2. Compute Compound Interest

Transcription

1 The Mathematcs of Fnance Careers and Mathematcs 9 In ths chapter, we wll dscuss the mathematcs of finance the rules that govern nvestng and borrowng money. 9.1 Interest Actuary Actuares use ther broad knowledge of statstcs, finance, and busness to desgn nsurance polces, penson plans, and other financal strateges, and ensure that these plans are mantaned on a sound financal bass. They assemble and analyze data to estmate the probablty and lkely cost of an event such as death, sckness, njury, dsablty, or loss of property. Most actuares are employed n the nsurance ndustry, specalzng n ether lfe and health nsurance or property and casualty nsurance. They produce probablty tables or use modelng technques that determne the lkelhood that a potental event wll generate a clam. From these, they estmate the amount a company can expect to pay n clams. Actuares ensure that the premums charged for such nsurance wll enable the company to cover clams and other expenses. 9.2 Annutes and Future Value 9.3 Present Value of an Annuty; Amortzaton Chapter Revew Chapter Test Cumulatve Revew Exercses Actuares held about 18,000 jobs n Educaton Actuares need a strong background n mathematcs and general busness. Actuares usually earn an undergraduate degree n mathematcs, statstcs, or actuaral scence, or a busness-related field such as finance, economcs or busness. Actuares must pass a seres of examnatons to gan full professonal status. Job Outlook Employment of actuares For a sample applcaton, see Example 3 n Secton 9.3. For more nformaton, see Alexander Walter/Getty Images s expected to ncrease by about 24 percent through Medan annual earnngs of actuares were $82,800 n

2 Chapter 9 The Mathematcs of Fnance 9.1 Interest Objectves Francsco Martnez/Alamy Compute Smple Interest Compute Compound Interest Borrow Money Usng Bank Notes Compute Effectve Rate of Interest Compute Present Value Wages, rent, and nterest are three common ways to earn money: A wage refers to money receved for lettng someone use your labor. Rent refers to money receved for lettng someone use your property, especally real estate. Interest refers to money receved for lettng someone use your money. Few people become wealthy by recevng wages. Unless you receve a large hourly rate of pay, there wll not be enough left after daly lvng expenses to amass true wealth. You wll have a better chance of becomng wealthy by supplementng wages wth rent. For example, f you borrow money to buy an apartment buldng, the rent receved from tenants can pay off the loan, and eventually you wll own the buldng wthout spendng your own money. Perhaps the easest way to buld wealth s to use money to earn nterest. If you can earn a good rate of nterest, compounded contnuously, and keep the nvestment for a long tme, t s amazng how large an nvestment can grow. In fact, t s sad that compound nterest s the eghth wonder of the world. In ths first secton, we wll dscuss ths mportant money-makng tool: nterest. When money s borrowed, the lender expects to be pad back the amount of the loan plus an addtonal charge for the use of the money. Ths addtonal charge s called nterest. When money s deposted n a bank, the bank pays the depostor for the use of the money. The money the depost earns s also called nterest. Interest can be computed n two ways: ether as smple nterest or as compound nterest. 1. Compute Smple Interest Smple nterest s computed by findng the product of the prncpal (the amount of money on depost), the rate of nterest (usually wrtten as a decmal), and the tme (usually expressed n years). Interest prncpal ⴢ rate ⴢ tme Ths word equaton suggests the followng formula. Smple Interest The smple nterest I earned on a prncpal P n an account payng an annual nterest rate r for a length of tme t s gven by the formula I Prt

3 9.1 Interest EXAMPLE 1 Soluton Fnd the smple nterest earned on a depost of $5,750 that s left on depost for 32 1 years and earns an annual nterest rate of 42%. 1 1 We wrte 32 and 42% as decmals and substtute the gven values n the formula for smple nterest. I Prt I 5,750 ⴢ ⴢ 3.5 I Ths s the formula for smple nterest. Substtute 5,750 for P, for r, and 3.5 for t. Perform the multplcatons. In 312 years, the account wll earn $ n smple nterest. Self Check 1 Fnd the smple nterest earned on a depost of $12,275 that s left on depost for 514 years and earns an annual nterest rate of 334%. EXAMPLE 2 Three years after nvestng $15,000, a retred couple receved a check for $3,375 n smple nterest. Fnd the annual nterest rate ther money earned durng that tme. Soluton The couple nvested $15,000 (the prncpal) for 3 years (the tme) and earned $3,375 (the smple nterest). We must find the annual nterest rate r. To do so, we substtute the gven numbers nto the smple nterest formula and solve for r. I Prt 3,375 15,000 ⴢ r ⴢ 3 3,375 45,000r 3,375 45,000 r 45,000 45, r r 7.5% Substtute 3,375 for I, 15,000 for P, and 3 for t. Multply. Dvde both sdes by 45,000. Perform the dvsons. Wrte as a percent. The couple receved an annual rate of 7.5% for the 3-year perod. Self Check 2 Fnd the length of tme t wll take for the nterest to grow to $9, Compute Compound Interest When nterest s left n an account and also earns nterest, we say that the account earns compound nterest. EXAMPLE 3 Soluton A woman deposts $10,000 n a savngs account payng 6% nterest, compounded annually. Fnd the balance n her account after each of the first three years. At the end of the first year, the nterest earned s 6% of the $10,000, or 0.06($10,000) $600 Ths nterest s added to the $10,000 to get a new balance. After one year, ths balance wll be $10,600. The second year s earned nterest s 6% of $10,600, or 0.06($10,600) $636 Ths nterest s added to $10,600, gvng a second-year balance of $11,236.

