Mathematics of Finance

Transcription

1 CHAPTER 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 Revew Exercses 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. ISBN:

2 ISBN: Chapter 5 Mathematcs of Fnance Whether you are n a poston to nvest money or to borrow money, t s mportant for both consumers and busness managers to understand nterest. The formulas for nterest are developed n ths chapter. 5.1 SIMPLE AND COMPOUND INTEREST THINK ABOUT IT If you can borrow money at 11% nterest compounded annually or at 10.8% compounded monthly, whch loan would cost less? We shall see how to make such comparsons n ths secton. 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 percent per year, expressed as a 1 decmal. For example, 6% 5.06 and 11 2% The tme the money s earnng nterest s calculated n years. Smple nterest s the product of the prncpal, rate, and tme. SIMPLE 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, Jennfer Wall borrowed $5000 at 8% smple nterest for 11 months. How much nterest wll she pay? Soluton From the formula, I 5 Prt, wth P , r 5.08, and t 5 11/12 (n years). The total nterest she wll pay s or $ I / , 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. 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.

3 5.1 Smple and Compound Interest 205 FUTURE OR MATURITY VALUE FOR SIMPLE 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 9.2% Soluton 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 or $ (The answer s rounded to the nearest cent, as s customary n fnancal problems.) Of ths maturty value, represents nterest. (b) A loan of $11,280 for 85 days at 11% nterest Soluton 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 or $11, A c a 2 3 bd A < < , $ $ $ A 5 11,280 c1 1.11a 85 bd < 11,572.97, 360 CAUTION When usng the formula for future value, as well as all other formulas n ths chapter, we 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 $ /122 < $19.17, rounded to the nearest penny. After 8 months, the total s 81$ $153.36, whch s 3 more than we computed n the example. ISBN: In part (b) of Example 2 we assumed 360 days n a year. 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 ISBN: Chapter 5 Mathematcs of Fnance EXAMPLE 3 Smple Interest Carter Fenton wants to borrow $8000 from Chrstne O Bren. He s wllng to pay back $8380 n 6 months. What nterest rate wll he pay? Soluton Use the formula for future value, wth t 5 6/12 5.5, and solve for r. A , P , A 5 P11 1 rt r r r r Thus, the nterest rate s 9.5%. Dstrbutve property Subtract Dvde by 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 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. 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. $ $1100. A 5 P11 1 r P11 1 r2 A 5 3P11 1 r r P11 1 r2 2 P11 1 r2 3. A 5 P11 1 r2 t,

5 5.1 Smple and Compound Interest 207 NOTE Compare ths formula for compound nterest wth the formula for smple nterest. Compound nterest Smple nterest 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. 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. Soluton In the formula above, P , /1, and n The compound amount s ISBN: A 5 P n A Usng a calculator, we get A < $ , the compound amount.

6 ISBN: Chapter 5 Mathematcs of Fnance (b) Fnd the amount of nterest earned. Soluton Subtract the ntal depost from the compound amount. Amount of nterest 5 $ $ $ EXAMPLE 5 Compound Interest Fnd the amount of nterest earned by a depost of $2450 for 6.5 years at 5.25% compounded quarterly. Soluton 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 /42 26 < Rounded to the nearest cent, the compound amount s $ , so the nterest s $ $ $ 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. 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 at compound nterest, $1000 grows to $ , whereas wth smple nterest, t amounts to $ , a dfference of $ Spreadsheets are deal for performng fnancal calculatons. Fgure 2 (on the next page) 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 the calculator results n Fgure 1. For more detals on the use of spreadsheets n the mathematcs of fnance, see The Spreadsheet Manual that s avalable wth ths book. 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 Compound Interest Smple Interest 20 FIGURE 1

7 5.1 Smple and Compound Interest 209 FIGURE 2 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 r e for the effectve rate. EXAMPLE 6 Effectve Rate Fnd the effectve rate correspondng to a stated rate of 6% compounded semannually. Soluton Here, r/m 5 6%/2 5 3% 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. EFFECTIVE 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 b m 2 1. ISBN: *When appled to consumer fnance, the effectve rate s called the annual percentage rate, APR, or annual percentage yeld, APY.

8 ISBN: Chapter 5 Mathematcs of Fnance EXAMPLE 7 Effectve Rate A bank pays nterest of 4.9% compounded monthly. Fnd the effectve rate. Soluton Use the formula gven above wth r and m The effectve rate s r e 5 a b , or 5.01%. EXAMPLE 8 Effectve Rate Joe Vetere needs to borrow money. Hs neghborhood bank charges 11% nterest compounded semannually. A downtown bank charges 10.8% nterest compounded monthly. At whch bank wll Joe pay the lesser amount of nterest? Soluton Compare the effectve rates. Neghborhood bank: Downtown bank: r e 5 a b < 11.3% r e 5 a b < < 11.4% The neghborhood bank has the lower effectve rate, although t has a hgher stated rate. 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? Soluton Here A , 5.062, n 5 5, and P s unknown. Substtutng these values nto the formula for the compound amount gves P 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 $ , 5.062, and n 5 5. You should get A 5 $

9 5.1 Smple and Compound Interest 211 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 The present value of A dollars compounded at an nterest rate per perod for n perods s P 5 A n or P 5 A n. EXAMPLE 10 Present Value Fnd the present value of $16,000 n 9 years f money can be deposted at 6% compounded semannually. Soluton 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, 5.03, and n P 5 A ,000 n 18 < A depost of $ today, at 6% compounded semannually, wll produce a total of $16,000 n 9 years. We can solve the compound amount formula for n also, as the followng example shows. EXAMPLE 11 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. Soluton 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. ISBN: By tryng varous values of n, we fnd that n 5 9 s approxmately correct, because < 2. The exact value of n can be found quckly by usng logarthms, but that s beyond the scope of ths chapter. Thus, the overall level of prces wll double n about 9 years.

