Mathematics of Finance


 Eleanor Rodgers
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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 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 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 360day 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 365day 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 160year 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 1year 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 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. 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 5year 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 sanglen 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 20year 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 TI83/84 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 TI83/84 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 TI83/84 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
0.02t if 0 t 3 δ t = 0.045 if 3 < t
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