Mathematics of Finance

Transcription

1 Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY 5 Contnuous Compoundng Most people must take out a loan for a bg purchase, such as a car, a major applance, or a house. People who carry a balance on ther credt cards are, n effect, also borrowng money. Loan payments must be accurately determned, and t may take some work to f nd the best deal. See Exercse 54 on page 262 and Exercse 57 on page 242. We must all plan for eventual retrement, whch usually nvolves savngs accounts and nvestments n stocks, bonds, and annutes to fund 401K accounts or ndvdual retrement accounts (IRAs). See Exercses 40 and 41 on page 250. It s mportant for both busnesspersons and consumers to understand the mathematcs of fnance n order to make sound fnancal decsons. Interest formulas for borrowng and nvestng money are ntroduced n ths chapter. NOTE We try to present realstc, up-to-date applcatons n ths text. Because nterest rates change so frequently, however, t s very unlkely that the rates n effect when ths chapter was wrtten are the same as the rates today when you are readng t. Fortunately, the mathematcs of f nance s the same regardless of the level of nterest rates. So we have used a varety of rates n the examples and exercses. Some wll be realstc and some won t by the tme you see them but all of them have occurred n the past several decades. 5.1 Smple Interest and Dscount Interest s the fee pad to use someone else s money. Interest on loans of a year or less s frequently calculated as smple nterest, whch s pad only on the amount borrowed or nvested and not on past nterest. The amount borrowed or deposted s called the 225

2 226 CHAPTER 5 Mathematcs of Fnance prncpal. The rate of nterest s gven as a percent per year, expressed as a decmal. For example, 6% =.06 and % =.115. The tme durng whch the money s accrung nterest s calculated n years. Smple nterest s the product of the prncpal, rate, and tme. Smple Interest The smple nterest I on P dollars at a rate of nterest r per year for t years s I = Prt. It s customary n fnancal problems to round nterest to the nearest cent. Example 1 To furnsh her new apartment, Magge Chan borrowed $4000 at 3% nterest from her parents for 9 months. How much nterest wll she pay? Soluton Use the formula I = Prt, wth P = 4000, r = 0.03, and t = 9>12 = 3>4 years: I = Prt Checkpont 1 Fnd the smple nterest for each loan. (a) $2000 at 8.5% for 10 months (b) $3500 at 10.5% for years Answers to Checkpont exercses are found at the end of the secton. I = 4000 *.03 * 3 4 = 90. Thus, Magge pays a total of $90 n nterest. 1 Smple nterest s normally used only for loans wth a term of a year or less. A sgnfcant excepton s the case of corporate bonds and smlar fnancal nstruments. A typcal bond pays smple nterest twce a year for a specfed length of tme, at the end of whch the bond matures. At maturty, the company returns your ntal nvestment to you. Checkpont 2 For the gven bonds, fnd the semannual nterest payment and the total nterest pad over the lfe of the bond. (a) $7500 Tme Warner Cable, Inc. 30-year bond at 7.3% annual nterest. (b) $15,000 Clear Channel Communcatons 10-year bond at 9.0% annual nterest. Example 2 Fnance On January 8, 2013, Bank of Amerca ssued 10-year bonds at an annual smple nterest rate of 3.3%, wth nterest pad twce a year. John Altere buys a $10,000 bond. (Data from: (a) How much nterest wll he earn every sx months? Soluton Use the nterest formula, I = Prt, wth P = 10,000, r =.033, and t = 1 2 : I = Prt = 10,000 *.033 * 1 2 = $165. (b) How much nterest wll he earn over the 10-year lfe of the bond? Soluton Ether use the nterest formula wth t = 10, that s, I = Prt = 10,000 *.033 * 10 = $3300, or take the answer n part (a), whch wll be pad out every sx months for 10 years for a total of twenty tmes. Thus, John would obtan $165 * 20 = $

3 5.1 Smple Interest and Dscount 227 Future Value If you depost P dollars at smple nterest rate r for t years, then the future value (or maturty value ) A of ths nvestment s the sum of the prncpal P and the nterest I t has earned: A = Prncpal + Interest = P + I The followng box summarzes ths result. = P + Prt I = Prt. = P(1 + rt). Factor out P. Future Value (or Maturty Value) for Smple Interest The future value (maturty value) A of P dollars for t years at nterest rate r per year s A = P + I, or A = P(1 + rt). Example 3 Fnd each maturty value and the amount of nterest pad. (a) Rck borrows $20,000 from hs parents at 5.25% to add a room on hs house. He plans to repay the loan n 9 months wth a bonus he expects to receve at that tme. Soluton The loan s for 9 months, or 9>12 of a year, so t =.75, P = 20,000, and r = Use the formula to obtan A = P(1 + rt) = 20,000[ (.75)] 20,787.5, Use a calculator. or $20, The maturty value A s the sum of the prncpal P and the nterest I, that s, A = P + I. To fnd the amount of nterest pad, rearrange ths equaton: I = A - P I = $20, $20,000 = $ (b) A loan of $11,280 for 85 days at 9% nterest. Soluton Use the formula A = P(1 + rt), wth P = 11,280 and r =.09. Unless stated otherwse, we assume a 365-day year, so the perod n years s t = 85>365. The maturty value s Checkpont 3 Fnd each future value. (a) $1000 at 4.6% for 6 months (b) $8970 at 11% for 9 months (c) $95,106 at 9.8% for 76 days As n part (a), the nterest s A = P(1 + rt) A = 11,280a * b 11,280( ) $11, I = A - P = $11, $11,280 = $

4 228 CHAPTER 5 Mathematcs of Fnance Example 4 Suppose you borrow $15,000 and are requred to pay $15,315 n 4 months to pay off the loan and nterest. What s the smple nterest rate? Soluton One way to fnd the rate s to solve for r n the future-value formula when P = 15,000, A = 15,315, and t = 4>12 = 1>3: P(1 + rt) = A 15,000a1 + r * 1 3 b = 15,315 Checkpont 4 You lend a frend $500. She agrees to pay you $520 n 6 months. What s the nterest rate? 15, ,000r 3 = 15,315 Multply out left sde. 15,000r 3 = 315 Subtract 15,000 from both sdes. 15,000r = 945 Multply both sdes by 3. r = 945 = ,000 Dvde both sdes by 15,000. Therefore, the nterest rate s 6.3%. 4 Present Value A sum of money that can be deposted today to yeld some larger amount n the future s called the present value of that future amount. Present value refers to the prncpal to be nvested or loaned, so we use the same varable P as we dd for prncpal. In nterest problems, P always represents the amount at the begnnng of the perod, and A always represents the amount at the end of the perod. To fnd a formula for P, we begn wth the future-value formula: A = P(1 + rt). Dvdng each sde by 1 + rt gves the followng formula for the present value. Present Value for Smple Interest The present value P of a future amount of A dollars at a smple nterest rate r for t years s P = A 1 + rt. Checkpont 5 Fnd the present value of the gven future amounts. Assume 6% nterest. (a) $7500 n 1 year (b) $89,000 n 5 months (c) $164,200 n 125 days Example 5 Soluton P = Fnd the present value of $32,000 n 4 months at 9% nterest. A 1 + rt = 32, (.09)a 4 12 b = 32, = 31, A depost of $31, today at 9% nterest would produce $32,000 n 4 months. These two sums, $31, today and $32, n 4 months, are equvalent (at 9%) because the frst amount becomes the second amount n 4 months. 5

5 5.1 Smple Interest and Dscount 229 Checkpont 6 Jerrell Davs s owed $19,500 by Chrstne O Bren. The money wll be pad n 11 months, wth no nterest. If the current nterest rate s 10%, how much should Davs be wllng to accept today n settlement of the debt? Example 6 Because of a court settlement, Jeff Wedenaar owes $5000 to Chuck Synovec. The money must be pad n 10 months, wth no nterest. Suppose Wedenaar wants to pay the money today and that Synovec can nvest t at an annual rate of 5%. What amount should Synovec be wllng to accept to settle the debt? Soluton The $5000 s the future value n 10 months. So Synovec should be wllng to accept an amount that wll grow to $5000 n 10 months at 5% nterest. In other words, he should accept the present value of $5000 under these crcumstances. Use the present-value formula wth A = 5000, r =.05, and t = 10>12 = 5>6: P = A 1 + rt = * 5 6 = Synovec should be wllng to accept $4800 today n settlement of the debt. 6 Example 7 Larry Parks owes $6500 to Vrgna Donovan. The loan s payable n one year at 6% nterest. Donovan needs cash to pay medcal blls, so four months before the loan s due, she sells the note (loan) to the bank. If the bank wants a return of 9% on ts nvestment, how much should t pay Donovan for the note? Soluton Frst fnd the maturty value of the loan the amount (wth nterest) that Parks must pay Donovan: A = P(1 + rt) Maturty-value formula = 6500( * 1) Let P = 6500, r =.06, and t = 1. = 6500(1.06) = $6890. In four months, the bank wll receve $6890. Snce the bank wants a 9% return, compute the present value of ths amount at 9% for four months: Checkpont 7 A frm accepts a $21,000 note due n 8 months, wth nterest of 10.5%. Two months before t s due, the frm sells the note to a broker. If the broker wants a 12.5% return on hs nvestment, how much should he pay for the note? P = = Dscount A 1 + rt 6890 Present-value formula a 4 12 b = $ Let A = 6890, r =.09, and t = 4,12. The bank pays Donovan $ and four months later collects $6890 from Parks. 7 The precedng examples dealt wth loans n whch money s borrowed and smple nterest s charged. For most loans, both the prncpal (amount borrowed) and the nterest are pad at the end of the loan perod. Wth a corporate bond (whch s a loan to a company by the nvestor who buys the bond), nterest s pad durng the lfe of the bond and the prncpal s pad back at maturty. In both cases, the borrower receves the prncpal, but pays back the prncpal plus the nterest. In a smple dscount loan, however, the nterest s deducted n advance from the amount of the loan and the balance s gven to the borrower. The full value of the loan must be pad back at maturty. Thus, the borrower receves the prncpal less the nterest, but pays back the prncpal.

6 230 CHAPTER 5 Mathematcs of Fnance The most common examples of smple dscount loans are U.S. Treasury blls (T-blls), whch are essentally short-term loans to the U.S. government by nvestors. T-blls are sold at a dscount from ther face value and the Treasury pays back the face value of the T-bll at maturty. The dscount amount s the nterest deducted n advance from the face value. The Treasury receves the face value less the dscount, but pays back the full face value. Checkpont 8 The maturty tmes and dscount rates for $10,000 T-blls sold on March 7, 2013, are gven. Fnd the dscount amount and the prce of each T-bll. (a) one year;.15% (b) sx months;.12% (c) three months;.11% Checkpont 9 Fnd the actual nterest rate pad by the Treasury for each T-bll n Checkpont 8. Example 8 Fnance An nvestor bought a sx-month $8000 treasury bll on February 28, 2013 that sold at a dscount rate of.135%. What s the amount of the dscount? What s the prce of the T-bll? (Data from: Soluton The dscount rate on a T-bll s always a smple annual nterest rate. Consequently, the dscount (nterest) s found wth the smple nterest formula, usng P = 8000 (face value), r = (dscount rate), and t =.5 (because 6 months s half a year): Dscount = Prt = 8000 * *.5 = $5.40. So the prce of the T-bll s Face Value - Dscount = = $ In a smple dscount loan, such as a T-bll, the dscount rate s not the actual nterest rate the borrower pays. In Example 8, the dscount rate.135% was appled to the face value of $8000, rather than the $ that the Treasury (the borrower) receved. Example 9 Fnance Fnd the actual nterest rate pad by the Treasury n Example 8. Soluton Use the formula for smple nterest, I = Prt wth r as the unknown. Here, P = (the amount the Treasury receved) and I = 5.40 (the dscount amount). Snce ths s a sx-month T-bll, t =.5, and we have I = Prt 5.40 = (r)(.5) 5.40 = r Multply out rght sde. r = Dvde both sdes by So the actual nterest rate s.13509% Exercses Unless stated otherwse, nterest means smple nterest, and nterest rate and dscount rate refer to annual rates. Assume 365 days n a year. 1. What factors determne the amount of nterest earned on a fxed prncpal? Fnd the nterest on each of these loans. (See Example 1.) 2. $35,000 at 6% for 9 months 3. $2850 at 7% for 8 months 4. $1875 at 5.3% for 7 months 5. $3650 at 6.5% for 11 months 6. $5160 at 7.1% for 58 days 7. $2830 at 8.9% for 125 days 8. $8940 at 9%; loan made on May 7 and due September $5328 at 8%; loan made on August 16 and due December $7900 at 7%; loan made on July 7 and due October 25 Fnance For each of the gven corporate bonds, whose nterest rates are provded, fnd the semannual nterest payment and the total nterest earned over the lfe of the bond. (See Example 2, Data from: $5000 IBM, 3-year bond; 1.25% 12. $9000 Barrck Gold Corp., 10-year bond; 3.85% 13. $12,500 Morgan Stanley, 10-year bond; 3.75%

7 5.1 Smple Interest and Dscount $4500 Goldman Sachs, 3-year bond; 6.75% 15. $6500 Amazon.com Corp, 10-year bond; 2.5% 16. $10,000 Wells Fargo, 10-year bond; 3.45% Fnd the future value of each of these loans. (See Example 3.) 17. $12,000 loan at 3.5% for 3 months 18. $3475 loan at 7.5% for 6 months 19. $6500 loan at 5.25% for 8 months 20. $24,500 loan at 9.6% for 10 months 21. What s meant by the present value of money? 22. In your own words, descrbe the maturty value of a loan. Fnd the present value of each future amount. (See Examples 5 and 6.) 23. $15,000 for 9 months; money earns 6% 24. $48,000 for 8 months; money earns 5% 25. $15,402 for 120 days; money earns 6.3% 26. $29,764 for 310 days; money earns 7.2% Fnance The gven treasury blls were sold on Aprl 4, Fnd (a) the prce of the T-bll, and (b) the actual nterest rate pad by the Treasury. (See Examples 8 and 9. Data from: www. treasurydrect.gov.) 27. Three-month $20,000 T-bll wth dscount rate of.075% 28. One-month $12,750 T-bll wth dscount rate of.070% 29. Sx-month $15,500 T-bll wth dscount rate of.105% 30. One-year $7000 T-bll wth dscount rate of.140% Fnance Hstorcally, treasury blls offered hgher rates. On March 9, 2007 the dscount rates were substantally hgher than n Aprl, For the followng treasury blls bought n 2007, fnd (a) the prce of the T-bll, and (b) the actual nterest rate pad by the Treasury. (See Examples 8 and 9. Data from: www. treasury.gov.) 31. Three-month $20,000 T-bll wth dscount rate of 4.96% 32. One-month $12,750 T-bll wth dscount rate of 5.13% 33. Sx-month $15,500 T-bll wth dscount rate of 4.93% 34. Sx-month $9000 T-bll wth dscount rate of 4.93% Fnance Work the followng appled problems. 35. In March 1868, Wnston Churchll s grandfather, L.W. Jerome, ssued $1000 bonds (to pay for a road to a race track he owned n what s now the Bronx). The bonds carred a 7% annual nterest rate payable semannually. Mr. Jerome pad the nterest untl March 1874, at whch tme New York Cty assumed responsblty for the bonds (and the road they fnanced). (Data from: New York Tmes, February 13, 2009.) (a) The frst of these bonds matured n March At that tme, how much nterest had New York Cty pad on ths bond? (b) Another of these bonds wll not mature untl March 2147! At that tme, how much nterest wll New York Cty have pad on t? 36. An accountant for a corporaton forgot to pay the frm s ncome tax of $725, on tme. The government charged a penalty of 9.8% nterest for the 34 days the money was late. Fnd the total amount (tax and penalty) that was pad. 37. Mke Branson nvested hs summer earnngs of $3000 n a savngs account for college. The account pays 2.5% nterest. How much wll ths amount to n 9 months? 38. To pay for textbooks, a student borrows $450 from a credt unon at 6.5% smple nterest. He wll repay the loan n 38 days, when he expects to be pad for tutorng. How much nterest wll he pay? 39. An account nvested n a money market fund grew from $67, to $67, n a month. What was the nterest rate, to the nearest tenth? 40. A $100,000 certfcate of depost held for 60 days s worth $101, To the nearest tenth of a percent, what nterest rate was earned? 41. Dave took out a $7500 loan at 7% and eventually repad $7675 (prncpal and nterest). What was the tme perod of the loan? 42. What s the tme perod of a $10,000 loan at 6.75%, n whch the total amount of nterest pad was $618.75? 43. Tuton of $1769 wll be due when the sprng term begns n 4 months. What amount should a student depost today, at 3.25%, to have enough to pay the tuton? 44. A f rm of accountants has ordered 7 new computers at a cost of $5104 each. The machnes wll not be delvered for 7 months. What amount could the frm depost n an account payng 6.42% to have enough to pay for the machnes? 45. John Sun Yee needs $6000 to pay for remodelng work on hs house. A contractor agrees to do the work n 10 months. How much should Yee depost at 3.6% to accumulate the $6000 at that tme? 46. Lore Relly decdes to go back to college. For transportaton, she borrows money from her parents to buy a small car for $7200. She plans to repay the loan n 7 months. What amount can she depost today at 5.25% to have enough to pay off the loan? 47. A sx-month $4000 Treasury bll sold for $3930. What was the dscount rate? 48. A three-month $7600 Treasury bll carres a dscount of $ What s the dscount rate for ths T-bll? Fnance Work the next set of problems, n whch you are to fnd the annual smple nterest rate. Consder any fees, dvdends, or profts as part of the total nterest. 49. A stock that sold for $22 at the begnnng of the year was sellng for $24 at the end of the year. If the stock pad a dvdend of $.50 per share, what s the smple nterest rate on an nvestment n ths stock? ( Hnt: Consder the nterest to be the ncrease n value plus the dvdend.)

