360 Chapter 10 Annuities 10.3 Future Value and Present Value of an Ordinary General Annuity 29. In an ordinary general annuity, payments are made at the end of each payment period and the compounding period is not equal to the payment period. In this case, if we find the equivalent periodic rate, i 2, that matches the payment period, then using this equivalent periodic rate we can use the ordinary simple annuity formula to calculate the future value and present value. To calculate this, we first find the number of compounding periods per payment period (c) and then the equivalent periodic interest rate per payment period (i 2 ) using the following two formulas: Formula 10.3(a) Number of Compounding Periods per Payment Period c = Formula 10.3(b) Equivalent Periodic Interest Rate per Payment Period i 2 = (1 + i) c - 1 Note: The value of 'c' as calculated from Formula 10.3(a) is used in calculating 'i 2 ' in Formula 10.3(b). Substitute the value of 'i 2 ' for 'i' in the ordinary simple annuity formulas, Formula 10.2(a) and Formula 10.2(b), to solve for 'FV' and 'PV' of an ordinary general annuity: n n ^1+ h - 1 1- ^1+ h FV = PMT ; E - and PV = PMT ; E Example 10.3(a) Calculating the Compounding Periods per Payment Period (c) and the Equivalent Periodic Interest Rate (i 2 ) that Matches the Payment Period Complete the table by calculating the number of compounding periods per payment period (c) and the equivalent periodic interest rate (i 2 ) that matches the payment period for the related interest rates: Solution Interest Rate 8% compounded quarterly Payment Period c i 2 Monthly = 0.006622... per month 9% compounded semi-annually Quarterly = 0.022252... per quarter 6% compounded monthly Semiannually = 0.030377... per half year 5% compounded annually Monthly = 0.004074... per month
Chapter 10 Annuities 361 The following two examples will demonstrate future value and present value calculations of an ordinary general annuity. Example 10.3(b) Calculating the Future Value of an Ordinary General Annuity Rachel would like to save $100 every month for the next four years in a savings account at 2% compounded quarterly. (i) What would be the accumulated value of the investments at the end of four years? (ii) What would be the amount of interest earned? Solution This is an ordinary general annuity as: Payments are made at the end of each payment period (monthly) Compounding period (quarterly) payment period (monthly) (i) Using Formula 10.3(a), Number of compounding periods per year 4 c = = Number of payments per year 12 Using Formula 10.3(b), i 2 = (1 + i) c - 1 = (1 + 0.005) (4/12) - 1 = 0.001663 per month Using Formula 10.2(a) and substituting i 2 for i, n ^1+ h - 1 FV = PMT ; E 48 ^1+ 0001663.... h - 1 = 100 ; E 0001663.... = 100 [49.925669...] = 4992.566985... = $4992.57 Therefore, the accumulated value of her investments at the end of four years would be $4992.57. (ii) Interest Earned = FV - n(pmt) = $4992.566985... - 48(100) = 192.566985... = $192.57 Enter payment periods per year first Next, Enter compoundings per year Therefore, the amount of interest earned over the time period would be $192.57.
362 Chapter 10 Annuities Example 10.3(c) Calculating the Present Value of an Ordinary General Annuity Joseph borrowed money from a bank at 6% compounded annually. He settled the loan by repaying $500 at the end of every month for six years. (i) What was the loan amount received? (ii) What was the amount of interest charged? Solution This is an ordinary general annuity as: Payments are made at the end of each payment period (monthly) Compounding period (annually) payment period (monthly) (i) Using Formula 10.3(a), Number of compounding periods per year 1 c = = Number of payments per year 12 Using Formula 10.3(b), i 2 = (1 + i) c - 1 = (1 + 0.06) (1/12) - 1 = 0.004867... per month Using Formula 10.2(b) and substituting i 2 for i, - n 1- ^1+ h PV = PMT ; E -72 1- ^1+ 0004867.... h = 500 ; E 0004867.... = 500[60.61537...] = 30,306.768845... Therefore, the loan amount received was $30,306.77. (ii) Interest Charged = n(pmt) - PV = 72(500) - 30,306.768845... = 5693.231164... = $5693.23 Therefore, the amount of interest charged was $5693.23 10.3 Exercises Answers to the odd-numbered problems are available at the end of the textbook 1. Calculate the number of compounding periods per payment period (expressed as a fraction, wherever applicable) and the equivalent periodic interest rate per payment period (rounded to six decimal places, wherever applicable) that matches the payment period for each of the following: a. Interest rate is 5% compounded quarterly. Payment period is semi-annually. b. Interest rate is 4.2% compounded daily. Payment period is monthly. c. Interest rate is 4.8% compounded monthly. Payment period is semi-annually. d. Interest rate is 4.9% compounded semi-annually. Payment period is quarterly.
