Mathematics of Finance

Transcription

1 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car usually requres both some savngs for a down payment and a loan for the balance. An exercse n Secton 2 calculates the regular deposts that would be needed to save up the full purchase prce, and other exercses and examples n ths chapter compute the payments requred to amortze a loan. 187

2 188 CHAPTER 5 Mathematcs of Fnance Everybody uses money. Sometmes you work for your money and other tmes your money works for you. For example, unless you are attendng college on a full scholarshp, t s very lkely that you and your famly have ether saved money or borrowed money, or both, to pay for your educaton.when we borrow money, we normally have to pay nterest for that prvlege.when we save money, for a future purchase or retrement, we are lendng money to a fnancal nsttuton and we expect to earn nterest on our nvestment.we wll develop the mathematcs n ths chapter to understand better the prncples of borrowng and savng.these deas wll then be used to compare dfferent fnancal opportuntes and make nformed decsons. 5.1 APPLY IT Smple and Compound Interest If you can borrow money at 8% nterest compounded annually or at 7.9% compounded monthly, whch loan would cost less? In ths secton we wll learn how to compare dfferent nterest rates wth dfferent compoundng perods. The queston above wll be answered n Example 8. Smple Interest Interest on loans of a year or less s frequently calculated as smple nterest, a type of nterest that s charged (or pad) only on the amount borrowed (or nvested) and not on past nterest. The amount borrowed s called the prncpal. The rate of nterest s gven as a percentage per year, expressed as a decmal. For example, 6% and % The tme the money s earnng nterest s calculated n years. One year s nterest s calculated by multplyng the prncpal tmes the nterest rate, or Pr. If the tme that the money earns nterest s other than one year, we multply the nterest for one year by the number of years, or Prt. Smple Interest where P s the prncpal; r s the annual nterest rate; t s the tme n years. I 5 Prt EXAMPLE 1 Smple Interest To buy furnture for a new apartment, Candace Cooney borrowed $5000 at 8% smple nterest for 11 months. How much nterest wll she pay? SOLUTION Use the formula I 5 Prt, wth P , r , and t 5 11/12 (n years). The total nterest she wll pay s or $ I /12 2 < , A depost of P dollars today at a rate of nterest r for t years produces nterest of I 5 Prt. The nterest, added to the orgnal prncpal P, gves P 1 Prt 5 P11 1 rt2.

3 5.1 Smple and Compound Interest 189 Ths amount s called the future value of P dollars at an nterest rate r for tme t n years. When loans are nvolved, the future value s often called the maturty value of the loan. Ths dea s summarzed as follows. Future or Maturty Value for Smple Interest The future or maturty value A of P dollars at a smple nterest rate r for t years s A 5 P1 1 1 rt 2. EXAMPLE 2 Maturty Values Fnd the maturty value for each loan at smple nterest. (a) A loan of $2500 to be repad n 8 months wth nterest of 4.3% SOLUTION The loan s for 8 months, or 8/12 5 2/3 of a year. The maturty value s A 5 P1 1 1 rt 2 A c a 2 3 bd YOUR TURN 1 Fnd the maturty value for a $3000 loan at 5.8% nterest for 100 days. or $ (The answer s rounded to the nearest cent, as s customary n fnancal problems.) Of ths maturty value, represents nterest. A < , $ $ $71.67 (b) A loan of $11,280 for 85 days at 7% nterest SOLUTION It s common to assume 360 days n a year when workng wth smple nterest. We shall usually make such an assumpton n ths book. The maturty value n ths example s A 5 11,280 c a 85 bd < 11,466.43, 360 or $11, TRY YOUR TURN 1 CAUTION When usng the formula for future value, as well as all other formulas n ths chapter, we often neglect the fact that n real lfe, money amounts are rounded to the nearest penny. As a consequence, when the amounts are rounded, ther values may dffer by a few cents from the amounts gven by these formulas. For nstance, n Example 2(a), the nterest n each monthly payment would be $ /12 2 < $8.96, rounded to the nearest penny. After 8 months, the total s 81 $ $71.68, whch s 1 more than we computed n the example. In part (b) of Example 2 we assumed 360 days n a year. Hstorcally, to smplfy calculatons, t was often assumed that each year had twelve 30-day months, makng a year 360 days long. Treasury blls sold by the U.S. government assume a 360-day year n calculatng nterest. Interest found usng a 360-day year s called ordnary nterest and nterest found usng a 365-day year s called exact nterest. The formula for future value has four varables, P, r, t, and A. We can use the formula to fnd any of the quanttes that these varables represent, as llustrated n the next example.

4 190 CHAPTER 5 Mathematcs of Fnance YOUR TURN 2 Fnd the nterest rate f $5000 s borrowed, and $ s pad back 9 months later. EXAMPLE 3 Smple Interest Theresa Cortesn wants to borrow $8000 from Chrstne O Bren. She s wllng to pay back $8180 n 6 months. What nterest rate wll she pay? SOLUTION Use the formula for future value, wth A , P , t 5 6/ , and solve for r. A 5 P11 1 rt r r r r Dstrbutve property Subtract Dvde by Thus, the nterest rate s 4.5%. TRY YOUR TURN 2 When you depost money n the bank and earn nterest, t s as f the bank borrowed the money from you. Reversng the scenaro n Example 3, f you put $8000 n a bank account that pays smple nterest at a rate of 4.5% annually, you wll have accumulated $8180 after 6 months. Compound Interest As mentoned earler, smple nterest s normally used for loans or nvestments of a year or less. For longer perods compound nterest s used. Wth compound nterest, nterest s charged (or pad) on nterest as well as on prncpal. For example, f $1000 s deposted at 5% nterest for 1 year, at the end of the year the nterest s $ $50. The balance n the account s $ $50 5 $1050. If ths amount s left at 5% nterest for another year, the nterest s calculated on $1050 nstead of the orgnal $1000, so the amount n the account at the end of the second year s $ $ $ Note that smple nterest would produce a total amount of only The addtonal $2.50 s the nterest on $50 at 5% for one year. To fnd a formula for compound nterest, frst suppose that P dollars s deposted at a rate of nterest r per year. The amount on depost at the end of the frst year s found by the smple nterest formula, wth t 5 1. A 5 P11 1 r P11 1 r2 If the depost earns compound nterest, the nterest earned durng the second year s pad on the total amount on depost at the end of the frst year. Usng the formula A 5 P11 1 rt2 agan, wth P replaced by P11 1 r2 and t 5 1, gves the total amount on depost at the end of the second year. In the same way, the total amount on depost at the end of the thrd year s Generalzng, n t years the total amount on depost s called the compound amount. NOTE $ $1100. A 5 3P11 1 r r P11 1 r2 2 Compare ths formula for compound nterest wth the formula for smple nterest. Compound nterest Smple nterest P11 1 r2 3. A 5 P11 1 r2 t, A 5 P11 1 r2 t A 5 P1 1 1 rt 2 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.

5 5.1 Smple and Compound Interest 191 Interest can be compounded more than once per year. Common compoundng perods nclude semannually (two perods per year), quarterly (four perods per year), monthly (twelve perods per year), or daly (usually 365 perods per year). The nterest rate per perod,, s found by dvdng the annual nterest rate, r, by the number of compoundng perods, m, per year. To fnd the total number of compoundng perods, n, we multply the number of years, t, by the number of compoundng perods per year, m. The followng formula can be derved n the same way as the prevous formula. Compound Amount A 5 P n where 5 r and n 5 mt, m A s the future (maturty) value; P s the prncpal; r s the annual nterest rate; m s the number of compoundng perods per year; t s the number of years; n s the number of compoundng perods; s the nterest rate per perod. EXAMPLE 4 Compound Interest Suppose $1000 s deposted for 6 years n an account payng 4.25% per year compounded annually. (a) Fnd the compound amount. SOLUTION In the formula for the compound amount, P , /1, and n The compound amount s Usng a calculator, we get the compound amount. A 5 P n A A < $ , (b) Fnd the amount of nterest earned. SOLUTION Subtract the ntal depost from the compound amount. Amount of nterest 5 $ $ $ YOUR TURN 3 Fnd the amount of nterest earned by a depost of $1600 for 7 years at 4.2% compounded monthly. EXAMPLE 5 Compound Interest Fnd the amount of nterest earned by a depost of $2450 for 6.5 years at 5.25% compounded quarterly. SOLUTION Interest compounded quarterly s compounded 4 tmes a year. In 6.5 years, there are perods. Thus, n Interest of 5.25% per year s 5.25%/4 per quarter, so /4. Now use the formula for compound amount. A 5 P n A / < Rounded to the nearest cent, the compound amount s $ , so the nterest s $ $ $ TRY YOUR TURN 3

6 192 CHAPTER 5 Mathematcs of Fnance CAUTION As shown n Example 5, compound nterest problems nvolve two rates the annual rate r and the rate per compoundng perod. Be sure you understand the dstncton between them. When nterest s compounded annually, these rates are the same. In all other cases, 2 r. Smlarly, there are two quanttes for tme: the number of years t and the number of compoundng perods n. When nterest s compounded annually, these varables have the same value. In all other cases, n 2 t. It s nterestng to compare loans at the same rate when smple or compound nterest s used. Fgure 1 shows the graphs of the smple nterest and compound nterest formulas wth P at an annual rate of 10% from 0 to 20 years. The future value after 15 years s shown for each graph. After 15 years of compound nterest, $1000 grows to $ , whereas wth smple nterest, t amounts to $ , a dfference of $ A A = A = 1000(1.1) t 3500 Compound Interest A = A = 1000( t) 1000 Smple Interest 500 TECHNOLOGY NOTE FIGURE 1 Spreadsheets are deal for performng fnancal calculatons. Fgure 2 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%. Compare row 16 wth Fgure 1. For more detals on the use of spreadsheets n the mathematcs of fnance, see the Graphng Calculator and Excel Spreadsheet Manual avalable wth ths book. t A B perod compound smple C FIGURE 2

7 5.1 Smple and Compound Interest 193 We can also solve the compound amount formula for the nterest rate, as n the followng example. YOUR TURN 4 Fnd the annual nterest rate f $6500 s worth $ after beng nvested for 8 years n an account that compounded nterest monthly. EXAMPLE 6 Compound Interest Rate Suppose Carol Merrgan nvested $5000 n a savngs account that pad quarterly nterest. After 6 years the money had accumulated to $ What was the annual nterest rate? SOLUTION Because m 5 4 and t 5 6, the number of compoundng perods s n Usng ths value along wth P and A n the formula for compound amount, we have r/ r/ r/ /24 < r/ r Dvde both sdes by Take both sdes to the 1/24 power. Subtract 1 from both sdes. Multply both sdes by 4. The annual nterest rate was 4.5%. TRY YOUR TURN 4 Effectve Rate If $1 s deposted at 4% compounded quarterly, a calculator can be used to fnd that at the end of one year, the compound amount s $1.0406, an ncrease of 4.06% over the orgnal $1. The actual ncrease of 4.06% n the money s somewhat hgher than the stated ncrease of 4%. To dfferentate between these two numbers, 4% s called the nomnal or stated rate of nterest, whle 4.06% s called the effectve rate.* To avod confuson between stated rates and effectve rates, we shall contnue to use r for the stated rate and we wll use for the effectve rate. r E EXAMPLE 7 Effectve Rate Fnd the effectve rate correspondng to a stated rate of 6% compounded semannually. SOLUTION Here, 5 r/m / for m 5 2 perods. Use a calculator to fnd that < , whch shows that $1 wll ncrease to $ , an actual ncrease of 6.09%. The effectve rate s r E %. Generalzng from ths example, the effectve rate of nterest s gven by the followng formula. Effectve Rate The effectve rate correspondng to a stated rate of nterest r compounded m tmes per year s r E 5 a1 1 r m. m b 2 1 EXAMPLE 8 Effectve Rate Joe Vetere needs to borrow money. Hs neghborhood bank charges 8% nterest compounded semannually. A downtown bank charges 7.9% nterest compounded monthly. At whch bank wll Joe pay the lesser amount of nterest? *When appled to consumer fnance, the effectve rate s called the annual percentage rate, APR, or annual percentage yeld, APY.

