# Problem Set: Annuities and Perpetuities (Solutions Below)

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1 Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save \$300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years that costs \$1,000 and you can save \$150 per year, what interest rate would you need? 3. If you invest \$1,000 per year in a stock portfolio with a return of 8%, how much would you expect to have in 7 years? 4. How long would it take you to save \$1,000 if you invested \$200 per year, and the interest rate is 10%? 5. If you need \$10,000 to pay for your first year of graduate school in 3 years and you get an interest rate of 9%, how much must you invest each of the next three years? 6. If 6 years ago you invested \$500 and received an interest rate of 4% (compounded monthly), how much would you now have? 7. You borrowed \$100 from a friend, who said you need to pay back \$300 in 5 years, what rate are you being charged if it is compounded weekly? 8. How many years would it take you to have \$2,500 if you saved \$100 each month at 15%?

2 9. To have \$6000 in 7 years what interest rate would you need if you saved \$200 every quarter? 10. If you win a lottery worth \$1,000,000 payable in 15 years and the interest rate is 8% (compounded annually), what is this worth today? Compounded quarterly? Compounded monthly? Compounded weekly? 11. How long does it take for an investment to quadruple in value if the investment yields 6% per year (compounded monthly)? 12. What are the payments on a \$40,000 loan repaid monthly for six year (r = 7%)? 13. If I invest \$100 today and every quarter for 3 years in an account earning 11%, how much will I have at the end of three years? 14. Suppose that I am trying to borrow money from you to finance my business, and I promise to repay you \$1,000 quarterly for two years. If your opportunity cost of funds is 10%, how much are you willing to lend me? 15. Jim makes a deposit of \$120 every week (beginning next week). The deposit is to earn interest annually at the rate of 9 percent. How much will Jim have on deposit at the end of seven years? 16. How long will it take to repay a loan of \$150, if I pay \$1 per week and the rate on my loan is 4%? 17. Value an annuity of \$300 per month for 7 years (r = 12.3%). 18. Suppose you have the opportunity to make an investment expects to pay investors \$7,000 per year for next eight years. If the cost is \$50,000, what return would you receive?

3 19. If a two year weekly annuity is worth \$5000 and r = 9.8%, what is the weekly cash flow? 20. Which grows to a larger future value, \$1000 invested for 2 years a) at 10 percent compounded weekly, or b) at 11 percent compounded semi-annually? 21. Value an annuity of \$40 per year for ten years (r = 13%). 22. You want to save for your retirement in 50 years. How much do you need to save from your biweekly paycheck to have \$5 million if you expect a return is 7%? 23. If an investment is expected to pay \$400 per month for the next 14 months, how much should you be willing to pay for that asset if your cost of capital is 8%? 24. You have borrowed \$35,000 at an interest rate of 9%. If you plan to pay the loan off in annual installments of \$4,000 (beginning next year), when can you pay back the loan? 25. The type of house you would like to buy requires a down-payment of \$50,000. You plan to make that downpayment six years from now. How much do you need to save per week (beginning next week), if your money gets 7% (annually)? 26. You hope to go to graduate school, and the tuition will be \$50,000 for the one-year M.B.A. program. If you can only afford to save \$3,000/quarter and the interest rate is 9%, how long will you need to save? 27. The house you plan to buy will require a downpayment of \$40,000 in two years. How much do you need to save per month (beginning next month), if your savings gets 8% (annually)?

4 28. You have borrowed \$10,000 at an interest rate of 8.7%. If you plan to pay the loan off in quarterly installments of \$1,000 (beginning next quarter), how long will it take you to pay back the loan? 29. Value a perpetuity of \$400 per year (r = 14.9%). 30. If a perpetuity is worth \$1,000 and r = 15.5%, what is the cash flow?

