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(i) Calculate the combined present value of an immediate annuity payable monthly in arrears such that payments are (1,000 pa for the first 6 years and $400 pa for the next 4 years, together with a lump sum of f2,000 at the end of the 10 years. [3] (ii) Calculate the amount of the level annuity payable continuously for 10 years having the same present value as the payments in (i). [3] (iii) Calculate the accumulated values of the first 7 years' payments at the end of the 7th year for the payments in (i) and (ii). [3] Basis: Assume an interest rate of 12% pa convertible monthly. [Total 9](i) Define the accumulation factor A(1,/ + h) and give a formula for the force of interest 6() per unit time in terms of the accumulation factor. [2] (ii) The force of interest 6(/) at time / (measured in years) is given by 6(1) = 0.01 +0.04. (a) Calculate the corresponding nominal rate of interest for the period / = 1 to / = 2. (b) If an investment of 1 is made at time /= - , calculate the value to which it will have accumulated by time / =6. [6] (iii) Calculate the accumulated value after 6 months of an investment of $100 at time 0 at the following rates of interest: (a) a force of interest of 0.05 pa (b) a rate of interest of 5% pa convertible monthly (c) an effective rate of interest of 5% pa. [3] [Total 1 1]Let X denote the present value of an annuity consisting of payments of (2,000 payable at the end of each of the next 8 years, valued using an interest rate of 8% pa convertible quarterly and let Y denote the present value of an annuity consisting of payments of $4,000 payable at the end of every fourth year for the next 16 years, valued using an interest rate of 8% pa convertible half yearly. Calculate the ratio X / Y. [6]