Mathematics of Finance


 Jonathan Arnold
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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 30day months, makng a year 360 days long. Treasury blls sold by the U.S. government assume a 360day year n calculatng nterest. Interest found usng a 360day year s called ordnary nterest and nterest found usng a 365day 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 360day 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 365day 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 360day 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 5year 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 greatgreatgrandfather, 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 6month 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 1year 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 6month 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 sanglen 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 TI84 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 20year 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 PMT5200 FV P/Y512 C/Y512 PMT: END BEGIN FIGURE 9 Method 2 TVM Solver Usng the TVM Solver under the FINANCE menu on the TI84 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 TI84 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 TI84 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
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