4 Chapter 9 The Mathematcs of Fnance The nterest earned durng the thrd year s 6% of $11,236, or 0.06($11,236) $ Ths nterest s added to $11,236 to gve the woman a balance of $11,910.16, after three years. Self Check 3 Fnd the balance n the woman s account after two more years. We can generalze the method used n Example 3 to fnd a formula for compound nterest calculatons. Suppose that the orgnal depost n the account s A 0 dollars, that nterest s pad at an annual rate r, and that the accumulated amount or the future value n the account at the end of the frst year s A 1. Then the nterest earned that year s A 0 r, and The amount after one year A 1 A 0 A 0 r A 0 (1 r) equals the orgnal depost Factor out the common factor,. plus The amount, A 1, at the end of the frst year s the balance n the account at the begnnng of the second year. So, the amount at the end of the second year,, s A 0 the nterest earned on the orgnal depost. A 2 The amount after two years equals plus A 2 A 1 A 1 r A 1 (1 r) Factor out the common factor, A 1. A 0 (1 r)(1 r) Substtute A 0 (1 r) for A 1. A 0 (1 r) 2 Smplfy. By the end of the thrd year, the amount wll be A 3 A 0 (1 r) 3 the amount after one year The pattern contnues wth the followng result. the nterest earned on the amount after one year. Compound Interest (Annual Compoundng) A sngle depost A 0, earnng compound nterest for n years at an annual rate r, wll grow to a future value accordng to the formula A n A 0 (1 r) n A n EXAMPLE 4 For ther newborn chld, parents depost $10,000 n a college account that pays 8% nterest, compounded annually. How much wll be n the account on the chld s 17th brthday? Soluton We substtute A 0 10,000, r 0.08, and n 17 nto the compound nterest formula to fnd the future value A 17. A n A 0 (1 r) n A 17 10,000(1 0.08) 17 10,000(1.08) 17 37, Use a calculator.

5 9.1 Interest To the nearest cent, $37, wll be avalable on the chld s 17th brthday. Self Check 4 If the parents leave the money on depost for two more years, what amount wll be avalable? Interest compounded once each year s compounded annually. Many fnancal nsttutons compound nterest more often. For example, nstead of payng an annual rate of 8% once a year, a bank mght pay 4% twce each year, or 2% four tmes each year. The annual rate, 8%, s also called the nomnal rate, and the tme between nterest calculatons s called the converson perod. If there are k perods each year, nterest s pad at the perodc rate gven by the followng formula. Perodc Rate Perodc rate Ths formula s often wrtten as annual rate number of perods per year r k where s the perodc nterest rate, r s the annual rate, and k s the number of tmes nterest s pad each year. If nterest s calculated k tmes each year, n n years there wll be kn conversons. Each converson s at the perodc rate. Ths leads to another form of the compound nterest formula. Compound Interest Formula An amount A 0, earnng nterest compounded k tmes a year for n years at an annual rate r, wll grow to the future value accordng to the formula A n A 0 (1 ) kn A n where r s the perodc nterest rate. k Interest pad twce each year s called semannual compoundng, four tmes each year quarterly compoundng, twelve tmes each year monthly compoundng, and 360 or 365 tmes each year daly compoundng. EXAMPLE 5 If the parents of Example 4 nvested that $10,000 n an account payng 8%, compounded quarterly, how much more money would they have after 17 years? Soluton We frst calculate the perodc rate,. r k Substtute r 0.08 and k 4. We then substtute A 0 10,000, 0.02, k 4, and n 17 nto the compound nterest formula.

6 Chapter 9 The Mathematcs of Fnance A n A 0 (1 ) kn A 17 10,000(1 0.02) ,000(1.02) 68 38, Use a calculator. To the nearest cent, $38, wll be avalable, an ncrease of $1, over annual compoundng. Self Check 5 a. What would $10,000 become n 17 years f compounded monthly at a nomnal rate of 8%? b. How does ths compare wth quarterly compoundng? Accent on Technology Growth of Money We can use a graphng calculator to fnd the tme t would take a $10,000 nvestment to trple, assumng an 8% annual rate, compounded quarterly. In n years, $10,000 earnng 8% nterest, compounded quarterly, wll become the future value 10,000(1.02) 4n To watch ths value grow, we enter the functon Y *1.02 (4*X) n a graphng calculator, and set the wndow to 0 X 10 (for 10 years) and 0 Y (for the dollar amount). The graph appears n Fgure 9-1(a). To fnd the tme t would take for the nvestment to trple, we use TRACE to move to the pont wth a Y-value close to 30,000. The X-value n Fgure 9-1(b) shows that the nvestment would trple n about 13.9 years. Y1 = ^(4 X) X = Y = (a) Fgure 9-1 (b) 3. Borrow Money Usng Bank Notes When a customer borrows money from a bank, the bank s makng an nvestment n that person. The amount of the loan s the bank s depost, and the bank expects to be repad wth nterest n a sngle balloon payment at a later date. These loans, called notes, are based on a 360-day year, and they are usually wrtten for 30 days, 90 days, or 180 days. We use the formula for compound nterest to calculate the terms of the loan.

7 9.1 Interest EXAMPLE 6 Soluton Self Check 6 A student needs $4,000 for tuton. If hs bank wrtes a 9%, 180-day note, wth nterest compounded daly, what wll he owe at the end of 180 days? In grantng the loan, the bank nvests $4,000. The amount to be repad s the expected future value A n A 0 (1 ) kn where A 0 s $4,000, the frequency of compoundng k s 360, the perodc rate s , and the term n s 0.5 (180 days s one-half of 360 days). To determne what the student wll owe, we substtute these numbers nto the compound nterest formula and solve for A n. A n A 0 (1 ) kn A 0.5 4,000( ) ,000( ) 180 4, Use a calculator. 4, Round to the nearest cent. The student must repay $4, A woman borrows $7,500 for 90 days at 12%. If nterest s compounded daly, how much wll she owe at the end of 90 days? 4. Compute Effectve Rate of Interest The true performance of an nvestment depends on both the frequency of compoundng and the annual rate. To help nvestors compare dfferent savngs plans, fnancal nsttutons are requred by law to provde the effectve rate the rate that, f compounded annually, would provde the same yeld as a plan that s compounded more frequently. To derve a formula for an effectve rate, we assume that A 0 dollars are nvested for n years at an annual rate r, compounded k tmes per year. That same nvestment of A 0 dollars, compounded annually at the effectve rate R, would produce the same accumulated value. Snce these amounts are to be equal, we have the equaton Accumulated amount at effectve rate R, compounded annually equals accumulated amount at annual rate r, compounded k tmes per year. 0 A 0 (1 R) n A 0 (1 ) kn s the perodc rate, r k. We can solve ths equaton for R. A 0 (1 R) n A 0 (1 ) kn (1 R) n (1 ) kn Dvde both sdes by A. [(1 R) n ] 1n [(1 ) kn ] 1n Rase both sdes to the 1/n power. 1 R (1 ) k Multply the exponents. R (1 ) k 1 Subtract 1 from both sdes. Ths result establshes the followng formula.

