Could someone help me with problems 13, 14, 15, and 16 on the attached PDF? I need to show work for all, so explanations would be really helpful. Thank you!

1. Assume the following rates are being utilized for the given yearly periods. i 8% for 0 t 2 t 0.015t for 2 t 5 d 6% for 5 t 8 i (4) 10% for 8 t Find the present value of an asset which will payout a single cash-flow amount of 1000 at time t = 10. (a) 439 (b) 459 (c) 479 (d) 499 (e) 519 Answer: ____________________ 2. Eileen buys a perpetuity-immediate with annual payments for a purchase price of X. The first payment is 1.02, the 2nd payment is 3% greater than the first, the third payment is 2% greater than the second, and so on, with the increase in future payments alternating between 3% and 2%. At the purchase price of X, Eileen's effective annual interest yield rate is 3%. Determine X. Hint: I thought of the even-year payments and odd-year payments separately. (a) 102 (b) 103 (c) 204 (d) 205 (e) 206 Answer: ____________________ 3. Account A offers a simple interest rate of 5% per year and Account B offers a force of interest of t 0.05t where t is the time in years. Suppose that 1000 is invested into each account at time t = 0. What is the sum of the interest gained from both accounts between times t = 1 and t = 2? (a) 120 (b) 130 Answer: ____________________ (c) 140 (d) 150 (e) 160 4. Which of the following notations given below is equivalent to a10 i ? Note: Assume that i > 0. (b) 1 v 9 s9 i (a) v10a10 i (c) s10 i a10 i (d) (1 i)a9 i 1 (e) v s9 i Answer: ____________________ 5. A 39-year annuity-immediate will pay 13 in each of the first 3 years, 12 in each of the next 3 years, and so on, until payments of 1 are made in each of the last 3 years. The present value of the annuity at an annual effective interest rate of 3% is X. Determine X. (a) 59 (b) 165 (c) 177 (d) 184 (e) 187 Answer: ____________________ 6. At an annual effective rate of interest i > 0, payments of 100 now, 200 two years from now and 100 four years from now have a total present value of 300. Calculate i (a) 11.7% (b) 13.0% (c) 14.5% (d) 15.8% (e) 16.9% Answer: ____________________ 7. You are given the following information. X is the value at the end of year two of a 20-year annuity-due of 1 per year. 1 The annual effective interest rate for year t is 8t Which of the summations given below is equal to the value of X? 28 28 29 29 30 10 11 10 11 10 (a) (b) (c) (d) (e) t 9 t t 9 t t 10 t t 10 t t 11 t Answer: ____________________ 8. At an effective annual interest rate of i, where i > 0, each of the following two sets of payments has present value K. A payment of 121 immediately and another payment of 121 at the end of one year. A payment of 144 at the end of two years and another payment of 144 at the end of three years. Calculate K. (a) 237 (b) 232 (c) 227 (d) 222 (e) 217 Answer: ____________________ 9. Ralph buys a perpetuity-due paying 500 annually. He deposits the payments into a savings account earning interest at an effective annual rate of 10%. Ten years later, before receiving the eleventh payment, Ralph sells the remaining payments of the perpetuity based on an annual effective interest rate of 10%. Using the proceeds from the sale plus the money in the savings account, Ralph purchases an annuity-due paying X per year for 20 years at an effective annual interest rate of 10%. Calculate X. (a) 1145 (b) 1260 (c) 1385 (d) 1525 (e) 1675 Answer: ____________________ 10. A perpetuity pays 1 at the end of the first two years (i.e. at time t = 1 and t = 2), pays 2 at the end of the second two years (i.e. at time t = 3 and t = 4), 3 at the end of the third two years (i.e. at time t = 5 and t = 6), and so on. Assuming an annual effective interest rate of 8%, find the present value of the perpetuity. Give your answer rounded to the nearest whole number. Answer: ____________________ 11. At time t = 0, Donald puts 1000 into a fund crediting interest at an annual nominal interest rate of i compounded semiannually. 1 At time t = 2, Lewis puts 1000 into a different fund crediting interest at a force of t for all t. 5t At time t = 16, the amounts in each fund will be equal. Note: In this problem, t is representing time in years. Calculate i. Give your answer as a percentage rounded to one decimal place. Answer: ____________________ 12. Jeff deposits 11 into an account on day 5, 12 on day 10, 13 on day 15 and so on. The account gains interest via an annual interest rate of 3.65% compounded daily. Find the future value of the account on day 600, immediately after the deposit of 130 is made. Round your answer to the nearest whole number. Note: There are 365 days in a year. Answer: ____________________ 13. Jim buys a perpetuity-immediate which pays 180 each year and has an annual effective interest rate of 2i. Tina buys a perpetuity-immediate which pays 60 at the end of the first year, and each subsequent payment increases by 5% from the previous payment. Tina's annuity has an annual effective interest rate of i. Both annuities have the same present value. Find i. Give your answer as a percentage rounded to the nearest whole number. Answer: ____________________ 14. At the start of the year you decide to provide yourself with a retirement account. You begin by depositing 100 into the account at the end of each month, and each year you increase this amount by 10%. Thus, for the first year, you make 12 deposits of size 100, for the second year you make 12 deposits of size 110, for the third year you make 12 deposits of size 121, and so on. Suppose that you make these deposits for 35 years, upon which time you retire. After retirement, you will make an annual withdrawal from the account at the start of each year for the next 30 years. Each withdrawal will be of equal size X. If the account gains interest at an annual rate of 8% compounded monthly, and we assume that after 65 years, the account has a balance of zero, then find X. Round your answer to two decimal places. Answer: ____________________ 15. An annuity pays 1 at the start of each year for 4n years. Using an annual effective interest rate of i, the accumulated/discounted value of the entire 4n-year annuity at time 2n is 637.72. If it is also known that (1 i)n 8.7384, then find n. Round your answer to the nearest whole number. Hint: Break the annuity into four n-year annuities and find the value of each of these annuities at time 2n. The sum of these values will equal 637.72. Answer: ____________________ 16. Bob buys a perpetuity at time t = 0 years for P. The perpetuity makes annual payments beginning at t = 8. The first payment is for 500. Each annual payment thereafter is decreased by 10 until a payment of 200 is reached. Subsequent payments remain level at 200. If the annual effective interest rate is 4%, find the value of P. Round your answer to the nearest whole number. Answer: ____________________ 17. Iggy borrows X at an annual effective interest rate of 6% and he agrees to pay it back within 10 years. If he pays the principal and accumulated interest in one lump sum at the end of the 10 years, he would pay 356.54 more in interest than if he repaid the loan with 10 level payments at the end of each year for the next 10 years. Calculate X. Round your answer to the nearest whole number. Hint: You need to find the total amount that Iggy pays in each scenario. Answer: ____________________