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Chapter 4

Chapter 4. Annuities. START. EXIT. Chapter Outline. 4.1 What Is an Annuity? 4.2 Future Values of Annuities 4.3 Sinking Funds 4.4 Present Values of Annuities 4.5 Amortization Tables Chapter Summary Chapter Exercises. 4.1 What Is an Annuity?.

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Chapter 4

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  1. Chapter 4 Annuities START EXIT

  2. Chapter Outline 4.1 What Is an Annuity? 4.2 Future Values of Annuities 4.3 Sinking Funds 4.4 Present Values of Annuities 4.5 Amortization Tables Chapter Summary Chapter Exercises

  3. 4.1 What Is an Annuity? • All of the situations we have considered so far, whether simple interest, simple discount, or compound interest, have had something in common. • In every case, a sum of money is lent and then the loan is repaid in full all at once at the maturity date. • However, take a car loan, for example. Suppose you borrow $14,500 at 8% compounded monthly for 5 years to buy a car. • It’s highly unlikely that you will get $14,500 up front, do nothing for 5 years, and then repay the entire loan. Instead, you would be making monthly payments.

  4. 4.1 What Is an Annuity? • Definition 4.1.1 An annuity is any collection of equal payments made at regular time intervals. • Examples • Car loan • Student loan • Mortgage • Paycheck (salary) • Rent • Pension or Social Security • These are NOT annuities: • Monthly credit card payment • Utility bills

  5. 4.1 What Is an Annuity? • Definition 4.1.2 A sum of money to which an annuity’s payments and interest accumulate in the end is called the annuity’s future value. A sum of money paid at the beginning of an annuity, to which the annuity’s payments are accepted as equivalent, is called the annuity’s present value.

  6. 4.1 What Is an Annuity? Example 4.1.1 • Problem • Dylan deposits $25 from each paycheck into a 401(k) savings plan at work. He will keep this up for the next 40 years, at which time he plans to retire, hopefully having accumulated a large balance in his account. Since equal payments are being made into the account at regular intervals, this is an annuity. Is the value of the account when Dylan retires a present value or a future value? • Solution • The value of the account when Dylan reaches retirement would be the future value.

  7. 4.1 What Is an Annuity? Example 4.1.1 Cont. • Problem • Terri borrowed $160,000 to buy a house. To pay off this mortgage loan, she agreed to make payments of $1,735.52 per month for 30 years. Since her mortgage payments are all equal and are made at regular intervals, they constitute an annuity. Was the amount she borrowed this annuity’s present value or future value? • Solution • The $160,000 was received at the start of the annuity payments. Therefore, it would be the present value of the annuity.

  8. 4.1 What Is an Annuity? • Definition 4.1.3 An ordinary annuity is an annuity whose payments are made at the end of each time period. However, end of the month refers to a month counted from the date the annuity begins, not necessarily the end of the calendar month. An annuity due is an annuity whose payments are made at the beginning of each time period.

  9. 4.1 What Is an Annuity? Example 4.1.2 • Problem • John took out a car loan on May 7. Payments will be made monthly. The first payment is not due until June 7 (the second will be due on July 7, etc.). Is this an ordinary annuity or an annuity due? • Solution • Because his first payment is not made until the end of the first month, the second at the end of the second month, and so on, his car payments are an ordinary annuity.

  10. 4.1 What Is an Annuity? Example 4.1.2 Cont. • Problem • Jenna won a lottery jackpot which will pay her $35,000 per year for the next 26 years. She does not have to wait an entire year to get her first check – she will be paid the first $35,000 right away. Her second payment will come a year from now, at the start of the second year, and so on. Is this an ordinary annuity or an annuity due? • Solution • Since the payments come at the start of each year, her prize payout is an annuity due.