8 232 CHAPTER 5 Mathematcs of Fnance 50. Jerry Ryan borrowed $8000 for nne months at an nterest rate of 7%. The bank also charges a $100 processng fee. What s the actual nterest rate for ths loan? 51. You are due a tax refund of $760. Your tax preparer offers you a no-nterest loan to be repad by your refund check, whch wll arrve n four weeks. She charges a $60 fee for ths servce. What actual nterest rate wll you pay for ths loan? ( Hnt: The tme perod of ths loan s not 4>52, because a 365-day year s 52 weeks and 1 day. So use days n your computatons.) 52. Your cousn s due a tax refund of $400 n sx weeks. Hs tax preparer has an arrangement wth a bank to get hm the $400 now. The bank charges an admnstratve fee of $29 plus nterest at 6.5%. What s the actual nterest rate for ths loan? (See the hnt for Exercse 51.) Fnance Work these problems. (See Example 7.) 53. A buldng contractor gves a $13,500 promssory note to a plumber who has loaned hm $13,500. The note s due n nne months wth nterest at 9%. Three months after the note s sgned, the plumber sells t to a bank. If the bank gets a 10% return on ts nvestment, how much wll the plumber receve? Wll t be enough to pay a bll for $13,650? 54. Shala Johnson owes $7200 to the Eastsde Musc Shop. She has agreed to pay the amount n seven months at an nterest rate of 10%. Two months before the loan s due, the store needs $7550 to pay a wholesaler s bll. The bank wll buy the note, provded that ts return on the nvestment s 11%. How much wll the store receve? Is t enough to pay the bll? 55. Let y 1 be the future value after t years of $100 nvested at 8% annual smple nterest. Let y 2 be the future value after t years of $200 nvested at 3% annual smple nterest. (a) Thnk of y 1 and y 2 as functons of t and wrte the rules of these functons. (b) Wthout graphng, descrbe the graphs of y 1 and y 2. (c) Verfy your answer to part (b) by graphng y 1 and y 2 n the frst quadrant. (d) What do the slopes and y -ntercepts of the graphs represent (n terms of the nvestment stuaton that they descrbe)? 56. I f y = 16.25t and y s the future value after t years of P dollars at nterest rate r, what are P and r? ( Hnt: See Exercse 55.) Checkpont Answers 1. (a) $ (b) $ (a) $273.75; $16,425 (b) $675; $13, (a) $1023 (b) $ (c) $97, % 5. (a) $ (b) $86, (c) $160, $17, $22, (a) $15; $9985 (b) $6; $9994 (c) $2.75; $ (a) About.15023% (b) About.12007% (c) About.11003% 5.2 Compound Interest Wth annual smple nterest, you earn nterest each year on your orgnal nvestment. Wth annual compound nterest, however, you earn nterest both on your orgnal nvestment and on any prevously earned nterest. To see how ths process works, suppose you depost $1000 at 5% annual nterest. The followng chart shows how your account would grow wth both smple and compound nterest: End of Year SIMPLE INTEREST Interest Earned Balance COMPOUND INTEREST Interest Earned Balance Orgnal Investment: $1000 Orgnal Investment: $ (.05) = $50 $ (.05) = $50 $ (.05) = $50 $ (.05) = $52.50 $ (.05) = $50 $ (.05) = $55.13 * $ As the chart shows, smple nterest s computed each year on the orgnal nvestment, but compound nterest s computed on the entre balance at the end of the precedng year. So smple nterest always produces $50 per year n nterest, whereas compound nterest *Rounded to the nearest cent.

9 5.2 Compound Interest 233 Checkpont 1 Extend the chart n the text by fndng the nterest earned and the balance at the end of years 4 and 5 for (a) smple nterest and (b) compound nterest. produces $50 nterest n the frst year and ncreasngly larger amounts n later years (because you earn nterest on your nterest). 1 Example 1 If $7000 s deposted n an account that pays 4% nterest compounded annually, how much money s n the account after nne years? Soluton After one year, the account balance s % of 7000 = (.04)7000 = 7000(1 +.04) Dstrbutve property = 7000(1.04) = $7280. The ntal balance has grown by a factor of At the end of the second year, the balance s % of 7280 = (.04)7280 = 7280(1 +.04) Dstrbutve property = 7280(1.04) = Once agan, the balance at the begnnng of the year has grown by a factor of Ths s true n general: If the balance at the begnnng of a year s P dollars, then the balance at the end of the year s So the account balance grows lke ths: P + 4% of P = P +.04P = P(1 +.04) = P(1.04). Year 1 Year 2 Year S 7000(1.04) S [7000(1.04)](1.04) S [7000(1.04)(1.040)](1.04) S g. 7000(1.04) (1.04) 3 $''''%''''& At the end of nne years, the balance s $'''''%'''''& 7000(1.04) 9 = $ (rounded to the nearest penny). The argument used n Example 1 apples n the general case and leads to ths concluson. Compound Interest If P dollars are nvested at nterest rate per perod, then the compound amount (future value) A after n compoundng perods s A = P(1 + ) n. In Example 1, for nstance, we had P = 7000, n = 9, and =.04 (so that 1 + = = 1.04). NOTE Compare ths future value formula for compound nterest wth the one for smple nterest from the prevous secton, usng t as the number of years: Compound nterest A = P(1 + r) t ; Smple nterest A = P(1 + rt). The mportant dstncton between the two formulas s that, n the compound nterest formula, the number of years, t, s an exponent, so that money grows much more rapdly when nterest s compounded.

10 234 CHAPTER 5 Mathematcs of Fnance Checkpont 2 Suppose $17,000 s deposted at 4% compounded annually for 11 years. (a) Fnd the compound amount. (b) Fnd the amount of nterest earned. Example 2 Suppose $1000 s deposted for sx years n an account payng 8.31% per year compounded annually. (a) Fnd the compound amount. Soluton In the formula above, P = 1000, =.0831, and n = 6. The compound amount s A = P(1 + ) n A = 1000(1.0831) 6 A = $ (b) Fnd the amount of nterest earned. Soluton Subtract the ntal depost from the compound amount: Amount of nterest = $ $1000 = $ TECHNOLOGY TIP Spreadsheets are deal for performng f nancal calculatons. Fgure 5.1 shows a Mcrosoft Excel spreadsheet wth the formulas for compound and smple nterest used to create columns B and C, respectvely, when $1000 s nvested at an annual rate of 10%. Notce how rapdly the com pound amount ncreases compared wth the maturty value wth smple nterest. For more detals on the use of spreadsheets n the mathematcs of f nance, see the Spread sheet Manual that s avalable wth ths text. Fgure 5.1 Example 3 If a $16,000 nvestment grows to $50,000 n 18 years, what s the nterest rate (assumng annual compoundng)? Soluton Use the compound nterest formula, wth P = 16,000, A = 50,000, and n = 18, and solve for : P(1 + ) n = A 16,000(1 + ) 18 = 50,000 (1 + ) 18 = 50,000 = ,000 Dvde both sdes by 16, (1 + ) 18 = Take 18th roots on both sdes. 1 + = = Subtract 1 from both sdes. So the nterest rate s about 6.535%.

11 5.2 Compound Interest 235 Interest can be compounded more than once a year. Common compoundng perods nclude semannually (2 perods per year), quarterly (4 perods per year), monthly (12 perods per year), and daly (usually 365 perods per year). When the annual nterest s compounded m tmes per year, the nterest rate per perod s understood to be >m. Example 4 Fnance In Aprl 2013, advertsed a oneyear certfcate of depost (CD) for GE Captal Retal Bank at an nterest rate of 1.05%. Fnd the value of the CD f $10,000 s nvested for one year and nterest s compounded accordng to the gven perods. (a) Annually Soluton Apply the formula A = P(1 + ) n wth P = 10,000, =.0105, and n = 1: A = P(1 + ) n = 10,000( ) 1 = 10,000(1.0105) = $10,105. (b) Semannually Soluton Use the same formula and value of P. Here nterest s compounded twce a year, so the number of perods s n = 2 and the nterest rate per perod s = : A = P(1 + ) n = 10,000 a b 2 = $10, (c) Quarterly Soluton Proceed as n part (b), but now nterest s compounded 4 tmes a year, and so n = 4 and the nterest rate per perod s = : (d) Monthly A = P(1 + ) n = 10,000 a b 4 = $10, Soluton Interest s compounded 12 tmes a year, so n = 12 and = : (e) Daly A = P(1 + ) n = 10,000 a b 12 = $10, Soluton Interest s compounded 365 tmes a year, so n = 365 and = : A = P(1 + ) n = 10,000 a b 365 = $10, Example 5 Fnance The gven CDs were advertsed onlne by varous banks n Aprl Fnd the future value of each one. (Data from: cdrates.bankaholc.com.)

12 236 CHAPTER 5 Mathematcs of Fnance Checkpont 3 Fnd the future value for these CDs. (a) Natonal Republc Bank of Chcago: $1000 at 1.3% compounded monthly for 3 years. (b) Dscover Bank: $2500 at.8% compounded daly for 9 months (assume 30 days n each month). (a) Natonwde Bank: $100,000 for 5 years at 1.73% compounded daly. Soluton Use the compound nterest formula wth P = 100,000. Interest s compounded 365 tmes a year, so the nterest rate per perod s = Snce there are fve years, the 365 number of perods n 5 years s n = 365(5) = The future value s A = P(1 + ) n = 100,000 a b 1825 = $109, (b) Calforna Frst Natonal Bank: $5000 for 2 years at 1.06% compounded monthly. Soluton Use the compound nterest formula wth P = Interest s compounded 12 tmes a year, so the nterest rate per perod s = Snce there are two years, the 12 number of perods n 2 years s n = 12(2) = 24. The future value s A = P(1 + ) n = 5000a b 24 = $ Example 4 shows that the more often nterest s compounded, the larger s the amount of nterest earned. Snce nterest s rounded to the nearest penny, however, there s a lmt on how much can be earned. In Example 4, part (e), for nstance, that lmt of $10, has been reached. Nevertheless, the dea of compoundng more and more frequently leads to a method of computng nterest called contnuous compoundng that s used n certan fnancal stuatons The formula for contnuous compoundng s developed n Case 5, but the formula s gven n the followng box where e = , whch was ntroduced n Chapter 4. Contnuous Compound Interest The compound amount A for a depost of P dollars at an nterest rate r per year compounded contnuously for t years s gven by A = Pe rt. Checkpont 4 Fnd the compound amount for $7500 nvested at an annual nterest rate of 2.07% compounded contnuously for 3 years. Example 6 Suppose that $5000 s nvested at an annual nterest rate of 3.1% compounded contnuously for 4 years. Fnd the compound amount. Soluton In the formula for contnuous compoundng, let P = 5000, r =.031, and t = 4. Then a calculator wth an e x key shows that A = Pe rt = 5000e.031(4) = $ TECHNOLOGY TIP TI-84+ and most Casos have a TVM solver for f nancal computatons (n the TI APPS/f nancal menu or the Caso man menu); a smlar one can be downloaded for TI-89. Fgure 5.2 shows the soluton of Example 4 (e) on such a solver (FV means future value). The use of these solvers s explaned n the next secton. Most of the problems n ths secton can be solved just as quckly wth an ordnary calculator. Fgure 5.2 Ordnary corporate or muncpal bonds usually make semannual smple nterest payments. Wth a zero-coupon bond, however, there are no nterest payments durng the lfe of the bond. The nvestor receves a sngle payment when the bond matures, consstng of

13 5.2 Compound Interest 237 hs orgnal nvestment and the nterest (compounded semannually) that t has earned. Zero-coupon bonds are sold at a substantal dscount from ther face value, and the buyer receves the face value of the bond when t matures. The dfference between the face value and the prce of the bond s the nterest earned. Checkpont 5 Fnd the face value of the zero coupon. (a) 30-year bond at 6% sold for $2546 (b) 15-year bond at 5% sold for $16,686 Example 7 Doug Payne bought a 15-year zero-coupon bond payng 4.5% nterest (compounded semannually) for $12, What s the face value of the bond? Soluton Use the compound nterest formula wth P = 12, Interest s pad twce a year, so the rate per perod s =.045>2, and the number of perods n 15 years s n = 30. The compound amount wll be the face value: A = P(1 + ) n = 12,824.50( >2) 30 = 24, Roundng to the nearest cent, we see that the face value of the bond n $25, Example 8 Suppose that the nflaton rate s 3.5% (whch means that the overall level of prces s rsng 3.5% a year). How many years wll t take for the overall level of prces to double? Soluton We want to fnd the number of years t wll take for $1 worth of goods or servces to cost $2. Thnk of $1 as the present value and $2 as the future value, wth an nterest rate of 3.5%, compounded annually. Then the compound amount formula becomes whch smplfes as P(1 + ) n = A 1( ) n = 2, n = 2. We must solve ths equaton for n. There are several ways to do ths. Graphcal Use a graphng calculator (wth x n place of n ) to fnd the ntersecton pont of the graphs of y 1 = x and y 2 = 2. Fgure 5.3 shows that the ntersecton pont has (approxmate) x -coordnate So t wll take about years for prces to double. Fgure 5.3 Checkpont 6 Usng a calculator, fnd the number of years t wll take for $500 to ncrease to $750 n an account payng 6% nterest compounded semannually. Algebrac The same answer can be obtaned by usng natural logarthms, as n Secton 4.4 : n = 2 ln n = ln 2 Take the logarthm of each sde. n ln = ln 2 Power property of logarthms. n = ln 2 ln Dvde both sdes by ln n Use a calculator. 6 Effectve Rate (APY) If you nvest $100 at 9%, compounded monthly, then your balance at the end of one year s A = P(1 + ) n = 100a b = $ You have earned $9.38 n nterest, whch s 9.38% of your orgnal $100. In other words, $100 nvested at 9.38% compounded annually wll produce the same amount of nterest