8 194 CHAPTER 5 Mathematcs of Fnance APPLY IT YOUR TURN 5 Fnd the effectve rate for an account that pays 2.7% compounded monthly. SOLUTION Compare the effectve rates. Neghborhood bank: Downtown bank: r E 5 a b % r E 5 a b < < 8.19% The neghborhood bank has the lower effectve rate, although t has a hgher stated rate. TRY YOUR TURN 5 Present Value The formula for compound nterest, A 5 P 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. EXAMPLE 9 Present Value Rachel Reeve 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? SOLUTION Here A , , n 5 5, and P s unknown. Substtutng these values nto the formula for the compound amount gves P or $ If Rachel 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 5 $ , , and n 5 5. You should get A 5 $ As Example 9 shows, $6000 n 5 years s approxmately the same as $ today (f money can be deposted at 6.2% compounded annually). An amount that can be deposted today to yeld a gven sum n the future s called the present value of the future sum. Generalzng from Example 9, by solvng A 5 P n for P, we get the followng formula for present value. Present Value for Compound Interest P < , The present value of A dollars compounded at an nterest rate per perod for n perods s A P or P 5 A n. n YOUR TURN 6 Fnd the present value of $10,000 n 7 years f money can be deposted at 4.25% compounded quarterly. EXAMPLE 10 Present Value Fnd the present value of $16,000 n 9 years f money can be deposted at 6% compounded semannually. SOLUTION In 9 years there are semannual perods. A rate of 6% per year s 3% n each semannual perod. Apply the formula wth A 5 16,000, , and n A P ,000 n 18 < A depost of $ today, at 6% compounded semannually, wll produce a total of $16,000 n 9 years. TRY YOUR TURN 6 We can solve the compound amount formula for n also, as the followng example shows.

9 5.1 Smple and Compound Interest 195 Method 1 Graphng Calculator EXAMPLE 11 Compoundng Tme Suppose the $2450 from Example 5 s deposted at 5.25% compounded quarterly untl t reaches at least $10,000. How much tme s requred? SOLUTION Graph the functons y /4 2 x and y 5 10,000 n the same wndow, and then fnd the pont of ntersecton. As Fgure 3 shows, the functons ntersect at x Note, however, that nterest s only added to the account every quarter, so we must wat 108 quarters, or 108/ years, for the money to be worth at least $10, ,000 Intersecton 0 X Y FIGURE 3 FOR REVIEW Method 2 Usng Logarthms (Optonal) For a revew of logarthmc functons, please refer to Appendx B f you are usng Fnte Mathematcs, or to Secton 10.5 f you are usng Fnte Mathematcs and Calculus wth Applcatons. The only property of logarthms that s needed to fnd the compoundng tme s logx r 5 rlogx. Logarthms may be used n base 10, usng the LOG button on a calculator, or n base e, usng the LN button. YOUR TURN 7 Fnd the tme needed for $3800 deposted at 3.5% compounded semannually to be worth at least $7000. The goal s to solve the equaton Dvde both sdes by 2450, and smplfy the expresson n parentheses to get Now take the logarthm (ether base 10 or base e) of both sdes to get log n 2 5 log1 10,000/ n log log1 10,000/ n 5 log1 10,000 / log < Use logarthm property log x r 5 r log x. Dvde both sdes by log As n Method 1, ths means that we must wat 108 quarters, or 108/ years, for the money to be worth at least $10,000. EXAMPLE /4 2 n 5 10, n 5 10, Prce Doublng Suppose the general level of nflaton n the economy averages 8% per year. Fnd the number of years t would take for the overall level of prces to double. SOLUTION To fnd the number of years t wll take for $1 worth of goods or servces to cost $2, fnd n n the equaton n, where A 5 2, P 5 1, and Ths equaton smplfes to n. Solvng ths equaton usng ether a graphng calculator or logarthms, as n Example 11, shows that n Thus, the overall level of prces wll double n about 9 years. TRY YOUR TURN 7 You can quckly estmate how long t takes a sum of money to double, when compounded annually, by usng ether the rule of 70 or the rule of 72. The rule of 70 (used for

10 196 CHAPTER 5 Mathematcs of Fnance small rates of growth) says that for # r, 0.05, the value of 70/100r gves a good approxmaton of the doublng tme. The rule of 72 (used for larger rates of growth) says that for 0.05 # r # 0.12, the value of 72/100r approxmates the doublng tme well. In Example 12, the nflaton rate s 8%, so the doublng tme s approxmately 72/8 5 9 years.* Contnuous Compoundng Suppose that a bank, n order to attract more busness, offers to not just compound nterest every quarter, or every month, or every day, or even every hour, but constantly? Ths type of compound nterest, n whch the number of tmes a year that the nterest s compounded becomes nfnte, s known as contnuous compoundng. To see how t works, look back at Example 5, where we found that $2450, when deposted for 6.5 years at 5.25% compounded quarterly, resulted n a compound amount of $ We can fnd the compound amount f we compound more often by puttng dfferent values of n n the formula A /n 2 6.5n, as shown n the followng table. Compoundng n Tmes Annually n Type of Compoundng Compound Amount 4 quarterly $ monthly $ daly $ every hour $ Notce that as n becomes larger, the compound amount also becomes larger, but by a smaller and smaller amount. In ths example, ncreasng the number of compoundng perods a year from 360 to 8640 only earns 8 more. It s shown n calculus that as n becomes nfntely large, P1 1 1 r/n 2 nt gets closer and closer to Pe rt, where e s a very mportant rratonal number whose approxmate value s To calculate nterest wth contnuous compoundng, use the e x button on your calculator. You wll learn more about the number e f you study calculus, where e plays as mportant a role as p does n geometry. Contnuous Compoundng If a depost of P dollars s nvested at a rate of nterest r compounded contnuously for t years, the compound amount s A 5 Pe rt dollars. EXAMPLE 13 Contnuous Compoundng Suppose that $2450 s deposted at 5.25% compounded contnuously. (a) Fnd the compound amount and the nterest earned after 6.5 years. SOLUTION Usng the formula for contnuous compoundng wth P , r , and t 5 6.5, the compound amount s A e < The compound amount s $ , whch s just a penny more than f t had been compounded hourly, or 9 more than daly compoundng. Because t makes so lttle dfference, contnuous compoundng has dropped n popularty n recent years. The nterest n ths case s $ $996.43, or $7.65 more than f t were compounded quarterly, as n Example 5. *To see where the rule of 70 and the rule of 72 come from, see the secton on Taylor Seres n Calculus wth Applcatons by Margaret L. Lal, Raymond N. Greenwell, and Nathan P. Rtchey, Pearson, 2012.

11 5.1 Smple and Compound Interest 197 YOUR TURN 8 Fnd the nterest earned on $5000 deposted at 3.8% compounded contnuously for 9 years. (b) Fnd the effectve rate. SOLUTION As n Example 7, the effectve rate s just the amount of nterest that $1 would earn n one year, or e < , or 5.39%. In general, the effectve rate for nterest compounded contnuously at a rate r s just e r 2 1. (c) Fnd the tme requred for the orgnal $2450 to grow to $10,000. SOLUTION Smlar to the process n Example 11, we must solve the equaton 10, e t. Dvde both sdes by 2450, and solve the resultng equaton as n Example 11, ether by takng logarthms of both sdes or by usng a graphng calculator to fnd the ntersecton of the graphs of y e t and y 5 10,000. If you use logarthms, you can take advantage of the fact that ln1 e x 2 5 x, where ln x represents the logarthm n base e. In ether case, the answer s years. Notce that unlke n Example 11, you don t need to wat untl the next compoundng perod to reach ths amount, because nterest s beng added to the account contnuously. TRY YOUR TURN 8 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 5 prncpal or present value A 5 future or maturty value r 5 annual (stated or nomnal) nterest rate t 5 number of years m 5 number of compoundng perods per year 5 nterest rate per perod 5 r/m n 5 total number of compoundng perods n 5 tm r E 5 effectve rate Smple Interest Compound Interest Contnuous Compoundng A 5 P11 1 rt2 P 5 A 1 1 rt A 5 P n P 5 A 5 A n n r E 5 a1 1 r m b m 2 1 A 5 Pe rt P 5 Ae 2rt r E 5 e r EXERCISES 1. What factors determne the amount of nterest earned on a fxed prncpal? 2. In your own words, descrbe the maturty value of a loan. 3. What s meant by the present value of money? 4. We calculated the loan n Example 2(b) assumng 360 days n a year. Fnd the maturty value usng 365 days n a year. Whch s more advantageous to the borrower? Fnd the smple nterest. 5. $25,000 at 3% for 9 months 6. $4289 at 4.5% for 35 weeks 7. $1974 at 6.3% for 25 weeks 8. $6125 at 1.25% for 6 months

12 198 CHAPTER 5 Mathematcs of Fnance Fnd the smple nterest. Assume a 360-day year. 9. $ at 3.1% for 72 days 10. $ at 4.25% for 30 days Fnd the maturty value and the amount of smple nterest earned. 11. $3125 at 2.85% for 7 months 12. $12,000 at 5.3% for 11 months 13. If $1500 earned smple nterest of $56.25 n 6 months, what was the smple nterest rate? 14. If $23,500 earned smple nterest of $ n 9 months, what was the smple nterest rate? 15. Explan the dfference between smple nterest and compound nterest. 16. What s the dfference between r and? 17. What s the dfference between t and n? 18. In Fgure 1, one lne s straght and the other s curved. Explan why ths s, and whch represents each type of nterest. Fnd the compound amount for each depost and the amount of nterest earned. 19. $1000 at 6% compounded annually for 8 years 20. $1000 at 4.5% compounded annually for 6 years 21. $470 at 5.4% compounded semannually for 12 years 22. $15,000 at 6% compounded monthly for 10 years 23. $8500 at 8% compounded quarterly for 5 years 24. $9100 at 6.4% compounded quarterly for 9 years Fnd the nterest rate for each depost and compound amount. 25. $8000 accumulatng to $11,672.12, compounded quarterly for 8 years 26. $12,500 accumulatng to $20,077.43, compounded quarterly for 9 years 27. $4500 accumulatng to $ , compounded monthly for 5 years 28. $6725 accumulatng to $10,353.47, compounded monthly for 7 years Fnd the effectve rate correspondng to each nomnal rate % compounded quarterly 30. 6% compounded quarterly % compounded semannually % compounded semannually Fnd the present value (the amount that should be nvested now to accumulate the followng amount) f the money s compounded as ndcated. 33. $12, at 4.8% compounded annually for 6 years 34. $36, at 5.3% compounded annually for 10 years 35. $2000 at 6% compounded semannually for 8 years 36. $2000 at 7% compounded semannually for 8 years 37. $8800 at 5% compounded quarterly for 5 years 38. $7500 at 5.5% compounded quarterly for 9 years 39. How do the nomnal or stated nterest rate and the effectve nterest rate dffer? 40. If nterest s compounded more than once per year, whch rate s hgher, the stated rate or the effectve rate? Usng ether logarthms or a graphng calculator, fnd the tme requred for each ntal amount to be at least equal to the fnal amount. 41. $5000, deposted at 4% compounded quarterly, to reach at least $ $8000, deposted at 3% compounded quarterly, to reach at least $23, $4500, deposted at 3.6% compounded monthly, to reach at least $11, $6800, deposted at 5.4% compounded monthly, to reach at least $15,000 Fnd the doublng tme for each of the followng levels of nflaton usng (a) logarthms or a graphng calculator, and (b) the rule of 70 or 72, whchever s approprate % % For each of the followng amounts at the gven nterest rate compounded contnuously, fnd (a) the future value after 9 years, (b) the effectve rate, and (c) the tme to reach $10, $5500 at 3.1% 48. $4700 at 4.65% APPLICATIONS Busness and Economcs 49. Loan Repayment Tanya Kerchner borrowed $7200 from her father to buy a used car. She repad hm after 9 months, at an annual nterest rate of 6.2%. Fnd the total amount she repad. How much of ths amount s nterest? 50. Delnquent Taxes An accountant for a corporaton forgot to pay the frm s ncome tax of $321, on tme. The government charged a penalty based on an annual nterest rate of 13.4% for the 29 days the money was late. Fnd the total amount (tax and penalty) that was pad. (Use a 365-day year.) 51. Savngs A $1500 certfcate of depost held for 75 days was worth $ To the nearest tenth of a percent, what nterest rate was earned? Assume a 360-day year. 52. Bond Interest A bond wth a face value of $10,000 n 10 years can be purchased now for $ What s the smple nterest rate? 53. Stock Growth 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 $0.50 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.) 54. Wealth A 1997 artcle n The New York Tmes dscussed how long t would take for Bll Gates, the world s second rchest person at the tme (behnd the Sultan of Brune), to become the world s frst trllonare. Hs brthday s October 28, 1955, and on

