Provide solutions to the questions in the attachments below. Kindly ensure precision in your answers.

William, aged 75, and Laura aged 80, are the guardians of a child. They take out an annual premium life assurance policy that provides a payment of $25,000 payable immediately when the last guardian dies. (i) State the conditions under which premiums would normally be payable. [1] (ii) State with reasons which, if any, of the guardians' lives should be examined medically. [1] (iii) Calculate the premium, assuming that premium payments are limited to a maximum of 10 years, the medical examinations were satisfactory and the lives are subject to PA92C20 mortality. Assume 4% pa interest and allow for expenses of 5% of each premium and an initial expense of $250. [9] [Total 11]In any year, the rate of interest on funds invested with a particular company has mean value / and standard deviation s, and is independent of the rates of interest in all previous years. (i) Derive formulae for the mean and the variance of the accumulated value after n years of a single investment of 1 at time 0. [5] (ii) Let i, be the rate of interest earned in the /th year. Each year the value of (1 + i, ) is lognormally distributed, with parameters # = 0.04 and oz =0.09. (a) Show that n, the number of years that must elapse before the accumulation of a lump sum invested at time 0 has a 75% probability of at least doubling in size, satisfies: 0.04n - 0.2024vn - In 2 =0 (b) Hence, or otherwise, calculate the value of n. [8] [Total 13]