10 ISBN: Chapter 5 Mathematcs of Fnance 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 A 5 P11 1 rt2 P 5 A 1 1 rt P 5 Compound Interest A 5 P n A 5 A n n r e 5 a1 1 r m b m EXERCISES 1. What s the dfference between r and? between t and n? 2. 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? 3. What factors determne the amount of nterest earned on a fxed prncpal? 4. In your own words, descrbe the maturty value of a loan. 5. What s meant by the present value of money? Fnd the smple nterest. 6. $25,000 at 7% for 9 months 7. $3850 at 9% for 8 months 8. $1974 at 6.3% for 7 months 9. $3724 at 8.4% for 11 months Fnd the smple nterest. Assume a 360-day year. 10. $ at 10.1% for 58 days 11. $ at 11.9% for 123 days 12. Explan the dfference between smple nterest and compound nterest. 13. In Fgure 1, one graph s a straght lne and the other s curved. Explan why ths s, and whch represents each type of nterest. Fnd the compound amount for each depost. 14. $1000 at 6% compounded annually for 8 years 15. $1000 at 7% compounded annually for 10 years 16. $470 at 10% compounded semannually for 12 years 17. $15,000 at 6% compounded semannually for 11 years 18. $6500 at 12% compounded quarterly for 6 years 19. $9100 at 8% compounded quarterly for 4 years Fnd the amount that should be nvested now to accumulate the followng amounts, f the money s compounded as ndcated. 20. $15, at 9.8% compounded annually for 7 years 21. $27, at 12.3% compounded annually for 11 years

11 5.1 Smple and Compound Interest $2000 at 9% compounded semannually for 8 years 23. $2000 at 11% compounded semannually for 8 years 24. $8800 at 10% compounded quarterly for 5 years 25. $7500 at 12% compounded quarterly for 9 years 26. How do the nomnal or stated nterest rate and the effectve nterest rate dffer? 27. If nterest s compounded more than once per year, whch rate s hgher, the stated rate or the effectve rate? Fnd the effectve rate correspondng to each nomnal rate % compounded quarterly 29. 8% compounded quarterly % compounded semannually % compounded semannually Applcatons BUSINESS AND ECONOMICS 32. Loan Repayment Susan Carsten borrowed $25,900 from her father to start a flower shop. She repad hm after 11 mo, wth nterest of 8.4%. Fnd the total amount she repad. 33. Delnquent Taxes An accountant for a corporaton forgot to pay the frm s ncome tax of $725, on tme. The government charged a penalty of 12.7% nterest for the 34 days the money was late. Fnd the total amount (tax and penalty) that was pad. (Use a 365-day year.) 34. Savngs A $100,000 certfcate of depost held for 60 days was worth $101, To the nearest tenth of a percent, what nterest rate was earned? 35. Savngs A frm of accountants has ordered 7 new IBM computers at a cost of $5104 each. The machnes wll not be delvered for 7 months What amount could the frm depost n an account payng 6.42% to have enough to pay for the machnes? 36. 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 $.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.) 37. 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? 38. Loan Interest A small busness borrows $50,000 for expanson at 12% compounded monthly. The loan s due n 4 years. How much nterest wll the busness pay? 39. 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 Octo- ber 28, 1955, and on July 16, 1997, he was worth $42 bllon. (Note: A trllon dollars s 1000 bllon dollars.) 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 10.9% growth, the average return on all stocks snce 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. An artcle on September 19, 1999, gave Bll Gates s wealth as roughly $90 bllon. What was the rate of growth of hs wealth between the 1997 and 1999 artcles? 40. Loan Interest A developer needs $80,000 to buy land. He s able to borrow the money at 10% per year compounded quarterly. How much wll the nterest amount to f he pays off the loan n 5 years? 41. PayngOffaLawsut Acompanyhasagreedtopay$2.9mllon n 5 years to settle a lawsut. How much must they nvest now n an account payng 8% compounded monthly to have that amount when t s due? 42. Buyng a House George Duda wants to have $20,000 avalable n 5 years for a down payment on a house. He has nherted $15,000. How much of the nhertance should he nvest now to accumulate $20,000, f he can get an nterest rate of 8% compounded quarterly? ISBN: *The New York Tmes, July 20, 1997, Sec. 4, p. 2. The New York Tmes, Sept. 19, 1999, WK Rev., p. 2. To fnd the current net worth of Bll Gates, see

12 ISBN: Chapter 5 Mathematcs of Fnance 43. 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? 44. 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? 45. Comparng CD Rates A Vrgna bank offered the followng specal on CD (certfcate 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. Term 6 mo 1 yr 18 mo 2 yr 3 yr APY(%) Effectve Rate An advertsement for E*TRADE Bank boasted We re ahead of banks that had a 160-year start, wth an APY (or effectve rate) of 2.01%.* The actual rate was not stated. Gven that nterest was compounded monthly, fnd the actual rate. 47. Effectve Rate Accordng to a fnancal Web ste, on July 23, 2003, Countrywde Bank of Alexandra, Vrgna, pad 2.03% nterest, compounded daly, on a 1-year CD, whle New South Federal Savngs of Brmngham, Alabama, pad 2.05% compounded semannually. What are the effectve rates for the two CDs, and whch bank pays a hgher effectve rate? 48. 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. 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? Doublng Tme Use the deas from Example 11 to fnd the tme t would take for the general level of prces n the economy to double at each average annual nflaton rate % 50. 5% 51. 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. 52. 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. 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 55. 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 greatgreat-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. 56. 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? 57. Investments Sun Kang borrowed $5200 from hs frend Hop Fong Yee to pay for remodelng work on hs house. He repad the loan 10 months later wth smple nterest at 7%. Yee then nvested the proceeds n a 5-year certfcate of depost payng 6.3% compounded quarterly. How much wll *The New York Tmes, June 2, 2003, p. C5. The New York Tmes, Dec. 31, 1995, Sec. 3. p. 5. The New York Tmes, March 30, 1995.