13 5.1 Smple and Compound Interest 199 July 16, 1997, he was worth $42 bllon. (Note: A trllon dollars s 1000 bllon dollars.) Source: The New York Tmes. a. Assume that Bll Gates s fortune grows at an annual rate of 58%, the hstorcal growth rate through 1997 of Mcrosoft stock, whch made up most of hs wealth n Fnd the age at whch he becomes a trllonare. (Hnt: Use the formula for nterest compounded annually, A 5 P n, wth P Graph the future value as a functon of n on a graphng calculator, and fnd where the graph crosses the lne y ) b. Repeat part a usng 10.5% growth, the average return on all stocks snce Source: CNN. c. What rate of growth would be necessary for Bll Gates to become a trllonare by the tme he s elgble for Socal Securty on January 1, 2022, after he has turned 66? d. Forbes magazne s lstngs of bllonares for 2006 and 2010 have gven Bll Gates s worth as roughly $50.0 bllon and $53.0 bllon, respectvely. What was the rate of growth of hs wealth between 2006 and 2010? Source: Forbes. 55. Student Loan Upon graduaton from college, Kelly was able to defer payment on hs $40,000 subsdzed Stafford student loan for 6 months. Snce the nterest wll no longer be pad on hs behalf, t wll be added to the prncpal untl payments begn. If the nterest s 6.54% compounded monthly, what wll the prncpal amount be when he must begn repayng hs loan? Source: SalleMae. 56. Comparng Investments 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? 57. Comparng Investments 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? 58. Retrement Savngs The pe graph below shows the percent of baby boomers aged who sad they had nvestments wth a total value as shown n each category. Source: The New York Tmes. Don't know or no answer 28% More than $1 mllon 3% $150,000 to $1 mllon 13% Less than $10,000 30% $10,000 to $149,000 29% Fgures add to more than 100% because of roundng. Note that 30% have saved less than $10,000. Assume the money s nvested at an average rate of 8% compounded quarterly. What wll the top numbers n each category amount to n 20 years, when ths age group wll be ready for retrement? Negatve Interest Under certan condtons, Swss banks pay negatve nterest: they charge you. (You ddn t thnk all that secrecy was free?) Suppose a bank pays 22.4% nterest compounded annually. Fnd the compound amount for a depost of $150,000 after each perod years years 61. Investment In the New Testament, Jesus commends a wdow who contrbuted 2 mtes to the temple treasury (Mark 12: 42 44). A mte was worth roughly 1/8 of a cent. Suppose the temple nvested those 2 mtes at 4% nterest compounded quarterly. How much would the money be worth 2000 years later? 62. Investments Erc Cobbe borrowed $5200 from hs frend Frank Cronn to pay for remodelng work on hs house. He repad the loan 10 months later wth smple nterest at 7%. Frank then nvested the proceeds n a 5-year certfcate of depost payng 6.3% compounded quarterly. How much wll he have at the end of 5 years? (Hnt: You need to use both smple and compound nterest.) 63. Investments Suppose $10,000 s nvested at an annual rate of 5% for 10 years. Fnd the future value f nterest s compounded as follows. a. Annually b. Quarterly c. Monthly d. Daly (365 days) e. Contnuously 64. Investments In Exercse 63, notce that as the money s compounded more often, the compound amount becomes larger and larger. Is t possble to compound often enough so that the compound amount s $17,000 after 10 years? Explan. The followng exercse s from an actuaral examnaton. Source: The Socety of Actuares. 65. Savngs On January 1, 2000, Jack deposted $1000 nto bank X to earn nterest at a rate of j per annum compounded semannually. On January 1, 2005, he transferred hs account to bank Y to earn nterest at the rate of k per annum compounded quarterly. On January 1, 2008, the balance of bank Y s $ If Jack could have earned nterest at the rate of k per annum compounded quarterly from January 1, 2000, through January 1, 2008, hs balance would have been $ Calculate the rato k/j. 66. Interest Rate In 1995, O. G. McClan of Houston, Texas, maled a $100 check to a descendant of Texas ndependence hero Sam Houston to repay a $100 debt of McClan s great-great-grandfather, who ded n 1835, to Sam Houston. A bank estmated the nterest on the loan to be $420 mllon for the 160 years t was due. Fnd the nterest rate the bank was usng, assumng nterest s compounded annually. Source: The New York Tmes. 67. Comparng CD Rates Marne Bank offered the followng CD (Certfcates of Depost) rates. The rates are annual percentage yelds, or effectve rates, whch are hgher than the correspondng nomnal rates. Assume quarterly compoundng. Solve for r to approxmate the correspondng nomnal rates to the nearest hundredth. Source: Marne Bank. Term 6 mo Specal! 9 mo 1 yr 2 yr 3 yr APY%

14 200 CHAPTER 5 Mathematcs of Fnance 68. Effectve Rate A Web ste for E*TRADE Fnancal clams that they have one of the hghest yelds n the naton on a 6-month CD. The stated yeld was 5.46%; the actual rate was not stated. Assumng monthly compoundng, fnd the actual rate. Source: E*TRADE. 69. Effectve Rate On August 18, 2006, Centennal Bank of Fountan Valley, Calforna, pad 5.5% nterest, compounded monthly, on a 1-year CD, whle Frst Source Bank of South Bend, Indana, pad 5.63% compounded annually. What are the effectve rates for the two CDs, and whch bank pays a hgher effectve rate? Source: Bankrate.com. 70. Savngs A department has ordered 8 new Dell computers at a cost of $2309 each. The order wll not be delvered for 6 months. What amount could the department depost n a specal 6-month CD payng 4.79% compounded monthly to have enough to pay for the machnes at tme of delvery? 71. Buyng a House Steve May wants to have $30,000 avalable n 5 years for a down payment on a house. He has nherted $25,000. How much of the nhertance should he nvest now to accumulate $30,000, f he can get an nterest rate of 5.5% compounded quarterly? 72. Rule of 70 On the day of ther frst grandchld s brth, a new set of grandparents nvested $10,000 n a trust fund earnng 4.5% compounded monthly. a. Use the rule of 70 to estmate how old the grandchld wll be when the trust fund s worth $20,000. b. Use your answer to part a to determne the actual amount that wll be n the trust fund at that tme. How close was your estmate n part a? Doublng Tme Use the deas from Example 12 to fnd the tme t would take for the general level of prces n the economy to double at each average annual nflaton rate % 74. 5% 75. Doublng Tme 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 utltes wll need to double ther generatng capacty. 76. Doublng Tme 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 utltes wll need to double generatng capacty. 77. Mtt Romney Accordng to The New York Tmes, Durng the fourteen years [Mtt Romney] ran t, Ban Captal s nvestments reportedly earned an annual rate of return of over 100 percent, potentally turnng an ntal nvestment of $1 mllon nto more than $14 mllon by the tme he left n Source: The New York Tmes. a. What rate of return, compounded annually, would turn $1 mllon nto $14 mllon by 1998? b. The actual rate of return of Ban Captal durng the 14 years that Romney ran t was 113%. Source: The Amercan. How much would $1 mllon, compounded annually at ths rate, be worth after 14 years? YOUR TURN ANSWERS 1. $ % 3. $ % % 6. $ years 8. $ APPLY IT Future Value of an Annuty If you depost $1500 each year for 6 years n an account payng 8% nterest compounded annually, how much wll be n your account at the end of ths perod? In ths secton and the next, we develop future value and present value formulas for such perodc payments. To develop these formulas, we must frst dscuss sequences. Geometrc Sequences If a and r are nonzero real numbers, the nfnte lst of numbers a, ar, ar 2, ar 3, ar 4, *, ar n, * s called a geometrc sequence. For example, f a 5 3 and r 522, we have the sequence 3, , , , , *, or 3, 26, 12, 224, 48, *. In the sequence a, ar, ar 2, ar 3, ar 4, *, the number a s called the frst term of the sequence, ar s the second term, ar 2 s the thrd term, and so on. Thus, for any n $ 1, ar n21 s the nth term of the sequence.

15 5.2 Future Value of an Annuty 201 Each term n the sequence s r tmes the precedng term. The number r s called the common rato of the sequence. CAUTION Do not confuse r, the rato of two successve terms n a geometrc seres, wth r, the annual nterest rate. Dfferent letters mght have been helpful, but the usage n both cases s almost unversal. EXAMPLE 1 Geometrc Sequence Fnd the seventh term of the geometrc sequence 5, 20, 80, 320, *. SOLUTION The frst term n the sequence s 5, so a 5 5. The common rato, found by dvdng the second term by the frst, s r 5 20/ We want the seventh term, so n 5 7. Use ar n21, wth a 5 5, r 5 4, and n 5 7. Next, we need to fnd the sum If r 5 1, then If r 2 1, multply both sdes of equaton (1) by r to get of the frst n terms of a geometrc sequence, where Now subtract correspondng sdes of equaton (1) from equaton (2). Ths result s summarzed below. Sum of Terms S n 1 r a1 r n ar n ,480 S n S n 5 a 1 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n21. S n 5 a 1 a 1 a 1 a 1 ) 1 a 5 na. (''''''')'''''''* rs n 5 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n. rs n 5 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n21 1 ar n 2S n 521 a 1 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n ar n rs n 2 S n 52a 1 ar n S n 5 a1 rn r 2 1 n terms Factor. Dvde both sdes by r 2 1. If a geometrc sequence has frst term a and common rato r, then the sum n terms s gven by S n 5 a1 r n 2 1 2, r u 1. r 2 1 S n (1) (2) of the frst EXAMPLE 2 Sum of a Geometrc Sequence Fnd the sum of the frst sx terms of the geometrc sequence 3, 12, 48, *. SOLUTION Here a 5 3, r 5 4, and n 5 6. Fnd by the formula above. S 6 YOUR TURN 1 Fnd the sum of the frst 9 terms of the geometrc seres 4, 12, 36,.... S n 5 6, a 5 3, r 5 4. TRY YOUR TURN 1

16 202 CHAPTER 5 Mathematcs of Fnance APPLY IT Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. If the payments are made at the end of the tme perod, and f the frequency of payments s the same as the frequency of compoundng, the annuty s called an ordnary 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. The future value of the annuty, the fnal sum on depost, s defned as the sum of the compound amounts of all the payments, compounded to the end of the term. Two common uses of annutes are to accumulate funds for some goal or to wthdraw funds from an account. For example, an annuty may be used to save money for a large purchase, such as an automoble, a college educaton, or a down payment on a home. An annuty also may be used to provde monthly payments for retrement. We explore these optons n ths and the next secton. For example, suppose $1500 s deposted at the end of each year for the next 6 years n an account payng 8% per year compounded annually. Fgure 4 shows ths annuty. To fnd the future value of the annuty, look separately at each of the $1500 payments. The frst of these payments wll produce a compound amount of 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. FIGURE 4 Use 5 as the exponent nstead of 6, snce the money s deposted at the end of the frst year and earns nterest for only 5 years. The second payment of $1500 wll produce a compound amount of As shown n Fgure 5, the future value of the annuty s (The last payment earns no nterest at all.) Year Depost $1500 $1500 $1500 $1500 $1500 $1500 FIGURE 5 $ (1.08) 1500(1.08) 1500(1.08) 1500(1.08) 1500(1.08) Readng ths sum n reverse order, we see that t s the sum of the frst sx terms of a geometrc sequence, wth a , r , and n 5 6. Thus, the sum equals a1r n 2 12 r < $11, To generalze ths result, suppose that payments of R dollars each are deposted nto an account at the end of each perod for n perods, at a rate of nterest per perod. The frst payment of R dollars wll produce a compound amount of R n21 dollars, the second