14 238 CHAPTER 5 Mathematcs of Fnance (namely, $100 *.0938 = $9.38) as does 9% compounded monthly. In ths stuaton, 9% s called the nomnal or stated rate, whle 9.38% s called the effectve rate or annual percentage yeld (APY). In the dscusson that follows, the nomnal rate s denoted r and the APY (effectve rate) s denoted r E. Effectve Rate (r E ) or Annual Percentage Yeld (APY) The APY r E s the annual compoundng rate needed to produce the same amount of nterest n one year, as the nomnal rate does wth more frequent compoundng. Example 9 Fnance In Aprl 2013, Natonwde Bank offered ts customers a 5-year $100,000 CD at 1.73% nterest, compounded daly. Fnd the APY. (Data from: cdrates.bankaholc.com.) Soluton The box gven prevously means that we must have the followng: $100,000 at rate r E $100,000 at 1.73%, = compounded annually compounded daly 100,000(1 + r E ) 1 = 100,000a b Compound nterest formula. (1 + r E ) = a b 365 r E = a b r E So the APY s about 1.75%. Dvde both sdes by 100, Subtract 1 from both sdes. The argument n Example 9 can be carred out wth 100,000 replaced by P,.0173 by r, and 365 by m. The result s the effectve-rate formula. Effectve Rate (APY) The effectve rate (APY) correspondng to a stated rate of nterest r compounded m tmes per year s r E = a1 + r m b m - 1. Example 10 Fnance When nterest rates are low (as they were when ths text went to press), the nterest rate and the APY are nsgnfcantly dfferent. To see when the dfference s more pronounced, we wll fnd the APY for each of the gven money market checkng accounts (wth balances between $50,000 and $100,000), whch were advertsed n October 2008 when offered rates were hgher. (a) Imperal Captal Bank: 3.35% compounded monthly. Soluton Use the effectve-rate formula wth r =.0335 and m = 12: r E = a1 + r m m b - 1 = a b - 1 = So the APY s about 3.40%, a slght ncrease over the nomnal rate of 3.35%.

15 5.2 Compound Interest 239 (b) U.S. Bank: 2.33% compounded daly. Soluton Use the formula wth r =.0233 and m = 365: Checkpont 7 Fnd the APY correspondng to a nomnal rate of (a) 12% compounded monthly; (b) 8% compounded quarterly. TECHNOLOGY TIP Effectve rates (APYs) can be computed on TI-84+ by usng Eff n the APPS f nancal menu, as shown n Fgure 5.4 for Example 11. r E = a1 + r m b m The APY s about 2.36%. 7-1 = a b = Example 11 Bank A s now lendng money at 10% nterest compounded annually. The rate at Bank B s 9.6% compounded monthly, and the rate at Bank C s 9.7% compounded quarterly. If you need to borrow money, at whch bank wll you pay the least nterest? Soluton Compare the APYs: Bank A: a b - 1 =.10 = 10%; Bank B: a b = %; Bank C: a b = %. The lowest APY s at Bank A, whch has the hghest nomnal rate. 8 Fgure 5.4 Checkpont 8 Fnd the APY correspondng to a nomnal rate of (a) 4% compounded quarterly; (b) 7.9% compounded daly. NOTE Although you can f nd both the stated nterest rate and the APY for most certf - cates of depost and other nterest-bearng accounts, most bank advertsements menton only the APY. Present Value for Compound Interest The formula for compound nterest, A = P(1 + ) n, has four varables: A, P,, and n. Gven the values of any three of these varables, the value of the fourth can be found. In partcular, f A (the future amount),, and n are known, then P can be found. Here, P s the amount that should be deposted today to produce A dollars n n perods. Checkpont 9 Fnd P n Example 12 f the nterest rate s (a) 6%; (b) 10%. Example 12 Kesha Jones must pay a lump sum of $6000 n 5 years. What amount deposted today at 6.2% compounded annually wll amount to $6000 n 5 years? Soluton Here, A = 6000, =.062, n = 5, and P s unknown. Substtutng these values nto the formula for the compound amount gves 6000 = P(1.062) 5 P = 6000 (1.062) 5 = , or $ If Jones leaves $ for 5 years n an account payng 6.2% compounded annually, she wll have $6000 when she needs t. To check your work, use the compound nterest formula wth P = $ , =.062, and n = 5. You should get A = $ As Example 12 shows, $6000 n 5 years s (approxmately) the same as $ today (f money can be deposted at 6.2% annual nterest). An amount that can be deposted today to yeld a gven amount n the future s called the present value of the future amount. By solvng A = P(1 + ) n for P, we get the followng general formula for present value.

16 240 CHAPTER 5 Mathematcs of Fnance Present Value for Compound Interest The present value of A dollars compounded at an nterest rate per perod for n perods s P = A (1 + ) n, or P = A(1 + ) n. Checkpont 10 Fnd the far prce (present value) n Example 13 f the nterest rate s 7.5%. Checkpont 11 What dd a $1000 tem sell for 5 years pror f the annual nflaton rate has been 3.2%? Example 13 A zero-coupon bond wth face value $15,000 and a 6% nterest rate (compounded semannually) wll mature n 9 years. What s a far prce to pay for the bond today? Soluton Thnk of the bond as a 9-year nvestment payng 6%, compounded semannually, whose future value s $15,000. Its present value (what t s worth today) would be a far prce. So use the present value formula wth A = 15,000. Snce nterest s compounded twce a year, the nterest rate per perod s =.06>2 =.03 and the number of perods n nne years s n = 9(2) = 18. Hence, Summary At ths pont, t seems helpful to summarze the notaton and the most mportant formulas for smple and compound nterest. We use the followng varables: P = prncpal or present value; A = future or maturty value; r = annual (stated or nomnal) nterest rate; t = number of years ; m = number of compoundng perods per year; = nterest rate per perod; n = total number of compoundng perods; r E = effectve rate (A P Y ). Smple Interest Compound Interest Contnuous Compoundng A = P(1 + rt) A = P(1 + ) n A = Pe rt P = A 1 + rt P = A (1 + ) n = 15, (1 +.03) So a far prce would be the present value of $ Example 14 Economcs The average annual nflaton rate for the years was 2.29%. How much dd an tem that sells for $1000 n early 2013 cost three years before? (Data from: nflatondata.com.) Soluton Thnk of the prce three years pror as the present value P and $1000 as the future value A. Then =.0229, n = 3, and the present value s A P = (1 + ) n = 1000 ( ) 3 = $ So the tem cost $ three years pror. 11 P = A (1 + ) n = A(1 + ) n P = A e rt r E = a1 + r m b m 1

17 5.2 Compound Interest Exercses Interest on the zero-coupon bonds here s compounded semannually. 1. In the precedng summary what s the dfference between r and? between t and n? 2. Explan the dfference between smple nterest and compound nterest. 3. What factors determne the amount of nterest earned on a fxed prncpal? 4. In your own words, descrbe the maturty value of a loan. 5. What s meant by the present value of money? 6. If nterest s compounded more than once per year, whch rate s hgher, the stated rate or the effectve rate? Fnd the compound amount and the nterest earned for each of the followng deposts. (See Examples 1, 2, 4, and 5.) 7. $1000 at 4% compounded annually for 6 years 8. $1000 at 6% compounded annually for 10 years 9. $470 at 8% compounded semannually for 12 years 10. $15,000 at 4.6% compounded semannually for 11 years 11. $6500 at 4.5% compounded quarterly for 8 years 12. $9100 at 6.1% compounded quarterly for 4 years Fnance The followng CDs were avalable on com on Aprl 13, Fnd the compound amount and the nterest earned for each of the followng. (See Example 5.) 13. Vrtual Bank: $10,000 at.9% compounded daly for 1 year 14. AloStar Bank of Commerce: $1000 at.85% compounded daly for 1 year 15. USAA: $5000 at.81% compounded monthly for 2 years 16. Centennal Bank: $20,000 at.45% compounded monthly for 2 years 17. E-LOAN: $100,000 at 1.52% compounded daly for 5 years 18. Thrd Federal Savngs and Loans: $150,000 at 1.15% compounded quarterly for 5 years Fnd the nterest rate (wth annual compoundng) that makes the statement true. (See Example 3.) 19. $3000 grows to $3606 n 5 years 20. $2550 grows to $3905 n 11 years 21. $8500 grows to $12,161 n 7 years 22. $9000 grows to $17,118 n 16 years Fnd the compound amount and the nterest earned when the followng nvestments have contnuous compoundng. (See Example 6.) 23. $20,000 at 3.5% for 5 years 24. $15,000 at 2.9% for 10 years 25. $30,000 at 1.8% for 3 years 26. $100,000 at 5.1% for 20 years Fnd the face value (to the nearest dollar) of the zero-coupon bond. (See Example 7.) year bond at 5.2%; prce $ year bond at 4.1%; prce $13, year bond at 3.5%; prce $ How do the nomnal, or stated, nterest rate and the effectve nterest rate (APY) dffer? Fnd the APY correspondng to the gven nomnal rates. (See Examples 9 11 ) % compounded semannually 32. 6% compounded quarterly 33. 5% compounded quarterly % compounded semannually Fnd the present value of the gven future amounts. (See Example 12.) 35. $12,000 at 5% compounded annually for 6 years 36. $8500 at 6% compounded annually for 9 years 37. $17,230 at 4% compounded quarterly for 10 years 38. $5240 at 6% compounded quarterly for 8 years What prce should you be wllng to pay for each of these zerocoupon bonds? (See Example 13.) year $5000 bond; nterest at 3.5% year $10,000 bond; nterest at 4% year $20,000 bond; nterest at 4.7% year $15,000 bond; nterest at 5.3% Fnance For Exercses 43 and 44, assume an annual nflaton rate of 2.07% (the annual nflaton rate of 2012 accordng to ). Fnd the prevous prce of the followng tems. (See Example 14.) 43. How much dd an tem that costs $5000 now cost 4 years pror? 44. How much dd an tem that costs $7500 now cost 5 years pror? 45. If the annual nflaton rate s 3.6%, how much dd an tem that costs $500 now cost 2 years pror? 46. If the annual nflaton rate s 1.18%, how much dd an tem that costs $1250 now cost 6 years pror? 47. If money can be nvested at 8% compounded quarterly, whch s larger, $1000 now or $1210 n 5 years? Use present value to decde. 48. If money can be nvested at 6% compounded annually, whch s larger, $10,000 now or $15,000 n 6 years? Use present value to decde.

18 242 CHAPTER 5 Mathematcs of Fnance Fnance Work the followng appled problems. 49. A small busness borrows $50,000 for expanson at 9% compounded monthly. The loan s due n 4 years. How much nterest wll the busness pay? 50. A developer needs $80,000 to buy land. He s able to borrow the money at 10% per year compounded quarterly. How much wll the nterest amount to f he pays off the loan n 5 years? 51. Lora Relly has nherted $10,000 from her uncle s estate. She wll nvest the money for 2 years. She s consderng two nvestments: a money market fund that pays a guaranteed 5.8% nterest compounded daly and a 2-year Treasury note at 6% annual nterest. Whch nvestment pays the most nterest over the 2-year perod? 52. Whch of these 20-year zero-coupon bonds wll be worth more at maturty: one that sells for $4510, wth a 6.1% nterest rate, or one that sells for $5809, wth a 4.8% nterest rate? 53. As the prze n a contest, you are offered $1000 now or $1210 n 5 years. If money can be nvested at 6% compounded annually, whch s larger? 54. Two partners agree to nvest equal amounts n ther busness. One wll contrbute $10,000 mmedately. The other plans to contrbute an equvalent amount n 3 years, when she expects to acqure a large sum of money. How much should she contrbute at that tme to match her partner s nvestment now, assumng an nterest rate of 6% compounded semannually? 55. In the Captal Apprecaton Fund, a mutual fund from T. Rowe Prce, a $10,000 nvestment grew to $11,115 over the 3-year perod Fnd the annual nterest rate, compounded yearly, that ths nvestment earned. 56. In the Vanguard Informaton Technology Index Fund, a $10,000 nvestment grew to $16, over the 10-year perod Fnd the annual nterest rate, compounded yearly, that ths nvestment earned. 57. The Flagstar Bank n Mchgan offered a 5-year certfcate of depost (CD) at 4.38% nterest compounded quarterly n June On the same day on the Internet, Prncpal Bank offered a 5-year CD at 4.37% nterest compounded monthly. Fnd the APY for each CD. Whch bank pad a hgher APY? 58. The Westfeld Bank n Oho offered the CD rates shown n the accompanyng table n October The APY rates shown assume monthly compoundng. Fnd the correspondng nomnal rates to the nearest hundredth. ( Hnt: Solve the effectverate equaton for r.) Term 6 mo 1 yr 2 yr 3 yr 5 yr APY (%) A company has agreed to pay $2.9 mllon n 5 years to settle a lawsut. How much must t nvest now n an account payng 5% nterest compounded monthly to have that amount when t s due? 60. Bll Poole wants to have $20,000 avalable n 5 years for a down payment on a house. He has nherted $16,000. How much of the nhertance should he nvest now to accumulate the $20,000 f he can get an nterest rate of 5.5% compounded quarterly? 61. If nflaton has been runnng at 3.75% per year and a new car costs $23,500 today, what would t have cost three years ago? 62. If nflaton s 2.4% per year and a washng machne costs $345 today, what dd a smlar model cost fve years ago? Economcs Use the approach n Example 8 to fnd the tme t would take for the general level of prces n the economy to double at the average annual nflaton rates n Exercses % 64. 4% 65. 5% % 67. The consumpton of electrcty has ncreased hstorcally at 6% per year. If t contnues to ncrease at ths rate ndefntely, fnd the number of years before the electrc utlty companes wll need to double ther generatng capacty. 68. Suppose a conservaton campagn coupled wth hgher rates causes the demand for electrcty to ncrease at only 2% per year, as t has recently. Fnd the number of years before the utlty companes wll need to double ther generatng capacty. 69. You decde to nvest a $16,000 bonus n a money market fund that guarantees a 5.5% annual nterest rate compounded monthly for 7 years. A one-tme fee of $30 s charged to set up the account. In addton, there s an annual admnstratve charge of 1.25% of the balance n the account at the end of each year. (a) How much s n the account at the end of the frst year? (b) How much s n the account at the end of the seventh year? 70. Joe Marusa decdes to nvest $12,000 n a money market fund that guarantees a 4.6% annual nterest rate compounded daly for 6 years. A one-tme fee of $25 s charged to set up the account. In addton, there s an annual admnstraton charge of.9% of the balance n the account at the end of each year. (a) How much s n the account at the end of the frst year? (b) How much s n the account at the end of the sxth year? The followng exercses are from professonal examnatons. 71. On January 1, 2002, Jack deposted $1000 nto Bank X to earn nterest at the rate of j per annum compounded semannually. On January 1, 2007, he transferred hs account to Bank Y to earn nterest at the rate of k per annum compounded quarterly. On January 1, 2010, the balance at Bank Y was $ If Jack could have earned nterest at the rate of k per annum compounded quarterly from January 1, 2002, through January 1, 2010, hs balance would have been $ Whch of the followng represents the rato k>j? (Depost of Jack n Bank X from Course 140 Examnaton, Mathematcs of Compound Interest. Copyrght Socety of Actuares. Reproduced by permsson of Socety of Actuares.) (a) 1.25 (b) 1.30 (c) 1.35 (d) 1.40 (e) On January 1, 2009, Tone Company exchanged equpment for a $200,000 non-nterest-bearng note due on January 1, The prevalng rate of nterest for a note of ths type on January 1, 2009, was 10%. The present value of $1 at 10% for three perods s What amount of nterest revenue should be ncluded n Tone s 2010 ncome statement? (Adapted from the Unform CPA Examnaton, Amercan Insttute of Certfed Publc Accountants.) (a) $7500 (b) $15,000 (c) $16,500 (d) $20,000

