Address the questions below with explanations.the questions are complete

An investigator wishes to construct a multiple decrement model of the mortality of a population, subdivided by the following causes of death. Cause 1: Cancer Cause 2: Heart disease Cause 3: All other causes You are given the following definitions: M's is the force of mortality due to cause i (i = 1, 2, 3 ) at exact age x "qx is the probability that a life at exact age x dies due to cause i (i = 1, 2, 3 ) before reaching exact age x tu (u 20) " Px is the probability that a life aged exactly x is still alive at exact age x tu (u > 0) You may assume that "4x _ u as u=> 0. (i) Derive an expression for , p. (t 2 0), in terms of the forces of mortality, using only the above functions in your derivation. [5](ii) Write down a formula for q, in terms of ,p, and the appropriate force(s) of mortality. (Note that q* = 19% . ) [2] (iii) By reference to the expressions obtained in (i) and (ii) above, explain briefly why q, is often referred to as a dependent rate of mortality. [2] (iv) Assuming that the force of mortality from each cause i is a constant / between integer ages x and x + 1, show that: = 3 xq, where q, = 1-Px [5] M i=1 (Note that Px = 1 Px-) [Total 14]An investor borrows money at an effective rate of interest of 10% pa to invest in a 6-year project. The cashflows for the project are: an initial outlay of f25,000 regular income of f10,000 pa during the first 5 years (assumed to be received continuously) regular expenditure of f2,000 pa during the first 5 years (assumed to be payable continuously) a decommissioning expense of f5,000 at the end of the 6th year. Calculate: the net present value of the project's cashflows (ii) the discounted payback period (iii) the internal rate of return. An investor borrows f1,000 by taking out an interest-only loan at an effective rate of interest of 6% pa, and invests the money in a project. The loan is to be repaid in full after 2 years (with no early repayment option) and interest on the money borrowed is paid at the end of each month. The project will provide income of f50 at the end of each month for 24 months and the investor can invest spare funds at an interest rate of 5% po effective. Calculate the accumulated profit at the end of 2 years. (i) Define: (a) the discounted payback period ( b ) the payback period of an investment project. [2] (ii) Describe the disadvantages of using these two measures for determining whether to proceed with an investment project. [3] [Total 5]