  11. Section 4.1 Exercises

  12. Problem 1 • You just bought a car and will make monthly payments for the next five years. Is this an annuity? CHECK YOUR ANSWER

  13. Solution 1 • You just bought a car and will make monthly payments for the next five years. Is this an annuity? • Yes, an annuity is any collection of equal payments made at regular time intervals. BACK TO GAME BOARD

  14. Problem 2 • You just bought a house and took out a mortgage loan, agreeing to pay $738 every month for the next 30 years. Is this a present or future value? CHECK YOUR ANSWER

  15. Solution 2 • You just bought a house and took out a mortgage loan, agreeing to pay $738 every month for the next 30 years. Is this a present or future value? • It is a present value because the loan has been received at the start of the annuity payments. BACK TO GAME BOARD

  16. Problem 3 • Your rent is due at the end of each month. Is this an ordinary annuity or an annuity due? CHECK YOUR ANSWER

  17. Solution 3 • Your rent is due at the end of each month. Is this an ordinary annuity or an annuity due? • It’s an ordinary annuity because payments are made at the end of each time period. BACK TO GAME BOARD

  18. Problem 4 • You sponsor an orphan by paying $32 every month to www.compassion.com. Is this an annuity? CHECK YOUR ANSWER

  19. Solution 4 • You sponsor an orphan by paying $32 every month to www.compassion.com. Is this an annuity? • Yes, equal payments are made at the regular time intervals. BACK TO GAME BOARD

  20. 4.2 Future Values of Annuities • Suppose that you deposit $1,200 at the end of each year into an investment account that earns 7.2% compounded annually. Assuming you keep the payments up, how much would your account be worth in 5 year? • Your annual payments constitute an ordinary annuity. • Since we are asking about their accumulated value with interest at the end, we are looking for the future value.

  21. 4.2 Future Values of Annuities • Approach 1: The Chronological Approach • A natural way to approach the problem described on a previous slide is to build up the account value year by year, crediting interest as it comes due and adding new payments as they are made. • At the end of the first year, $1,200 is deposited. No interest would be paid since there was no money in the account prior to this deposit. • At the end of the second year, 1 year’s worth of interest would be paid on the $1,200, raising the account value to $1,200(1.072) = $1,286.40. In addition, $1,200 comes in from the second-year deposit, bringing the total to $2,486.40. • It’s obvious, however, that it’s a very tedious process.

  22. 4.2 Future Values of Annuities • Approach 2: The Bucket Approach • Suppose now that you opened a new account for each year’s deposit, instead of making them all to the same account. • The bucket approach arises from an observation that keeping separate accounts would make no difference whatsoever in the total end result; the final balance would be the same. • The first payment of $1,200 is placed in the first bucket. This money is kept on deposit from the end of year 1 until the end of year 5, a total of 4 years. At the end of the fifth year, the bucket will contain a total of $1,200(1.072)4 = $1,584.75. • We do same for each of the other five payments. • However, this approach is still rather tedious if there are numerous payments.

  23. 4.2 Future Values of Annuities • Approach 3: The Annuity Factor Approach • There is a third alternative. Suppose that instead of payments of $1,200, you made a payment of $3,600, three times as much. So to find the future value of the $3,600 annuity, we wouldn’t need to start from scratch; we could just multiply the $1,200 annuity’s future value by 3 to get FV = 3($6,928.48) = $20,785.44 • If you have any doubts about this, you can verify it for yourself by working through the future value of a $3,600 annuity using either of the approaches we used above. • In general, a larger or smaller payment changes the future value proportionately. • Exploiting this, we can define the future value annuity factor.

  24. 4.2 Future Values of Annuities • Definition 4.2.1 • For a given interest rate, payment frequency, and number of payments, the future value annuity factor is the future value that would accumulate if each payment were $1. We denote this factor with they symbol sn/i where n is the number of payments and i is the interest rate per payment period. • When we find annuity factors, we carry the results out to more than the usual two decimal places, to avoid losing too much in the rounding. • It is conventional not to write a dollar sign in front of the annuity factor.

  25. 4.2 Future Values of Annuities FORMULA 4.2.1 The Future Value of an Ordinary Annuity FV = PMTsn/i where FV represents the FUTURE VALUE of the annuity PMT represents the AMOUNT OF EACH PAYMENT and sn/i is the ANNUITY FACTOR

  26. 4.2 Future Values of Annuities • Table of Annuity Factors • One way to find annuity factors is simply to have a table listing the factors for certain interest rates and numbers of payments. • See Sample Table of Annuity Factors on page 186 of your textbook. • This table shows us to look up the value of the appropriate annuity factor for any of the interest rates and values of n that the table includes. • For example, if we need the annuity factor for a 15-year annual annuity at 8%, we would look in the n=15 row and 8% column and see that the value is 27.15211393.