8 Chapter 9 The Mathematcs of Fnance Effectve Rate of Interest The effectve rate of nterest R for an account payng a nomnal rate r, compounded k tmes per year, s R (1 ) k 1 where s the perodc rate, r. k EXAMPLE 7 A bank offers the savngs plans shown n the table. Calculate the effectve nterest rates for each nvestment. a. Money b. Certfcate market fund of depost Annual rate Compoundng Effectve rate 6.5% 7% quarterly monthly Soluton a. For the money market fund, r and k 4, so r k To fnd the effectve rate, we substtute k 4 and n the formula for effectve rate. Use a calculator. Round to the nearest ten thousandth. As a percent, the effectve rate s 6.66%, or approxmately %. b. For the certfcate of depost, r 0.07 and k 12, so and R (1 ) k 1 R ( ) R ( ) Use a calculator Round to the nearest ten thousandth. As a percent, the effectve rate s 7.23%. Self Check 7 A passbook savngs account offers daly compoundng (365 days per year) at an annual rate of 6%. Fnd the effectve rate to the nearest hundredth. 5. Compute Present Value For an ntal depost A 0, the compound nterest formula gves the future value A n of the account after n years. Ths s the stuaton suggested by Fgure 9-2, where we know the begnnng amount and need to fnd ts future value.

9 9.1 Interest S n years? Known prncpal A o Unknown future value A n Fgure 9-2 Often, the stuaton s reversed: We need to make a depost now that wll become a specfc amount several years from now perhaps enough to buy a car or pay tuton. As Fgure 9-3 suggests, we need to know what sngle depost now wll accomplsh that goal: What present value wll yeld a specfc future value? A 0 A n? n years S Unknown present value A o Fgure 9-3 Known future value A n To derve the formula for present value, we solve the compound nterest formula for. A 0 A n A 0 (1 ) kn The orgnal depost s the present value, A 0. A n Dvde both sdes by (1 ) kn. (1 ) A 0(1 ) kn kn (1 ) kn A n (1 ) kn Dvde on both sdes: 1. kn (1 ) A kn 0 (1 ) 1 A n (1 ) kn A 0 Use the defnton of negatve exponent: x a ax. Ths result establshes the followng formula. Present Value The present value A 0n s the amount that must be deposted now to provde a future value A n after years s gven by the formula A 0 A n (1 ) kn where nterest s compounded k tmes per year at an annual rate r 1 s the r perodc rate,. k2 EXAMPLE 8 When a medcal student graduates n 8 years, she wll need $25,700 to buy furnture for her medcal offce. What amount must she depost now (at 8%, compounded twce per year) to meet ths future oblgaton? Soluton Use the annual rate (r 0.08) and the frequency of compoundng (k 2) to fnd the perodc rate: r k In the present value formula, we substtute

10 Chapter 9 The Mathematcs of Fnance the number of years, n 8, the perodc rate, 0.04, the frequency of compoundng, k 2, and the future value n 8 years, A 8 25,700. A 0 A n (1 ) kn A 0 25,700(1 0.04) 28 A n A 8 25,700 25,700(1.04) 16 Use a calculator. 13, Round to the nearest cent. She must depost $13, now to have $25,700 n 8 years. Self Check 8 If the student decdes to take two extra years to complete medcal school, her oblgaton wll be $27,000. What present value wll meet her goal? Self Check Answers 1. $2, yr 3. $13, $43, a. $38, b. $ more than wth quarterly compoundng 6. $7, % 8. $12, Exercses Vocabulary and Concepts Fll n the blanks. 1. A bank pays for the prvlege of usng your money. 2. If nterest s left on depost to earn more nterest, the account earns nterest. 3. Interest compounded once each year s called compoundng. 4. The ntal depost s called the or the value. 5. After a specfc tme, the prncpal grows to a value. 6. Interest s calculated as a of the amount on depost. 7. Future value prncpal earned 8. The annual rate also s called the rate. annual rate 9. number of perods per year 10. The tme between nterest calculatons s the perod. 11. In the future value formula A n A 0 (1 ) kn, A 0 s the, s the, k s the, and n s the. 12. In the present value formula A 0 A n (1 ) kn, A n s the, s the, k s the, and n s the. 13. To help consumers compare savngs plans, banks advertse the rate of nterest. 14. If after one year, $100 grows to $110, the effectve rate s %. Practce 15. Fnd the smple nterest earned n an account where $4,500 s on depost for 4 years at % annual nterest. 16. Fnd the smple nterest earned n an account where $12,400 s on depost for 81 years at % annual nterest. 17. Fnd the prncpal necessary to earn $814 n smple nterest f the money s to be left on depost for 4 years and earns % annual nterest. 18. Fnd the tme necessary for a depost of $11,500 to earn $3,450 n smple nterest f the money s to earn % annual nterest.

