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Let X be a random variable representing the present value of the benefits of a whole of life assurance, and Y be a random variable representing the present value of the benefits of a temporary assurance with a term of a years. Both assurances have a sum assured of 1 payable at the end of the year of death and were issued to the same life aged x. (1) Describe the benefits provided by the contract which has a present value represented by the random variable X - Y. [1] (ii) Show that: cov(X, I) = ZAG - Ax And and hence or otherwise that: var(X - Y) = 'Ax -(m/4x)= _ 2Al where the functions A are determined using an interest rate of /, and functions 2 A are determined using an interest rate of i- + 2i. [7] [Total 8]Let X be a random variable representing the present value of the benefits of a whole of life assurance, and Y be a random variable representing the present value of the benefits of a temporary assurance with a term of a years. Both assurances have a sum assured of 1 payable at the end of the year of death and were issued to the same life aged x. (1) Describe the benefits provided by the contract which has a present value represented by the random variable X - Y. [1] (ii) Show that: cov(X, I) = ZAG - Ax And and hence or otherwise that: var(X - Y) = 'Ax -(m/4x)= _ 2Al where the functions A are determined using an interest rate of /, and functions 2 A are determined using an interest rate of i- + 2i. [7] [Total 8](i) Define the accumulation factor A(t,/+h) and give a formula for the force of interest 6(t) per unit time in terms of the accumulation factor. [2] (ii) The force of interest 6() at time r (measured in years) is given by 5(1) =0.01/ +0.04. (a) Calculate the corresponding nominal rate of interest for the period / = 1 to 1 = 2. (b) If an investment of 1 is made at time /=, calculate the value to which it will have accumulated by time r =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](i) Define the accumulation factor A(t,/+h) and give a formula for the force of interest 6(t) per unit time in terms of the accumulation factor. [2] (ii) The force of interest 6() at time r (measured in years) is given by 5(1) =0.01/ +0.04. (a) Calculate the corresponding nominal rate of interest for the period / = 1 to 1 = 2. (b) If an investment of 1 is made at time /=, calculate the value to which it will have accumulated by time r =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]