9.4 Annuities. Objectives. 1. Calculate the future value of an ordinary annuity. 2. Perform calculations regarding sinking funds.

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1 9.4 Annuities Objectives 1. Calculate the futue value of an odinay annuity. 2. Pefo calculations egading sinking funds. Soewhee ove the ainbow... skies ae blue,... and the deas that you dae to dea...eally do coe tue. Lyan Fank Bau (Autho of The Wizad of Oz) What ae you financial deas? Do you dea about owning a beautiful house? Visiting soe exotic fa-away place? Poviding a college education fo you childen? Retiing happily in cofotable suoundings? If you have any long-te plans such as these, to achieve you deas you will need to have a lage su of oney in the futue. In this section, we will discuss how you can do just that by aking a seies of egula payents ove any yeas to accuulate the oney that you will need. This type of investent is called an annuity. Copyight 2010 Peason Education, Inc.

2 9.4 y Annuities 423 KEY POINT We ake egula payents into an annuity. Annuities An annuity is an inteest-beaing account into which we ake a seies of payents of the sae size. If one payent is ade at the end of evey copounding peiod, the annuity is called an odinay annuity. The futue value of an annuity is the aount in the account, including inteest, afte aking all payents. To illustate the futue value of an annuity, suppose that in Januay you begin aking payents of at the end of each onth into an account paying 12% yealy inteest copounded onthly. How uch oney will be in this account fo a sue vacation beginning on July 1? This poble is diffeent fo those in Section 9.2. In the ealie pobles, we deposited a lup su that eaned a stated inteest ate fo an entie peiod. In this poble, we ae depositing a seies of payents, and each payent eans inteest fo a diffeent nube of peiods. The Januay deposit eans inteest fo Febuay, Mach, Apil, May, and June. Using the foula fo copound inteest, this deposit will gow to 100( ) 5 = $ Howeve, the May deposit eans inteest fo only 1 onth and theefoe gows to only 100( ) 1 = $101. The June deposit eans no inteest at all. We illustate this patten with the tieline in Figue 9.3. Januay Febuay Mach Apil May June July Eans inteest. Eans no inteest. FIGURE 9.3 Tieline fo odinay annuity with deposits at end of Januay,...,June. If we copute how uch each deposit contibutes to the account and su these aounts, we will have the value of the annuity on July 1. Januay 100(1.01) 5 = $ Febuay 100(1.01) 4 = $ Mach 100(1.01) 3 = $ Apil 100(1.01) 2 = $ May 100(1.01) 1 = $ June 100(1.01) 0 =.00 Total = $ We can expess the value of this annuity as 100(1.01) (1.01) (1.01) (1.01) (1.01) Copyight 2010 Peason Education, Inc.

