Microeconomics questions
Exercise 5.9 Consider the random variables Y = a7, and Z = ulx. (a) Explain briefly why the covariance of Y and Z is negative. (b) Derive an expression for the covariance, in terms of standard actuarial functions. (c) Show that the covariance is negative. Exercise 5.10 Find, and simplify where possible: (a) dar, and Exercise 5.11 Consider the following portfolio of annuities-due currently being paid from the assets of a pension fund. Number of Age annuitants 60 40 70 30 80 10 Each annuity has an annual payment of $10 000 as long as the annuitant survives. The lives are assumed to be independent. Calculate (a) the expected present value of the total outgo on annuities, (b) the standard deviation of the present value of the total outgo on annu- ities, and (c) the 95th percentile of the distribution of the present value of the total outgo on annuities using a Normal approximation. Exercise 5.12 Consider the quantities o (m) and B(m) in formula (5.38). By expressing i, i("), d and d'() in terms of 8, show that m - a(m) ~ 1 and B(m) ~ 2m Exercise 5.13 Using a spreadsheet, calculate the mean and variance of the present value of (a) an arithmetically increasing term annuity-due payable to a life aged 50 for at most 10 years under which the payment at time t is t + 1 for t = 0, 1, . ...9, and (b) a geometrically increasing term annuity-due payable to a life aged 50 for at most 10 years under which the payment at time t is 1.03' fort = 0, 1, ....9.Exercise 5.3 Given that l = 3.2066, l = 1329?, and lpj = |[1.9195, what is the effective rate of interest per year? Exercise 5.4 Given that {1,53 = 19.996, .1151 = 19.256, .1151 1115119 and i = 9.136, calculate 2pm . Exercise 5.5 You are given the following extract from a select life table. [I] 5m I[n+1 6+2 1+2 413' 33 519 33 435 33444] 42 41 33 46? 33 423 33 323 43 42 33 44}? 33 365 33 309 44 43 33 344] 33 294 33 231 45 44 33 265 33 213 33 143 46 Calculate the following, assuming an interest rate of 6% per year: (a) ammoP {b} ammu -" (C) {III} [ml :- (d) {IA} [4mm . (e) the standard deviation of the present value of a four-year tenn anrmity-due, with annual payment $100!}, payable to a select life age 41, and (i) Define the size and power of a statistical test. [2] An actuary is comparing a set of observed mortality rates with the values predicted by a certain formula. He has calculated the "standardised residuals" for 25 different ages. If the formula is satisfactory these residuals should have a standard normal distribution. He is assuming that the residuals X1..... X 25 are independent identically distributed random variables from a /(#,1) distribution and he wishes to carry out a test of the hypotheses Ho : / =0 versus H : //#0. (ii) Write down the test criterion the actuary should use for a symmetrical test of size 5%. [21 (iii) Calculate the power of this test if the true value of / is: (a) 0 (b) 0.5 (c) -0.2 [3] (iv) Hence sketch a graph of the power function for this test as a function of /. [2] [Total 9]