13 5.2 Future Value of an Annuty 215 he have at the end of 5 years? (Hnt: You need to use both smple and compound nterest.) 58. 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) 59. Investments In Exercse 58, 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.* 60. Savngs On January 1, 1980, Jack deposted $1000 nto bank X to earn nterest at a rate of j per annum compounded semannually. On January 1, 1985, he transferred hs account to bank Y to earn nterest at the rate of k per annum compounded quarterly. On January 1, 1988, the balance of bank Y s $ If Jack could have earned nterest at the rate of k per annum compounded quarterly from January 1, 1980, through January 1, 1988, hs balance would have been $ Calculate the rato k/j. 5.2 FUTURE VALUE OF AN ANNUITY THINK ABOUT IT 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 or 3, 31222, , , , *, 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. Each term n the sequence s r tmes the precedng term. The number r s called the common rato of the sequence. EXAMPLE 1 Geometrc Sequence Fnd the seventh term of the geometrc sequence 5, 20, 80, 320, *. ISBN: *Problem 5 from Course 140 Examnaton, Mathematcs of Compound Interest of the Educaton and Examnaton Commttee of The Socety of Actuares. Reprnted by permsson of The Socety of Actuares.

14 ISBN: Chapter 5 Mathematcs of Fnance Soluton Here, a 5 5 and 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. ar n ,480 EXAMPLE 2 Geometrc Sequence Fnd the frst fve terms of the geometrc sequence wth a 5 10 and r 5 2. Soluton The frst fve terms are 10, 10122, , , , or 10, 20, 40, 80, 160. Next, we need to fnd the sum where If r 5 1, then of the frst n terms of a geometrc sequence, S n 5 a 1 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n21. If r 2 1, multply both sdes of equaton (1) by r to get Now subtract correspondng sdes of equaton (1) from equaton (2). rs n 5 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 1 r a1 r n S n 5 a1 rn r 2 1 Ths result s summarzed below. S n S n 5 a 1 a 1 a 1 a 1 ) 1 a 5 na. (''''''')'''''''* n terms rs n 5 ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n. ar 1 ar 2 1 ar 3 1 ar 4 1 ) 1 ar n21 1 ar n Factor Dvde both sdes by r 2 1. (1) (2) SUM OF TERMS If a geometrc sequence has frst term a and common rato r, then the sum the frst n terms s gven by S n 5 a1 r n 2 1 2, r u 1. r 2 1 S n of EXAMPLE 3 Sum of a Geometrc Sequence Fnd the sum of the frst sx terms of the geometrc sequence 3, 12, 48, *.

15 5.2 Future Value of an Annuty 217 Soluton Here a 5 3 and r 5 4. Fnd by the formula above. S 6 S n 5 6, a 5 3, r 5 4. 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, an expensve trp, 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 3 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 3 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 4 on the next page, the future value of the annuty s ISBN: (The last payment earns no nterest at all.)

16 ISBN: Chapter 5 Mathematcs of Fnance Year Depost $1500 $1500 $1500 $1500 $1500 $1500 $ (1.08) 1500(1.08) 1500(1.08) 1500(1.08) 1500(1.08) FIGURE 4 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 2 1 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 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 5 below), S 5 R n21 1 R n22 1 R n23 1 ) 1 R R, or, wrtten n reverse order, < $11, S 5 R 1 R R ) 1 R n21. Perod 1 2 n 1 Depost $R $R $R $R $R 3 n FIGURE 5 A depost of $R becomes R R(1 + ) n 3 R(1 + ) n 2 R(1 + ) n 1 R(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 R n 2 14 The quantty n brackets s commonly wrtten S 5 R. s n0. s n0 5 R c n 2 1 d. (read s-angle-n at ), so that

17 5.2 Future Value of an Annuty 219 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 ORDINARY ANNUITY S 5 R c n 2 1 d or S 5 Rs n0 where S s the future value; R s the payment; s the nterest rate per perod; n s the number of perods. A calculator wll be very helpful n computatons wth annutes. The TI- 83/84 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 Manual avalable for ths text. EXAMPLE 4 Ordnary Annuty Karen Scott 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? Soluton Her payments form an ordnary annuty, wth r 5 22,000, n 5 7, and The future value of ths annuty (by the formula above) s S 5 22,000 c d < 184,664.43,.06 or $184, Snkng Funds A fund set up to receve perodc payments as n Example 4 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. ISBN: *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 ISBN: Chapter 5 Mathematcs of Fnance 150,000 EXAMPLE 5 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. Sarah Santora, 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? Soluton Ths savngs plan s an annuty wth R 5 200, 5.072/12, and n The future value s S c / d < 106,752.47,.072/12 FIGURE 6 or $106, Fgure 6 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 5.072, shown at the bottom of the wndow, agrees wth our result above. (b) Sarah beleves she needs to accumulate $130,000 n the 20-year perod to have enough for retrement. What nterest rate would provde that amount? Method 1: Graphng Calculator Method 2: TVM Solver Soluton 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 6. 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 7 below. Usng the TVM Solver under the FINANCE menu on the TI-83/84 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 8, ndcates that an nterest rate of 8.79% s needed. 150, FIGURE 7 FIGURE 8