17 5.2 Future Value of an Annuty 203 payment wll produce R n22 dollars, and so on; the fnal payment earns no nterest and contrbutes just R dollars to the total. If S represents the future value (or sum) of the annuty, then (as shown n Fgure 6), S 5 R n21 1 R n22 1 R n23 1 ) 1 R R, or, wrtten n reverse order, S 5 R 1 R R ) 1 R n21. Perod 1 2 Depost $R $R $R $R $R 3 n 1 n FIGURE 6 R R(1 + )... A depost of $R becomes R(1 + ) R(1 + ) R(1 + ) n 3 n 2 n 1 The sum of these s the amount of the annuty. Ths result s the sum of the frst n terms of the geometrc sequence havng frst term R and common rato 1 1. Usng the formula for the sum of the frst n terms of a geometrc sequence, S 5 R n The quantty n brackets s commonly wrtten (read s-angle-n at ), so that Values of s n0 can be found wth a calculator. A formula for the future value of an annuty S of n payments of R dollars each at the end of each consecutve nterest perod, wth nterest compounded at a rate per perod, follows.* Recall that ths type of annuty, wth payments at the end of each tme perod, s called an ordnary annuty. Future Value of an Ordnary Annuty where S s the future value; R s the perodc payment; s the nterest rate per perod; n s the number of perods. 5 R n 2 14 s n0 S 5 R. s n0. 5 R c n 2 1 d. S 5 R c n 2 1 d or S 5 Rs n0 TECHNOLOGY NOTE A calculator wll be very helpful n computatons wth annutes. The TI-84 Plus graphng calculator has a specal FINANCE menu that s desgned to gve any desred result after enterng the basc nformaton. If your calculator does not have ths feature, many calculators can easly be programmed to evaluate the formulas ntroduced n ths secton and the next. We nclude these programs n the Graphng Calculator and Excel Spreadsheet Manual avalable for ths text. *We use S for the future value here, nstead of A as n the compound nterest formula, to help avod confusng the two formulas.

18 204 CHAPTER 5 Mathematcs of Fnance YOUR TURN 2 Fnd the accumulated amount after 11 years f $250 s deposted every month n an account payng 3.3% nterest compounded monthly. EXAMPLE 3 Ordnary Annuty Bethany Ward s an athlete who beleves that her playng career wll last 7 years. To prepare for her future, she deposts $22,000 at the end of each year for 7 years n an account payng 6% compounded annually. How much wll she have on depost after 7 years? SOLUTION Her payments form an ordnary annuty, wth R 5 22,000, n 5 7, and Usng the formula for future value of an annuty, S 5 22,000 c d < 184,664.43, 0.06 or $184, Note that she made 7 payments of $22,000, or $154,000. The nterest that she earned s $184, $154,000 5 $30, TRY YOUR TURN 2 Snkng Funds A fund set up to receve perodc payments as n Example 3 s called a snkng fund. The perodc payments, together wth the nterest earned by the payments, are desgned to produce a gven sum at some tme n the future. For example, a snkng fund mght be set up to receve money that wll be needed to pay off the prncpal on a loan at some future tme. If the payments are all the same amount and are made at the end of a regular tme perod, they form an ordnary annuty. 150,000 Y1 200(((1 (X/12))^ Intersecton 0 X.072 Y ,000 FIGURE 7 Intersecton 0 X Y FIGURE 8 Method 1 Graphng Calculator EXAMPLE 4 Snkng Fund Experts say that the baby boom generaton (Amercans born between 1946 and 1960) cannot count on a company penson or Socal Securty to provde a comfortable retrement, as ther parents dd. It s recommended that they start to save early and regularly. Nancy Hart, a baby boomer, has decded to depost $200 each month for 20 years n an account that pays nterest of 7.2% compounded monthly. (a) How much wll be n the account at the end of 20 years? SOLUTION Ths savngs plan s an annuty wth R 5 200, /12, and n The future value s S c / d < 106,752.47, 0.072/12 or $106, Fgure 7 shows a calculator graph of the functon S c x / d x/12 where r, the annual nterest rate, s desgnated x. The value of the functon at x , shown at the bottom of the wndow, agrees wth our result above. (b) Nancy beleves she needs to accumulate $130,000 n the 20-year perod to have enough for retrement. What nterest rate would provde that amount? SOLUTION One way to answer ths queston s to solve the equaton for S n terms of x wth S 5 130,000. Ths s a dffcult equaton to solve. Although tral and error could be used, t would be easer to use the graphng calculator graph n Fgure 7. Addng the lne y 5 130,000 to the graph and then usng the capablty of the calculator to fnd the ntersecton pont wth the curve shows the annual nterest rate must be at least 8.79% to the nearest hundredth. See Fgure 8.

19 5.2 Future Value of an Annuty 205 N5240 I% PV50 PMT5-200 FV P/Y512 C/Y512 PMT: END BEGIN FIGURE 9 Method 2 TVM Solver Usng the TVM Solver under the FINANCE menu on the TI-84 Plus calculator, enter 240 for N (the number of perods), 0 for PV (present value), 2200 for PMT (negatve because the money s beng pad out), for FV (future value), and 12 for P/Y (payments per year). Put the cursor next to I% (payment) and press SOLVE. The result, shown n Fgure 9, ndcates that an nterest rate of 8.79% s needed. In Example 4 we used snkng fund calculatons to determne the amount of money that accumulates over tme through monthly payments and nterest. We can also use ths formula to determne the amount of money necessary to perodcally nvest at a gven nterest rate to reach a partcular goal. Start wth the annuty formula S 5 R c n 2 1 d, and multply both sdes by / n 2 14 to derve the followng formula. Snkng Fund Payment S R n 2 1 where R s the perodc payment; S s the future value; s the nterest rate per perod; n s the number of perods. or R 5 S s n0 EXAMPLE 5 150,000 Snkng Fund Payment Suppose Nancy, n Example 4, cannot get the hgher nterest rate to produce $130,000 n 20 years. To meet that goal, she must ncrease her monthly payment. What payment should she make each month? SOLUTION Nancy s goal s to accumulate $130,000 n 20 years at 7.2% compounded monthly. Therefore, the future value s S 130,000, the monthly nterest rate s 0.072/12, and the number of perods s n 12(20). Use the snkng fund payment formula to fnd the payment R. R , / / < Nancy wll need payments of $ each month for 20 years to accumulate at least $130,000. Notce that $ s not qute enough, so round up here. Fgure 10(a) shows the pont of ntersecton of the graphs of Y 1 5 X c / d 0.072/12 X X5244 Y YOUR TURN 3 Fnd the quarterly payment needed to produce $13,500 n 14 years at Intersecton 3.75% nterest compounded 0 X Y quarterly. (a) (b) FIGURE 10

20 206 CHAPTER 5 Mathematcs of Fnance and Y ,000. The result agrees wth the answer we found analytcally. The table shown n Fgure 10(b) confrms that the payment should be between $243 and $244. TRY YOUR TURN 3 TECHNOLOGY NOTE We can also use a graphng calculator or spreadsheet to make a table of the amount n a snkng fund. In the formula for future value of an annuty, smply let n be a varable wth values from 1 to the total number of payments. Fgure 11(a) shows the begnnng of such a table generated on a TI-84 Plus for Example 5. Fgure 11(b) shows the begnnng of the same table usng Mcrosoft Excel. X X51 Y (a) n FIGURE Amount n Fund (b) Annutes Due The formula developed above s for ordnary annutes those wth payments made at the end of each tme perod. These results can be modfed slghtly to apply to annutes due annutes n whch payments are made at the begnnng of each tme perod. To fnd the future value of an annuty due, treat each payment as f t were made at the end of the precedng perod. That s, fnd s n0 for one addtonal perod; to compensate for ths, subtract the amount of one payment. Thus, the future value of an annuty due of n payments of R dollars each at the begnnng of consecutve nterest perods, wth nterest compounded at the rate of per perod, s S 5 R c n d 2 R or S 5 Rs n110 2 R. TECHNOLOGY NOTE The fnance feature of the TI-84 Plus can be used to fnd the future value of an annuty due as well as an ordnary annuty. If ths feature s not bult n, you may wsh to program your calculator to evaluate ths formula, too. YOUR TURN 4 Fnd the future value of an annuty due wth $325 made at the begnnng of each month for 5 years n an account payng 3.3% compounded monthly. EXAMPLE 6 Future Value of an Annuty Due Fnd the future value of an annuty due f payments of $500 are made at the begnnng of each quarter for 7 years, n an account payng 6% compounded quarterly. SOLUTION In 7 years, there are n 5 28 quarterly perods. Add one perod to get n , and use the formula wth / S c d < 17, The account wll have a total of $17, after 7 years. TRY YOUR TURN 4

21 5.2 Future Value of an Annuty EXERCISES Fnd the ffth term of each geometrc sequence. 1. a 5 3; r a 5 7; r a 528; r a 526; r a 5 1; r a 5 12; r a 5 256; r a 5 729; r Fnd the sum of the frst four terms for each geometrc sequence. 9. a 5 1; r a 5 4; r a 5 5; r a 5 6; r a 5 128; r a 5 64; r Explan how a geometrc sequence s related to an ordnary annuty. 16. Explan the dfference between an ordnary annuty and an annuty due. Fnd the future value of each ordnary annuty. Interest s compounded annually. 17. R 5 100; ; n R ; ; n R 5 25,000; ; n R 5 29,500; ; n 5 15 Fnd the future value of each ordnary annuty, f payments are made and nterest s compounded as gven. Then determne how much of ths value s from contrbutons and how much s from nterest. 21. R ; 10% nterest compounded semannually for 7 years 22. R ; 5% nterest compounded semannually for 18 years 23. R 5 800; 6.51% nterest compounded semannually for 12 years 24. R ; 8.73% nterest compounded quarterly for 9 years 25. R 5 12,000; 4.8% nterest compounded quarterly for 16 years 26. R 5 42,000; 10.05% nterest compounded semannually for 12 years 27. What s meant by a snkng fund? Gve an example of a snkng fund. 28. Lst some reasons for establshng a snkng fund. Determne the nterest rate needed to accumulate the followng amounts n a snkng fund, wth monthly payments as gven. 29. Accumulate $56,000, monthly payments of $300 over 12 years 30. Accumulate $120,000, monthly payments of $500 over 15 years Fnd the perodc payment that wll amount to each gven sum under the gven condtons. 31. S 5 $10,000; nterest s 5% compounded annually; payments are made at the end of each year for 12 years. 32. S 5 $150,000; nterest s 6% compounded semannually; payments are made at the end of each semannual perod for 11 years. Fnd the amount of each payment to be made nto a snkng fund so that enough wll be present to accumulate the followng amounts. Payments are made at the end of each perod. 33. $8500; money earns 8% compounded annually; there are 7 annual payments. 34. $2750; money earns 5% compounded annually; there are 5 annual payments. 35. $75,000; money earns 6% compounded semannually for years $25,000; money earns 5.7% compounded quarterly for 3 2 years $65,000; money earns 7.5% compounded quarterly for 2 2 years $9000; money earns 4.8% compounded monthly for 2 years. Fnd the future value of each annuty due. Assume that nterest s compounded annually. 39. R 5 600; ; n R ; ; n R 5 16,000; ; n R ; ; n 5 11 Fnd the future value of each annuty due. Then determne how much of ths value s from contrbutons and how much s from nterest. 43. Payments of $1000 made at the begnnng of each semannual perod for 9 years at 8.15% compounded semannually 44. $750 deposted at the begnnng of each month for 15 years at 5.9% compounded monthly 45. $250 deposted at the begnnng of each quarter for 12 years at 4.2% compounded quarterly 46. $1500 deposted at the begnnng of each semannual perod for 11 years at 5.6% compounded semannually APPLICATIONS Busness and Economcs 47. Comparng Accounts Laure Campbell deposts $12,000 at the end of each year for 9 years n an account payng 8% nterest compounded annually. a. Fnd the fnal amount she wll have on depost. b. Laure s brother-n-law works n a bank that pays 6% compounded annually. If she deposts money n ths bank nstead of the one above, how much wll she have n her account?