3 424 CHAPTER 9 y Consue Matheatics By factoing out the 100, we can wite the value of the annuity in the fo 100[(1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) 1 + 1]. (1) Notice that although you deposited at the end of each onth fo six onths, the fist deposit eaned inteest fo five onths, the second deposit eaned inteest fo fou onths, and so on. Following this patten, it is clea that if you had deposited the in a vacation savings account at the end of each onth fo 10 onths, the value of the annuity would have been 100[(1.01) 9 + (1.01) (1.01) 2 + (1.01) 1 + 1]. (2) Notice that in equation (1) we have an expession of the fo x 5 + x 4 + x 3 + x 2 + x 1 + 1, and in equation (2) we have an expession of the fo x 9 + x x 2 + x Fotunately, thee is a way to wite these lengthy expessions in a siple fo. Quiz Youself KEY POINT 14 Wite x 3 + x 2 + x as a quotient of two polynoials. (Do coputations that ae siila to those we did in Exaple 1.) The futue value of an annuity depends on the size of the payent, the inteest ate, and the nube of payents. EXAMPLE 1 Siplifying Annuity Coputations Show that x 5 + x 4 + x 3 + x 2 + x = x 6-1. x - 1 SOLUTION: To show this elationship, we ultiply the polynoials x 5 + x 4 + x 3 + x 2 + x and x - 1 in the usual way. x 5 + x 4 + x 3 + x 2 + x x 6 + x 5 + x 4 + x 3 + x 2 + x 1 * x x 5 - x 4 - x 3 - x 2 - x 1-1 x 6-1 This esult shows that (x 5 + x 4 + x 3 + x 2 + x 1 + 1)(x - 1) = x 6-1. Dividing this equation by x - 1, we obtain ou desied elationship. Now ty Execises 3 to 4. ] Following the patten of Exaple 1, we can pove that Retuning to ou vacation savings account exaple, we can think of 1.01 as x in equation (1). Then we can use equation (3) to siplify ou calculations. Because (1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) we can wite x n + x n -1 + x n-2 + Á + x 2 + x = x n+1-1 x [(1.01) 5 + (1.01) 4 + (1.01) 3 + (1.01) 2 + (1.01) 1 + 1] L 100(6.1520) = $ This is the sae aount that we found ealie. Doing siila coputations, we find that the 10-onth vacation savings account annuity has a value of 100[(1.01) 9 + (1.01) 8 + Á + (1.01) 2 + (1.01) 1 + 1] 14 = (1.01) = = L , We can genealize the pattens that we have just seen in a foula fo finding the futue value of an annuity. (3) = 100c (1.01)10-1 d L 100( ) = $1, Copyight 2010 Peason Education, Inc.

4 9.4 y Annuities 425 FORMULA FOR FINDING THE FUTURE VALUE OF AN ORDINARY ANNUITY Assue that we ae aking n egula payents, R, into an odinay annuity. The inteest is being copounded ties a yea and deposits ae ade at the end of each copounding peiod. The futue value (o aount), A, of this annuity at the end of the n peiods is given by the equation A = R a1 + b n - 1. To calculate this expession, you should do the following steps: 1st: Find and add 1. 2nd: Raise 1 + to the n powe and then subtact 1. 3d: Divide the aount that you found in step 2 by. 4th: Multiply the quantity that you found in step 3 by R. EXAMPLE 2 Finding the Futue Value of an Odinay Annuity Assue that we ake a payent of $50 at the end of each onth into an account paying a 6% annual inteest ate, copounded onthly. How uch will be in that account afte 3 yeas? SOLUTION: This account is an odinay annuity. The payent R is 50, the onthly ate 6% is 12 = = 0.005, and the nube of payents n is 3 * 12 = 36. Using the foula fo finding the futue value of an odinay annuity, we get TI-83 sceen veifies calculations in Exaple 2. Quiz Youself 15 Redo Exaple 2, except now assue that you ae depositing $75 pe onth fo 2 yeas. 1 + n (1.005) A R ( ) $1, * Now ty Execises 7 to 16. ] If you ake egula payents into an annuity fo any yeas, the value of the annuity can becoe enoous due to the copounding of inteest. Figue 9.4 shows that if the annual inteest ate is 6.6%, then in oughly 19 yeas, the aount of inteest that the annuity has eaned exceeds the aount of the deposits. The fact that the inteest cuve is ising so apidly indicates that the futue value of you account is also gowing apidly. KEY POINT With a sinking fund, we ake payents to save a specified aount. Sinking Funds You ay want to save egulaly to have a fixed aount available in the futue. Fo exaple, you ay want to save $1,800 to tavel to Jaaica in 2 yeas. The question is, how uch should you save each onth to accoplish this? The account that you establish fo *When using you calculato, you should hold off on ounding you answes as long as possible. If you ound off too soon, you answes will diffe slightly fo the answes in this text. Copyight 2010 Peason Education, Inc.