11 9.1 Interest Fnd the annual rate necessary for a depost of $50,000 to earn $7,500 n smple nterest f the money s to be left on depost for years. 20. Fnd the tme necessary for a depost of $5,000 to double n an account payng % smple nterest. Assume that $1,200 s deposted n an account n whch nterest s compounded annually at a rate of 8%. Fnd the accumulated amount after the gven number of years year years years years Assume that $1,200 s deposted n an account n whch nterest s compounded annually at the gven rate. Fnd the accumulated amount after 10 years % 26. 5% 27. 9% % Assume that $1,200 s deposted n an account n whch nterest s compounded at the gven frequency, at an annual rate of 6%. Fnd the accumulated amount after 15 years. 29. k k k k 365 Fnd the effectve nterest rate wth the gven annual rate r and compoundng frequency k. 33. r 6%, k r 8%, k r %, k r 10%, k 360 Fnd the present value of $20,000 due n 6 years, at the gven annual rate and compoundng frequency %, semannually 38. 8%, quarterly 39. 9%, monthly 40. 7%, daly (360 days/year) Applcatons 41. Small busness To start a moble dog-groomng servce, a woman borrowed $2,500. If the loan was for 2 years and the amount of nterest was $175, what smple nterest rate was she charged? 42. Bankng Three years after openng an account that pad 6.45% smple nterest, a depostor wthdrew the $3,483 n nterest earned. How much money was left n the account? 43. Savng for college At the brth of ther chld, the Feldsons deposted $7,000 n an account payng 6% nterest, compounded quarterly. How much wll be avalable when the chld turns 18? 44. Plannng a celebraton When the Fernandez famly made reservatons at the end of 2008 for the December 2014 New Year s celebraton n Pars, they placed $5,700 nto an account payng 8% nterest, compounded monthly. What amount wll be avalable at the tme of the celebraton? 45. Plannng for retrement When Jm retres n 12 years, he expects to lve lavshly on the money n a retrement account that s earnng % nterest, compounded semannually. If the account now contans $147,500, how much wll be avalable at retrement? 46. Penson fund management The managers of a penson fund nvested $3 mllon n government bonds payng 8.73% annual nterest, compounded semannually. After 8 years, what wll the nvestment be worth? 47. Real estate nvestng Property values n the suburbs have been apprecatng about 11% annually. If ths trend contnues, what wll a $137,000 home be worth n four years? Gve the result to the nearest dollar. 48. Real estate nvestng Property n suburbs closer to the cty s apprecatng about 8.5% annually. If ths trend contnues, what wll a $47,000 one-acre lot be worth n fve years? Gve the result to the nearest dollar. 49. Gas consumpton The gas utltes expect natural gas consumpton to ncrease at 7.2% per year for the next decade. Monthly consumpton for one county s currently 4.3 mllon cubc feet. What monthly demand for gas s expected n ten years? 50. Comparng banks Bank One offers a passbook account wth 4.35% annual rate, compounded quarterly. Bank Two offers a money market account at 4.3%, compounded monthly. Whch account provdes the better growth? (Hnt: Fnd the effectve rates.)

12 Chapter 9 The Mathematcs of Fnance 51. Comparng accounts A savngs and loan offers the two accounts shown n the table. Fnd the effectve rates. Annual Effectve rate Compoundng rate NOW 7.2% quarterly account Money 6.9% monthly market 52. Comparng accounts A credt unon offers the two accounts shown n the table. Fnd the effectve rates. Annual Effectve rate Compoundng rate Certfcate 6.2% semannually of depost Passbook 5.25% quarterly 53. Car repar Crag borrows $1,230 for unexpected car repar costs. Hs bank wrtes a 90-day note at 12%, wth nterest compounded daly. What wll Crag owe? 54. Fly now, pay later For a 7-day Hawa vacaton, Beth borrowed $2,570 for 9 months at an annual rate of 11.4%, compounded monthly. What dd she owe? 55. Buyng a computer A man estmates that the computer he plans to buy n 18 months wll cost $4,200. To meet ths goal, how much should he depost n an account payng 5.75%, compounded monthly? 56. Buyng a coper An accountng frm plans to depost enough money now n an account payng 7.6% nterest, compounded quarterly, to fnance the purchase of a $2,780 coper n 18 months. What should be the amount of that depost? Dscovery and Wrtng 57. Addng to an nvestment To prepare for hs retrement n 14 years, Jay deposted $12,000 n an account payng 7.5% annual nterest, compounded monthly. Ten years later, he deposted another $12,000. How much wll be avalable at retrement? 58. Changng rates Ten years ago, a man nvested $1,100 n a 5-year certfcate of depost payng 10%, compounded monthly. When the CD matured, he nvested the proceeds n another 5-year CD payng 8%, compounded semannually. How much s avalable now? 59. The power of tme A young person s most powerful money-makng scheme, sad an nvestment advsor, s tme. Wrte a paragraph explanng what the advsor meant. 60. Watchng money grow $10,000 s nvested at 10%, compounded annually. Use a graphng calculator to fnd how long t wll take for the accumulated value to exceed $1 mllon. 61. Explan why the compound nterest formula on page 727 s equvalent to the one on page Explan why the present value formula on page 731 s equvalent to the compound nterest formula on page 727. Revew Smplfy each expresson. Assume that all varables represent postve numbers. x 2 2x x 2 9x x(x 2 5) (x 3 2x) (3 x)(x 3) 65. x x 2 6x 9 (x 3)

13 9.2 Annutes and Future Value Annutes and Future Value Corbs Super RF/Alamy Objectves 1. Fnd the Future Value of an Annuty 2. Work wth Snkng Funds Now that we understand how nterest works, we wll dscuss how to use nterest to buld wealth to pay for retrement, a vacaton, or some other specal event. Two nstruments for dong ths are annutes and snkng funds. 1. Fnd the Future Value of an Annuty Fnancal plans that nvolve a seres of payments are called annutes. Monthly mortgage payments, for example, are part of an annuty, as are regular contrbutons to a retrement plan. Annuty A plan nvolvng payments made at regular ntervals s called an annuty. Future Value The future value of an annuty s the sum of all the payments and the nterest those payments earn. Term The tme over whch the payments are made s called the term of the annuty. Ordnary Annuty In an ordnary annuty, the payments are made at the end of each tme nterval. In ths book, we wll consder only ordnary annutes wth equal perodc payments made for a fixed term. To understand how an annuty works, assume that a savngs account pays 12% 12% annual nterest, compounded monthly. Its perodc rate s 12, or 1%. Also assume that each month for the next year, $100 wll be deposted n that account. To determne the future value of ths annuty, we thnk of each monthly payment as a one-tme ntal contrbuton to a compound-nterest savngs account. The future value of the annuty s the sum of the accumulated values of 12 ndvdual accounts. As Fgure 9-4 suggests, the first depost earns nterest for 11 months, the second for 10 months, and so on. Because the last depost s made at the end of the year, t earns no nterest. A few mnute s work wth a calculator wll show that the total value of the account s $1,