19 5.2 Future Value of an Annuty 221 EXAMPLE 6 Snkng Fund Suppose Sarah, n Example 5, 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? Soluton Start wth the annuty formula S 5 R c n 2 1 d. Solve for R by multplyng both sdes by / n ,000 R 5 S n 2 1 Now substtute S 5 130,000, 5.072/12, and n to fnd R. R , / / Sarah 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 9 shows the pont of ntersecton of the graphs of Y 1 5 X c / d.072/12 FIGURE 9 and Y ,000. The result agrees wth the answer we found above analytcally. The table shown n Fgure 9 confrms that the payment should be between $243 and $244. 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 10(a) (a) (b) ISBN: FIGURE 10

20 ISBN: Chapter 5 Mathematcs of Fnance shows the begnnng of such a table generated on a TI-83/84 Plus for Example 6. Fgure 10(b) shows the begnnng of the same table usng Mcrosoft Excel. 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. The fnance feature of the TI-83/84 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. EXAMPLE 7 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. Soluton In 7 years, there are n 5 28 quarterly perods. Add one perod to get n , and use the formula wth 5 6%/ %. S c d < 17, The account wll have a total of $17, after 7 years. 5.2 EXERCISES Fnd the ffth term of each geometrc sequence. 1. a 5 3; r a 5 5; r a 528; r a 526; r a 5 1; r a 5 12; r a ; r a 5 729; r Fnd the sum of the frst four terms for each geometrc sequence. 9. a 5 1; r a 5 3; r a 5 5; r a 5 6; r a 5 128; r a 5 81; r

21 5.2 Future Value of an Annuty 223 Fnd each value. 15. s s Lst some reasons for establshng a snkng fund. 20. Explan the dfference between an ordnary annuty and an annuty due. Fnd the future value of each ordnary annuty. Interest s compounded annually. 21. R 5 100; 5.06; n R ; 5.06; n R 5 46,000; 5.063; n R 5 29,500; 5.058; n 5 15 Fnd the future value of each ordnary annuty, f payments are made and nterest s compounded as gven. 25. R ; 10% nterest compounded semannually for 7 years 26. R ; 8% nterest compounded semannually for 11 years 27. R 5 800; 6.51% nterest compounded semannually for 12 years 28. R ; 8.73% nterest compounded quarterly for 9 years 29. R 5 15,000; 12.1% nterest compounded quarterly for 6 years 30. R 5 42,000; 10.05% nterest compounded semannually for 12 years Fnd the future value of each annuty due. Assume that nterest s compounded annually. 31. R 5 600; 5.06; n R ; 5.08; n R 5 20,000; 5.08; n R ; 5.06; n 5 11 Fnd the future value of each annuty due. 35. Payments of $1000 made at the begnnng of each semannual perod for 9 years at 8.15% compounded semannually 36. $750 deposted at the begnnng of each month for 15 years at 5.9% compounded monthly 37. $100 deposted at the begnnng of each quarter for 9 years at 12.4% compounded quarterly 38. $1500 deposted at the begnnng of each semannual perod for 11 years at 5.6% compounded semannually Fnd the perodc payment that wll amount to each gven sum under the gven condtons. 39. S 5 $10,000; nterest s 5% compounded annually; payments are made at the end of each year for 12 years 40. S 5 $100,000; nterest s 8% compounded semannually; payments are made at the end of each semannual perod for 9 years 41. What s meant by a snkng fund? Gve an example of a snkng fund. 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. 42. $8500; money earns 8% compounded annually; 7 annual payments 43. $2000; money earns 6% compounded annually; 5 annual payments $75,000; money earns 6% compounded semannually for 4 2 years $25,000; money earns 5.7% compounded quarterly for 3 2 years $50,000; money earns 7.9% compounded quarterly for 2 2 years $9000; money earns 12.23% compounded monthly for 2 years 2 s s ISBN:

22 ISBN: Chapter 5 Mathematcs of Fnance Applcatons BUSINESS AND ECONOMICS 48. Comparng Accounts Alex Leverng 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. Alex 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? c. How much would Alex lose over 9 years by usng her brother-n-law s bank? 49. Savngs Tom De Marco s savng for a computer. At the end of each month he puts $60 n a savngs account that pays 8% nterest compounded monthly. How much s n the account after 3 years? 50. Savngs Hass s pad on the frst day of the month and $80 s automatcally deducted from hs pay and deposted n a savngs account. If the account pays 7.5% nterest compounded monthly, how much wll be n the account after 3 years and 9 months? 51. Savngs A typcal pack-a-day smoker spends about $55 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? 52. 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 twenty-frst brthday. If the account pays 9.5% nterest compounded annually, how much s n the account at the end of the day on the daughter s twenty-frst brthday? 53. Retrement Plannng A 45-year-old man puts $1000 n a retrement account at the end of each quarter untl he reaches the age of 60 and makes no further deposts. If the account pays 8% nterest compounded quarterly, how much wll be n the account when the man retres at age 65? 54. Retrement Plannng At the end of each quarter a 50-yearold woman puts $1200 n a retrement account that pays 7% nterest compounded quarterly. When she reaches age 60, she wthdraws the entre amount and places t n a mutual fund that pays 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? 55. Savngs Jasspreet Kaur 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. 56. Savngs Chuck Hckman deposts $10,000 at the begnnng of each year for 12 years n an account payng 5% compounded annually. He then puts the total amount on depost n another account payng 6% compounded semannually for another 9 years. Fnd the fnal amount on depost after the entre 21-year perod. 57. 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. 58. 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? 59. Buyng a Car Susan Laferrere wants to buy an $18,000 car n 6 years. How much money must she depost at the end of each quarter n an account payng 5% compounded quarterly so that she wll have enough to pay for her car? Indvdual Retrement Accounts Suppose a 40-year-old person deposts $2000 per year n an Indvdual Retrement Account untl age 65. Fnd the total n the account wth the followng assumptons of nterest rates. (Assume semannual compoundng, wth payments of $1000 made at the end of each semannual perod.) 60. 6% 61. 8% 62. 4% %

