lender, who has agreed to lend Merchant plc f5 million at an interest rate of 7% p.a. This loan will be secured on Merchant pic's existing assets because the milling machine is deemed to be too specialised to be a suitable form of security. The sales director proposes that the project should be evaluated at a required rate of return of 7% p.a. because it will be financed by means of a self-contained loan package upon which that rate will be charged. The finance director believes that the project should be evaluated at a much higher rate than 7% p.a., but cannot state the specific rate that should be charged without conducting a further analysis