22 208 CHAPTER 5 Mathematcs of Fnance c. How much would Laure lose over 9 years by usng her brother-n-law s bank? 48. Savngs Matthew Paster s savng for a Plasma HDTV. At the end of each month he puts $100 n a savngs account that pays 2.25% nterest compounded monthly. How much s n the account after 2 years? How much dd Matthew depost? How much nterest dd he earn? 49. Savngs A typcal pack-a-day smoker spends about $ per month on cgarettes. Suppose the smoker nvests that amount each month n a savngs account at 4.8% nterest compounded monthly. What would the account be worth after 40 years? Source: MSN. 50. Retrement Plannng A 45-year-old man puts $2500 n a retrement account at the end of each quarter untl he reaches the age of 60, then makes no further deposts. If the account pays 6% nterest compounded quarterly, how much wll be n the account when the man retres at age 65? 51. Retrement Plannng At the end of each quarter, a 50-yearold woman puts $3000 n a retrement account that pays 5% nterest compounded quarterly. When she reaches 60, she wthdraws the entre amount and places t n a mutual fund that pays 6.9% nterest compounded monthly. From then on she deposts $300 n the mutual fund at the end of each month. How much s n the account when she reaches age 65? Indvdual Retrement Accounts Suppose a 40-year-old person deposts $4000 per year n an Indvdual Retrement Account untl age 65. Fnd the total n the account wth the followng assumptons of nterest rates. (Assume quarterly compoundng, wth payments of $1000 made at the end of each quarter perod.) Fnd the total amount of nterest earned % 53. 8% 54. 4% % 56. Savngs Greg Tobn needs $10,000 n 8 years. a. What amount should he depost at the end of each quarter at 8% compounded quarterly so that he wll have hs $10,000? b. Fnd Greg s quarterly depost f the money s deposted at 6% compounded quarterly. 57. Buyng Equpment Harv, the owner of Harv s Meats, knows that he must buy a new deboner 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? 58. Buyng a Car Amanda Perdars wants to have a $20,000 down payment when she buys a new car n 6 years. How much money must she depost at the end of each quarter n an account payng 3.2% compounded quarterly so that she wll have the down payment she desres? 59. Savngs Stacy Schrank s pad on the frst day of the month and $80 s automatcally deducted from her pay and deposted n a savngs account. If the account pays 2.5% nterest compounded monthly, how much wll be n the account after 3 years and 9 months? 60. Savngs A father opened a savngs account for hs daughter on the day she was born, depostng $1000. Each year on her brthday he deposts another $1000, makng the last depost on her 21st brthday. If the account pays 5.25% nterest compounded annually, how much s n the account at the end of the day on hs daughter s 21st brthday? How much nterest has been earned? 61. Savngs Beth Dahlke deposts $2435 at the begnnng of each semannual perod for 8 years n an account payng 6% compounded semannually. She then leaves that money alone, wth no further deposts, for an addtonal 5 years. Fnd the fnal amount on depost after the entre 13-year perod. 62. Savngs Davd Kurzawa 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 semannually for another 9 years. Fnd the fnal amount on depost after the entre 21-year perod. In Exercses 63 and 64, use a graphng calculator to fnd the value of that produces the gven value of S. (See Example 4(b).) 63. Retrement 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? 64. Rate of Return Carolne DTullo made payments of $250 per month at the end of each month to purchase a pece of property. At the end of 30 years, she completely owned the property, whch she sold for $330,000. What annual nterest rate would she need to earn on an annuty for a comparable rate of return? 65. Lottery 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. Source: The Washngton Post. 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%? 66. Buyng Real Estate Vck Kakouns 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 on the $60,000. b. The buyer sets up a snkng fund so that enough money wll be present to pay off the $60,000. The buyer wll make semannual payments nto the snkng fund; the account pays 6% compounded semannually. Fnd the amount of each payment nto the fund.

23 67. Buyng Rare Stamps Phl Weaver 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 on the $4000. a. Fnd the amount of each semannual nterest payment. b. Paul sets up a snkng fund so that enough money wll be present to pay off the $4000. He wll make annual payments nto the fund. The account pays 8% compounded annually. Fnd the amount of each payment. 68. Down Payment A conventonal loan, such as for a car or a house, s smlar to an annuty but usually ncludes a down Present Value of an Annuty; Amortzaton 209 payment. Show that f a down payment of D dollars s made at the begnnng of the loan perod, the future value of all the payments, ncludng the down payment, s S 5 D n 1 R c n 2 1 d. YOUR TURN ANSWERS 1. 39, $39, $ $21, Present Value of an Annuty; Amortzaton FOR REVIEW APPLY IT Recall from Secton R.6 that for any nonzero number a, a Also, by the product rule for exponents, a x. a y 5 a x1y. In partcular, f a s any nonzero number, a n. a 2n 5 a n112n2 5 a What monthly payment wll pay off a $17,000 car loan n 36 monthly payments at 6% annual nterest? The answer to ths queston s gven n Example 2 n ths secton. We shall see that t nvolves fndng the present value of an annuty. Suppose that at the end of each year, for the next 10 years, $500 s deposted n a savngs account payng 7% nterest compounded annually. Ths s an example of an ordnary annuty. The present value of an annuty s the amount that would have to be deposted n one lump sum today (at the same compound nterest rate) n order to produce exactly the same balance at the end of 10 years. We can fnd a formula for the present value of an annuty as follows. Suppose deposts of R dollars are made at the end of each perod for n perods at nterest rate per perod. Then the amount n the account after n perods s the future value of ths annuty: S 5 R. s n0 5 R c n 2 1 d. On the other hand, f P dollars are deposted today at the same compound nterest rate, then at the end of n perods, the amount n the account s P n. If P s the present value of the annuty, ths amount must be the same as the amount S n the formula above; that s, P n 5 R c n 2 1 d. To solve ths equaton for P, multply both sdes by n. P 5 R n c n 2 1 d Use the dstrbutve property; also recall that n n 5 1. P 5 R c n n n n d 5 R c d The amount P s the present value of the annuty. The quantty n brackets s abbrevated as a n0, so n a n0 5. (The symbol a n0 s read a-angle-n at. Compare ths quantty wth s n0 n the prevous secton.) The formula for the present value of an annuty s summarzed on the next page.

24 210 CHAPTER 5 Mathematcs of Fnance Present Value of an Ordnary Annuty The present value P of an annuty of n payments of R dollars each at the end of consecutve nterest perods wth nterest compounded at a rate of nterest per perod s P 5 R c n d or P 5 Ra n0. CAUTION Don t 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 n, but n the future value formula, t s n 2 1. TECHNOLOGY NOTE The fnancal feature of the TI-84 Plus calculator can be used to fnd the present value of an annuty by choosng that opton from the menu and enterng the requred nformaton. If your calculator does not have ths bult-n feature, t wll be useful to store a program to calculate present value of an annuty n your calculator. A program s gven n the Graphng Calculator and Excel Spreadsheet Manual avalable wth ths book. YOUR TURN 1 Fnd the present value of an annuty of $120 at the end of each month put nto an account yeldng 4.8% compounded monthly for 5 years. EXAMPLE 1 Present Value of an Annuty John Cross and Wendy Mears are both graduates of the Brsbane Insttute of Technology (BIT). They both agree to contrbute to the endowment fund of BIT. John says that he wll gve $500 at the end of each year for 9 years. Wendy prefers to gve a lump sum today. What lump sum can she gve that wll equal the present value of John s annual gfts, f the endowment fund earns 7.5% compounded annually? SOLUTION Here, R 5 500, n 5 9, and , and we have P 5 R a c d < Therefore, Wendy must donate a lump sum of $ today. TRY YOUR TURN 1 One of the most mportant uses of annutes s n determnng the equal monthly payments needed to pay off a loan, as llustrated n the next example. APPLY IT YOUR TURN 2 Fnd the car payment n Example 2 f there are 48 equal monthly payments and the nterest rate s 5.4%. EXAMPLE 2 Car Payments A car costs $19,000. After a down payment of $2000, the balance wll be pad off n 36 equal monthly payments wth nterest of 6% per year on the unpad balance. Fnd the amount of each payment. SOLUTION A sngle lump sum payment of $17,000 today would pay off the loan. So, $17,000 s the present value of an annuty of 36 monthly payments wth nterest of 6%/ % per month. Thus, P 5 17,000, n 5 36, , and we must fnd the monthly payment R n the formula P 5 R c n d ,000 5 R c d R < A monthly payment of $ wll be needed. TRY YOUR TURN 2

25 5.3 Present Value of an Annuty; Amortzaton 211 Each payment n Example 2 ncludes nterest on the unpad balance, wth the remander gong to reduce the loan. For example, the frst payment of $ ncludes nterest of $17, $85 and s dvded as follows. monthly nterest to reduce payment due the balance $ $85 5 $ At the end of ths secton, amortzaton schedules show that ths procedure does reduce the loan to $0 after all payments are made (the fnal payment may be slghtly dfferent). Amortzaton A loan s amortzed f both the prncpal and nterest are pad by a sequence of equal perodc payments. In Example 2, a loan of $17,000 at 6% nterest compounded monthly could be amortzed by payng $ per month for 36 months. The perodc payment needed to amortze a loan may be found, as n Example 2, by solvng the present value equaton for R. 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 5 P c n d 5 P P or R 5 2n a n0. EXAMPLE 3 Home Mortgage The Perez famly buys a house for $275,000, wth a down payment of $55,000. They take out a 30-year mortgage for $220,000 at an annual nterest rate of 6%. (a) Fnd the amount of the monthly payment needed to amortze ths loan. SOLUTION Here P 5 220,000 and the monthly nterest rate s / * The number of monthly payments s Therefore, R 5 220,000 5 a , c d Monthly payments of $ are requred to amortze the loan. (b) Fnd the total amount of nterest pad when the loan s amortzed over 30 years. SOLUTION The Perez famly makes 360 payments of $ each, for a total of $474, Snce the amount of the loan was $220,000, the total nterest pad s $474, $220,000 5 $254, Ths large amount of nterest s typcal of what happens wth a long mortgage. A 15-year mortgage would have hgher payments but would nvolve sgnfcantly less nterest. (c) Fnd the part of the frst payment that s nterest and the part that s appled to reducng the debt. SOLUTION Durng the frst month, the entre $220,000 s owed. Interest on ths amount for 1 month s found by the formula for smple nterest, wth r 5 annual nterest rate and t 5 tme n years. 1 I 5 Prt 5 220, $1100 *Mortgage rates are quoted n terms of annual nterest, but t s always understood that the monthly rate s 1/12 of the annual rate and that nterest s compounded monthly.