23 5.3 Present Value of an Annuty; Amortzaton 225 In Exercses 64 and 65, use a graphng calculator to fnd the value of that produces the gven value of S. (See Example 5(b).) 64. 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? 65. Rate of Return Jennfer Wall 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? 66. 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. 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%? 67. Buyng Real Estate Marsa Raffaele 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. 68. Buyng Rare Stamps Paul Alter 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. 69. Down Payment A conventonal loan, such as for a car or a house, s smlar to an annuty, but usually ncludes a down 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. 5.3 PRESENT VALUE OF AN ANNUITY; AMORTIZATION THINK ABOUT IT What monthly payment wll pay off a $10,000 car loan n 36 monthly payments at 12% annual nterest? The answer to ths queston s gven later 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: ISBN: *The Washngton Post, March 10, 1992, p. A1.

24 ISBN: Chapter 5 Mathematcs of Fnance FOR REVIEW Recall that for any nonzero number a, a Also, by the product rule for exponents, a x. a y 5 a x1y. In partcular, for any nonzero number a, a n. a 2n 5 a n112n2 5 a 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 below. PRESENT VALUE OF AN ANNUITY 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 n P 5 R c 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. The fnancal feature of the TI-83/84 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 Manual that s avalable wth ths book. EXAMPLE 1 Present Value of an Annuty Mr. Bryer and Ms. Gonsalez are both graduates of the Brsbane Insttute of Technology. They both agree to contrbute to the endowment fund of BIT. Mr. Bryer

25 5.3 Present Value of an Annuty; Amortzaton 227 says that he wll gve $500 at the end of each year for 9 years. Ms. Gonsalez prefers to gve a lump sum today. What lump sum can she gve that wll equal the present value of Mr. Bryer s annual gfts, f the endowment fund earns 7.5% compounded annually? Soluton Here, R 5 500, n 5 9, and 5.075, and we have P 5 R. a c d < Therefore, Ms. Gonsalez must donate a lump sum of $ today. 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. EXAMPLE 2 Car Payments A car costs $12,000. After a down payment of $2000, the balance wll be pad off n 36 equal monthly payments wth nterest of 12% per year on the unpad balance. Fnd the amount of each payment. Soluton A sngle lump sum payment of $10,000 today would pay off the loan. So, $10,000 s the present value of an annuty of 36 monthly payments wth nterest of 12%/12 5 1% per month. Thus, P 5 10,000, n 5 36, 5.01, and we must fnd the monthly payment R n the formula n P 5 R c d 10,000 5 R c d.01 R < A monthly payment of $ wll be needed. 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.01 1$10, $100 and s dvded as follows. monthly nterest to reduce payment due the balance $ $100 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). ISBN: 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 $10,000 at 12% 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.

26 ISBN: Chapter 5 Mathematcs of Fnance AMORTIZATION 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 5 a n0 P n c d P n. EXAMPLE 3 Home Mortgage The Perez famly buys a house for $94,000 wth a down payment of $16,000. They take out a 30-year mortgage for $78,000 at an annual nterest rate of 9.6%. (a) Fnd the amount of the monthly payment needed to amortze ths loan. Soluton Here P 5 78,000 and the monthly nterest rate s * The number of monthly payments s %/ / Therefore, R 5 78,000 5 a , < c d.008 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. Soluton The Perez famly makes 360 payments of $ each, for a total of $238, Snce the amount of the loan was $78,000, the total nterest pad s $238, $78,000 5 $160, 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. Soluton Durng the frst month, the entre $78,000 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 78, At the end of the month, a payment of $ s made; snce $624 of ths s nterest, a total of $ $624 5 $37.56 s appled to the reducton of the orgnal debt. *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.

27 5.3 Present Value of an Annuty; Amortzaton 229 It can be shown that the unpad balance after x payments s approxmately gven by the functon n2x2 y 5 R c d. 80,000 For example, the unpad balance n Example 3 after 1 payment s y 5 $ c d < $77, Ths s very close to the amount left after deductng the $37.56 appled to the loan n part (c): $78,000 2 $ $77, FIGURE 11 A calculator graph of ths functon s shown n Fgure 11. 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 wndow n Fgure 12(a). You may be surprsed that the remanng balance on a $78,000 loan s as large as $75, 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 11/2278, ,000 to the fgure, we can fnd when half the loan has been repad. From Fgure 12(b) we see that 280 payments are requred. Note that only 80 payments reman at that pont, whch agan emphaszes the fact that the earler payments do lttle to reduce the loan. 80,000 80, (a) FIGURE 12 (b) 360 Amortzaton Schedules In the precedng example, 360 payments are made to amortze a $78,000 loan. The loan balance after the frst payment s reduced by only $37.56, whch s much less than 11/ ,0002 < $ 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. ISBN: EXAMPLE 4 Early Payment Susan Dratch borrows $1000 for 1 year at 12% annual nterest compounded monthly. Verfy that her monthly loan payment s $ After makng three payments, she decdes to pay off the remanng balance all at once. How much must she pay?