19 Checkpont Answers 1. (a) Year Interest Balance 4 $50 $ $50 $1250 (b) Year Interest Balance 4 $57.88 $ $60.78 $ (a) $26, (b) $ (a) $ (b) $ Annutes, Future Value, and Snkng Funds 4. $ (a) $15,000 (b) $35, About 7 years (n = 6.86) 7. (a) 12.68% (b) 8.24% 8. (a) 4.06% (b) 8.220% 9. (a) $ (b) $ $ $ Annutes, Future Value, and Snkng Funds So far n ths chapter, only lump-sum deposts and payments have been dscussed. Many fnancal stuatons, however, nvolve a sequence of payments at regular ntervals, such as weekly deposts n a savngs account or monthly payments on a mortgage or car loan. Such perodc payments are the subject of ths secton and the next. The analyss of perodc payments wll requre an algebrac technque that we now develop. Suppose x s a real number. For reasons that wll become clear later, we want to fnd the product (x - 1)(1 + x + x 2 + x x 11 ). Usng the dstrbutve property to multply ths expresson out, we see that all but two of the terms cancel: x(1 + x + x 2 + x x 11 ) - 1(1 + x + x 2 + x x 11 ) = (x + x 2 + x x 11 + x 12 ) x - x 2 - x x 11 = x Hence, (x - 1)(1 + x + x 2 + x x 11 ) = x Dvdng both sdes by x - 1, we have 1 + x + x 2 + x x 11 = x12-1 x - 1. The same argument, wth any postve nteger n n place of 12 and n - 1 n place of 11, produces the followng result: If x s a real number and n s a postve nteger, then 1 + x + x 2 + x x n - 1 = xn - 1 x - 1. For example, when x = 5 and n = 7, we see that = = 78,124 4 = 19,531. Fgure 5.5 A calculator can easly add up the terms on the left sde, but t s faster to use the formula (Fgure 5.5 ).

20 244 CHAPTER 5 Mathematcs of Fnance Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme from the begnnng of the frst payment perod to the end of the last perod s called the term of the annuty. Annutes can be used to accumulate funds for example, when you make regular deposts n a savngs account. Or they can be used to pay out funds as when you receve regular payments from a penson plan after you retre. Annutes that pay out funds are consdered n the next secton. Ths secton deals wth annutes n whch funds are accumulated by regular payments nto an account or nvestment that earns compound nterest. The future value of such an annuty s the fnal sum on depost that s, the total amount of all deposts and all nterest earned by them. We begn wth ordnary annutes ones where the payments are made at the end of each perod and the frequency of payments s the same as the frequency of compoundng the nterest. Example 1 $1500 s deposted at the end of each year for the next 6 years n an account payng 8% nterest compounded annually. Fnd the future value of ths annuty. Soluton Fgure 5.6 shows the stuaton schematcally. Term of annuty End of year Perod 1 Perod 2 Perod 3 Perod 4 Perod 5 Perod 6 $1500 $1500 $1500 $1500 $1500 $1500 The $1500 s deposted at the end of the year. Fgure 5.6 To fnd the future value of ths annuty, look separately at each of the $1500 payments. The frst $1500 s deposted at the end of perod 1 and earns nterest for the remanng 5 perods. From the formula n the box on page 233, the compound amount produced by ths payment s 1500(1 +.08) 5 = 1500(1.08) 5. The second $1500 payment s deposted at the end of perod 2 and earns nterest for the remanng 4 perods. So the compound amount produced by the second payment s 1500(1 +.08) 4 = 1500(1.08) 4. Contnue to compute the compound amount for each subsequent payment, as shown n Fgure 5.7. Note that the last payment earns no nterest. Year Depost $1500 $1500 $1500 $1500 $1500 $1500 Fgure 5.7 $ (1.08) 1500 (1.08) (1.08) (1.08) (1.08) 5 The sum of these s the total amount after 6 years. The last column of Fgure 5.7 shows that the total amount after 6 years s the sum # # # # # = 1500( ). (1)

21 5.3 Annutes, Future Value, and Snkng Funds 245 Checkpont 1 Complete these steps for an annuty of $2000 at the end of each year for 3 years. Assume nterest of 6% compounded annually. (a) The frst depost of $2000 produces a total of. (b) The second depost becomes. (c) No nterest s earned on the thrd depost, so the total n the account s. Now apply the algebrac fact n the box on page 243 to the expresson n parentheses (wth x = 1.08 and n = 6). It shows that the sum (the future value of the annuty) s 1500 # = $11, Example 1 s the model for fndng a formula for the future value of any annuty. Suppose that a payment of R dollars s deposted at the end of each perod for n perods, at an nterest rate of per perod. Then the future value of ths annuty can be found by usng the procedure n Example 1, wth these replacements: T R T T 1 + T n T n - 1 The future value S n Example 1 call t S s the sum (1), whch now becomes S = R[1 + (1 + ) + (1 + ) (1 + ) n-2 + (1 + ) n-1 ]. Apply the algebrac fact n the box on page 243 to the expresson n brackets (wth x = 1 + ). Then we have S = Rc (1 + )n - 1 (1 + ) - 1 d = Rc (1 + )n - 1 d. The quantty n brackets n the rght-hand part of the precedng equaton s sometmes wrtten s n (read s-angle-n at ). So we can summarze as follows. * Future Value of an Ordnary Annuty The future value S of an ordnary annuty used to accumulate funds s gven by S = Rc (1 + )n 1 d, or S = R # sn, where R s the payment at the end of each perod, s the nterest rate per perod, and n s the number of perods. Fgure 5.8 TECHNOLOGY TIP Most computatons wth annutes can be done quckly wth a spreadsheet program or a graphng calculator. On a calculator, use the TVM solver f there s one (see the Technology Tp on page 236 ); otherwse, use the programs n the Program Appendx. Fgure 5.8 shows how to do Example 1 on a TI-84+ TVM solver. Frst, enter the known quanttes: N = number of payments, l % = annual nterest rate, PV = present value, PMT = payment per perod (entered as a negatve amount), P>Y = number of payments per year, and C>Y = number of compoundngs per year. At the bottom of the screen, set PMT: to END for ordnary annutes. Then put the cursor next to the unknown amount FV (future value), and press SOLVE. Note: P>Y and C>Y should always be the same for problems n ths text. If you use the solver for ordnary compound nterest problems, set PMT = 0 and enter ether PV or FV (whchever s known) as a negatve amount. Example 2 A rooke player n the Natonal Football League just sgned hs frst 7-year contract. To prepare for hs future, he deposts $150,000 at the end of each year for 7 years n an account payng 4.1% compounded annually. How much wll he have on depost after 7 years? * We use S for the future value here nstead of A, as n the compound nterest formula, to help avod confusng the two formulas.

22 246 CHAPTER 5 Mathematcs of Fnance Checkpont 2 Johnson Buldng Materals deposts $2500 at the end of each year nto an account payng 8% per year compounded annually. Fnd the total amount on depost after (a) 6 years; (b) 10 years. Checkpont 3 Fnd the total value of the account n part (b) of Example 3 f the fund s return for the last 15 years s 8.72%, compounded monthly. Soluton Hs payments form an ordnary annuty wth R = 150,000, n = 7, and =.041. The future value of ths annuty (by the prevous formula) s Example 3 Allyson, a college professor, contrbuted $950 a month to the CREF stock fund (an nvestment vehcle avalable to many college and unversty employees). For the past 10 years ths fund has returned 4.25%, compounded monthly. (a) How much dd Allyson earn over the course of the last 10 years? Soluton Allyson s payments form an ordnary annuty, wth monthly payment R = 950. The nterest per month s =.0425, and the number of months n 10 years s n = * 12 = 120. The future value of ths annuty s S = Rc (1 + )n - 1 Snkng Funds S = 150,000c (1.041)7-1 d = $1,188, d = 950c ( >12)120-1 d = $141, >12 (b) As of Aprl 14, 2013, the year to date return was 9.38%, compounded monthly. If ths rate were to contnue, and Allyson contnues to contrbute $950 a month, how much would the account be worth at the end of the next 15 years? Soluton Deal separately wth the two parts of her account (the $950 contrbutons n the future and the $141, already n the account). The contrbutons form an ordnary annuty as n part (a). Now we have R = 950, =.0938>12, and n = 12 * 15 = 180. So the future value s S = Rc (1 + )n - 1 d = 950c ( >12)180-1 d = $372, >12 Meanwhle, the $141, from the frst 10 years s also earnng nterest at 9.38%, compounded monthly. By the compound amount formula ( Secton 5.2 ), the future value of ths money s 141,746.90( >12) 180 = $575, So the total amount n Allyson s account after 25 years s the sum $372, $575, = $947, A snkng fund s a fund set up to receve perodc payments. Corporatons and muncpaltes use snkng funds to repay bond ssues, to retre preferred stock, to provde for replacement of fxed assets, and for other purposes. If the payments are equal and are made at the end of regular perods, they form an ordnary annuty. Example 4 A busness sets up a snkng fund so that t wll be able to pay off bonds t has ssued when they mature. If t deposts $12,000 at the end of each quarter n an account that earns 5.2% nterest, compounded quarterly, how much wll be n the snkng fund after 10 years? Soluton The snkng fund s an annuty, wth R = 12,000, =.052>4, a n d n = 4(10) = 40. The future value s S = Rc (1 + )n - 1 So there wll be about $624,370 n the snkng fund. d = 12,000c ( >4)40-1 d = $624, >4

23 5.3 Annutes, Future Value, and Snkng Funds 247 Checkpont 4 Francsco Arce needs $8000 n 6 years so that he can go on an archaeologcal dg. He wants to depost equal payments at the end of each quarter so that he wll have enough to go on the dg. Fnd the amount of each payment f the bank pays (a) 12% nterest compounded quarterly; (b) 8% nterest compounded quarterly. Fgure 5.9 Fgure 5.10 Checkpont 5 Pete s Pzza deposts $5800 at the end of each quarter for 4 years. (a) Fnd the fnal amount on depost f the money earns 6.4% compounded quarterly. (b) Pete wants to accumulate $110,000 n the 4-year perod. What nterest rate (to the nearest tenth) wll be requred? Example 5 A frm borrows $6 mllon to buld a small factory. The bank requres t to set up a $200,000 snkng fund to replace the roof after 15 years. If the frm s deposts earn 6% nterest, compounded annually, fnd the payment t should make at the end of each year nto the snkng fund. Soluton Ths stuaton s an annuty wth future value S = 200,000, nterest rate =.06, and n = 15. Solve the future-value formula for R : Annutes Due S = Rc (1 + )n - 1 d 200,000 = Rc (1 +.06)15-1 d.06 Let S = 200,000, =.06, and n = ,000 = R[ ] Compute the quantty n brackets. R = 200,000 = $ Dvde both sdes by So the annual payment s about $ Example 6 As an ncentve for a valued employee to reman on the job, a company plans to offer her a $100,000 bonus, payable when she retres n 20 years. If the company deposts $200 a month n a snkng fund, what nterest rate must t earn, wth monthly compoundng, n order to guarantee that the fund wll be worth $100,000 n 20 years? Soluton The snkng fund s an annuty wth R = 200, n = 12(20) = 240, and future value S = 100,000. We must fnd the nterest rate. If x s the annual nterest rate n decmal form, then the nterest rate per month s = x>12. Insertng these values nto the futurevalue formula, we have Rc (1 + )n - 1 d = S 200c (1 + x>12)240-1 d = 100,000. x>12 Ths equaton s hard to solve algebracally. You can get a rough approxmaton by usng a calculator and tryng dfferent values for x. Wth a graphng calculator, you can get an accurate soluton by graphng y 1 = 200c (1 + x>12)240-1 d and y x>12 2 = 100,000 and fndng the x -coordnate of the pont where the graphs ntersect. Fgure 5.9 shows that the company needs an nterest rate of about 6.661%. The same answer can be obtaned on a TVM solver (Fgure 5.10 ). 5 The formula developed prevously s for ordnary annutes annutes wth payments at the end of each perod. The results can be modfed slghtly to apply to annutes due annutes where payments are made at the begnnng of each perod. An example wll llustrate how ths s done. Consder an annuty due n whch payments of $100 are made for 3 years, and an ordnary annuty n whch payments of $100 are made for 4 years, both wth 5% nterest, compounded annually. Fgure 5.11 computes the growth of each payment separately (as was done n Example 1 ).

24 248 CHAPTER 5 Mathematcs of Fnance Annuty Due (payments at begnnng of year for 3 years) Year 1 Year 2 Year 3 $100 $100 $ ( ) 100( ) 100( ) Future value s the sum of ths column Ordnary Annuty (payments at end of year for 4 years) Year 1 Year 2 Year 3 Year 4 $100 $100 $100 $ ( ) 100( ) 100( ) Future value s the sum of ths column Fgure 5.11 Fgure 5.11 shows that the future values are the same, except for one $100 payment on the ordnary annuty (shown n red). So we can use the formula on page 245 to fnd the future value of the 4-year ordnary annuty and then subtract one $100 payment to get the future value of the 3-year annuty due: Future value of = Future value of One payment 3-year annuty due 4-year ordnary annuty S = 100c d = $ Essentally the same argument works n the general case. Future Value of an Annuty Due The future value S of an annuty due used to accumulate funds s gven by S = Rc (1 + )n+1-1 d - R S = Future value of an ordnary annuty - One payment, of n + 1 payments where R s the payment at the begnnng of each perod, s the nterest rate per perod, and n s the number of perods. Example 7 Payments of $500 are made at the begnnng of each quarter for 7 years n an account payng 8% nterest, compounded quarterly. Fnd the future value of ths annuty due.