14 Chapter 9 The Mathematcs of Fnance Contrbuton made at the end of month number: $100 $100 $100 $100 $100 $100 $100 $100 $100 $100 $100 $ (1.01) month 100(1.01) months 100(1.01) months 100(1.01) Fgure months 100(1.01) months 100(1.01) months 100(1.01) Value at end of year 1, Future value of the annuty We wll now generalze ths example to derve a formula for the future value of an annuty. Consder an annuty wth regular deposts of P dollars and wth nterest compounded k tmes per year for n years, at an annual rate r 1perodc rate k2 r. Durng n years, there wll be kn perods and kn deposts. The last depost, made at the end of the last perod, earns no nterest. The next-to-last depost earns nterest for one perod, and so on. The frst depost, made at the end of the frst perod, earns compound nterest for kn 1 perods. future value of the last depost A n P P(1 ) 1 P(1 ) 2 P(1 ) 3 p P(1 ) kn1 Ths s a geometrc sequence wth frst term P and a common rato of (1 ). Its sum s gven by A n P[(1 )kn 1] We summarze ths result. future value of the next-to-last depost p future value of the frst depost. Recall that the sum of the terms of the geometrc sequence s S n a(rn 1) S n a ar ar 2 ar 3 p ar n1. r 1 Future Value of an Annuty The future value A n of an ordnary annuty wth deposts of P dollars made regularly k tmes each year for n years, wth nterest compounded k tmes per year at an annual rate r, s A n P[(1 )kn 1] where s the perodc rate, r. k EXAMPLE 1 Verfy the future value of the annuty outlned n Fgure 9-4. Soluton From Fgure 9-4, we fnd the term, n years: n 1 the frequency of compoundng: k 12

15 Annutes and Future Value 737 the annual rate: the regular depost: We then calculate the perodc nterest rate: r k , and substtute these numbers n the formula for the future value of an annuty. A n P[(1 )kn 1] r 0.12 P A 1 100[(1 0.01)121 1] [(1.01)12 1] , Use a calculator. 1, Round to the nearest cent. The annuty of Fgure 9-4 wll provde $1, by the end of the year. Self Check 1 Under the payroll savngs plan, a student contrbutes $50 a month to an ordnary annuty payng % annual nterest, compounded monthly. How much wll he have n 5 years? Accent on Technology Growth of Money The value of an annuty grows rapdly. To watch the value of the annuty of Example 1 grow, we graph the functon A n 100[(1.01)12n 1] ,000(1.0112n 1) We enter the functon Y *(1.01 (12*X) 1) ,000 on a graphng calculator, set the wndow values to be 0 X 50 and 0 Y 1,000,000, and graph t, as n Fgure 9-5(a). Note how the amount ncreases more rapdly as the years go by. Usng TRACE, as n Fgure 9-5(b), we see that over $1 mllon accumulates n just 39 years. Y1 = (1.01^(12 X) 1_ X = Y = (a) Fgure 9-5 (b) 2. Work wth Snkng Funds If we know the amount of each depost, we can calculate the future value of an annuty usng the formula on page 736. Ths stuaton s often reversed: What

16 Chapter 9 The Mathematcs of Fnance regular deposts, made over tme wll provde a specfc future amount? An annuty created to produce a fxed future value s called a snkng fund. To determne the requred perodc payment P, we solve the future value formula for P. A n A n P[(1 )kn 1] (1 ) kn 1 P[(1 )kn 1] (1 ) kn 1 A n (1 ) kn 1 P Ths result establshes the followng formula. To solate P, multply both sdes by (1 ) kn 1. Smplfy. Snkng Fund Payment For an annuty to provde a future value A n, regular payments P are made k tmes per year for n years, wth nterest compounded k tmes per year at an annual rate n. The payment P s gven by A n P (1 ) kn 1 where s the perodc rate, r. k EXAMPLE 2 Soluton An accountng frm wll need $17,000 n 5 years to replace ts computer system. What perodc deposts to a snkng fund payng quarterly nterest at a 9% annual rate wll acheve that goal? The snkng fund wll have the followng characterstcs: future value: A n $17,000 annual rate: r 0.09 term: n 5 number of perods per year: k 4 perodc rate: r k 4 We substtute the gven values n the formula to fnd the snkng fund payment. A n P (1 ) kn 1 (17,000)(0.0225) ( ) Use a calculator Round to the nearest cent. Quarterly payments of $ wll accumulate to $17,000 n 5 years. Self Check 2 What quarterly deposts to the above account are requred to rase the $50,000 startup cost of a branch offce n 7 years? Self Check Answers 1. $3, $1,301.26

17 Annutes and Future Value Exercses Vocabulary and Concepts Fll n the blanks. 1. Plans nvolvng payments made at regular ntervals are called. 2. In an ordnary annuty, payments are made at the of each perod. 3. The future value of an annuty s the sum of all the and. 4. The tme over whch the payments are made s the of the annuty. 5. In the future value formula, A n P[(1 )kn 1] P s the, s the, k s the, and n s the. 6. An annuty created to fund a specfc future oblgaton s a fund. Practce Assume that $100 s deposted at the end of each year n an account n whch nterest s compounded annually at a rate of 6%. Fnd the accumulated amount after the gven number of years years 8. 5 years 9. 3 years years Assume that $100 s deposted at the end of each year nto an account n whch nterest s compounded annually at the gven rate. Fnd the accumulated amount after 10 years % 12. 7% % % Assume that $100 s deposted at the end of each perod n an account n whch nterest s compounded at the gven frequency, at an annual rate of 8%. Fnd the accumulated amount after 15 years. 15. k k k k 1 Fnd the amount of each regular payment to provde $20,000 n 10 years, at the gven annual rate and compoundng frequency %, annually 20. 6%, quarterly 21. 9%, semannually 22. 8%, monthly Applcatons 23. Savng for a vacaton For next year s vacaton, the Phelps famly s savng $200 each month n an account payng 6% annual nterest, compounded monthly. How much wll be avalable a year from now? 24. Plannng for retrement Hank s regular $1,300 quarterly contrbutons to hs retrement account have earned 6.5% annual nterest, compounded quarterly, snce he started 21 years ago. How much s n hs account now? 25. Penson fund management The managers of a company s penson fund nvest the monthly employee contrbutons of $135,000 nto a government fund payng 8.7%, compounded monthly. To what value wll the fund grow n 20 years? 26. Savng for college A mother has been savng regularly for her daughter s college $25 each month for 11 years. The money has been earnng % annual nterest, compounded monthly. How much s now n the account? 27. Buyng offce machnes A company s new corporate headquarters wll be completed n years. At that tme, $750,000 wll be needed for offce equpment. How much should be nvested monthly to fund that expense? Assume 9.75% nterest, compounded monthly. 28. Retrement lfestyle A woman would lke to receve a $500,000 lump-sum dstrbuton from her retrement account when she retres n 25 years. She begns makng monthly contrbutons now to an annuty payng 8.5%, compounded monthly. Fnd the amount of that monthly contrbuton. 29. Comparng accounts Whch account wll requre the lower annual contrbutons to fund a $10,000 oblgaton n 20 years? (Hnt: Compare the yearly total contrbutons.) Bank A Bank B 5.5%; annually 5.35%; monthly