28 ISBN: Chapter 5 Mathematcs of Fnance Soluton 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 Susan s remanng balance, computed by ths method, s $ 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 2 cents because the monthly payment and the other calculatons were rounded to the nearest penny. Although most people would not qubble about a dfference of 2 cents 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 Susan Dratch n Example 4 owes after three monthly payments. Soluton An amortzaton table for the loan s shown on the next page. 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 dffers slghtly from that obtaned by ether method n Example 4.

29 5.3 Present Value of an Annuty; Amortzaton 231 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 $.88 $ 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. A graphng calculator program to produce an amortzaton schedule s avalable n The Graphng Calculator Manual that s avalable wth ths book. The TI-83/84 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 13 shows an Excel amortzaton table for Example 5. For more detals, see The Spreadsheet Manual, also avalable wth ths book. ISBN: FIGURE 13

30 ISBN: Chapter 5 Mathematcs of Fnance 5.3 EXERCISES 1. Whch of the followng s represented by a n0? n n n n a. b. c. d. 2. Whch of the choces n Exercse 1 represents s n0? Fnd each value a a 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? Fnd the present value of each ordnary annuty. 8. Payments of $890 each year for 16 years at 8% compounded annually 9. Payments of $1400 each year for 8 years at 8% compounded annually 10. Payments of $10,000 semannually for 15 years at 10% compounded semannually 11. Payments of $50,000 quarterly for 10 years at 8% compounded quarterly 12. Payments of $15,806 quarterly for 3 years at 10.8% compounded quarterly 13. Payments of $18,579 every 6 months for 8 years at 9.4% 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 % compounded annually 15. 6% compounded annually 16. What does t mean to amortze a loan? Fnd the payment necessary to amortze each loan. 17. $2500; 8% compounded quarterly; 6 quarterly payments 18. $41,000; 10% compounded semannually; 10 semannual payments 19. $90,000; 8% compounded annually; 12 annual payments 20. $140,000; 12% compounded quarterly; 15 quarterly payments 21. $7400; 8.2% compounded semannually; 18 semannual payments 22. $5500; 12.5% compounded monthly; 24 monthly payments a Use the amortzaton table n Example 5 to answer the questons n Exercses How much of the fourth payment s nterest? 24. How much of the eleventh payment s used to reduce the debt? 25. How much nterest s pad n the frst 4 months of the loan? 26. How much nterest s pad n the last 4 months of the loan? 27. 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? a

31 5.3 Present Value of an Annuty; Amortzaton 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. 29. $149,560 at 7.75% for 25 years 30. $170,892 at 8.11% for 30 years 31. $153,762 at 8.45% for 30 years 32. $96,511 at 9.57% for 25 years Applcatons BUSINESS AND ECONOMICS 33. 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. 35. Car Payments Hong Le buys a car costng $6000. He agrees to make payments at the end of each monthly perod for 4 years. He pays 12% nterest, compounded monthly. a. What s the amount of each payment? b. Fnd the total amount of nterest Le wll pay. 36. Land Purchase A speculator agrees to pay $15,000 for a parcel of land; ths amount, wth nterest, wll be pad over 4 years, wth semannual payments, at an nterest rate of 10% compounded semannually. Fnd the amount of each payment. 37. New Car General Motors Summerdrve 2 the Max advertsng campagn pledged a cash-back allowance of $3500 or 0% fnancng for 60 months for a 2003 Pontac Sunbrd car.* a. Determne the payments on a Sunbrd f a person chooses the 0% fnancng opton and needs to fnance $13,500 for 60 months. b. Determne the payments on a Sunbrd f a person chooses the cash-back opton and now needs to fnance only $10,000. Assume that the buyer s able to fnd fnancng from a local bank at 5.9% for 60 months, compounded monthly. c. Dscuss whch deal s best and why. d. Fnd the nterest rate at the bank that would make the other opton optmal. 38. New Truck General Motors Summerdrve 2 the Max advertsng campagn pledged a cash-back allowance of $4000 or 0% fnancng for 60 months for a 2003 Chevrolet Avalanche pckup truck. a. Determne the payments on a Avalanche f a person chooses the 0% fnancng opton and needs to fnance $25,000 for 60 months. b. If a person purchases an Avalanche and chooses the cashback opton, she wll need to fnance $21,500. Assume that she s able to choose between two optons at her local bank, 4.5% for 48 months or 6.9% for 60 months. Fnd the monthly payment and the total amount of money that she wll pay back to the bank on each opton. c. Of the three deals, dscuss whch s best and why. 39. 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. * Ibd. Gould, Los, Tcket to Trouble, The New York Tmes Magazne, Aprl 23, 1995, p. 39. ISBN:

32 ISBN: Chapter 5 Mathematcs of Fnance 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. Student Loans Student borrowers now have more optons to choose from when selectng repayment plans.* The standard plan repays the loan n 10 years wth equal monthly payments. The extended plan allows from 12 to 30 years to repay the loan. A student borrows $35,000 at 7.43% compounded monthly. 40. Fnd the monthly payment and total nterest pad under the standard plan. 41. Fnd the monthly payment and total nterest pad under the extended plan wth 20 years to pay off the loan. Installment Buyng In Exercses 42 44, prepare an amortzaton schedule showng the frst four payments for each loan. 42. 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. 43. Large semtraler trucks cost $72,000 each. Ace Truckng buys such a truck and agrees to pay for t by a loan that wll be amortzed wth 9 semannual payments at 10% compounded semannually. 44. One retaler charges $1048 for a computer montor. A frm of tax accountants buys 8 of these montors. They make a down payment of $1200 and agree to amortze the balance wth monthly payments at 12% compounded monthly for 4 years. 45. 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. 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. 46. 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 12% compounded semannually. a. Fnd the amount of each payment. b. Refer to the text and Fgures 11 and 12. 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? 47. House Payments Kareem Adagbo buys a house for $285,000. He pays $60,000 down and takes out a mortgage at 9.5% 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 Fgures 11 and 12. When wll half the 20-year loan n part b be pad off? 48. Inhertance Sand Goldsten 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. 49. Chartable Trust The trustees of a college have accepted a gft of $150,000. The donor has drected the trustees to *Hansell, Saul, Money and College, The New York Tmes, Aprl 2, 1995, p. 28. The New York Tmes, Nov. 12, 1996, pp. A1, A22.