  27. 4.2 Future Values of Annuities Example 4.2.1 • Problem • How much will I have as a future value if I deposit $3,000 at the end of each year into an account paying 6% compounded annually for 30 years? • Solution • The payments are equal and at regular intervals, and their timing is at the end of each period, so we have an ordinary annuity. • FV = PMTsni • FV = ($3,000)s30/0.06 = $3,000 x 79.05818622 = $237,174.56

  28. 4.2 Future Values of Annuities FORMULA 4.2.2 The Future Value Annuity Factor sn/i = where i represents the INTEREST RATE PER PAYMENT PERIOD and n represents the NUMBER OF PAYMENTS

  29. 4.2 Future Values of Annuities Example 4.2.2 • Problem • Find the future value annuity factor for a term of 20 years with an interest rate of 7.9% compounded annually. • Solution

  30. 4.2 Future Values of Annuities Example 4.2.3 • Problem • Suppose that $750 is deposited each year into an account paying 7.9% interest compounded annually. What will the future value of the account be? • Solution FV = PMTsn/i = $750 x 45.258204619 = $33,943.65

  31. 4.2 Future Values of Annuities Example 4.2.4 • Problem • Find the future value of quarterly payments of $750 for 5 years, assuming an 8% interest rate. • Solution i = 0.08/4 = 0.02 n = 5 x 4 = 20 FV = PMTsn/i = $750 x 24.2973698 = $18,223.03

  32. 4.2 Future Values of Annuities Example 4.2.5 • Problem • Find the future value annuity factor for a monthly annuity, assuming the term is 15 years and the interest rate is 7.1% compounded monthly. • Solution i = 0.071/12 = 0.005916667 n = 15 x 12 = 180

  33. 4.2 Future Values of Annuities Example 4.2.6 • Problem • Each month, Carrie deposits $250 into a savings account that pays 4.5% (compounded monthly). Assuming that she keeps this up, and that the interest rate does not change, how much will her deposits have grown to after 5 years? • Solution i = 0.045/12 = 0.00375 n = 5 x 12 = 60 FV = PMTsn/i = $250 x 67.14555214 = $16,786.39

  34. 4.2 Future Values of Annuities Example 4.2.7 • Problem • How much total interest did Carrie earn? • Solution • We first observe that the money in her account comes from two sources: the money she deposits and the interest she earns. • She made monthly deposits for 5 years, for a total of n = 60 deposits. • Each one was for $250, so in total she deposited 60 x $250 = $15,000. • The rest must have come from interest. • Carrie’s Total Interest = $16,786.39 -- $15,000 = $1,786.39

  35. 4.2 Future Values of Annuities • With an annuity due, payments are made at the beginning of each period rather than the end. • Each payment is made earlier, so it stands to reason that an annuity due would have a larger future value than an ordinary annuity, since the payments have longer to earn interest. • To see how much more, let’s revisit the 5 year, $1,200 per year annuity with 7.2% interest. • This time, though, as an annuity due. • The first payment would earn interest from the start of the first year until the end of the fifth year, for a total of 5 years. • The second payment would earn interest for 4 years, the third payment for 3 years, and so on. • Since there is an extra year to earn interest, the maturity value is $7,427.33 instead of $6,928.48. • See table on page 186 in your textbook.

  36. 4.2 Future Values of Annuities FORMULA 4.2.3 Future Value for an Annuity Due FV = PMTsn/i(1 + i) where FV represents the future value of the annuity PMT is the amount of each payment i is the interest rate per period and sn/i is the annuity factor

  37. 4.2 Future Values of Annuities Example 4.2.8 • Problem • On New Year’s Day 2004, Max resolved to deposit $3,000 at the start of each year into a retirement savings account. Assuming that he sticks to this resolution, and that his account earns 8 ¼% compounded annually, how much will he have after 40 years? • Solution • The payments are equal and made at the start of each year, so this is an annuity due. FV = PMTsn/i(1 + i) FV = $3,000 x 276.72205752 x (1 + 0.0825) = $898,654.88