Chapter 10 Annuities 363 2. Calculate the number of compounding periods per payment period (expressed as a fraction, wherever applicable) and the equivalent periodic interest rate per payment period (rounded to six decimal places, wherever applicable) that matches the payment period for each of the following: a. Interest rate is 4% compounded daily. Payment period is quarterly. b. Interest rate is 3.9% compounded quarterly. Payment period is monthly. c. Interest rate is 5% compounded semi-annually. Payment period is monthly. d. Interest rate is 4.9% compounded monthly. Payment period is quarterly. 3. Calculate the future value of each of the following ordinary general annuities: Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency a. $2200 Every year 15 years 4.80% Quarterly b. $1400 Every 6 months 16.5 years 4.00% Monthly c. $1000 Every 3 months 11 years and 3 months 3.75% Semi-annually d. $750 Every month 5 years 2 months 3.90% Daily 4. Calculate the future value of each of the following ordinary general annuities: Periodic Payment Payment Period Term of Annuity Interest Rate Compounding Frequency a. $1200 Every 3 months 8 years and 6 months 3.20% Monthly b. $400 Every month 15 years and 9 months 3.60% Semi-annually c. $1850 Every 6 months 7.5 years 4.40% Quarterly d. $3000 Every year 12 years 4.70% Daily 5. Calculate the present value of each of the ordinary general annuities in Problem 3. 6. Calculate the present value of each of the ordinary general annuities in Problem 4. 7. Adrian invested $100 at the end of every month into an RRSP for five years. If the RRSP was providing an interest rate of 5% compounded quarterly, how much did he have in the RRSP at the end of the five years? 8. Bina saved $250 at the end of every month for two years in a savings account that earns 5% compounded quarterly. How much would she have in the account at the end of two years and how much of this is the interest earned? 9. Calculate the accumulated value of end-of-quarter payments of $800 made at the following interest rates for five years: a. 6.23% compounded quarterly. b. 6.24% compounded semi-annually. 10. Calculate the future value of month-end payments of $180 made at the following interest rates for ten years: a. 7.8% compounded monthly. b. 8% compounded quarterly. 11. Barry has been contributing $1000 at the end of every three months into a retirement fund for the past 10 years. He decided to stop making payments and to allow his investment to grow for another 5 years. If money could earn 8% compounded semi-annually, how much interest would he have earned over the 15-year period? 12. Toby deposited $300 in a savings account at the end of every three months for five years. If the amount earned 5% compounded monthly, and if he left the accumulated money in the account to grow for another three years, calculate the balance in his savings account at the end of the period. What was his total deposit and total earnings?
364 Chapter 10 Annuities 13. Becca can afford to invest $400 at the end of every month in an RRSP for five years. By calculating the accumulated value of the payments in each of the following two investment options, identify the one that will give her the best return on her investment: RRSP 1: 3.55% compounded annually. RRSP 2: 3.50% compounded semi-annually. 14. If an annuity payment for five years is $200 at the end of every month, compare the future values if the investment earns 4% compounded monthly, 4% compounded quarterly, and 4% compounded semi-annually. 15. A credit union offers an interest rate of 4.5% compounded monthly for all investments. Which of the following would result in a higher future value and by how much more, when invested at the credit union? a. $18,250 invested for ten years. b. Annuity payments of $475 at the end of every quarter for ten years. 16. Calculate the accumulated value of the following investment options if the bank offers an interest rate of 5% compounded semi-annually: a. A single amount of $5000 saved for five years. b. A series of payments of $100 at the end of every month for five years. 17. Alejandro has $25,000 in a savings account that yields 5.75% compounded quarterly. He intends to use these savings for his retirement in five years. In addition to these savings, he intends to deposit $1250 at the end of every month in a mutual fund that yields the same return. How much money will he have available for his retirement in five years? 18. Lily has accumulated $90,000 in a mutual fund. If she continues to deposit $500 at the end of every month from her salary into the fund for the next ten years, how much money will she have at the end of ten years if the fund earns interest at 3.75% compounded quarterly? 19. Amanda invested $500 at the end of every month into an RRSP that had an interest rate of 3% compounded quarterly. Two years later, the interest rate on her RRSP increased to 3.25% compounded quarterly and remained the same, thereafter. What is the accumulated value of the RRSP in six years? 