33 Chapter Summary 235 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 Prepare an amortzaton schedule for each loan. 50. A loan of $37, wth nterest at 8.5% compounded annually, to be pad wth equal annual payments over 10 years. 51. A loan of $ at 9.25% nterest compounded semannually, to be repad n 5 years n equal semannual payments. 52. 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 an annuty s gven by P 5 R. 53. Perpetuty Usng the result of Exercse 52, 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 CHAPTER SUMMARY Ths chapter ntroduces the mathematcs of fnance. Smple nterest s the startng pont; when nterest s earned on nterest prevously earned, we have compound nterest. In an annuty, money contnues to be deposted at regular ntervals, and compound nterest s earned on that money. In an ordnary annuty, 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 annuty; a fund s set up to receve perodc payments, so 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. Ths dea leads to an amortzaton table for a loan, whch shows the payments, broken down nto nterest and prncpal, for a loan to be pad back after a specfed tme. A Strategy for Solvng Fnance Problems ISBN: 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?

34 ISBN: Chapter 5 Mathematcs of Fnance 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 Interest Future Value Present Value Smple Interest I 5 Prt A 5 P11 1 rt2 P 5 A 1 1 rt Effectve Rate Compound Interest I 5 A 2 P A 5 P n A 2n P 5 5 A n r e 5 a1 1 r m b m 2 1 Ordnary Annuty Annuty Due Future Value Present Value Future Value S 5 R c n 2 1 d 5 R. s n n P 5 R c d 5 R. a n0 S 5 R c n d 2 R KEY TERMS 5.1 smple nterest prncpal rate tme future value maturty value compound nterest compound amount nomnal (stated) rate effectve rate present value 5.2 geometrc sequence terms common rato annuty ordnary 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

35 Chapter 5 Revew Exercses 237 CHAPTER 5 REVIEW EXERCISES Fnd the smple nterest for each loan. 1. $15,903 at 8% for 8 months 2. $4902 at 9.5% for 11 months 3. $42,368 at 5.22% for 5 months 4. $3478 at 7.4% for 88 days (assume a 360-day year) 5. 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. 6. $2800 at 6% compounded annually for 10 years 7. $19, at 12% compounded semannually for 7 years 8. $ at 6% compounded semannually for 16 years 9. $57, at 12% compounded quarterly for 5 years Fnd the amount of nterest earned by each depost. 10. $3954 at 8% compounded annually for 12 years 11. $12, at 10% compounded semannually for 7 years 12. $12, at 10.37% compounded quarterly for 29 quarters 13. $34, at 9.72% compounded monthly for 32 months 14. What s meant by the present value of an amount A? Fnd the present value of each amount. 15. $42,000 n 7 years, 12% compounded monthly 16. $17,650 n 4 years, 8% compounded quarterly 17. $ n 3.5 years, 6.77% compounded semannually 18. $ n 44 months, 5.93% compounded monthly 19. 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. ISBN: $500 deposted at the end of each 6-month perod for 8 years; money earns 6% compounded semannually 29. $1288 deposted at the end of each year for 14 years; money earns 8% compounded annually 30. $4000 deposted at the end of each quarter for 7 years; money earns 6% compounded quarterly 31. $233 deposted at the end of each month for 4 years; money earns 12% compounded monthly 32. $672 deposted at the begnnng of each quarter for 7 years; money earns 8% compounded quarterly

36 ISBN: Chapter 5 Mathematcs of Fnance 33. $11,900 deposted at the begnnng of each month for 13 months; money earns 12% compounded monthly 34. 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. (Recall, n a snkng fund, payments are made at the end of every nterest perod.) 35. $6500; money earns 8% compounded annually; 6 annual payments $57,000; money earns 6% compounded semannually for 8 2 years $233,188; money earns 9.7% compounded quarterly for 7 4 years $1,056,788; money earns 8.12% compounded monthly for 4 years Fnd the present value of each ordnary annuty. 39. Deposts of $850 annually for 4 years at 8% compounded annually 40. Deposts of $1500 quarterly for 7 years at 8% compounded quarterly 41. Payments of $4210 semannually for 8 years at 8.6% compounded semannually 42. Payments of $ monthly for 17 months at 9.4% compounded monthly 43. Gve two examples of the types of loans that are commonly amortzed. Fnd the amount of the payment necessary to amortze each loan. 44. $80,000; 8% compounded annually; 9 annual payments 45. $3200; 8% compounded quarterly; 10 quarterly payments 46. $32,000; 9.4% compounded quarterly; 17 quarterly payments 47. $51,607; 13.6% compounded monthly; 32 monthly payments Fnd the monthly house payments for each mortgage. 48. $56,890 at 10.74% for 25 years 49. $77,110 at 11.45% 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. 2 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, $ $ $ $125,464.39