  38. Section 4.2 Exercises

  39. Problem 1 • You decided to invest $500 at the end of each year into an investment account that earns 5% interest compounded annually for 20 years. Find the future value of the annuity. CHECK YOUR ANSWER

  40. Solution 1 • You decided to invest $500 at the end of each year into an investment account that earns 5% interest compounded annually for 20 years. Find the future value of the annuity. • i = 0.05 • n = 20 FV = PMTsn/i FV = $500 x 33.0659541 = $16,532.98 BACK TO GAME BOARD

  41. Problem 2 • At the end of each month, Laura deposits $100 into her savings account that pays 2.5% interest. Assuming that she keeps doing it for the next 10 years, how much will her deposits grow to? CHECK YOUR ANSWER

  42. Solution 2 • At the end of each month, Laura deposits $100 into her savings account that pays 2.5% interest. Assuming that she keeps doing it for the next 10 years, how much will her deposits grow to? • i = 0.025/12 = 0.002083333 • n = 12 x 10 = 120 FV = PMTsn/i FV = $100 x 136.1719618 = $13,617.20 BACK TO GAME BOARD

  43. Problem 3 • How much total interest would Laura earn? CHECK YOUR ANSWER

  44. Solution 3 • How much total interest would Laura earn? • Total Payments = $100 x 120 = $12,000 • Future Value = $13,617.20 • Total Interest = $13,617.20 -- $12,000 = $1,617.20 BACK TO GAME BOARD

  45. Problem 4 • Suppose Laura deposits $100 into her savings account that pays 2.5% interest. However, she does it on the first day of each month. Assuming that she keeps doing it for the next 10 years, how much will her deposits grow to? CHECK YOUR ANSWER

  46. Solution 4 • Suppose Laura deposits $100 into her savings account that pays 2.5% interest. However, she does it on the first day of each month. Assuming that she keeps doing it for the next 10 years, how much more will she earn? • i = 0.025/12 = 0.002083333 • n = 12 x 10 = 120 FV = PMTsn/i(1 + i) FV = $100 x 136.1719618(1 + 0.002083333) = $13,645.57 $13,645.57 -- $13,617.20 = $28.37 BACK TO GAME BOARD

  47. 4.3 Sinking Funds • So far, we have looked at annuities from the point of view that the payments determine the future value. • If instead we set up an annuity with a future value goal in mind, we would need to look at things in the opposite direction. • We call such annuities sinking funds. • Definition • A sinking fund is an annuity for which the amount of the payments is determined by the future value desired.

  48. 4.3 Sinking Funds Example 4.3.1 • Problem • Suppose Calvin has set a goal of having $10,000 in a savings account in 5 years. He plans to make equal deposits to the account at the end of each month, and expects the account to earn 3.6% interest. How much should each of his deposits be? • Solution i = 0.036/12 = 0.003 n = 5 x 12 = 60 FV = PMTsn/i $10,000 = PMT x 65.63160098 PMT = $152.37

  49. 4.3 Sinking Funds Example 4.3.2 • Problem • Shannon owns a software development company, and as part of a new product she has licensed the right to include in it some code owned by her friend Elena. To allow Shannon time to develop and market the product, Elena has agreed to wait 2 years before getting the $10,000 Shannon has agreed to pay. If, in anticipation of paying Elena, she decides to make equal deposits at the start of each quarter into an account paying 4.8%, how much should each deposit be? • Solution i = 0.048/4 = 0.012 n = 2 x 4 = 8 FV = PMTsn/i(1 + i) $10,000 = PMT x 8.344186128 x (1 + 0.012) PMT = $1,184.23

  50. 4.3 Sinking Funds • Sinking Funds with Loans • Knowing that her company had a large expense ahead on the horizon, it was prudent of Shannon to set up a fund to be able to meet it, rather than just wait and then get hit with a major expense all at once. • She may or may not have been under any obligation to do this. • However, what if Elena is concerned with Shannon’s inability to come up with a payment at the end of 2 years? Given this possibility, Elena has a good reason to want Shannon to be building up that balance along the way, so it’s possible that Elena might insist that Shannon sets up a sinking fund and makes regular payments.

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