18 Chapter 9 The Mathematcs of Fnance 30. Avodng a balloon payment The last payment of a home mortgage s a balloon payment of $47,000, whch the owner s scheduled to pay n 12 years. How much extra should he start ncludng n each monthly payment to elmnate the balloon payment? Hs mortgage s at 10.2%, compounded monthly. Dscovery and Wrtng 31. Retrement strategy Jm wll retre n 30 years. He wll nvest $100 each month for 15 years and then let the accumulated value contnue to grow for the next 15 years. How much wll be avalable at retrement? Assume 8%, compounded monthly. 32. Retrement strategy (See Exercse 31.) Jm s brother Jack also wll retre n 30 years. He plans on dong nothng durng the first 15 years, then contrbutng twce as much $200 monthly to catch up. How much wll be avalable at retrement? Assume 8%, compounded monthly Changng plans A woman needs $13,500 n 10 years. She would lke to make regular annual contrbutons for the first 5 years and then let the amount grow at compound nterest for the next 5 years. What should her contrbutons be? Assume 9%, compounded annually. 34. Talkng financal sense How would you explan to a frend who has just been hred for her first job that now s the tme to start thnkng about retrement? Revew Solve each equaton. 2(5x 12) 2(5x 12) x x x x x 3 x Present Value of an Annuty; Amortzaton Objectves 1. Compute Present Value of an Annuty 2. Amortzaton Suppose you are lucky enough to come nto a great sum of money. Perhaps you wll receve an nhertance, sell a busness, or wn the lottery. Would you spend the money wsely, or would you waste t as have so many lottery wnners? In ths secton, we wll dscuss how to nvest a porton of ths money to guarantee that you wll receve regular payments n the future. 1. Compute Present Value of an Annuty Instead of usng an annuty to create a future value An, we mght ask, What sngle depost made now would create that same future value? The one depost that gves the same final result as an annuty s called the present value of that annuty. To find a formula for the present value of an annuty, we combne two prevous formulas. A seres of regular payments of P dollars for n years wll grow to a future value An gven by (1) An P[(1 )kn 1]

19 Present Value of an Annuty; Amortzaton 741 (2) From a formula n Secton 9.1, the present value of a future asset s gven by A 0 A n (1 ) kn We can fnd the present value of a seres of future payments by substtutng the rght sde of Equaton 1 nto Equaton 2. A 0 A n (1 ) kn A 0 P[(1 )kn 1] (1 ) kn P[(1 )kn (1 ) kn 1(1 ) kn ] P[1 (1 )kn ] Ths establshes the followng formula. Ths s Equaton 2. P[(1 ) kn 1] Substtute for A n n Equaton 2. Use the dstrbutve property. Smplfy: x m x m x 0 1. Present Value of an Annuty The present value A 0 of an annuty wth payments of P dollars made k tmes per year for n years, wth nterest compounded k tmes per year at an annual rate r, s A 0 P[1 (1 )kn ] where s the perodc rate, r. k EXAMPLE 1 Soluton To buy a boat n 2 years, the Hggns famly plans to save $200 a month n an account that pays 12% nterest, compounded monthly. a. Fnd the total amount of the payments. b. Fnd the value of the account n 2 years. c. Fnd the sngle depost n that account that would gve the same future value. a. At $200 per month for 24 months, the total amount contrbuted s $200(24) $4,800 b. To fnd the value after 2 years, we use the formula for future value of an annuty found on page 736: A n P[(1 )kn 1] A 2 200[ ] , Use a calculator. 5, Round to the nearest cent. c. To fnd the present value of the annuty, we substtute: the term, n years: the frequency of compoundng: n 2 k 12 the annual rate: r 0.12

20 Chapter 9 The Mathematcs of Fnance the payment: the perodc nterest rate: n the present value formula. P 200 r k A 0 P[1 (1 )kn ] A 0 200[1 (1 0.01)122 ] [1 (1.01)24 ] Smplfy , Use a calculator. 4, Round to the nearest cent. The present value of the annuty s $4, That one depost now wll provde the same fnal amount, $5,394.69, as the annuty. Self Check 1 For hs retrement n 30 years, a man plans to make monthly contrbutons of $25 to an ordnary annuty payng % annually, compounded monthly. a. Fnd the total amount of hs contrbutons. b. Fnd the sngle depost now that wll provde the same retrement beneft. State lottery wnnngs are usually pad as a 20-year annuty. That s to the state s advantage, because t can fund the annuty wth a sngle amount that s much smaller than the total prze. EXAMPLE 2 Soluton Brtta won the lottery. She wll receve $75,000 per month for the next 20 years a total of $18 mllon. What sngle depost should the lottery commsson make now to fund Brtta s annuty? Assume 8.4% annual nterest, compounded monthly. The lottery commsson fnds the present value of the annuty, wth the payment: the annual rate: the frequency of compoundng: the perodc rate: r k 12 the term, n years: n 20 These values are used n the formula for the present value of an annuty. A 0 P[1 (1 )kn ] P 75,000 r k 12 A 0 75,000[1 (1.007)1220 ] ,705, Use a calculator. 8,705, Round to the nearest cent. To fund the $18 mllon prze, the commsson must depost $8,705,

