# Mathematics of Finance

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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%

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)

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