20. What will be the future value of a series of $1000 deposits made at the end of each quarter for ten years, if the interest rate is 5% compounded monthly for the first five years and 4% compounded annually for the next five years? 21. What is the discounted value of annuity payments of $2000 made at the end of every year for five years at 6% compounded semi-annually? 22. How much should Gilbert pay for a ten-year annuity that provides month-end payments of $1800 at 4.5% compounded quarterly? 23. A lottery winner is offered a choice between $100,000 now and another $100,000 in five years or month-end payments of $2300 for eight years. If money can earn 3.75% compounded semi-annually, which alternative is economically better (in current value) for her and by how much more? 24. If you win a lottery that entitles you to receive $250,000 now and another $125,000 in three years or month-end payments of $8300 for five years, which offer is economically better (in current value) for you and by how much more, if money can earn 3.65% compounded daily? 25. Calculate the price of Troy's car if he has to make a down payment of $1000 at the time of purchase and payments of $330 at the end of every month for 3 years and 3 months at an interest rate of 6% compounded semi-annually. What is the total interest he would pay for the car? 26. Byron Manufacturing Inc. paid $30,000 as a down payment to purchase a machine. It received a loan for the remaining amount at 4.5% compounded quarterly. What is the purchase price of the machine and the total amount of interest paid if it settled the loan with payments of $4500 made at the end of every month for five years?
Chapter 10 Annuities 365 27. Two annuities that provide end-of-quarter payments of $850 for a period of five years have the following interest rates: Annuity A: 9.45% compounded monthly. Annuity B: 9.50% compounded semi-annually. Which one would have a cheaper purchase price and by how much? 28. Ellen wants to receive a retirement income of $3000 every month for 20 years from her savings. If a bank in London was offering 6% compounded semi-annually, how much would she have to invest to get her planned retirement income? If a bank in Toronto was offering 5.98% compounded quarterly, by how much would it be cheaper for Ellen to invest in this bank instead of investing in the bank in London? 29. What is the purchase price of an annuity that provides month-end payments of $500 for the first two years and $1000 for the next three years? Assume that the interest rate is 3% compounded quarterly throughout the time period. 30. How much should Emma pay for an annuity that would give her $5000 at the end of every year for the first seven years and $8000 at the end of every year for the next four years, if the interest rate is 6% compounded semi-annually throughout the time period? 31. Starting at age 35, Gabriella invested $400 into an RRSP at the end of every month until her 45 th birthday and left the fund to grow under compound interest until her 65 th birthday. Starting at age 50, Gerard deposited $800 into a similar fund at the end of each month until his 65 th birthday. Assuming that both funds earned 6.4% compounded quarterly, who had more money and how much more in his or her fund at age 65? 32. Harry deposited $1000 in a retirement account at the end of each quarter for 20 years until he reached the age of 55, and then made no further deposits. His wife deposited $1000 in a retirement fund at the end of each quarter for 30 years until she reached the age of 65. Assuming that both funds earned 4.65% compounded monthly, who had more money and how much more in his or her fund at age 65? 33. A loan is settled by making payments of $2000 at the end of every three months for four years and then $500 at the end of every month for the next six years. What was the loan amount if the interest rate was 4.5% compounded monthly? 34. Georgia borrowed money from a bank at 6% compounded quarterly. She settled the loan by repaying $500 at the end of every month for the first two years and $2000 at the end of every three months for the next two years. What was the loan amount received? 35. 10.4 Future Value and Present Value of a Simple Annuity Due In a simple annuity due, payments are made at the beginning of each payment period, and the compounding period is equal to the payment period. In this section, you will learn how to calculate the future value and present value of a simple annuity due. Future Value of a Simple Annuity Due In an annuity due, as each payment is made at the beginning of the payment period, you will notice that the future value (accumulated value) of each payment is a multiple of the future value (accumulated value) of each payment in an ordinary annuity by a factor of (1+i). Let us compare two examples of an annuity with five annual payments, where in the first example, payments are made at the beginning of each year (annuity due) and in the second example, payments are made at the end of each year (ordinary annuity).