37 Chapter 5 Revew Exercses 239 Use the table to answer the followng questons. 50. How much of the ffth payment s nterest? 51. How much of the twelfth payment s used to reduce the debt? 52. How much nterest s pad n the frst 3 months of the loan? 53. How much has the debt been reduced at the end of the frst year? Applcatons BUSINESS AND ECONOMICS 54. Personal Fnance Mchael Garbn owes $5800 to hs mother. He has agreed to repay the money n 10 months at an nterest rate of 10.3%. How much wll he owe n 10 months? How much nterest wll he pay? 55. Busness Fnancng John Remngton needs to borrow $9820 to buy new equpment for hs busness. The bank charges hm 12.1% for a 7-month loan. How much nterest wll he be charged? What amount must he pay n 7 months? 56. Busness Fnancng An accountant loans $28,000 at smple nterest to her busness. The loan s at 11.5% and earns $3255 nterest. Fnd the tme of the loan n months. 57. Busness Investment A developer deposts $84,720 for 7 months and earns $ n smple nterest. Fnd the nterest rate. 58. Personal Fnance In 3 years Joan McKee must pay a pledge of $7500 to her college s buldng fund. What lump sum can she depost today, at 10% compounded semannually, so that she wll have enough to pay the pledge? 59. 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? 60. 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? 61. Busness Investment A frm of attorneys deposts $5000 of proft-sharng money at the end of each semannual 1 perod for 7 2 years. Fnd the fnal amount n the account f the deposts earn 10% compounded semannually. Fnd the amount of nterest earned. 62. 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 1 quarter for 6 2 years at an nterest rate of 12% compounded quarterly. Fnd the amount of each payment. 63. 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 10%. Fnd the amount of each payment. 64. Personal Fnance To buy a new computer, Mark Nguyen borrows $3250 from a frend at 9% nterest compounded annually for 4 years. Fnd the compound amount he must pay back at the end of the 4 years. 65. Effectve Rate Accordng to a fnancal Web ste, on October 16, 2000, Guarantee Bank of Mlwaukee, Wsconsn, pad 6.90% nterest, compounded quarterly, on a 1-year CD, whle Captal Crossng Bank of Boston, Massachusetts, pad 6.88% compounded monthly. What are the effectve rates for the two CDs, and whch bank pays a hgher effectve rate? 66. Home Fnancng When the Lee famly bought ther home, they borrowed $115,700 at 10.5% 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.) 67. New Car Chrysler s Summer Sales Drve advertsng campagn pledged a cash back allowance of $1500 or 0% fnancng for 60 months for a 2003 PT Cruser. ISBN: * Pocket That Penson, Smart Money, Oct. 1994, p

38 ISBN: Chapter 5 Mathematcs of Fnance a. Determne the payments on a PT Cruser f a person chooses the 0% fnancng opton and needs to fnance $17,000 for 60 months. b. Determne the payments on a PT Cruser f a person chooses the cash back opton and now needs to fnance only $15,500. Assume that the buyer s able to fnd fnancng from a local bank at 6.5% for 60 months compounded monthly. c. Dscuss whch deal s best and why. d. Fnd the nterest rate at the bank that would make the other opton optmal. 68. New Van Chrysler s Summer Sales Drve advertsng campagn pledged a cash back allowance of $3500 or 0% fnancng for 60 months for a 2003 Town and Country van.* a. Determne the payments on a Town and Country van f a person chooses the 0% fnancng opton and needs to fnance $31,500 for 60 months. b. If a person purchases a Town and Country van and chooses the cash back opton, then she wll need to fnance $28,000. Assume that she s able to choose between two optons at her local bank, 4.9% for 48 months or 5.5% for 60 months. Fnd the monthly payment and the total amount of money that she wll pay back to the bank on each opton. c. Of the three deals, dscuss whch s the best and why. 69. Buyng and Sellng a House The Zambrano famly bought a house for $91,000. They pad $20,000 down and took out a 30-year mortgage for the balance at 9%. a. Fnd ther monthly payment. b. How much of the frst payment s nterest? After 180 payments, the famly sells ts house for $136,000. They must pay closng costs of $3700 plus 2.5% of the sale prce. 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. 70. 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. 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. 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. * Problem 16 from Course 140 Examnaton, Mathematcs of Compound Interest of the Educaton and Examnaton Commttee of The Socety of Actuares. Reprnted by permsson of The Socety of Actuares. The New York Tmes, Sept. 27, 1998, p. BU 10.

39 Extended Applcaton 241 EXTENDED APPLICATION: Tme, Money, and Polynomals* A tme 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 14. $6000 $5840 $4000 $5200 $29, t 0 t 1 t 2 t 3 t4 tme $3851 FIGURE 15 t 0 t 1 $1000 $2500 t 2 FIGURE 14 tme Assumng nterest s compounded annually at a rate, and usng the compound nterest formula, gves the followng descrpton of the YTM 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 Bll Poole 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 Poole would lke to know the YTM 5 for hs hard-earned retrement dollars. From a tme lne (see Fgure 15), we set up the followng equaton n 1 1 for Poole s savngs program. Let x We need to solve the fourth-degree polynomal equaton 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 f and f , Usng a graphng calculator, we fnd that there s one postve soluton to ths equaton, x , so 5 YTM %. Exercses , f 1x x x x x 229, Brenda Bravener 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, Jay Beckensten 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, Greg Odjakjan deposted bonuses of $1025, $2200, and $1850, respectvely, n an account. He receved no bonus the followng ISBN: *Copyrght COMAP Consortum COMAP, Inc. 57 Bedford Street #210, Lexngton, MA

40 ISBN: Chapter 5 Mathematcs of Fnance 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. 4. Pat Kelley 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. 5. People often lose money on nvestments. Jm Carlson nvested $50 at the begnnng of each of 2 years n a mutual fund, and at the end of 2 years hs 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. Drectons 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.