26 212 CHAPTER 5 Mathematcs of Fnance YOUR TURN 3 Fnd the monthly payment and total amount of nterest pad n Example 3 f the mortgage s for 15 years and the nterest rate s 7%. y 1 (1.005) (360 x) y = [ ] 220, , , ,000 60,000 20, FIGURE 12 x At the end of the month, a payment of $ s made; snce $1100 of ths s nterest, a total of $ $ $ s appled to the reducton of the orgnal debt. TRY YOUR TURN 3 It can be shown that the unpad balance after x payments s gven by the functon y 5 R c n2x2 d, although ths formula wll only gve an approxmaton f R s rounded to the nearest penny. For example, the unrounded value of R n Example 3 s When ths value s put nto the above formula, the unpad balance s found to be y c d 5 219,780.99, whle roundng R to n the above formula gves an approxmate balance of $219, A graph of ths functon s shown n Fgure 12. We can fnd the unpad balance after any number of payments, x, by fndng the y-value that corresponds to x. For example, the remanng balance after 5 years or 60 payments s shown at the bottom of the graphng calculator screen n Fgure 13(a). You may be surprsed that the remanng balance on a $220,000 loan s as large as $204, Ths s because most of the early payments on a loan go toward nterest, as we saw n Example 3(c). By addng the graph of y 5 1 1/2 2220, ,000 to the fgure, we can fnd when half the loan has been repad. From Fgure 13(b) we see that 252 payments are requred. Note that only 108 payments reman at that pont, whch agan emphaszes the fact that the earler payments do lttle to reduce the loan. 220, ,000 0 X 60 Y (a) FIGURE 13 0 X Y (b) Amortzaton Schedules In the precedng example, 360 payments are made to amortze a $220,000 loan. The loan balance after the frst payment s reduced by only $219.01, whch s much less than 1 1/ ,000 2 < $ Therefore, even though equal payments are made to amortze a loan, the loan balance does not decrease n equal steps. Ths fact s very mportant f a loan s pad off early. EXAMPLE 4 Early Payment Am Agen borrows $1000 for 1 year at 12% annual nterest compounded monthly. Verfy that her monthly loan payment s $ , whch s rounded to $ After makng three payments, she decdes to pay off the remanng balance all at once. How much must she pay? SOLUTION Snce nne payments reman to be pad, they can be thought of as an annuty consstng of nne payments of $88.85 at 1% nterest per perod. The present value of ths annuty s c d < So Am s remanng balance, computed by ths method, s $

27 5.3 Present Value of an Annuty; Amortzaton 213 YOUR TURN 4 Fnd the remanng balance n Example 4 f the balance was to be pad off after four months. Use the unrounded value for R of $ An alternatve method of fgurng the balance s to consder the payments already made as an annuty of three payments. At the begnnng, the present value of ths annuty was c d < So she stll owes the dfference $ $ $ Furthermore, she owes the nterest on ths amount for 3 months, for a total of < $ Ths balance due dffers from the one obtaned by the frst method by 1 cent because the monthly payment and the other calculatons were rounded to the nearest penny. If we had used the more accurate value of R and not rounded any ntermedate answers, both methods would have gven the same value of $ TRY YOUR TURN 4 Although most people would not qubble about a dfference of 1 cent n the balance due n Example 4, the dfference n other cases (larger amounts or longer terms) mght be more than that. A bank or busness must keep ts books accurately to the nearest penny, so t must determne the balance due n such cases unambguously and exactly. Ths s done by means of an amortzaton schedule, whch lsts how much of each payment s nterest and how much goes to reduce the balance, as well as how much s owed after each payment. EXAMPLE 5 Amortzaton Table Determne the exact amount Am Agen n Example 4 owes after three monthly payments. SOLUTION An amortzaton table for the loan s shown below. It s obtaned as follows. The annual nterest rate s 12% compounded monthly, so the nterest rate per month s 12%/12 5 1% When the frst payment s made, 1 month s nterest namely $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 chart. When payment 2 s made, 1 month s nterest on $ s owed, namely $9.21. Subtractng ths from the $88.85 payment leaves $79.64 to reduce the prncpal. Hence, the prncpal at the end of payment 2 s $ $ $ The nterest porton of payment 3 s based on ths amount, and the remanng lnes of the table are found n a smlar fashon. The schedule shows that after three payments, she stll owes $761.08, an amount that agrees wth the frst method n Example 4. Amortzaton Table Payment Amount of Interest Porton to Prncpal at Number Payment for Perod Prncpal End of Perod 0 $ $88.85 $10.00 $78.85 $ $88.85 $9.21 $79.64 $ $88.85 $8.42 $80.43 $ $88.85 $7.61 $81.24 $ $88.85 $6.80 $82.05 $ $88.85 $5.98 $82.87 $ $88.85 $5.15 $83.70 $ $88.85 $4.31 $84.54 $ $88.85 $3.47 $85.38 $ $88.85 $2.61 $86.24 $ $88.85 $1.75 $87.10 $ $88.84 $0.88 $87.96 $0.00

28 214 CHAPTER 5 Mathematcs of Fnance The amortzaton schedule n Example 5 s typcal. In partcular, note that all payments are the same except the last one. It s often necessary to adjust the amount of the fnal payment to account for roundng off earler and to ensure that the fnal balance s exactly 0. An amortzaton schedule also shows how the perodc payments are appled to nterest and prncpal. The amount gong to nterest decreases wth each payment, whle the amount gong to reduce the prncpal ncreases wth each payment. TECHNOLOGY NOTE A graphng calculator program to produce an amortzaton schedule s avalable n the Graphng Calculator and Excel Spreadsheet Manual avalable wth ths book. The TI-84 Plus ncludes a bult-n program to fnd the amortzaton payment. Spreadsheets are another useful tool for creatng amortzaton tables. Mcrosoft Excel has a bult-n feature for calculatng monthly payments. Fgure 14 shows an Excel amortzaton table for Example 5. For more detals, see the Graphng Calculator and Excel Spreadsheet Manual, also avalable wth ths book A B C D E F Pmt# Payment Interest Prncpal End Prncpl FIGURE 14 Payng Off a Loan Early Suppose that n Example 2, the car owner decdes that she can afford to make payments of $700 rather than $ How much earler would she pay off the loan? How much nterest would she save? SOLUTION Puttng R 5 700, P 5 17,000, and nto the formula for the present value of an annuty gves 17, c n d Multply both sdes by and dvde by 700 to get n, or EXAMPLE n Solve ths usng ether logarthms or a graphng calculator, as n Example 11 n Secton 5.1, to get n Ths means that 25 payments of $700, plus a fnal, smaller payment, would be suffcent to pay off the loan. Create an amortzaton table to verfy that the fnal payment would be $ (the sum of the prncpal after the penultmate payment plus the nterest on that prncpal for the fnal month). The loan would then be pad off after 26 months, or 10 months early. The orgnal loan requred 36 payments of $517.17, or 36(517.17) 5 $18,618.12, although the amount s actually $18, because the fnal payment was $517.29, as an amortzaton table would show. Wth larger payments, the car owner pad 25(700) $18, Therefore, the car owner saved $18, $18, $ n nterest by makng larger payments each month.

29 5.3 Present Value of an Annuty; Amortzaton EXERCISES 1. Explan the dfference between the present value of an annuty and the future value of an annuty. For a gven annuty, whch s larger? Why? 2. What does t mean to amortze a loan? Fnd the present value of each ordnary annuty. 3. Payments of $890 each year for 16 years at 6% compounded annually 4. Payments of $1400 each year for 8 years at 6% compounded annually 5. Payments of $10,000 semannually for 15 years at 5% compounded semannually 6. Payments of $50,000 quarterly for 10 years at 4% compounded quarterly 7. Payments of $15,806 quarterly for 3 years at 6.8% compounded quarterly 8. Payments of $18,579 every 6 months for 8 years at 5.4% compounded semannually 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. 9. 4% compounded annually 10. 6% compounded annually Fnd (a) the payment necessary to amortze each loan; (b) the total payments and the total amount of nterest pad based on the calculated monthly payments, and (c) the total payments and total amount of nterest pad based upon an amortzaton table. 11. $2500; 6% compounded quarterly; 6 quarterly payments 12. $41,000; 8% compounded semannually; 10 semannual payments 13. $90,000; 6% compounded annually; 12 annual payments 14. $140,000; 8% compounded quarterly; 15 quarterly payments 15. $7400; 6.2% compounded semannually; 18 semannual payments 16. $5500; 10% compounded monthly; 24 monthly payments Suppose that n the loans descrbed n Exercses 13 16, the borrower pad off the loan after the tme ndcated below. Calculate the amount needed to pay off the loan, usng ether of the two methods descrbed n Example After 3 years n Exercse After 5 quarters n Exercse After 3 years n Exercse After 7 months n Exercse 16 Use the amortzaton table n Example 5 to answer the questons n Exercses How much of the fourth payment s nterest? 22. How much of the eleventh payment s used to reduce the debt? 23. How much nterest s pad n the frst 4 months of the loan? 24. How much nterest s pad n the last 4 months of the loan? 25. 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? 26. 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 monthly house payments necessary to amortze each loan. Then calculate the total payments and the total amount of nterest pad. 27. $199,000 at 7.01% for 25 years 28. $175,000 at 6.24% for 30 years 29. $253,000 at 6.45% for 30 years 30. $310,000 at 5.96% for 25 years Suppose that n the loans descrbed n Exercses 13 16, the borrower made a larger payment, as ndcated below. Calculate (a) the tme needed to pay off the loan, (b) the total amount of the payments, and (c) the amount of nterest saved, compared wth part c of Exercses $16,000 n Exercse $18,000 n Exercse $850 n Exercse $400 n Exercse 16 APPLICATIONS Busness and Economcs 35. House Payments Calculate the monthly payment and total amount of nterest pad n Example 3 wth a 15-year loan, and then compare wth the results of Example Installment Buyng Stereo Shack 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. 37. Car Payments Hong Le buys a car costng $14,000. He agrees to make payments at the end of each monthly perod for 4 years. He pays 7% nterest, compounded monthly. a. What s the amount of each payment? b. Fnd the total amount of nterest Le wll pay. 38. Credt Card Debt Tom Shaffer charged $8430 on hs credt card to relocate for hs frst job. When he realzed that the nterest rate for the unpad balance was 27% compounded monthly, he decded not to charge any more on that account. He wants to have ths account pad off by the end of 3 years,

30 216 CHAPTER 5 Mathematcs of Fnance so he arranges to have automatc payments sent at the end of each month. a. What monthly payment must he make to have the account pad off by the end of 3 years? b. How much total nterest wll he have pad? 39. New Car In Sprng 2010, some dealers offered a cash-back allowance of $2250 or 0.9% fnancng for 36 months on an Acura TL. Source: cars.com. a. Determne the payments on an Acura TL f a buyer chooses the 0.9% fnancng opton and needs to fnance $30,000 for 36 months, compounded monthly. Fnd the total amount the buyer wll pay for ths opton. b. Determne the payments on an Acura TL f a buyer chooses the cash-back opton and now needs to fnance only $27,750. At the tme, t was possble to get a new car loan at 6.33% for 48 months, compounded monthly. Fnd the total amount the buyer wll pay for ths opton. c. Dscuss whch deal s best and why. 40. New Car In Sprng 2010, some dealers offered a cash-back allowance of $1500 or 1.9% fnancng for 36 months on a Volkswagen Tguan. Source: cars.com. a. Determne the payments on a Volkswagen Tguan f a buyer chooses the 1.9% fnancng opton and needs to fnance $25,000 for 36 months, compounded monthly. Fnd the total amount the buyer wll pay for ths opton. b. Determne the payments on a Volkswagen Tguan f a buyer chooses the cash-back opton and now needs to fnance only $23,500. At the tme, t was possble to get a new car loan at 6.33% for 48 months, compounded monthly. Fnd the total amount the buyer wll pay for ths opton. c. Dscuss whch deal s best and why. 41. Lottery Wnnngs In most states, the wnnngs of mllondollar lottery jackpots are dvded nto equal payments gven annually for 20 years. (In Colorado, the results are dstrbuted over 25 years.) Ths means that the present value of the jackpot s worth less than the stated prze, wth the actual value determned by the nterest rate at whch the money could be nvested. Source: The New York Tmes Magazne. a. Fnd the present value of a $1 mllon lottery jackpot dstrbuted n equal annual payments over 20 years, usng an nterest rate of 5%. b. Fnd the present value of a $1 mllon lottery jackpot dstrbuted n equal annual payments over 20 years, usng an nterest rate of 9%. c. Calculate the answer for part a usng the 25-year dstrbuton tme n Colorado. d. Calculate the answer for part b usng the 25-year dstrbuton tme n Colorado. The extended plan allows up to 25 years to repay the loan. Source: U.S. Department of Educaton. A student borrows $55,000 at 6.80% compounded monthly. 42. Fnd the monthly payment and total nterest pad under the standard plan over 10 years. 43. Fnd the monthly payment and total nterest pad under the extended plan over 25 years. Installment Buyng In Exercses 44 46, prepare an amortzaton schedule showng the frst four payments for each loan. 44. 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. 45. Large semtraler trucks cost $110,000 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 8% compounded semannually. 46. One retaler charges $1048 for a laptop computer. A frm of tax accountants buys 8 of these laptops. They make a down payment of $1200 and agree to amortze the balance wth monthly payments at 6% compounded monthly for 4 years. 47. Investment In 1995, Oseola McCarty donated $150,000 to the Unversty of Southern Msssspp to establsh a scholarshp fund. What s unusual about her s that the entre amount came from what she was able to save each month from her work as a washer woman, a job she began n 1916 at the age of 8, when she dropped out of school. Sources: The New York Tmes. a. How much would Ms. McCarty have to put nto her savngs account at the end of every 3 months to accumulate $150,000 over 79 years? Assume she receved an nterest rate of 5.25% compounded quarterly. b. Answer part a usng a 2% and a 7% nterest rate. Student Loans Student borrowers now have more optons to choose from when selectng repayment plans. The standard plan repays the loan n up to 10 years wth equal monthly payments.

