The questions below are economics model task.answer them

Evaluate (a) Aco and (b) Aco on the bases: (i) A1967-70 ultimate, 4% p.a. interest; (ii) E.L.T. No.12 - Males, 4% p.a. interest. 2 (i) Show that Ar = vqr + vPrArti (ii) Given that peo = 0.985, p61 = 0.98, i = 0.05 and A62 = 0.6 , evaluate Asi and Aco 3 Consider n lives now aged 21, 12, ...; In respectively. Let Z be the total present value of a contract providing the sum of fS, immediately on death of (r,), i = 1, 2, ..., n. The n lives are subject to the same non-select mortality table, Table B, and the interest is taken to be fixed at rate i p.a. (i) Show that E(Z) = SiAn + ..+ SnAr, on Table B with interest at rate i p.a. (ii) Assuming further that the future lifetimes T(x;) of the lives are independent variables, show that Var(Z) = _ SHIA: - (An,)"] where * refers to an interest rate of 2i + 12 p.a. 4 Using commutation functions or otherwise calculate the values of the following: (i) A(40]:10 on A1967-70, 4% p.a. interest; (ii) A30:20) on A1967-70 Ultimate, 4%; (iii) A30:20) on A1967-70 Ultimate, 4%; (iv) A30:20) on A1967-70 Ultimate, 4%; (v) A30:201 on English Life Table No.12 Males, 4%. 5 A life aged 50 who is subject to the mortality of the A1967-70 Select table, effects a pure endowment policy with a term of 20 years for a sum assured of f10,000. (i) Write down the present value of the benefits under this contract, regarded as a random variable. (ii) Assuming an effective rate of interest of 5% per annum, calculate the mean and the variance of the present value of the benefits available under this contract. 6 What are the random variables (in terms of K = curtate future lifetime of (x)) whose means are represented by the following symbols? (i) n Er (ii) (iii) Arm 7 A life aged exactly 60 wishes to arrange for a payment to be made to a charity in 10 years time. If he is still alive at that date the payment will be f1000. If he dies before the payment date, the amount given will be f500. Assuming an effective interest rate of 6% per annum andmortality according to ELT No.12-Males, calculate the standard deviation of the present value of the liability. 8 (Difficult) You are given that (i) 1000 (IA) 50 =4,996.75 , (ii) 1000 Al 50:1 =5.58. (iii) 1000 As = 249.05 and (iv) i =0.06. Calculate 1000(IA)51