5 Solutions 1. If you plan to save \$300 annually for 10 years and the discount rate is 15%, what is the future value? P/Y = 1; N = 10; I/Y = 15; PV = 0; PMT = -300; FV = \$6, If you want to buy a boat in 6 years that costs \$1,000 and you can save \$150 per year, what interest rate would you need? P/Y = 1; N = 6; I/Y = 4.20%; PV = 0; PMT = -150; FV = 1, If you invest \$1,000 per year in a stock portfolio with a return of 8%, how much would you expect to have in 7 years? P/Y = 1; N = 7; I/Y = 8; PV = 0; PMT = -1,000; FV = \$8, How long would it take you to save \$1,000 if you invested \$200 per year, and the interest rate is 10%? P/Y = 1; N = 4.25 years; I/Y = 10; PV = 0; PMT = 200; FV = - 1, x 12 = 3 4 years, 3 months NOTE: When the question involves time, you must convert the answer to x years and y units. 5. If you need \$10,000 to pay for your first year of graduate school in 3 years and you get an interest rate of 9%, how much must you invest each of the next three years? P/Y = 1; N = 3; I/Y = 9; PV = 0; PMT = \$3,050.55; FV = -10, If 6 years ago you invested \$500 and received an interest rate of 4% (compounded monthly), how much would you now have?

6 P/Y = 12; N = 72 (= 6 x 12); I/Y = 4; PV = -500; PMT = 0; FV = \$ You borrowed \$100 from a friend, who said you need to pay back \$300 in 5 years, what rate are you being charged if it is compounded weekly? P/Y = 52; N = 260 (= 5 x 52); I/Y = 22.02%; PV = -100; PMT = 0; FV = How many years would it take you to have \$2,500 if you saved \$100 each month at 15%? P/Y = 12; N = months; I/Y = 15; PV = 0; PMT = -100; FV = 2, months 1 year, 10 months NOTE: Since N is periods, the time unit is the payment period. 9. To have \$6000 in 7 years what interest rate would you need if you saved \$200 every quarter? P/Y = 4; N = 28 (= 7 x 4); I/Y = 2.02%; PV = 0; PMT = -200; FV = 6, If you win a lottery worth \$1,000,000 payable in 15 years and the interest rate is 8% (compounded annually), what is this worth today? Compounded quarterly? Compounded monthly? Compounded weekly? P/Y = 1; N = 15; I/Y = 8; PV = \$315,241.70; PMT = 0; FV = - 1,000,000 P/Y = 4; N = 60 (= 15 x 4); I/Y = 8; PV = \$304,782.27; PMT = 0; FV = -1,000,000 P/Y = 12; N = 180 (=15 x 12); I/Y = 8; PV = \$302,396.05; PMT = 0; FV = -1,000,000

7 P/Y = 52; N = 780 (= 15 x 52); I/Y = 8; PV = \$301,472.08; PMT = 0; FV = -1,000, How long does it take for an investment to quadruple in value if the investment yields 6% per year (compounded monthly)? P/Y = 12; N = months; I/Y = 6; PV = -1; PMT = 0; FV = months 23 years, 2 months 12. What are the payments on a \$40,000 loan repaid monthly for six year (r = 7%)? P/Y = 12; N = 72 (= 6 x 12); I/Y = 7; PV = -40,000; PMT = \$681.96; FV = If I invest \$100 today and every quarter for 3 years in an account earning 11%, how much will I have at the end of five years? P/Y = 4; N = 12 (= 3 x 4); I/Y = 11; PV = 0; PMT = -100; FV = \$1, Value = 1, \$100 = \$1, You add \$100 to account for the first payment coming now instead of one week from now. 14. Suppose that I am trying to borrow money from you to finance my business, and I promise to repay you \$1,000 quarterly for two years. If your opportunity cost of funds is 10%, how much are you willing to lend me? P/Y = 4; N = 8 (= 2 x 4); I/Y = 10; PV = \$7,170.14; PMT = - 1,000; FV = Jim makes a deposit of \$120 every week (beginning next week). The deposit is to earn interest annually at the