31 5.3 Present Value of an Annuty; Amortzaton Loan Payments When Nancy Hart 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 8% compounded semannually. a. Fnd the amount of each payment. b. Refer to the text and Fgure 13. When her loan had been reduced below $5000, Nancy receved a large tax refund and decded to pay off the loan. How many payments were left at ths tme? 49. House Payments Ian Desrosers buys a house for $285,000. He pays $60,000 down and takes out a mortgage at 6.5% 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. Refer to the text and Fgure 13. When wll half the 20-year loan n part b be pad off? 50. House Payments The Chavara famly buys a house for $225,000. They pay $50,000 down and take out a 30-year mortgage on the balance. Fnd ther monthly payment and the total amount of nterest they wll pay f the nterest rate s a. 6%; b. 6.5%; c. 7%. d. Refer to the text and Fgure 13. When wll half the 7% loan n part c be pad off? 51. Refnancng a Mortgage Ffteen years ago, the Buda famly bought a home and fnanced $150,000 wth a 30-year mortgage at 8.2%. a. Fnd ther monthly payment, the total amount of ther payments, and the total amount of nterest they wll pay over the lfe of ths loan. b. The Budas made payments for 15 years. Estmate the unpad balance usng the formula y 5 R c n2x2 d, and then calculate the total of ther remanng payments. c. Suppose nterest rates have dropped snce the Buda famly took out ther orgnal loan. One local bank now offers a 30-year mortgage at 6.5%. The bank fees for refnancng are $3400. If the Budas pay ths fee up front and refnance the balance of ther loan, fnd ther monthly payment. Includng the refnancng fee, what s the total amount of ther payments? Dscuss whether or not the famly should refnance wth ths opton. d. A dfferent bank offers the same 6.5% rate but on a 15-year mortgage. Ther fee for fnancng s $4500. If the Budas pay ths fee up front and refnance the balance of ther loan, fnd ther monthly payment. Includng the refnancng fee, what s the total amount of ther payments? Dscuss whether or not the famly should refnance wth ths opton. 52. Inhertance Deborah Harden has nherted $25,000 from her grandfather s estate. She deposts the money n an account offerng 6% nterest compounded annually. She wants to make equal annual wthdrawals from the account so that the money (prncpal and nterest) lasts exactly 8 years. a. Fnd the amount of each wthdrawal. b. Fnd the amount of each wthdrawal f the money must last 12 years. 53. Chartable Trust The trustees of a college have accepted a gft of $150,000. The donor has drected the trustees to depost the money n an account payng 6% per year, compounded semannually. The trustees may make equal wthdrawals at the end of each 6-month perod; the money must last 5 years. a. Fnd the amount of each wthdrawal. b. Fnd the amount of each wthdrawal f the money must last 6 years. Amortzaton loan. Prepare an amortzaton schedule for each 54. A loan of $37,948 wth nterest at 6.5% compounded annually, to be pad wth equal annual payments over 10 years. 55. A loan of $4836 at 7.25% nterest compounded sem-annually, to be repad n 5 years n equal semannual payments. 56. Perpetuty A perpetuty s an annuty n whch the payments go on forever. We can derve a formula for the present value of a perpetuty by takng the formula for the present value of an annuty and lookng at what happens when n gets larger and larger. Explan why the present value of a perpetuty s gven by P 5 R. 57. Perpetuty Usng the result of Exercse 56, fnd the present value of perpetutes for each of the followng. a. Payments of $1000 a year wth 4% nterest compounded annually b. Payments of $600 every 3 months wth 6% nterest compounded quarterly YOUR TURN ANSWERS 1. $ $ $ , $135, $679.84

32 218 CHAPTER 5 Mathematcs of Fnance 5 CHAPTER REVIEW SUMMARY In ths chapter we ntroduced the mathematcs of fnance. We frst extended smple nterest calculatons to compound nterest, whch s nterest earned on nterest prevously earned. We then developed the mathematcs assocated wth the followng fnancal concepts. In an annuty, money contnues to be deposted at regular ntervals, and compound nterest s earned on that money as well. In an ordnary annuty, payments are made at the end of each tme perod, and the compoundng perod s the same as the tme between payments, whch smplfes the calculatons. An annuty due s slghtly dfferent, n that the payments are made at the begnnng of each tme perod. A snkng fund s lke an ordnary annuty; a fund s set up to receve perodc payments. The payments plus the compound nterest wll produce a desred sum by a certan date. The present value of an annuty s the amount that would have to be deposted today to produce the same amount as the annuty at the end of a specfed tme. An amortzaton table shows how a loan s pad back after a specfed tme. It shows the payments broken down nto nterest and prncpal. 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 nterest rate? 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, what s beng sought: present value, future value, number of perods at nterest, or effectve rate? b. For an annuty,. Is 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, present value should always be less than future value. The amount of nterest or the payments n an annuty should be farly small compared to the total future value. Lst of Varables r s the annual nterest rate. s the nterest rate per perod. t s the number of years. n s the number of perods. m s the number of perods per year. 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. 5 r m n 5 tm

33 CHAPTER 5 Revew 219 Interest Future Value Present Value Smple Interest Compound Interest Contnuous Compoundng I 5 Prt I 5 A 2 P I 5 A 2 P A 5 P1 1 1 rt 2 A 5 P n A 5 Pe rt P 5 A 1 1 rt P 5 A 2n 5 A n P 5 Ae 2rt Effectve Rate r E 5 a1 1 r m b m 2 1 r E 5 e r21 Ordnary Annuty Annuty Due Snkng Fund Payment Amortzaton Payments Future Value S 5 R c n 2 1 d 5 R. s n n Present Value P 5 R c d 5 R. a n0 Future Value S 5 R c n d 2 R S R n S s n0 P P R n a n0 KEY TERMS 5.1 smple nterest prncpal rate tme future value maturty value compound nterest compound amount REVIEW EXERCISES CONCEPT CHECK nomnal (stated) rate effectve rate present value rule of 70 rule of 72 contnuous compoundng 5.2 geometrc sequence terms Determne whether each of the followng statements s true or false, and explan why. 1. For a partcular nterest rate, compound nterest s always better than smple nterest. 2. The sequence 1, 2, 4, 6, 8,... s a geometrc sequence. 3. If a geometrc sequence has frst term 3 and common rato 2, then the sum of the frst 5 terms s S The value of a snkng fund should decrease over tme. 5. For payments made on a mortgage, the (nonnterest) porton of the payment appled on the prncpal ncreases over tme. 6. On a 30-year conventonal home mortgage, at recent nterest rates, t s common to pay more money on the nterest on the loan than the actual loan tself. 7. One can use the amortzaton payments formula to calculate the monthly payment of a car loan. 8. The effectve rate formula can be used to calculate the present value of a loan. common rato annuty ordnary annuty payment perod future value of an annuty term of an annuty future value of an ordnary annuty snkng fund annuty due future value of an annuty due 5.3 present value of an annuty amortze a loan amortzaton schedule 9. The followng calculaton gves the monthly payment on a $25,000 loan, compounded monthly at a rate of 5% for a perod of sx years: 25,000 c / d. 0.05/ The followng calculaton gves the present value of an annuty of $5,000 payments at the end of each year for 10 years. The fund earns 4.5% compounded annually c d PRACTICE AND EXPLORATION Fnd the smple nterest for each loan. 11. $15,903 at 6% for 8 months 12. $4902 at 5.4% for 11 months 13. $42,368 at 5.22% for 7 months 14. $3478 at 6.8% for 88 days (assume a 360-day year)

34 220 CHAPTER 5 Mathematcs of Fnance 15. For a gven amount of money at a gven nterest rate for a gven tme perod, does smple nterest or compound nterest produce more nterest? Fnd the compound amount n each loan. 16. $2800 at 7% compounded annually for 10 years 17. $19, at 8% compounded semannually for 7 years 18. $ at 5.6% compounded semannually for 16 years 19. $57, at 6% compounded quarterly for 5 years Fnd the amount of nterest earned by each depost. 20. $3954 at 8% compounded annually for 10 years 21. $12, at 5% compounded semannually for 7 years 22. $12, at 6.4% compounded quarterly for 29 quarters 23. $34, at 4.8% compounded monthly for 32 months 24. What s meant by the present value of an amount A? Fnd the present value of each amount. 25. $42,000 n 7 years, 6% compounded monthly 26. $17,650 n 4 years, 4% compounded quarterly 27. $ n 3.5 years, 6.77% compounded semannually 28. $ n 44 months, 5.93% compounded monthly 29. Wrte the frst fve terms of the geometrc sequence wth a 5 2 and r Wrte the frst four terms of the geometrc sequence wth a 5 4 and r 5 1/ Fnd the sxth term of the geometrc sequence wth a 523 and r Fnd the ffth term of the geometrc sequence wth a 522 and r Fnd the sum of the frst four terms of the geometrc sequence wth a 523 and r Fnd the sum of the frst fve terms of the geometrc sequence wth a and r 521/ Fnd s Fnd s What s meant by the future value of an annuty? Fnd the future value of each annuty and the amount of nterest earned. 38. $500 deposted at the end of each 6-month perod for 10 years; money earns 6% compounded semannually. 39. $1288 deposted at the end of each year for 14 years; money earns 4% compounded annually. 40. $4000 deposted at the end of each quarter for 7 years; money earns 5% compounded quarterly. 41. $233 deposted at the end of each month for 4 years; money earns 4.8% compounded monthly. 42. $672 deposted at the begnnng of each quarter for 7 years; money earns 4.4% compounded quarterly. 43. $11,900 deposted at the begnnng of each month for 13 months; money earns 6% compounded monthly. 44. What s the purpose of a snkng fund? Fnd the amount of each payment that must be made nto a snkng fund to accumulate each amount. 45. $6500; money earns 5% compounded annually for 6 years $57,000; money earns 4% compounded semannually for 8 2 years $233,188; money earns 5.2% compounded quarterly for 7 4 years $1,056,788; money earns 7.2% compounded monthly for 4 years. Fnd the present value of each ordnary annuty. 49. Deposts of $850 annually for 4 years at 6% compounded annually 50. Deposts of $1500 quarterly for 7 years at 5% compounded quarterly 51. Payments of $4210 semannually for 8 years at 4.2% compounded semannually 52. Payments of $ monthly for 17 months at 6.4% compounded monthly 53. Gve two examples of the types of loans that are commonly amortzed. Fnd the amount of the payment necessary to amortze each loan. Calculate the total nterest pad. 54. $80,000; 5% compounded annually; 9 annual payments 55. $3200; 8% compounded quarterly; 12 quarterly payments 56. $32,000; 6.4% compounded quarterly; 17 quarterly payments 57. $51,607; 8% compounded monthly; 32 monthly payments Fnd the monthly house payments for each mortgage. Calculate the total payments and nterest. 58. $256,890 at 5.96% for 25 years 59. $177,110 at 6.68% for 30 years A porton of an amortzaton table s gven below for a $127,000 loan at 8.5% nterest compounded monthly for 25 years. Payment Amount of Interest Porton to Prncpal at Number Payment for Perod Prncpal End of Perod 1 $ $ $ $126, $ $ $ $126, $ $ $ $126, $ $ $ $126, $ $ $ $126, $ $ $ $126, $ $ $ $126, $ $ $ $125, $ $ $ $125, $ $ $ $125, $ $ $ $125, $ $ $ $125,

