Solve the questions below

1 (i) Explain what it means for the last survivor status 50:60 to remain active for at least 10 years. (ii) Calculate the probability that the event described in part (i) occurs, assuming the two lives are independent with respect to mortality and: (a) the mortality of each life follows the ELT15 (Males) table (b) each life is subject to a constant force of mortality of 0.025 pa. 2 (i) Explain what it means for the joint life status 50:60 to fail within the next 10 years. (ii) Calculate the probability that the event described in part (i) occurs, assuming the two lives are independent with respect to mortality and: (a) the mortality of each life follows the ELT15 (Females) table (b) each life is subject to a constant force of mortality of 0.025 pa. .3 Given that ,9, =0.3 and ,q, =0.5, calculate ,qxy and ,4xy . 4 Consider each of the symbols listed below: (a) P xy (b) Axy (c) Explain carefully the meaning of each of these symbols and calculate the value of each, assuming that: (x) is subject to a constant force of mortality of 0.01 pa (v) is subject to a constant force of mortality of 0.02 pa the force of interest is 0.04 pa. A life insurance company issues 1,000 last survivor annuities to pairs of lives aged 60. Each pair comprises one male and one female, and the annuity pays $5,000 po continuously until the second of the two lives dies. The single premium charged is $90,000. Calculate the expected present value of the profit to the life office and the standard deviation of this profit in respect of this group of policies. Basis: Mortality: PMA92C20 for the male life, PFA92C20 for the female life Interest: 4% pa effective Using this mortality assumption, Age =0.20021 at an interest rate of 8.16% pa.6 William, aged 75, and Laura, aged 80, are the guardians of a child. They take out a life assurance policy that provides a payment of f25,000 immediately when the second of them dies. Level annual premiums are payable in advance whilst the policy is in force. (i) Calculate the annual gross premium, using the basis given below. [4] (ii) Calculate the gross premium prospective reserve just before the sixth premium is paid, using the basis given below, assuming that both William and Laura are still alive at that time. [3] [Total 7] Basis: Mortality: PMA92C20 for William, PFA92C20 for Laura Interest: 4% pa effective Expenses: Initial: E250 Renewal: 5% of each premium, excluding the first A life insurance company issues an annuity to a male, aged 68, and a female, aged 65. The annuity of f10,000 po is payable annually in arrears and continues until both lives have died. The insurance company values this benefit using PMA92C20 mortality for the male life, PFA92C20 mortality for the female life and 4% pa interest. (1) Calculate the expected present value of this annuity. [2] (ii) Derive an expression for the variance of the present value of this annuity in terms of appropriate single life and joint life assurance functions. [4] Let X be the present value of the insurer's profit from this policy. (iii) If the insurance company charges a premium of f150,000 for this policy, calculate P(X > 0) . [4] [Total 10]