8 rate of 9 percent. How much will Jim have on deposit at the end of seven years? P/Y = 52; N = 364 (= 7 x 52); I/Y = 9; PV = 0; PMT = -120; FV = \$60, How long will it take to repay a loan of \$150, if I pay \$1 per week and the rate on my loan is 4%? P/Y = 52; N = ; I/Y = 4; PV = -150; PMT = 1; FV = weeks 3 years, 3 weeks 17. Value an annuity of \$300 per month for 7 years (r = 12.3%). P/Y = 12; N = 84 (= 7 x 12); I/Y = 12.3; PV = \$16,841.09; PMT = -300; FV = Suppose you have the opportunity to make an investment expects to pay investors \$7,000 per year for next eight years. If the cost is \$50,000, what return would you receive? P/Y = 1; N = 8; I/Y = 2.59%; PV = -50,000; PMT = 7,000; FV = If a two year weekly annuity is worth \$5000 and r = 9.8%, what is the weekly cash flow? P/Y = 52; N = 104 (= 2 x 52); I/Y = 9.8; PV = -5,000; PMT = \$52.99; FV = Which grows to a larger future value, \$1000 invested for 2 years a) at 10 percent compounded weekly, or b) at 11 percent compounded semi-annually? P/Y = 52; N = 104 (= 2 x 52); I/Y = 10; PV = -1,000; PMT = 0; FV = \$1, P/Y = 2; N = 4 (= 2 x 2); I/Y = 11; PV = -1,000; PMT = 0; FV = \$1, (better)

9 21. Value an annuity of \$40 per year for ten years (r = 13%). P/Y = 1; N = 10; I/Y = 13; PV = \$217.05; PMT = -40; FV = You want to save for your retirement in 50 years. How much do you need to save from your biweekly paycheck to have \$5 million if you expect a return is 7%? P/Y = 26; N = 1300 (= 50 x 26); I/Y = 7; PV = 0; PMT = \$421.20; FV = -5,000, If an investment is expected to pay \$400 per month for the next 14 months, how much should you be willing to pay for that asset if your cost of capital is 8%? P/Y = 12; N = 14; I/Y = 8; PV = \$5,329.68; PMT = -400; FV = You have borrowed \$35,000 at an interest rate of 9%. If you plan to pay the loan off in annual installments of \$4,000 (beginning next year), when can you pay back the loan? P/Y = 1; N = 17.97; I/Y = 9; PV = -35,000; PMT = 4,000; FV = years 25. The type of house you would like to buy requires a down-payment of \$50,000. You plan to make that downpayment six years from now. How much do you need to save per week (beginning next week), if your money gets 7% (annually)? P/Y = 52; N = 312 (= 6 x 52); I/Y = 7; PV = 0; PMT = \$129.06; FV = -50, You hope to go to graduate school, and the tuition will be \$50,000 for the one-year M.B.A. program. If you can only afford to save \$3,000/quarter and the interest rate is 9%, how long will you need to save?

10 P/Y = 4; N = 14.31; I/Y = 9; PV = 0; PMT = 3,000; FV = -50, quarters 3 years, 2 quarters or 3 years, 6 months 27. The house you plan to buy will require a downpayment of \$40,000 in two years. How much do you need to save per month (beginning next month), if your savings gets 8% (annually)? P/Y = 12; N = 24 (= 2 x 12); I/Y = 8; PV = 0; PMT = \$1,542.42; FV = -40, You have borrowed \$10,000 at an interest rate of 8.7%. If you plan to pay the loan off in quarterly installments of \$1,000 (beginning next quarter), how long will it take you to pay back the loan? P/Y = 4; N = 11.40; I/Y = 8.7; PV = 10,000; PMT = -1,000; FV = quarters 2 years, 3 quarters or 2 years, 9 months 29. Value a perpetuity of \$400 per year (r = 14.9%). C 400 Value r \$2, If a perpetuity is worth \$1,000 and r = 15.5%, what is the cash flow? C Value C Value r 1, r \$

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