35 CHAPTER 5 Revew 221 Use the table to answer the followng questons. 60. How much of the ffth payment s nterest? 61. How much of the twelfth payment s used to reduce the debt? 62. How much nterest s pad n the frst 3 months of the loan? 63. How much has the debt been reduced at the end of the frst year? APPLICATIONS Busness and Economcs 64. Personal Fnance Jane Flemng owes $5800 to her mother. She has agreed to repay the money n 10 months at an nterest rate of 5.3%. How much wll she owe n 10 months? How much nterest wll she pay? 65. Busness Fnancng Jule Ward needs to borrow $9820 to buy new equpment for her busness. The bank charges her 6.7% smple nterest for a 7-month loan. How much nterest wll she be charged? What amount must she pay n 7 months? 66. Busness Fnancng An accountant loans $28,000 at smple nterest to her busness. The loan s at 6.5% and earns $1365 nterest. Fnd the tme of the loan n months. 67. Busness Investment A developer deposts $84,720 for 7 months and earns $ n smple nterest. Fnd the nterest rate. 68. Personal Fnance In 3 years Beth Rechstener must pay a pledge of $7500 to her college s buldng fund. What lump sum can she depost today, at 5% compounded semannually, so that she wll have enough to pay the pledge? 69. Personal Fnance 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 the age of 65 f he nvests now, gven that he can earn 5% nterest compounded quarterly? 70. Pensons Penson experts recommend that you start drawng at least 40% of your full penson as early as possble. Suppose you have bult up a penson of $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 of 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? Source: Smart Money. 71. Busness Investment A frm of attorneys deposts $5000 of proft-sharng money at the end of each semannual perod for years. Fnd the fnal amount n the account f the deposts earn 10% compounded semannually. Fnd the amount of nterest earned. 72. Busness Fnancng A small resort must add a swmmng pool to compete wth a new resort bult nearby. The pool wll cost $28,000. The resort borrows the money and agrees to repay t wth equal payments at the end of each quarter for years at an nterest rate of 8% compounded quarterly. Fnd the amount of each payment. 73. Busness Fnancng The owner of Eastsde Hallmark borrows $48,000 to expand the busness. The money wll be repad n equal payments at the end of each year for 7 years. Interest s 6.5%. Fnd the amount of each payment and the total amount of nterest pad. 74. Personal Fnance To buy a new computer, Mark Nguyen borrows $3250 from a frend at 4.2% nterest compounded annually for 4 years. Fnd the compound amount he must pay back at the end of the 4 years. 75. Effectve Rate On May 21, 2010, Ascenca (a dvson of PBI Bank) pad 1.49% nterest, compounded monthly, on a 1-year CD, whle gantbank.com pad 1.45% compounded daly. What are the effectve rates for the two CDs, and whch bank pad a hgher effectve rate? Source: Bankrate.com. 76. Home Fnancng When the Lee famly bought ther home, they borrowed $315,700 at 7.5% compounded monthly for 25 years. If they make all 300 payments, repayng the loan on schedule, how much nterest wll they pay? (Assume the last payment s the same as the prevous ones.) 77. New Car In Sprng 2010, some dealers offered the followng optons on a Chevrolet HHR: a cash-back allowance of $4000, 0% fnancng for 60 months, or 3.9% fnancng for 72 months. Source: cars.com. a. Determne the payments on a Chevrolet HHR f a buyer chooses the 0% fnancng opton and needs to fnance $16,000 for 60 months. Fnd the total amount of the payments. b. Repeat part a for the 3.9% fnancng opton for 72 months. c. Determne the payments on a Chevrolet HHR f a buyer chooses the cash-back opton and now needs to fnance only $12,000. At the tme, t was possble to get a new car loan at 6.33% for 48 months, compounded monthly. Fnd the total amount of the payments. d. Dscuss whch deal s best and why. 78. New Car In Sprng 2010, some dealers offered a cash-back allowance of $5000 or 0% fnancng for 72 months on a Chevrolet Slverado. Source: cars.com. a. Determne the payments on a Chevrolet Slverado f a buyer chooses the 0% fnancng opton and needs to fnance $30,000 for 72 months. Fnd the total amount of the payments. b. Determne the payments on a Chevrolet Slverado f a buyer chooses the cash-back opton and now needs to fnance only $25,000. At the tme, t was possble to get a new car loan at 6.33% for 48 months, compounded monthly. Fnd the total amount of the payments. c. Dscuss whch deal s best and why. d. Fnd the nterest rate at the bank that would make the total amount of payments for the two optons equal. 79. Buyng and Sellng a House The Bahary famly bought a house for $191,000. They pad $40,000 down and took out a 30-year mortgage for the balance at 6.5%. a. Fnd ther monthly payment. b. How much of the frst payment s nterest? After 180 payments, the famly sells ts house for $238,000. They must pay closng costs of $3700 plus 2.5% of the sale prce.

36 222 CHAPTER 5 Mathematcs of Fnance c. Estmate the current mortgage balance at the tme of the sale usng one of the methods from Example 4 n Secton 3. d. Fnd the total closng costs. e. Fnd the amount of money they receve from the sale after payng off the mortgage. The followng exercse s from an actuaral examnaton. Source: The Socety of Actuares. 80. Death Beneft 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%. Calculate X. Choose one of the followng. a. 117 b. 118 c. 129 d. 135 e Investment The New York Tmes posed a scenaro wth two ndvduals, Sue and Joe, who each have $1200 a month to spend on housng and nvestng. Each takes out a mortgage for $140,000. Sue gets a 30-year mortgage at a rate of 6.625%. Joe gets a 15-year mortgage at a rate of 6.25%. Whatever money s left after the mortgage payment s nvested n a mutual fund wth a return of 10% annually. Source: The New york Tmes. a. What annual nterest rate, when compounded monthly, gves an effectve annual rate of 10%? b. What s Sue s monthly payment? c. If Sue nvests the remander of her $1200 each month, after the payment n part b, n a mutual fund wth the nterest rate n part a, how much money wll she have n the fund at the end of 30 years? d. What s Joe s monthly payment? e. You found n part d that Joe has nothng left to nvest untl hs mortgage s pad off. If he then nvests the entre $1200 monthly n a mutual fund wth the nterest rate n part a, how much money wll he have at the end of 30 years (that s, after 15 years of payng the mortgage and 15 years of nvestng)? f. Who s ahead at the end of the 30 years and by how much? g. Dscuss to what extent the dfference found n part f s due to the dfferent nterest rates or to the dfferent amounts of tme. EXTENDED APPLICATION TIME, MONEY, AND POLYNOMIALS t 0 t 1 $1000 $2500 $3851 Atme lne s often helpful for evaluatng complex nvestments. For example, suppose you buy a $1000 CD at tme t 0. After one year $2500 s added to the CD at t 1. By tme t 2, after another year, your money has grown to $3851 wth nterest. What rate of nterest, called yeld to maturty (YTM), dd your money earn? A tme lne for ths stuaton s shown n Fgure 15. tme Assumng nterest s compounded annually at a rate, and usng the compound nterest formula, gves the followng descrpton of the YTM. Source: COMAP t 2 FIGURE 15 To determne the yeld to maturty, we must solve ths equaton for. Snce the quantty 1 1 s repeated, let x and frst solve the second-degree (quadratc) polynomal equaton for x. 1000x x We can use the quadratc formula wth a , b , and c x " We get x and x Snce x 5 1 1, the two values for are % and %. We reject the negatve value because the fnal accumulaton s greater than the sum of the deposts. In some applcatons, however, negatve rates may be meanngful. By checkng n the frst equaton, we see that the yeld to maturty for the CD s 7.67%. Now let us consder a more complex but realstc problem. Suppose Austn Caperton has contrbuted for 4 years to a retrement fund. He contrbuted $6000 at the begnnng of the frst year. At the begnnng of the next 3 years, he contrbuted $5840, $4000, and $5200, respectvely. At the end of the fourth year, he had $29, n hs fund. The nterest rate earned by the fund vared between 21% and 23%, so Caperton would lke to know the YTM 5 for hs hardearned retrement dollars. From a tme lne (see Fgure 16), we set up the followng equaton n 1 1 for Caperton s savngs program ,912.38

37 $6000 $5840 $4000 $5200 $29, t 0 t 1 t 2 t 3 t4 FIGURE 16 tme Let x We need to solve the fourth-degree polynomal equaton f 1x x x x x There s no smple way to solve a fourth-degree polynomal equaton, so we wll use a graphng calculator. We expect that 0,, 1, so that 1, x, 2. Let us calculate f112 and f122. If there s a change of sgn, we wll know that there s a soluton to f1x2 5 0 between 1 and 2. We fnd that Usng a graphng calculator, we fnd that there s one postve soluton to ths equaton, x , so 5 YTM %. EXERCISES 229, f and f , Lorr Morgan receved $50 on her 16th brthday, and $70 on her 17th brthday, both of whch she mmedately nvested n the bank, wth nterest compounded annually. On her 18th brthday, she had $ n her account. Draw a tme lne, set up a polynomal equaton, and calculate the YTM. 2. At the begnnng of the year, Blake Allvne nvested $10,000 at 5% for the frst year. At the begnnng of the second year, he added $12,000 to the account. The total account earned 4.5% for the second year. a. Draw a tme lne for ths nvestment. b. How much was n the fund at the end of the second year? c. Set up and solve a polynomal equaton and determne the YTM. What do you notce about the YTM? 3. On January 2 each year for 3 years, Mchael Baley deposted bonuses of $1025, $2200, and $1850, respectvely, n an account. He receved no bonus the followng year, so he made no depost. At the end of the fourth year, there was $ n the account. a. Draw a tme lne for these nvestments. b. Wrte a polynomal equaton n x 1x and use a graphng calculator to fnd the YTM for these nvestments. c. Go to the webste WolframAlpha.com, and ask t to solve the polynomal from part b. Compare ths method of solvng the equaton wth usng a graphng calculator. 4. Don Bevlle nvested yearly n a fund for hs chldren s college educaton. At the begnnng of the frst year, he nvested $1000; at the begnnng of the second year, $2000; at the thrd through the sxth, $2500 each year, and at the begnnng of the seventh, he nvested $5000. At the begnnng of the eghth year, there was $21,259 n the fund. a. Draw a tme lne for ths nvestment program. b. Wrte a seventh-degree polynomal equaton n 1 1 that gves the YTM for ths nvestment program. c. Use a graphng calculator to show that the YTM s less than 5.07% and greater than 5.05%. d. Use a graphng calculator to calculate the soluton for 1 1 and fnd the YTM. e. Go to the webste WolframAlpha.com, and ask t to solve the polynomal from part b. Compare ths method of solvng the equaton wth usng a graphng calculator. 5. People often lose money on nvestments. Melssa Fscher nvested $50 at the begnnng of each of 2 years n a mutual fund, and at the end of 2 years her nvestment was worth $90. a. Draw a tme lne and set up a polynomal equaton n 1 1. Solve for. b. Examne each negatve soluton (rate of return on the nvestment) to see f t has a reasonable nterpretaton n the context of the problem. To do ths, use the compound nterest formula on each value of to trace each $50 payment to maturty. DIRECTIONS FOR GROUP PROJECT Assume that you are n charge of a group of fnancal analysts and that you have been asked by the broker at your frm to develop a tme lne for each of the people lsted n the exercses above. Prepare a report for each clent that presents the YTM for each nvestment strategy. Make sure that you descrbe the methods used to determne the YTM n a manner that the average clent should understand. 223