21 9.3 Present Value of an Annuty; Amortzaton Self Check 2 The lottery pays a total prze of $120,000 n monthly nstallments, as a 10-year annuty. Assumng 8.4% nterest, compounded monthly, what current depost s needed to fund the annuty? EXAMPLE 3 As a settlement n an automoble njury lawsut, Robyn wll receve $30,000 each year for the next 25 years, for a total of $750,000. The nsurance company s offerng a one-payment settlement of $300,000, now. Should she accept? Assume that the money can be nvested at 9% annual nterest. Soluton Robyn should calculate the present value of an annuty wth: the payment: the annual rate: the frequency of compoundng: the perodc rate: the term, n years: P 30,000 r 0.09 k 1 (annual) r k 1 n 25 She should use these values n the formula for the present value of an annuty. P[1 (1 ) kn] 30,000[1 (1.09) 1ⴢ25] A , , A0 Use a calculator. Round to the nearest cent. Snce the annuty s worth $294, and the company s offerng $300,000, Robyn should accept the $300,000. Self Check 3 If Robyn could nvest the settlement at 8% nterest, should she stll accept the lump-sum offer? When a worker s employed, regular contrbutons are usually made to a retrement fund. After retrement, those funds are gven back, ether as an annuty or as a lump-sum dstrbuton. EXAMPLE 4 Soluton Carlos wants to fund an annuty to supplement hs retrement ncome. How much should he depost now to generate retrement ncome of $1,000 a month for the 3 next 20 years? Assume that he can get 94% nterest, compounded monthly. Carlos must calculate the present value of a future stream of ncome, wth: the payment: the annual rate: the frequency of compoundng: the perodc rate: the term, n years: P 1,000 r k r k 12 n 20 He should use these values n the formula for the present value of an annuty.

22 Chapter 9 The Mathematcs of Fnance A 0 P[1 (1 )kn ] A 0 1,000[1 ( )1220 ] ,428 If $105,428 s deposted now, Carlos wll receve $1,000 per month n retrement ncome for 20 years. Self Check 4 If Carlos can nvest at %, what depost s needed now? 2. Amortzaton Before a bank wll lend money, you must sgn a promssory note ndcatng that you wll pay the money back. We dscussed one-payment notes n Secton 9.1. Most loans, however, are repad n nstallments nstead of all at once. Spreadng the repayment over several equal payments s called amortzaton. When a such a loan s made, the bank s buyng an annuty from the borrower, and the bank pays the borrower a certan amount and expects regular payments n return. To calculate the amount of these regular nstallment payments, we solve the present value formula for P to get A 0 P[1 (1 )kn ] A 0 P[1 (1 ) kn ] Multply both sdes by. A 0 P Dvde both sdes by 1 (1 ) kn. 1 (1 ) kn In ths context, the present value s the amount of the loan. A 0 Installment Payments The perodc payment P requred to repay an amount s gven by where A 0 P 1 (1 ) kn r k s the annual rate, s the frequency of compoundng (usually monthly), A 0 s the perodc rate, r, and k n s the term of the loan. EXAMPLE 5 Soluton The Almond famly takes a 15-year mortgage of $200,000 for ther new home, at 10.8%, compounded monthly. a. Fnd ther monthly payments. b. Fnd the total of ther payments over the full term. a. The mortgage has the followng characterstcs. the amount: A 0 200,000 the annual rate: r 0.108

23 Present Value of an Annuty; Amortzaton 745 the frequency of compoundng: k 12 the perodc rate: r k 12 the term, n years: n 15 We substtute these values nto the formula for nstallment payments to get A 0 P 1 (1 ) kn P (200,000)(0.009) 1 (1.009) , , Each monthly mortgage payment wll be $2, b. There are payments of $2, each, for a total of $404,665 more than twce the amount borrowed! Self Check 5 Instead of a 15-year mortgage, the Almonds consdered a 30-year mortgage. Answer the prevous two questons agan. Everyday Connectons Mortgage Rates LIBOR s an abbrevaton for London Interbank Offered Rate, and s the nterest rate offered by a specfc group of London banks for U.S. dollar deposts of a stated maturty. LIBOR s used as a base ndex for settng rates of some adjustable rate fnancal nstruments, ncludng Adjustable Rate Mortgages (ARMs) and other loans. One week n 2000, the average rate on a oneyear adjustable mortgage surged to 6.51 percent, the hghest snce January 2001, from 5.84 percent the pror week. The rate also surpassed the cost of a 30-year fxed loan for the frst tme. Suppose a prospectve homeowner obtans a mortgage loan wth the followng terms: Mortgage amount $324,000 Mortgage term 25 years Annual nterest rate (fxed) 5.64% Interest Rate 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 1 Year LIBOR Index Jan 97 Jan 98 Jan 99 Jan 00 Tme Jan 01 Jan 02 Jan 03 Jan 04 Jan 05 Jan 06 Calculate the annual monthly mortgage payment (prncpal and nterest). Source: Self Check Answers 1. a. $9,000 b. $3, about $81, No; the annuty s now worth more than $320, $113, a. $1, b. $674,814