25 Checkpont 6 (a) Ms. Black deposts $800 at the begnnng of each 6-month perod for 5 years. Fnd the fnal amount f the account pays 6% compounded semannually. (b) Fnd the fnal amount f ths account were an ordnary annuty. TECHNOLOGY TIP When a TVM solver s used for annutes due, the PMT: settng at the bottom of the screen should be BEGIN. See Fgure 5.12, whch shows the soluton of Example 8. Fgure Annutes, Future Value, and Snkng Funds Soluton In 7 years, there are n = 28 quarterly perods. For an annuty due, add one perod to get n + 1 = 29, and use the formula wth =.08>4 =.02: S = Rc (1 + )n+1-1 d - R = 500c (1 +.02)29-1 d = $18, After 7 years, the account balance wll be $18, Example 8 Jay Rechten plans to have a fxed amount from hs paycheck drectly deposted nto an account that pays 5.5% nterest, compounded monthly. If he gets pad on the frst day of the month and wants to accumulate $13,000 n the next three-anda-half years, how much should he depost each month? Soluton Jay s deposts form an annuty due whose future value s S = 13,000. The nterest rate s =.055>12. There are 42 months n three-and-a-half years. Snce ths s an annuty due, add one perod, so that n + 1 = 43. Then solve the future-value formula for the payment R : Rc (1 + )n d - R = S Rc ( >12)43-1 Let =.055>12, n = 43, d - R = 13, >12 and S = 13,000. Rac ( >12)43-1 d - 1b = 13, >12 Factor out R on left sde. R( ) = 13,000 Compute left sde. R = 13,000 = Dvde both sdes by Jay should have $ deposted from each paycheck. 5.3 Exercses Note : Unless stated otherwse, all payments are made at the end of the perod. Fnd each of these sums (to 4 decmal places) Fnd the future value of the ordnary annutes wth the gven payments and nterest rates. (See Examples 1, 2, 3 (a), and 4.) 3. R = $12,000, 6.2% nterest compounded annually for 8 years 4. R = $20,000, 4.5% nterest compounded annually for 12 years 5. R = $865, 6% nterest compounded semannually for 10 years 6. R = $7300, 9% nterest compounded semannually for 6 years 7. R = $1200, 8% nterest compounded quarterly for 10 years 8. R = $20,000, 6% nterest compounded quarterly for 12 years Fnd the fnal amount (rounded to the nearest dollar) n each of these retrement accounts, n whch the rate of return on the account and the regular contrbuton change over tme. (See Example 3.) 9. $400 per month nvested at 4%, compounded monthly, for 10 years; then $600 per month nvested at 6%, compounded monthly, for 10 years. 10. $500 per month nvested at 5%, compounded monthly, for 20 years; then $1000 per month nvested at 8%, compounded monthly, for 20 years. 11. $1000 per quarter nvested at 4.2%, compounded quarterly, for 10 years; then $1500 per quarter nvested at 7.4%, compounded quarterly, for 15 years. 12. $1500 per quarter nvested at 7.4%, compounded quarterly, for 15 years; then $1000 per quarter nvested at 4.2%, compounded quarterly, for 10 years. (Compare wth Exercse 11.) Fnd the amount of each payment to be made nto a snkng fund to accumulate the gven amounts. Payments are made at the end of each perod. (See Example 5.) 13. $11,000; money earns 5% compounded semannually for 6 years

26 250 CHAPTER 5 Mathematcs of Fnance 14. $65,000; money earns 6% compounded semannually for years 15. $50,000; money earns 8% compounded quarterly for years 16. $25,000; money earns 9% compounded quarterly for years 17. $6000; money earns 6% compounded monthly for 3 years 18. $9000; money earns 7% compounded monthly for years Fnd the nterest rate needed for the snkng fund to reach the requred amount. Assume that the compoundng perod s the same as the payment perod. (See Example 6.) 19. $50,000 to be accumulated n 10 years; annual payments of $ $100,000 to be accumulated n 15 years; quarterly payments of $ $38,000 to be accumulated n 5 years; quarterly payments of $ $77,000 to be accumulated n 20 years; monthly payments of $ What s meant by a snkng fund? Lst some reasons for establshng a snkng fund. 24. Explan the dfference between an ordnary annuty and an annuty due. Fnd the future value of each annuty due. (See Example 7.) 25. Payments of $500 for 10 years at 5% compounded annually 26. Payments of $1050 for 8 years at 3.5% compounded annually 27. Payments of $16,000 for 11 years at 4.7% compounded annually 28. Payments of $25,000 for 12 years at 6% compounded annually 29. Payments of $1000 for 9 years at 8% compounded semannually 30. Payments of $750 for 15 years at 6% compounded semannually 31. Payments of $100 for 7 years at 9% compounded quarterly 32. Payments of $1500 for 11 years at 7% compounded quarterly Fnd the payment that should be used for the annuty due whose future value s gven. Assume that the compoundng perod s the same as the payment perod. (See Example 8.) 33. $8000; quarterly payments for 3 years; nterest rate 4.4% 34. $12,000; annual payments for 6 years; nterest rate 5.1% 35. $55,000; monthly payments for 12 years; nterest rate 5.7% 36. $125,000; monthly payments for 9 years; nterest rate 6% Fnance Work the followng appled problems. 37. A typcal pack-a-day smoker n Oho spends about $170 per month on cgarettes. Suppose the smoker nvests that amount at the end of each month n an nvestment fund that pays a return of 5.3% compounded monthly. What would the account be worth after 40 years? (Data from: A typcal pack-a-day smoker n Illnos spends about $ per month on cgarettes. Suppose the smoker nvests that amount at the end of each month n an nvestment fund that pays a return of 4.9% compounded monthly. What would the account be worth after 40 years? (Data from: com.) 39. The Vanguard Explorer Value fund had as of Aprl 2013 a 10-year average return of 10.99%. (Data from: com.) (a) If Becky Anderson deposted $800 a month n the fund for 10 years, fnd the fnal value of the amount of her nvestments. Assume monthly compoundng. (b) If Becky had nvested nstead wth the Vanguard Growth and Income fund, whch had an average annual return of 7.77%, what would the fnal value of the amount of her nvestments be? Assume monthly compoundng. (c) How much more dd the Explorer Value fund generate than the Growth and Income fund? 40. The Janus Enterprse fund had as of Aprl 2013 a 10-year average return of 12.54%. (Data from: (a) If Elane Chuha deposted $625 a month n the fund for 8 years, fnd the fnal value of the amount of her nvestments. Assume monthly compoundng. (b) If Elane had nvested nstead wth the Janus Twenty fund, whch had an average annual return of 10.63%, what would the fnal value of the amount of her nvestments be? Assume monthly compoundng. (c) How much more dd the Janus Enterprse fund generate than the Janus Twenty fund? 41. Bran Fester, a 25-year-old professonal, nvests $200 a month n the T. Rowe Prce Captal Opportunty fund, whch has a 10-year average return of 8.75%. (Data from: (a) Bran wants to estmate what he wll have for retrement when he s 60 years old f the rate stays constant. Assume monthly compoundng. (b) If Bran makes no further deposts and makes no wthdrawals after age 60, how much wll he have for retrement at age 65? 42. Ian Morrson, a 30-year-old professonal, nvests $250 a month n the T. Rowe Prce Equty Income fund, whch has a 10-year average return of 9.04%. (Data from: (a) Ian wants to estmate what he wll have for retrement when he s 65 years old f the rate stays constant. Assume monthly compoundng. (b) If Ian makes no further deposts and makes no wthdrawals after age 65, how much wll he have for retrement at age 75? Assume monthly compoundng. 43. A mother opened an nvestment account for her son on the day he was born, nvestng $1000. Each year on hs brthday, she deposts another $1000, makng the last depost on hs 18 th brthday. If the account pad a return rate of 5.6% compounded annually, how much s n the account at the end of the day on the son s 18 th brthday?

27 5.3 Annutes, Future Value, and Snkng Funds A grandmother opens an nvestment account for her only granddaughter on the day she was born, nvestng $500. Each year on her brthday, she deposts another $500, makng the last depost on her 25 th brthday. If the account pad a return rate of 6.2% compounded annually, how much s n the account at the end of the day on the granddaughter s 25 th brthday? 45. Chuck Hckman deposts $10,000 at the begnnng of each year for 12 years n an account payng 5% compounded annually. He then puts the total amount on depost n another account payng 6% compounded sem-annually for another 9 years. Fnd the fnal amount on depost after the entre 21-year perod. 46. Suppose that the best rate that the company n Example 6 can fnd s 6.3%, compounded monthly (rather than the 6.661% t wants). Then the company must depost more n the snkng fund each month. What monthly depost wll guarantee that the fund wll be worth $100,000 n 20 years? 47. Davd Horwtz needs $10,000 n 8 years. (a) What amount should he depost at the end of each quarter at 5% compounded quarterly so that he wll have hs $10,000? (b) Fnd Horwtz s quarterly depost f the money s deposted at 5.8% compounded quarterly. 48. Harv s Meats knows that t must buy a new machne n 4 years. The machne costs $12,000. In order to accumulate enough money to pay for the machne, Harv decdes to depost a sum of money at the end of each 6 months n an account payng 6% compounded semannually. How much should each payment be? 49. Barbara Margolus wants to buy a $24,000 car n 6 years. How much money must she depost at the end of each quarter n an account payng 5% compounded quarterly so that she wll have enough to pay for her car? 50. The Chnns agree to sell an antque vase to a local museum for $19,000. They want to defer the recept of ths money untl they retre n 5 years (and are n a lower tax bracket). If the museum can earn 5.8%, compounded annually, fnd the amount of each annual payment t should make nto a snkng fund so that t wll have the necessary $19,000 n 5 years. 51. Dane Gray sells some land n Nevada. She wll be pad a lump sum of $60,000 n 7 years. Untl then, the buyer pays 8% smple nterest quarterly. (a) Fnd the amount of each quarterly nterest payment. (b) The buyer sets up a snkng fund so that enough money wll be present to pay off the $60,000. The buyer wants to make semannual payments nto the snkng fund; the account pays 6% compounded semannually. Fnd the amount of each payment nto the fund. 52. Joe Senw bought a rare stamp for hs collecton. He agreed to pay a lump sum of $4000 after 5 years. Untl then, he pays 6% smple nterest semannually. (a) Fnd the amount of each semannual nterest payment. (b) Senw sets up a snkng fund so that enough money wll be present to pay off the $4000. He wants to make annual payments nto the fund. The account pays 8% compounded annually. Fnd the amount of each payment. 53. To save for retrement, Karla Harby put $300 each month nto an ordnary annuty for 20 years. Interest was compounded monthly. At the end of the 20 years, the annuty was worth $147,126. What annual nterest rate dd she receve? 54. Jennfer Wall made payments of $250 per month at the end of each month to purchase a pece of property. After 30 years, she owned the property, whch she sold for $330,000. What annual nterest rate would she need to earn on an ordnary annuty for a comparable rate of return? 55. When Joe and Sarah graduate from college, each expects to work a total of 45 years. Joe begns savng for retrement mmedately. He plans to depost $600 at the end of each quarter nto an account payng 8.1% nterest, compounded quarterly, for 10 years. He wll then leave hs balance n the account, earnng the same nterest rate, but make no further deposts for 35 years. Sarah plans to save nothng durng the frst 10 years and then begn depostng $600 at the end of each quarter n an account payng 8.1% nterest, compounded quarterly, for 35 years. (a) Wthout dong any calculatons, predct whch one wll have the most n hs or her retrement account after 45 years. Then test your predcton by answerng the followng questons (calculaton requred to the nearest dollar). (b) How much wll Joe contrbute to hs retrement account? (c) How much wll be n Joe s account after 45 years? (d) How much wll Sarah contrbute to her retrement account? (e) How much wll be n Sarah s account after 45 years? 56. In a 1992 Vrgna lottery, the jackpot was $27 mllon. An Australan nvestment frm tred to buy all possble combnatons of numbers, whch would have cost $7 mllon. In fact, the frm ran out of tme and was unable to buy all combnatons, but ended up wth the only wnnng tcket anyway. The frm receved the jackpot n 20 equal annual payments of $1.35 mllon. Assume these payments meet the condtons of an ordnary annuty. (Data from: Washngton Post, March 10, 1992, p. A1.) (a) Suppose the frm can nvest money at 8% nterest compounded annually. How many years would t take untl the nvestors would be further ahead than f they had smply nvested the $7 mllon at the same rate? ( Hnt: Experment wth dfferent values of n, the number of years, or use a graphng calculator to plot the value of both nvestments as a functon of the number of years.) (b) How many years would t take n part (a) at an nterest rate of 12%? Checkpont Answers 1. (a) $ (b) $ (c) $ (a) $18, (b) $36, $872, (a) $ (b) $ (a) $104, (b) 8.9% 6. (a) $ (b) $

28 252 CHAPTER 5 Mathematcs of Fnance 5.4 Annutes, Present Value, and Amortzaton In the annutes studed prevously, regular deposts were made nto an nterest-bearng account and the value of the annuty ncreased from 0 at the begnnng to some larger amount at the end (the future value). Now we expand the dscusson to nclude annutes that begn wth an amount of money and make regular payments each perod untl the value of the annuty decreases to 0. Examples of such annutes are lottery jackpots, structured settlements mposed by a court n whch the party at fault (or hs or her nsurance company) makes regular payments to the njured party, and trust funds that pay the recpents a fxed amount at regular ntervals. In order to develop the essental formula for dealng wth payout annutes, we need another useful algebrac fact. If x s a nonzero number and n s a postve nteger, verfy the followng equalty by multplyng out the rght-hand sde: * x -1 + x -2 + x x -(n-1) + x -n = x -n (x n-1 + x n-2 + x n x 1 + 1). Now use the sum formula n the box on page 243 to rewrte the expresson n parentheses on the rght-hand sde: x -1 + x -2 + x x -(n-1) + x -n = x -n a xn - 1 x - 1 b We have proved the followng result: = x-n (x n - 1) x - 1 = x0 - x -n x - 1 = 1 - x-n x - 1. If x s a nonzero real number and n s a postve nteger, then x -1 + x -2 + x x -n = 1 - x-n x - 1. Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today (at the same nterest rate) n order to produce A dollars n n perods. Smlarly, the present value of an annuty s the amount that must be deposted today (at the same compound nterest rate as the annuty) to provde all the payments for the term of the annuty. It does not matter whether the payments are nvested to accumulate funds or are pad out to dsperse funds; the amount needed to provde the payments s the same n ether case. We begn wth ordnary annutes. Example 1 Your rch aunt has funded an annuty that wll pay you $1500 at the end of each year for sx years. If the nterest rate s 8%, compounded annually, fnd the present value of ths annuty. Soluton Look separately at each payment you wll receve. Then fnd the present value of each payment the amount needed now n order to make the payment n the future. The sum of these present values wll be the present value of the annuty, snce t wll provde all of the payments. * Remember that powers of x are multpled by addng exponents and that x n x -n = x n - n = x 0 = 1.

