A life insurance company sells with-profit whole life policies, with the sum assured payable immediately on the death of the life assured and with premiums payable annually in advance ceasing with the policyholder's death or on reaching age 65 if earlier. The company markets two versions of this policy, one with simple reversionary bonuses and the other with compound reversionary bonuses. In both cases the bonuses are added at the end of the policy year. The company prices the products using the following basis: AM92 Select 4% per annum Mortality Interest Expenses Bonuses (1) initial renewal claim simple compound 250 2% of second and subsequent premiums 150 at termination of contract 6% of basic sum assured per annum 4% of accumulated sum assured and bonuses per annum Write down an expression for the gross future loss at the point of sale for each of these policies, assuming they are sold to a life aged x exact (x < 65) at outset. Write the expression in terms of functions of the random variables T, and K. which represent the exact future lifetime and the curtate future lifetime of (x). respectively. [5] (ii) Calculate the gross premium required for each of the two policies for a sum assured of 200,000 and a life aged 40 exact at outset, using the equivalence principle. [8] (iii) After 10 years, bonuses totalling 90,000 have been declared for the compound reversionary bonus contract. Calculate the net premium reserve for that policy at that time, using AM92 ultimate mortality and interest of 4% per annum. [4] [Total 17]