24 Chapter 9 The Mathematcs of Fnance 9.3 Exercses Vocabulary and Concepts Fll n the blanks. 1. The current worth of a future stream of ncome s the of an annuty. 2. The amount requred now to produce a future stream of ncome s the of an annuty. 3. A loan s called a because you promse to repay t. 4. Often, loan repayment s spread out over several. 5. Spreadng repayment of a loan over several equal payments s called the loan. 6. An amortzed loan s also called a. Practce Fnd the present value of an annuty wth the gven terms. 7. Annual payments of $3,500 at 5.25%, compounded annually for 25 years 8. Semannual payments of $375 at a 4.92% annual rate, compounded semannually for 10 years Fnd the perodc payment requred to repay a loan wth the gven terms. 9. $25,000 repad over 15 years, wth monthly payments at a 12% annual rate 10. $1,750 repad n 18 monthly nstallments, at an annual rate of 19% 11. Fundng retrement Instead of makng quarterly contrbutons of $700 to a retrement fund for the next 15 years, a man would rather make only one contrbuton, now. How much should that be? Assume % annual nterest, compounded quarterly. 12. Fundng a lottery To fund Jame s lottery wnnngs of $15,000 per month for the next 20 years, the lottery commsson needs to make a sngle depost now. Assumng 9.2% compounded monthly, what should the depost be? 13. Money up front Instead of recevng an annuty of $12,000 each year for the next 15 years, a young woman would lke a one-tme payment, now. Assumng she could nvest the proceeds at %, what would be a far amount? 14. Fundng retrement What sngle amount deposted now nto an account payng % annual nterest, compounded quarterly, would fund an annuty payng $5,000 quarterly for the next 25 years? 15. Buyng a car The Jepsens are buyng a $21,700 car and fnancng t over the next 4 years. They secure an 8.4% loan. What wll ther monthly payments be? 16. Total cost of buyng a car What wll be the total amount the Jepsens wll pay over the lfe of the loan? (See Exercse 15.) 17. Choosng a mortgage One lender offers two mortgages a 15-year mortgage at 12%, and a 20-year mortgage at 11%. For each, fnd the monthly payment to repay $130, Total cost of a mortgage For each of the mortgages n Exercse 17, fnd the total of the monthly payments. Dscovery and Wrtng 19. Gettng an early start As Jorge starts workng now at the age of 20, he decdes to make regular contrbutons to a savngs account. He wants to accumulate enough by age 55 to fund an annuty of $5,000 per month untl age 80. What should hs monthly contrbutons be? Assume that both accounts pay 8.75%, compounded monthly. 20. Comparng annutes Whch of these 20-year plans s best, and why? All are at 8% annually. a. $1,000 each year for 10 years, and then let the accumulated amount grow for 10 years b. $500 each year for 20 years c. Do nothng for 10 years, and then contrbute $2,000 each year for 10 years d. One payment of $8,000 now, and let t grow 21. Changng the payment A woman contrbuted $500 per quarter for the frst 10 years of an annuty, but changed to quarterly payments of $1,500 for the last 10 years. Assumng % annual nterest compounded quarterly, what s her accumulated value? 22. Changng the rate A woman contrbuted $150 per month for 10 years to an account that pad 5% for the frst 5 years, but 6.5% for the last 5 years. How much has she saved?

25 9-25 Chapter Revew 747 Revew Smplfy each expresson. Assume that all varables represent postve numbers x 3 y x 5220x 26. B x 5 y 6 CHAPTER REVIEW 9.1 Interest Defntons and Concepts If funds n a savngs account earn smple nterest at an annual rate r, the amount deposted s the prncpal P, and the length of tme s t, the amount of nterest I earned s gven by the formula I Prt Compound nterest, annual compoundng: A sngle depost A 0 earnng compound nterest for n years at an annual rate r, wll grow to a future value accordng to the formula A n A n A 0 (1 r) n Compound nterest formulas: An amount A 0, earnng nterest compounded k tmes a year for n years at an annual rate r, wll grow to a future value accordng to the formula A n A n A 0 (1 ) kn where r k s the perodc nterest rate. Examples Fnd the smple nterest on a depost of $8,000 that s left on depost for 15 years at an annual rate of 4.5%. I Prt I 8, ,400 The nterest earned s $5,400 and the account wll contan $13,400. Fnd the amount n an account where $8,000 s left on depost for 15 years at an annual rate of 4.5%, compounded annually. A n A 0 (1 r) n A 15 8,000( ) 15 15, The amount wll be $15, Ths s $2, more than when the money was deposted at smple nterest. Fnd the amount n an account n whch $8,000 s left on depost for 15 years at an annual rate of 4.5%, compounded monthly. r The perodc nterest rate s k A n A 0 (1 ) kn A n 8,000( ) , The amount wll be $15,692.44, $ more than annual compoundng.

26 Chapter 9 The Mathematcs of Fnance The effectve rate R s used to compare dfferent savngs plans. R (1 ) k 1 The present value A 0 s the sngle depost now that wll yeld a specfc future value, A n. A 0 A n (1 ) kn where nterest s compounded k tmes a year at an annual rate r. 1 s the perodc rate r.2 k Exercses 1. $2,000 s deposted n an account that earns 9% smple nterest. Fnd the value of the account n 5 years. 2. $2,000 s deposted n an account n whch nterest s compounded annually at 9%. Fnd the value n 5 years. 3. Bran borrows $2,350 for medcal blls. The bank wrtes a 60-day note at 14%, wth nterest compounded daly. What wll Bran owe? Fnd the effectve rate n the example above. R (1 ) k 1 ( ) 12 1 ( ) The effectve rate s about 4.6%. Fnd the amount that must be deposted now to grow to be $15, n 15 years n an account earnng 4.5%, compounded monthly. As shown above, the perodc nterest rate s A 0 A n (1 ) nk A 0 15,692.44( ) , The present value s $8,000. In the example above, we saw that $8,000 grew to be $15, So t s expected that the present value of $15, s $8, $2,000 earns nterest, compounded quarterly, at an annual rate of 7.6% for 16 years. Fnd the future value. 5. BgBank advertses a savngs account at a 6.3% rate, compounded quarterly. BestBank offers 6.21%, compounded daly. Calculate each effectve rate and choose the better account. 6. What amount deposted now n an account payng 5.75% nterest, compounded semannually, wll yeld $7,900 n 6 years? 9.2 Annutes and Future Value Defntons and Concepts Examples An annuty s a seres of payments P made at regular Under a company savngs plan, a worker contrbutes ntervals. Its future value A n s the sum of all the payments and the nterest those payments earn. The tme compounded monthly. How much wll the annuty be $100 a month to an ordnary annuty payng 8%, over whch the payments are made s called the term worth n 50 years? of the annuty. In an ordnary annuty, the payments The perodc nterest rate s r are made at the end of each tme nterval. k A n The future value of an ordnary annuty wth deposts of P dollars made regularly k tmes each year for n years, wth nterest compounded k tmes per year at an annual rate r s A n P[(1 )kn 1] where s the perodc rate, r k. So, A n P[(1 )kn 1] A [( )1250 1] , The future value s $805,