29 5.4 Annutes, Present Value, and Amortzaton 253 To fnd the frst $1500 payment (due n one year), the present value of $1500 at 8% annual nterest s needed now. Accordng to the present-value formula for compound nterest on page 240 (wth A = 1500, =.08, and n = 1), ths present value s = = 1500( ) $ Ths amount wll grow to $1500 n one year. For the second $1500 payment (due n two years), we need the present value of $1500 at 8% nterest, compounded annually for two years. The present-value formula for compound nterest (wth A = 1500, =.08, and n = 2) shows that ths present value s 1500 (1 +.08) 2 = = 1500( ) $ Less money s needed for the second payment because t wll grow over two years nstead of one. A smlar calculaton shows that the thrd payment (due n three years) has present value $1500( ). Contnue n ths manner to fnd the present value of each of the remanng payments, as summarzed n Fgure Year Payment $1500 $1500 $1500 $1500 $1500 $1500 $1500( ) $1500( ) The sum of these s the present value. $1500( ) $1500( ) $1500( ) $1500( ) Fgure 5.13 The left-hand column of Fgure 5.13 shows that the present value s Checkpont 1 Show that $ wll provde all the payments n Example 1 as follows: (a) Fnd the balance at the end of the frst year after the nterest has been added and the $1500 payment subtracted. (b) Repeat part (a) to fnd the balances at the ends of years 2 through # # # # # # = 1500( ). (1) Now apply the algebrac fact n the box on page 252 to the expresson n parentheses (wth x = 1.08 and n = 6). It shows that the sum (the present value of the annuty) s 1500c d = 1500c d = $ Ths amount wll provde for all sx payments and leave a zero balance at the end of sx years (gve or take a few cents due to roundng to the nearest penny at each step). 1 Example 1 s the model for fndng a formula for the future value of any ordnary annuty. Suppose that a payment of R dollars s made at the end of each perod for n

30 254 CHAPTER 5 Mathematcs of Fnance perods, at nterest rate per perod. Then the present value of ths annuty can be found by usng the procedure n Example 1, wth these replacements: T T T T R 1 + n The future value n Example 1 s the sum n equaton (1), whch now becomes P = R[(1 + ) -1 + (1 + ) -2 + (1 + ) (1 + ) -n ]. Apply the algebrac fact n the box on page 252 to the expresson n brackets (wth x = 1 + ). Then we have P = Rc 1 - (1 + )-n (1 + ) (1 + )-n d = Rc d. The quantty n brackets n the rght-hand part of the precedng equaton s sometmes wrtten a n (read a-angle-n at ). So we can summarze as follows. Present Value of an Ordnary Annuty The present value P of an ordnary annuty s gven by P = Rc 1 (1 + ) n d, or P = R # an, where R s the payment at the end of each perod, s the nterest rate per perod, and n s the number of perods. CAUTION Do not confuse the formula for the present value of an annuty wth the one for the future value of an annuty. Notce the dfference: The numerator of the fracton n the present-value formula s 1 - (1 + ) -n, but n the future-value formula, t s (1 + ) n - 1. Checkpont 2 An nsurance company offers to pay Jane Parks an ordnary annuty of $1200 per quarter for fve years or the present value of the annuty now. If the nterest rate s 6%, fnd the present value. Example 2 Jm Rles was n an auto accdent. He sued the person at fault and was awarded a structured settlement n whch an nsurance company wll pay hm $600 at the end of each month for the next seven years. How much money should the nsurance company nvest now at 4.7%, compounded monthly, to guarantee that all the payments can be made? Soluton The payments form an ordnary annuty. The amount needed to fund all the payments s the present value of the annuty. Apply the present-value formula wth R = 600, n = 7 # 12 = 84, and =.047>12 (the nterest rate per month). The nsurance company should nvest 1 - (1 + )-n 1 - ( >12)-84 P = Rc d = 600c.047>12 d = $42, Example 3 To supplement hs penson n the early years of hs retrement, Ralph Taylor plans to use $124,500 of hs savngs as an ordnary annuty that wll make monthly payments to hm for 20 years. If the nterest rate s 5.2%, how much wll each payment be?

31 5.4 Annutes, Present Value, and Amortzaton 255 Soluton The present value of the annuty s P = $124,500, the monthly nterest rate s =.052>12, and n = 12 # 20 = 240 (the number of months n 20 years). Solve the present-value formula for the monthly payment R : Checkpont 3 Carl Dehne has $80,000 n an account payng 4.8% nterest, compounded monthly. He plans to use up all the money by makng equal monthly wthdrawals for 15 years. If the nterest rate s 4.8%, fnd the amount of each wthdrawal. Example 4 Surnder Snah and Mara Gonzalez are graduates of Kenyon College. They both agree to contrbute to an endowment fund at the college. Snah says he wll gve $500 at the end of each year for 9 years. Gonzalez prefers to gve a sngle donaton today. How much should she gve to equal the value of Snah s gft, assumng that the endowment fund earns 7.5% nterest, compounded annually? Soluton Snah s gft s an ordnary annuty wth annual payments of $500 for 9 years. Its future value at 7.5% annual compound nterest s S = Rc (1 + )n (1 + )-n P = Rc d 1 - ( >12) ,500 = Rc d.052>12 124,500 R = = $ ( >12)-240 c d.052>12 Taylor wll receve $ a month (about $10,026 per year) for 20 years. 3 d = 500c ( ) d = 500c d = $ We clam that for Gonzalez to equal ths contrbuton, she should today contrbute an amount equal to the present value of ths annuty, namely, 1 - (1 + )-n P = Rc d = 500c 1 - ( ) d = 500c d = $ To confrm ths clam, suppose the present value P = $ s deposted today at 7.5% nterest, compounded annually for 9 years. Accordng to the compound nterest formula on page 233, P wll grow to ( ) 9 = $ , the future value of Snah s annuty. So at the end of 9 years, Gonzalez and Snah wll have made dentcal gfts. Example 4 llustrates the followng alternatve descrpton of the present value of an accumulaton annuty. Checkpont 4 What lump sum deposted today would be equvalent to equal payments of (a) $650 at the end of each year for 9 years at 4% compounded annually? (b) $1000 at the end of each quarter for 4 years at 4% compounded quarterly? The present value of an annuty for accumulatng funds s the sngle depost that would have to be made today to produce the future value of the annuty (assumng the same nterest rate and perod of tme.) 4 Corporate bonds, whch were ntroduced n Secton 5.1, are routnely bought and sold n fnancal markets. In most cases, nterest rates when a bond s sold dffer from the nterest rate pad by the bond (known as the coupon rate ). In such cases, the prce of a bond wll not be ts face value, but wll nstead be based on current nterest rates. The next example shows how ths s done. Example 5 A 15-year $10,000 bond wth a 5% coupon rate was ssued fve years ago and s now beng sold. If the current nterest rate for smlar bonds s 7%, what prce should a purchaser be wllng to pay for ths bond?

32 256 CHAPTER 5 Mathematcs of Fnance Soluton Accordng to the smple nterest formula (page 226 ), the nterest pad by the bond each half-year s I = Prt = 10,000 #.05 # 1 2 = $250. Checkpont 5 Suppose the current nterest rate for bonds s 4% nstead of 7% when the bond n Example 5 s sold. What prce should a purchaser be wllng to pay for t? Thnk of the bond as a two-part nvestment: The frst s an annuty that pays $250 every sx months for the next 10 years; the second s the $10,000 face value of the bond, whch wll be pad when the bond matures, 10 years from now. The purchaser should be wllng to pay the present value of each part of the nvestment, assumng 7% nterest, compounded semannually. * The nterest rate per perod s =.07>2, and the number of sx-month perods n 10 years s n = 20. So we have: Present value of annuty 1 - (1 + )-n P = Rc d 1 - (1 +.07>2)-20 = 250c d.07>2 = $ Loans and Amortzaton Present value of $10,000 n 10 years P = A(1 + ) -n = 10,000(1 +.07>2) -20 = $ So the purchaser should be wllng to pay the sum of these two present values: $ $ = $ NOTE Example 5 and Checkpont 5 llustrate the nverse relaton between nterest rates and bond prces: If nterest rates rse, bond prces fall, and f nterest rates fall, bond prces rse. If you take out a car loan or a home mortgage, you repay t by makng regular payments to the bank. From the bank s pont of vew, your payments are an annuty that s payng t a fxed amount each month. The present value of ths annuty s the amount you borrowed. Example 6 Fnance Chase Bank n Aprl 2013 advertsed a new car auto loan rate of 2.23% for a 48-month loan. Shelley Fasulko wll buy a new car for $25,000 wth a down payment of $4500. Fnd the amount of each payment. (Data from: Soluton After a $4500 down payment, the loan amount s $20,500. Use the presentvalue formula for an annuty, wth P = 20,500, n = 48, and =.0223>12 (the monthly nterest rate). Then solve for payment R. Checkpont 6 Suzanne Belln uses a Chase auto loan to purchase a used car prced at $28,750 at an nterest rate of 3.64% for a 60-month loan. What s the monthly payment? 1 - (1 + )-n P = Rc d 1 - ( >12)-48 20,500 = Rc d.0223>12 20,500 R = 1 - ( >12)-48 c d.0223>12 R = $ Solve for R. A loan s amortzed f both the prncpal and nterest are pad by a sequence of equal perodc payments. The perodc payment needed to amortze a loan may be found, as n Example 6, by solvng the present-value formula for R. * The analyss here does not nclude any commssons or fees charged by the fnancal nsttuton that handles the bond sale.

33 5.4 Annutes, Present Value, and Amortzaton 257 Amortzaton Payments A loan of P dollars at nterest rate per perod may be amortzed n n equal perodc payments of R dollars made at the end of each perod, where R = c P 1 (1 + ) n d = P 1 (1 + ) n. Example 7 Fnance In Aprl 2013, the average rate for a 30-year fxed mortgage was 3.43%. Assume a down payment of 20% on a home purchase of $272,900. (Data from: Fredde Mac.) (a) Fnd the monthly payment needed to amortze ths loan. Soluton The down payment s.20(272,900) = $54,580. Thus, the loan amount P s $272,900-54,580 = $218,320. We can now apply the formula n the precedng box, wth n = 12(30) = 360 (the number of monthly payments n 30 years), and monthly nterest rate =.0343>12. * R = P (218,320)(.0343>12) -n = = $ (1 + ) ( >12) Monthly payments of $ are requred to amortze the loan. (b) After 10 years, approxmately how much s owed on the mortgage? Soluton You may be tempted to say that after 10 years of payments on a 30-year mortgage, the balance wll be reduced by a thrd. However, a sgnfcant porton of each payment goes to pay nterest. So, much less than a thrd of the mortgage s pad off n the frst 10 years, as we now see. After 10 years (120 payments), the 240 remanng payments can be thought of as an annuty. The present value for ths annuty s the (approxmate) remanng balance on the mortgage. Hence, we use the present-value formula wth R = , =.0343>12, and n = 240: Checkpont 7 Fnd the remanng balance after 20 years. 1 - ( >12)-240 P = c d = $168, (.0343>12) So the remanng balance s about $168, The actual balance probably dffers slghtly from ths fgure because payments and nterest amounts are rounded to the nearest penny. 7 TECHNOLOGY TIP A TVM solver on a graphng calculator can f nd the present value of an annuty or the payment on a loan: Fll n the known nformaton, put the cursor next to the unknown tem (PV or PMT), and press SOLVE. Fgure 5.14 shows the soluton to Example 7 (a) on a TVM solver. Alternatvely, you can use the program n the Program Appendx. Fgure 5.14 Example 7 (b) llustrates an mportant fact: Even though equal payments are made to amortze a loan, the loan balance does not decrease n equal steps. The method used to estmate the remanng balance n Example 7 (b) works n the general case. If n payments * Mortgage rates are quoted n terms of annual nterest, but t s always understood that the monthly rate s 1 12 of the annual nterest rate and that nterest s compounded monthly.

34 258 CHAPTER 5 Mathematcs of Fnance are needed to amortze a loan and x payments have been made, then the remanng payments form an annuty of n - x payments. So we apply the present-value formula wth n - x n place of n to obtan ths result. Remanng Balance If a loan can be amortzed by n payments of R dollars each at an nterest rate per perod, then the approxmate remanng balance B after x payments s 1 - (1 + )-(n-x) B = Rc d. Amortzaton Schedules The remanng-balance formula s a quck and convenent way to get a reasonable estmate of the remanng balance on a loan, but t s not accurate enough for a bank or busness, whch must keep ts books exactly. To determne the exact remanng balance after each loan payment, fnancal nsttutons normally use an amortzaton schedule, whch lsts how much of each payment s nterest, how much goes to reduce the balance, and how much s stll owed after each payment. Example 8 compounded monthly. Beth Hll borrows $1000 for one year at 12% annual nterest, (a) Fnd her monthly payment. Soluton Apply the amortzaton payment formula wth P = 1000, n = 12, and monthly nterest rate =.12>12 =.01. Her payment s R = P 1 - (1 + ) -n = 1000(.01) = $ (1 +.01) (b) After makng fve payments, Hll decdes to pay off the remanng balance. Approxmately how much must she pay? Soluton Apply the remanng-balance formula just gven, wth R = 88.85, =.01, and n - x = 12-5 = 7. Her approxmate remanng balance s - x) 1 - (1 + )-(n 1 - (1 +.01)-7 B = Rc d = 88.85c d = $ (c) Construct an amortzaton schedule for Hll s loan. Soluton An amortzaton schedule for the loan s shown n the table on the next page. It was obtaned as follows: The annual nterest rate s 12% compounded monthly, so the nterest rate per month s 12%>12 = 1% =.01. When the frst payment s made, one month s nterest, namely,.01(1000) = $10, s owed. Subtractng ths from the $88.85 payment leaves $78.85 to be appled to repayment. Hence, the prncpal at the end of the frst payment perod s = $921.15, as shown n the payment 1 lne of the table. When payment 2 s made, one month s nterest on the new balance of $ s owed, namely,.01(921.15) = $9.21. Contnue as n the precedng paragraph to compute the entres n ths lne of the table. The remanng lnes of the table are found n a smlar fashon.

35 5.4 Annutes, Present Value, and Amortzaton 259 Payment Number Amount of Payment Interest for Perod Porton to Prncpal Prncpal at End of Perod 0 $ $88.85 $10.00 $ Note that Hll s remanng balance after fve payments dffers slghtly from the estmate made n part (b). The fnal payment n the amortzaton schedule n Example 8 (c) dffers from the other payments. It often happens that the last payment needed to amortze a loan must be adjusted to account for roundng earler and to ensure that the fnal balance wll be exactly 0. TECHNOLOGY TIP Most Caso graphng calculators can produce amortzaton schedules. For other calculators, use the amortzaton table program n the Program Appendx. Spreadsheets are another useful tool for creatng amortzaton tables. Mcrosoft Excel (Mcrosoft Corporaton Excel 2013) has a bult-n feature for calculatng monthly payments. Fgure 5.15 shows an Excel amortzaton table for Example 8. For more detals, see the Spreadsheet Manual, also avalable wth ths text. Annutes Due Fgure 5.15 We want to fnd the present value of an annuty due n whch 6 payments of R dollars are made at the begnnng of each perod, wth nterest rate per perod, as shown schematcally n Fgure R R R R R R Fgure 5.16

36 260 CHAPTER 5 Mathematcs of Fnance The present value s the amount needed to fund all 6 payments. Snce the frst payment earns no nterest, R dollars are needed to fund t. Now look at the last 5 payments by themselves n Fgure R R R R R Fgure 5.17 If you thnk of these 5 payments as beng made at the end of each perod, you see that they form an ordnary annuty. The money needed to fund them s the present value of ths ordnary annuty. So the present value of the annuty due s gven by Present value of the ordnary R + annuty of 5 payments 1 - (1 + )-5 R + Rc d. Replacng 6 by n and 5 by n - 1, and usng the argument just gven, produces the general result that follows. Present Value of an Annuty Due The present value P of an annuty due s gven by where 1 - (1 + )-(n-1) P = R + Rc d, P = One payment R s the payment at the begnnng of each perod, s the nterest rate per perod, and n s the number of perods. + Present value of an ordnary annuty of n - 1 payments Checkpont 8 What s the cash value for a Lotto jackpot of $25 mllon f the Illnos Lottery can earn 6.2% annual nterest? Example 9 Fnance The Illnos Lottery Wnner s Handbook dscusses the optons of how to receve the wnnngs for a $12 mllon Lotto jackpot. One opton s to take 26 annual payments of approxmately $461,538.46, whch s $12 mllon dvded nto 26 equal payments. The other opton s to take a lump-sum payment (whch s often called the cash value ). If the Illnos lottery commsson can earn 4.88% annual nterest, how much s the cash value? Soluton The yearly payments form a 26-payment annuty due. An equvalent amount now s the present value of ths annuty. Apply the present-value formula wth R = 461,538.46, =.0488, and n = 26: 1 - (1 + )-(n-1) 1 - ( )-25 P = R + Rc d = 461, ,538.46c d.0488 = $7,045, The cash value s $7,045, Fgure 5.18 TECHNOLOGY TIP Fgure 5.18 shows the soluton of Example 9 on a TVM solver. Snce ths s an annuty due, the PMT: settng at the bottom of the screen s BEGIN.

37 5.4 Annutes, Present Value, and Amortzaton Exercses Unless noted otherwse, all payments and wthdrawals are made at the end of the perod. 1. Explan the dfference between the present value of an annuty and the future value of an annuty. Fnd the present value of each ordnary annuty. (See Examples 1, 2, and 4.) 2. Payments of $890 each year for 16 years at 6% compounded annually 3. Payments of $1400 each year for 8 years at 6% compounded annually 4. Payments of $10,000 semannually for 15 years at 7.5% compounded semannually 5. Payments of $50,000 quarterly for 10 years at 5% compounded quarterly 6. Payments of $15,806 quarterly for 3 years at 6.8% compounded quarterly Fnd the amount necessary to fund the gven wthdrawals. (See Examples 1 and 2.) 7. Quarterly wthdrawals of $650 for 5 years; nterest rate s 4.9%, compounded quarterly. 8. Yearly wthdrawals of $1200 for 14 years; nterest rate s 5.6%, compounded annually. 9. Monthly wthdrawals of $425 for 10 years; nterest rate s 6.1%, compounded monthly. 10. Semannual wthdrawals of $3500 for 7 years; nterest rate s 5.2%, compounded semannually. Fnd the payment made by the ordnary annuty wth the gven present value. (See Example 3.) 11. $90,000; monthly payments for 22 years; nterest rate s 4.9%, compounded monthly. 12. $45,000; monthly payments for 11 years; nterest rate s 5.3%, compounded monthly. 13. $275,000; quarterly payments for 18 years; nterest rate s 6%, compounded quarterly. 14. $330,000; quarterly payments for 30 years; nterest rate s 6.1% compounded quarterly. Fnd the lump sum deposted today that wll yeld the same total amount as payments of $10,000 at the end of each year for 15 years at each of the gven nterest rates. (See Example 4 and the box followng t.) 15. 3% compounded annually 16. 4% compounded annually 17. 6% compounded annually 18. What sum deposted today at 5% compounded annually for 8 years wll provde the same amount as $1000 deposted at the end of each year for 8 years at 6% compounded annually? 19. What lump sum deposted today at 8% compounded quarterly for 10 years wll yeld the same fnal amount as deposts of $4000 at the end of each 6-month perod for 10 years at 6% compounded semannually? Fnd the prce a purchaser should be wllng to pay for the gven bond. Assume that the coupon nterest s pad twce a year. (See Example 5.) 20. $20,000 bond wth coupon rate 4.5% that matures n 8 years; current nterest rate s 5.9%. 21. $15,000 bond wth coupon rate 6% that matures n 4 years; current nterest rate s 5%. 22. $25,000 bond wth coupon rate 7% that matures n 10 years; current nterest rate s 6%. 23. $10,000 bond wth coupon rate 5.4% that matures n 12 years; current nterest rate s 6.5%. 24. What does t mean to amortze a loan? Fnd the payment necessary to amortze each of the gven loans. (See Examples 6, 7 (a), and 8 (a).) 25. $2500; 8% compounded quarterly; 6 quarterly payments 26. $41,000; 9% compounded semannually; 10 semannual payments 27. $90,000; 7% compounded annually; 12 annual payments 28. $140,000; 12% compounded quarterly; 15 quarterly payments 29. $7400; 8.2% compounded semannually; 18 semannual payments 30. $5500; 9.5% compounded monthly; 24 monthly payments Fnance In Aprl 2013, the mortgage nterest rates lsted n Exercses for the gven companes were lsted at Fnd the monthly payment necessary to amortze the gven loans. (See Example 7 (a).) 31. $225,000 at 3.25% for 30 years from Amersave 32. $330,000 at 3.125% for 20 years from Qucken Loans 33. $140,000 at 2.375% for 15 years from Dscover Home Loans 34. $180,000 at 2.25% for 10 years from Roundpont Mortgage Company Fnance Fnd the monthly payment and estmate the remanng balance (to the nearest dollar). Assume nterest s on the unpad balance. The nterest rates are from natonal averages from www. bankrate.com n Aprl (See Examples 7 and 8.) 35. Four-year new car loan for $26,799 at 3.13%; remanng balance after 2 years 36. Three-year used car loan for $15,875 at 2.96%; remanng balance after 1 year 37. Thrty-year mortgage for $210,000 at 3.54%; remanng balance after 12 years

38 262 CHAPTER 5 Mathematcs of Fnance 38. Ffteen-year mortgage for $195,000 at 2.78%; remanng balance after 4.5 years Use the amortzaton table n Example 8 (c) to answer the questons n Exercses How much of the 5th payment s nterest? 40. How much of the 10th payment s used to reduce the debt? 41. How much nterest s pad n the frst 5 months of the loan? 42. How much nterest s pad n the last 5 months of the loan? Fnd the cash value of the lottery jackpot (to the nearest dollar). Yearly jackpot payments begn mmedately (26 for Mega Mllons and 30 for Powerball). Assume the lottery can nvest at the gven nterest rate. (See Example 9.) 43. Powerball: $57.6 mllon; 5.1% nterest 44. Powerball: $207 mllon; 5.78% nterest 45. Mega Mllons: $41.6 mllon; 4.735% nterest 46. Mega Mllons: $23.4 mllon; 4.23% nterest Fnance Work the followng appled problems. 47. An auto stereo dealer sells a stereo system for $600 down and monthly payments of $30 for the next 3 years. If the nterest rate s 1.25% per month on the unpad balance, fnd (a) the cost of the stereo system; (b) the total amount of nterest pad. 48. John Kushda buys a used car costng $6000. He agrees to make payments at the end of each monthly perod for 4 years. He pays 12% nterest, compounded monthly. (a) What s the amount of each payment? (b) Fnd the total amount of nterest Kushda wll pay. 49. A speculator agrees to pay $15,000 for a parcel of land; ths amount, wth nterest, wll be pad over 4 years wth semannual payments at an nterest rate of 10% compounded semannually. Fnd the amount of each payment. 50. Alan Stasa buys a new car costng $26,750. What s the monthly payment f the nterest rate s 4.2%, compounded monthly, and the loan s for 60 months? Fnd the total amount of nterest Alan wll pay. Fnance A student educaton loan has two repayment optons. The standard plan repays the loan n 10 years wth equal monthly payments. The extended plan allows from 12 to 30 years to repay the loan. A student borrows $35,000 at 7.43% compounded monthly. 51. Fnd the monthly payment and total nterest pad under the standard plan. 52. Fnd the monthly payment and total nterest pad under the extended plan wth 20 years to pay off the loan. Fnance Use the formula for the approxmate remanng balance to work each problem. (See Examples 7 (b) and 8 (b).) 53. When Teresa Flores opened her law offce, she bought $14,000 worth of law books and $7200 worth of offce furnture. She pad $1200 down and agreed to amortze the balance wth semannual payments for 5 years at 12% compounded semannually. (a) Fnd the amount of each payment. (b) When her loan had been reduced below $5000, Flores receved a large tax refund and decded to pay off the loan. How many payments were left at ths tme? 54. Kareem Adams buys a house for $285,000. He pays $60,000 down and takes out a mortgage at 6.9% on the balance. Fnd hs monthly payment and the total amount of nterest he wll pay f the length of the mortgage s (a) 15 years; (b) 20 years; (c) 25 years. (d) When wll half the 20-year loan be pad off? 55. Susan Carver wll purchase a home for $257,000. She wll use a down payment of 20% and fnance the remanng porton at 3.9%, compounded monthly for 30 years. (a) What wll be the monthly payment? (b) How much wll reman on the loan after makng payments for 5 years? (c) How much nterest wll be pad on the total amount of the loan over the course of 30 years? 56. Mohsen Manouchehr wll purchase a $230,000 home wth a 20-year mortgage. If he makes a down payment of 20% and the nterest rate s 3.3%, compounded monthly, (a) what wll the monthly payment be? (b) how much wll he owe after makng payments for 8 years? (c) how much n total nterest wll he pay over the course of the 20-year loan? Work each problem. 57. Elzabeth Bernard and her employer contrbute $400 at the end of each month to her retrement account, whch earns 7% nterest, compounded monthly. When she retres after 45 years, she plans to make monthly wthdrawals for 30 years. If her account earns 5% nterest, compounded monthly, then when she retres, what s her maxmum possble monthly wthdrawal (wthout runnng out of money)?

39 CHAPTER 5 Summary and Revew Jm Mllken won a $15,000 prze. On March 1, he deposted t n an account earnng 5.2% nterest, compounded monthly. On March 1 one year later, he begns to wthdraw the same amount at the begnnng of each month for a year. Assumng that he uses up all the money n the account, fnd the amount of each monthly wthdrawal. 59. Catherne Dohanyos plans to retre n 20 years. She wll make 20 years of monthly contrbutons to her retrement account. One month after her last contrbuton, she wll begn the frst of 10 years of wthdrawals. She wants to wthdraw $2500 per month. How large must her monthly contrbutons be n order to accomplsh her goal f the account earns nterest of 7.1% compounded monthly for the duraton of her contrbutons and the 120 months of wthdrawals? 60. Davd Turner plans to retre n 25 years. He wll make 25 years of monthly contrbutons to hs retrement account. One month after hs last contrbuton, he wll begn the frst of 10 years of wthdrawals. He wants to wthdraw $3000 per month. How large must hs monthly contrbutons be n order to accomplsh hs goal f the account earns nterest of 6.8% compounded monthly for the duraton of hs contrbutons and the 120 months of wthdrawals? 61. Wllam Blake plans to retre n 20 years. Wllam wll make 10 years (120 months) of equal monthly payments nto hs account. Ten years after hs last contrbuton, he wll begn the frst of 120 monthly wthdrawals of $3400 per month. Assume that the retrement account earns nterest of 8.2% compounded monthly for the duraton of hs contrbutons, the 10 years n between hs contrbutons and the begnnng of hs wthdrawals, and the 10 years of wthdrawals. How large must Wllam s monthly contrbutons be n order to accomplsh hs goal? 62. Gl Stevens plans to retre n 25 years. He wll make 15 years (180 months) of equal monthly payments nto hs account. Ten years after hs last contrbuton, he wll begn the frst of 120 monthly wthdrawals of $2900 per month. Assume that the retrement account earns nterest of 5.4% compounded monthly for the duraton of hs contrbutons, the 10 years n between hs contrbutons and the begnnng of hs wthdrawals, and the 10 years of wthdrawals. How large must Gl s monthly contrbutons be n order to accomplsh hs goal? Fnance In Exercses 63 66, prepare an amortzaton schedule showng the frst four payments for each loan. (See Example 8 (c).) 63. An nsurance frm pays $4000 for a new prnter for ts computer. It amortzes the loan for the prnter n 4 annual payments at 8% compounded annually. 64. Large semtraler trucks cost $72,000 each. Ace Truckng buys such a truck and agrees to pay for t by a loan that wll be amortzed wth 9 semannual payments at 6% compounded semannually. 65. One retaler charges $1048 for a certan computer. A frm of tax accountants buys 8 of these computers. It makes a down payment of $1200 and agrees to amortze the balance wth monthly payments at 12% compounded monthly for 4 years. 66. Joan Varozza plans to borrow $20,000 to stock her small boutque. She wll repay the loan wth semannual payments for 5 years at 7% compounded semannually. Checkpont Answers 1. (a) $ (b) $ ; $ ; $ ; $ ; $ $20, $ (a) $ (b) $14, $10, $ $98, $13,023, CHAPTER 5 Summary and Revew Key Terms and Symbols 5.1 smple nterest prncpal rate tme future value (maturty value) present value dscount and T-blls 5.2 compound nterest compound amount compoundng perod nomnal rate (stated rate) effectve rate (APY) present value 5.3 annuty payment perod term of an annuty ordnary annuty future value of an ordnary annuty snkng fund annuty due future value of an annuty due 5.4 present value of an ordnary annuty amortzaton payments remanng balance amortzaton schedule present value of an annuty due

40 264 CHAPTER 5 Mathematcs of Fnance Chapter 5 Key Concepts A Strategy for Solvng Fnance Problems Key Formulas We have presented a lot of new formulas n ths chapter. By answerng the followng questons, you can decde whch formula to use for a partcular problem. 1. Is smple or compound nterest nvolved? Smple nterest s normally used for nvestments or loans of a year or less; compound nterest s normally used n all other cases. 2. If smple nterest s beng used, what s beng sought nterest amount, future value, present value, or dscount? 3. If compound nterest s beng used, does t nvolve a lump sum (sngle payment) or an annuty (sequence of payments)? (a) For a lump sum, () s ordnary compound nterest nvolved? () what s beng sought present value, future value, number of perods, or effectve rate (APY)? (b) For an annuty, () s t an ordnary annuty (payment at the end of each perod) or an annuty due (payment at the begnnng of each perod)? () what s beng sought present value, future value, or payment amount? Once you have answered these questons, choose the approprate formula and work the problem. As a fnal step, consder whether the answer you get makes sense. For nstance, the amount of nterest or the payments n an annuty should be farly small compared wth the total future value. Lst of Varables r s the annual nterest rate. m s the number of perods per year. s the nterest rate per perod. t s the number of years. n s the number of perods. P s the prncpal or present value. A s the future value of a lump sum. S s the future value of an annuty. R s the perodc payment n an annuty. B s the remanng balance on a loan. = r m n = tm Interest Smple Interest Compound Interest Interest I = Prt I = A - P Future value A = P(1 + rt) A = P(1 + ) n Present value P = A 1 + rt P = A n = A(1 + )-n (1 + ) Effectve rate (or APY) r E = a1 + r m b m - 1 Dscount Contnuous Interest If D s the dscount on a T-bll wth face value P at smple nterest rate r for t years, then D = Prt. If P dollars are deposted for t years at nterest rate r per year, compounded contnuously, the compound amount (future value) s A = Pe rt. The present value P of A dollars at nterest rate r per year compounded contnuously for t years s P = A e rt.

41 CHAPTER 5 Summary and Revew 265 Annutes Ordnary annuty Future value S = Rc (1 + )n - 1 d = R # sn 1 - (1 + )-n Present value P = Rc d = R # an Annuty due Future value S = Rc (1 + )n+1-1 d - R -1) 1 - (1 + )-(n Present value P = R + Rc d Chapter 5 Revew Exercses Fnd the smple nterest for the followng loans. 1. $4902 at 6.5% for 11 months 2. $42,368 at 9.22% for 5 months 3. $3478 at 7.4% for 88 days 4. $2390 at 8.7% from May 3 to July 28 Fnd the semannual (smple) nterest payment and the total nterest earned over the lfe of the bond. 5. $12,000 Merck Company 6-year bond at 4.75% annual nterest 6. $20,000 General Electrc 9-year bond at 5.25% annual nterest Fnd the maturty value for each smple nterest loan. 7. $7750 at 6.8% for 4 months 8. $15,600 at 8.2% for 9 months 9. What s meant by the present value of an amount A? Fnd the present value of the gven future amounts; use smple nterest. 10. $ n 7 months; money earns 5.5% 11. $80,612 n 128 days; money earns 6.77% 12. A 9-month $7000 Treasury bll sells at a dscount rate of 3.5%. Fnd the amount of the dscount and the prce of the T-bll. 13. A 6-month $10,000 T-bll sold at a 4% dscount. Fnd the actual rate of nterest pad by the Treasury. 14. For a gven amount of money at a gven nterest rate for a gven perod greater than 1 year, does smple nterest or compound nterest produce more nterest? Explan. Fnd the compound amount and the amount of nterest earned n each of the gven scenaros. 15. $2800 at 6% compounded annually for 12 years 16. $57, at 4% compounded quarterly for 6 years 17. $12, at 6.37% compounded quarterly for 29 quarters 18. $ at 4.57% compounded monthly for 32 months Fnd the amount of compound nterest earned by each depost. 19. $22,000 at 5.5%, compounded quarterly for 6 years 20. $2975 at 4.7%, compounded monthly for 4 years Fnd the face value (to the nearest dollar) of the zero-coupon bond year bond at 3.9%; prce $12, year bond at 5.2%; prce $11,575 Fnd the APY correspondng to the gven nomnal rate % compounded semannually % compounded daly Fnd the present value of the gven amounts at the gven nterest rate. 25. $42,000 n 7 years; 12% compounded monthly 26. $17,650 n 4 years; 8% compounded quarterly 27. $ n 3.5 years; 6.2% compounded semannually 28. $ n 44 months; 5.75% compounded monthly Fnd the prce that a purchaser should be wllng to pay for these zero-coupon bonds year $15,000 bond; nterest at 4.4% year $30,000 bond; nterest at 6.2% 31. What s meant by the future value of an annuty? Fnd the future value of each annuty. 32. $1288 deposted at the end of each year for 14 years; money earns 7% compounded annually 33. $4000 deposted at the end of each quarter for 8 years; money earns 6% compounded quarterly 34. $233 deposted at the end of each month for 4 years; money earns 6% compounded monthly 35. $672 deposted at the begnnng of each quarter for 7 years; money earns 5% compounded quarterly 36. $11,900 deposted at the begnnng of each month for 13 months; money earns 7% compounded monthly 37. What s the purpose of a snkng fund? Fnd the amount of each payment that must be made nto a snkng fund to accumulate the gven amounts. Assume payments are made at the end of each perod. 38. $6500; money earns 5% compounded annually; 6 annual payments 39. $57,000; money earns 6% compounded semannually for years

42 266 CHAPTER 5 Mathematcs of Fnance 40. $233,188; money earns 5.7% compounded quarterly for years 41. $56,788; money earns 6.12% compounded monthly for years Fnd the present value of each ordnary annuty. 42. Payments of $850 annually for 4 years at 5% compounded annually 43. Payments of $1500 quarterly for 7 years at 8% compounded quarterly 44. Payments of $4210 semannually for 8 years at 5.6% compounded semannually 45. Payments of $ monthly for 17 months at 6.4% compounded monthly Fnd the amount necessary to fund the gven wthdrawals (whch are made at the end of each perod). 46. Quarterly wthdrawals of $800 for 4 years wth nterest rate 4.6%, compounded quarterly 47. Monthly wthdrawals of $1500 for 10 years wth nterest rate 5.8%, compounded monthly 48. Yearly wthdrawals of $3000 for 15 years wth nterest rate 6.2%, compounded annually Fnd the payment for the ordnary annuty wth the gven present value. 49. $150,000; monthly payments for 15 years, wth nterest rate 5.1%, compounded monthly 50. $25,000; quarterly payments for 8 years, wth nterest rate 4.9%, compounded quarterly 51. Fnd the lump-sum depost today that wll produce the same total amount as payments of $4200 at the end of each year for 12 years. The nterest rate n both cases s 4.5%, compounded annually. 52. If the current nterest rate s 6.5%, fnd the prce (to the nearest dollar) that a purchaser should be wllng to pay for a $24,000 bond wth coupon rate 5% that matures n 6 years. Fnd the amount of the payment necessary to amortze each of the gven loans. 53. $32,000 at 8.4% compounded quarterly; 10 quarterly payments 54. $5607 at 7.6% compounded monthly; 32 monthly payments Fnd the monthly house payments for the gven mortgages. 55. $95,000 at 3.67% for 30 years 56. $167,000 at 2.91% for 15 years 57. Fnd the approxmate remanng balance after 5 years of payments on the loan n Exercse Fnd the approxmate remanng balance after 7.5 years of payments on the loan n Exercse 56. Fnance Accordng to n 2013 the nterest rate for a drect unsubsdzed student loan was 6.8%. A porton of an amortzaton table s gven here for a $15,000 drect unsubsdzed student loan compounded monthly to be pad back n 10 years. Use the table to answer Exercses Payment Number Amount of Payment Interest for Perod Porton to Prncpal Prncpal at End of Perod 0 15, , , , , , , , , How much of the seventh payment s nterest? 60. How much of the fourth payment s used to reduce the debt? 61. How much nterest s pad n the frst 6 months of the loan? 62. How much has the debt been reduced at the end of the frst 8 months? Fnance Work the followng appled problems. 63. In February 2013, a Vrgna famly won a Powerball lottery prze of $217,000,000. (a) If they had chosen to receve the money n 30 yearly payments, begnnng mmedately, what would have been the amount of each payment? (b) The famly chose the one-tme lump-sum cash opton. If the nterest rate s 3.58%, approxmately how much dd they receve? 64. A f rm of attorneys deposts $15,000 of proft-sharng money n an account at 6% compounded semannually for years. Fnd the amount of nterest earned. 65. Tom, a graduate student, s consderng nvestng $500 now, when he s 23, or watng untl he s 40 to nvest $500. How much more money wll he have at age 65 f he nvests now, gven that he can earn 5% nterest compounded quarterly? 66. Accordng to a fnancal Web ste, on June 15, 2005, Frontenac Bank of Earth Cty, Mssour, pad 3.94% nterest, compounded quarterly, on a 2-year CD, whle E*TRADE Bank of Arlngton, Vrgna, pad 3.93% compounded daly. What was the effectve rate for the two CDs, and whch bank pad a hgher effectve rate? (Data from: Chalon Brdges deposts semannual payments of $3200, receved n payment of a debt, n an ordnary annuty at 6.8% compounded semannually. Fnd the fnal amount n the account and the nterest earned at the end of 3.5 years. 68. Each year, a frm must set asde enough funds to provde employee retrement benefts of $52,000 n 20 years. If the frm can nvest money at 7.5% compounded monthly, what

43 CASE STUDY 5 Contnuous Compoundng 267 amount must be nvested at the end of each month for ths purpose? 69. A benefactor wants to be able to leave a bequest to the college she attended. If she wants to make a donaton of $2,000,000 n 10 years, how much each month does she need to place n an nvestment account that pays an nterest rate of 5.5%, compounded monthly? 70. Suppose you have bult up a penson wth $12,000 annual payments by workng 10 years for a company when you leave to accept a better job. The company gves you the opton of collectng half the full penson when you reach age 55 or the full penson at age 65. Assume an nterest rate of 8% compounded annually. By age 75, how much wll each plan produce? Whch plan would produce the larger amount? 71. In 3 years, Ms. Nguyen must pay a pledge of $7500 to her favorte charty. What lump sum can she depost today at 10% compounded semannually so that she wll have enough to pay the pledge? 72. To f nance the $15,000 cost of ther ktchen remodelng, the Chews wll make equal payments at the end of each month for 36 months. They wll pay nterest at the rate of 7.2% compounded monthly. Fnd the amount of each payment. 73. To expand her busness, the owner of a small restaurant borrows $40,000. She wll repay the money n equal payments at the end of each semannual perod for 8 years at 9% nterest compounded semannually. What payments must she make? 74. The Fx famly bought a house for $210,000. They pad $42,000 down and took out a 30-year mortgage for the balance at 3.75%. (a) Fnd the monthly payment. (b) How much of the frst payment s nterest? After 15 years, the famly sold ther house for $255,000. (c) Estmate the current mortgage balance at the tme of the sale. (d) Fnd the amount of money they receved from the sale after payng off the mortgage. 75. Over a 20-year perod, the class A shares of the Davs New York Venture mutual fund ncreased n value at the rate of 11.2%, compounded monthly. If you had nvested $250 at the end of each month n ths fund, what would have the value of your account been at the end of those 20 years? 76. Dan Hook deposts $400 a month to a retrement account that has an nterest rate of 3.1%, compounded monthly. After makng 60 deposts, Dan changes hs job and stops makng payments for 3 years. After 3 years, he starts makng deposts agan, but now he deposts $525 per month. What wll the value of the retrement account be after Dan makes hs $525 monthly deposts for 5 years? 77. The proceeds of a $10,000 death beneft are left on depost wth an nsurance company for 7 years at an annual effectve nterest rate of 5%.* The balance at the end of 7 years s pad to the benefcary n 120 equal monthly payments of X, wth the frst payment made mmedately. Durng the payout perod, nterest s credted at an annual effectve nterest rate of 3%. Whch of the followng s the correct value of X? (a) 117 (b) 118 (c) 129 (d) 135 (e) Eleen Ganods wants to retre on $75,000 per year for her lfe expectancy of 20 years after she retres. She estmates that she wll be able to earn an nterest rate of 10.1%, compounded annually, throughout her lfetme. To reach her retrement goal, Eleen wll make annual contrbutons to her account for the next 30 years. One year after makng her last depost, she wll receve her frst retrement check. How large must her yearly contrbutons be? *The Proceeds of Death Beneft Left on Depost wth an Insurance Company from Course 140 Examnaton, Mathematcs of Compound Interest. Copyrght Socety of Actuares. Reproduced by permsson of Socety of Actuares. Case Study 5 Contnuous Compoundng Informally, you can thnk of contnuous compoundng as a process n whch nterest s compounded very frequently (for nstance, every nanosecond). You wll occasonally see an ad for a certfcate of depost n whch nterest s compounded contnuously. That s pretty much a gmmck n most cases, because t produces only a few more cents than daly compoundng. However, contnuous compoundng does play a serous role n certan fnancal stuatons, notably n the prcng of dervatves. * So let s see what s nvolved n contnuous compoundng. As a general rule, the more often nterest s compounded, the better off you are as an nvestor. (See Example 4 of Secton * Dervatves are complcated fnancal nstruments. But nvestors have learned the hard way that they can sometmes cause serous problems as was the case n the recesson that began n 2008, whch was blamed n part on the msuse of dervatves. 5.2.) But there s, alas, a lmt on the amount of nterest, no matter how often t s compounded. To see why ths s so, suppose you have $1 to nvest. The Exponental Bank offers to pay 100% annual nterest, compounded n tmes per year and rounded to the nearest penny. Furthermore, you may pck any value for n that you want. Can you choose n so large that your $1 wll grow to $5 n a year? We wll test several values of n n the formula for the compound amount, wth P = 1. In ths case, the annual nterest rate (n decmal form) s also 1. If there are n perods n the year, the nterest rate per perod s = 1>n. So the amount that your dollar grows to s: A = P(1 + ) n = 1a1 + 1 n b n.

44 268 CHAPTER 5 Mathematcs of Fnance A computer gves the followng results for varous values of n : Interest Is Compounded... n a1 + 1 n b n Annually 1 a b Semannually 2 a b Quarterly 4 a b Monthly 12 a b = 2 = Daly 365 a b Hourly 8760 a1 + Every mnute 525,600 a1 + Every second 31,536,000 a b , ,600 b ,536, ,536,000 b Because nterest s rounded to the nearest penny, the compound amount never exceeds $2.72, no matter how bg n s. (A computer was used to develop the table, and the fgures n t are accurate. If you try these computatons wth your calculator, however, your answers may not agree exactly wth those n the table because of round-off error n the calculator.) The precedng table suggests that as n takes larger and larger values, the correspondng values of a1 + 1 n b n get closer and closer to a specfc real number whose decmal expanson begns g. Ths s ndeed the case, as s shown n calculus, and the number g s denoted e. Ths fact s sometmes expressed by wrtng lm ns a1 + 1 n n b = e, whch s read the lmt of a1 + 1 n b n as n approaches nfnty s e. The precedng example s typcal of what happens when nterest s compounded n tmes per year, wth larger and larger values of n. It can be shown that no matter what nterest rate or prncpal s used, there s always an upper lmt (nvolvng the number e ) on the compound amount, whch s called the compound amount from contnuous compoundng. Contnuous Compoundng The compound amount A for a depost of P dollars at an nterest rate r per year compounded contnuously for t years s gven by A = Pe rt. Most calculators have an e x key for computng powers of e. See the Technology Tp on page 188 for detals on usng a calculator to evaluate e x. Example 1 Suppose $5000 s nvested at an annual rate of 4% compounded contnuously for 5 years. Fnd the compound amount. Soluton In the formula for contnuous compoundng, let P = 5000, r =.04, and t = 5. Then a calculator wth an e x k ey shows that A = 5000e (.04)5 = 5000e.2 $ You can readly verfy that daly compoundng would have produced a compound amount about 6 less. Exercses 1. Fnd the compound amount when $20,000 s nvested at 6% compounded contnuously for (a) 2 years; (b) 10 years; (c) 20 years. 2. (a) Fnd the compound amount when $2500 s nvested at 5.5%, compounded monthly for two years. (b) Do part (a) when the nterest rate s 5.5% compounded contnuously. 3. Determne the compounded amount from a depost of $25,000 when t s nvested at 5% for 10 years and compounded n the followng tme perods: (a) annually (b) quarterly (c) monthly (d) daly (e) contnuously

45 CASE STUDY 5 Contnuous Compoundng Determne the compounded amount from a depost of $250,000 when t s nvested at 5% for 10 years and compounded n the followng tme perods: (a) annually (b) quarterly (c) monthly (d) daly (e) contnuously 5. It can be shown that f nterest s compounded contnuously at nomnal rate r, then the effectve rate r E s e r - 1. Fnd the effectve rate of contnuously compounded nterest f the nomnal rate s (a) 4.5%; (b) 5.7%; (c) 7.4%. 6. Suppose you wn a court case and the defendant has up to 8 years to pay you $5000 n damages. Assume that the defendant wll wat untl the last possble mnute to pay you. (a) If you can get an nterest rate of 3.75% compounded contnuously, what s the present value of the $5000? [ Hnt: Solve the contnuous-compoundng formula for P.] (b) If the defendant offers you $4000 mmedately to settle hs debt, should you take the deal? Extended Projects 1. Investgate the nterest rates for the subsdzed and unsubsdzed student loans. If you have taken out student loans or plan to take out student loans before graduatng, calculate your own monthly payment and how much nterest you wll pay over the course of the repayment perod. If you have not taken out, and you do not plan to take out a student loan, contact the fnancal ad offce of your college and unversty to determne the medan amount borrowed wth student loans at your nsttuton. Determne the monthly payment and how much nterest s pad durng repayment for the typcal borrowng student. 2. Determne the best nterest rate for a new car purchase for a 48-month loan at a bank near you. If you fnance $25,999 wth such a loan, determne the payment and the total nterest pad over the course of the loan. Also, determne the best nterest rate for a new car purchase for a 48-month loan at a credt unon near you. Determne the monthly payment and total nterest pad f the same auto loan s fnanced through the credt unon. Is t true that the credt unon would save you money?