5 426 CHAPTER 9 y Consue Matheatics HIGHLIGHT Between the Nubes Who Do You Tust? The tust that we ae efeing to is the Social Secuity tust fund. When you fist stated woking, you ay have been disayed to see a deduction fo you fist paycheck labeled FICA, which is an acony fo the Fedeal Insuance Contibutions Act. This law andates that wokes ust contibute a cetain aount of thei wages to the Social Secuity tust fund. When you contibute to Social Secuity, you ae not actually saving you oney fo you etieent, but you taxes ae paying fo soeone else s etieent. The idea is that when you etie, younge wokes will then pay fo you etieent. Howeve, soe see a huge poble with this. Right now, oughly 50 illion Aeicans eceive Social Secuity benefits with contibutions suppoted by appoxiately 200 illion U.S. wokes. If we think of this as a atio, evey peson on Social Secuity is suppoted by 4 wokes. In 1950, the atio was 16 wokes fo evey Social Secuity beneficiay. It is pojected that by 2030, the atio will be 2 wokes fo evey beneficiay and the fund will be in touble. So what can you do to potect you etieent? The govenent has been encouaging people to ake plans fo thei etieent by establishing tax-defeed annuities to guaantee that when they etie they will have oney to suppleent Social Secuity benefits. Tax defeed eans that the oney you set aside in the annuity is not taxed now but at a late date when you stat withdawing fo the annuity. As you will see in the execises, thee can be a huge financial benefit to saving this way. 250 Thousands of dollas Inteest exceeds deposits Inteest Deposits Months FIGURE 9.4 At an annual ate of 6.6%, the aount of inteest eaned in an odinay annuity exceeds the aount of deposits in about 230 onths. you deposits is called a sinking fund. You could estiate the aount to save each onth 1,800 by siply dividing 1,800 by 24 onths to get 24 = $75 pe onth. Because you estiate ignoes the inteest that you deposits will geneate, the actual aount you would need to put aside each onth is soewhat less. Knowing exactly how uch you need to save each onth could be ipotant if you wee on a tight budget. Because a sinking fund is a special type of annuity, it is not necessay to find a new foula to answe this question. We can use the foula fo calculating the futue value of an odinay annuity that we have stated ealie. In this case, we know the value of A and we want to find R.* EXAMPLE 3 Calculating Payents fo a Sinking Fund Assue that you wish to save $1,800 in a sinking fund in 2 yeas. The account pays 6% copounded quately and you will also ake payents quately. What should be you onthly payent? *In aking payents into a sinking fund, we will always ound the payent up to the next cent. Copyight 2010 Peason Education, Inc.

6 9.4 y Annuities 427 SOLUTION: Recall the foula fo finding the futue value of an odinay annuity: TI-83 sceen veifies calculations in Exaple 3. Quiz Youself 16 What payents ust you ake to a sinking fund that pays 9% yealy inteest copounded onthly if you want to save $2,500 in 2 yeas? KEY POINT We use the log function to find how long it takes fo an annuity to accuulate a specified value. a1 + n b - 1 A = R. (4) We want the value A of the annuity to be $1,800, the onthly ate is 4 = 0.015, and the nube of payents n is 2 * 4 = 8. Substituting these values in equation (4), we get Dividing both sides of the equation by , we get the onthly payent Now ty Execises 17 to 20. ] Soe Good Advice You ay be tepted to eoize a new foula to solve sinking fund pobles. This is not necessay, because once you have leaned to solve annuity pobles, you can use the sae foula (and a little bit of algeba) to solve sinking fund pobles. Soeties in woking with annuities, we want to know how long it will take to save a cetain aount. That is, in the annuity foula we want to find n. This poble is a little oe coplicated than those we have solved so fa. To solve such pobles, we will use the exponent popety fo the log function, which we intoduced in Section 9.3. We will show you how to use this popety in Exaple 4. EXAMPLE 4 1,800 = R R = 1, L $ = R( ). Finding the Tie Requied to Accuulate $1,000,000 6% 4 = 0.06 Suppose you have decided to etie as soon as you have saved $1,000,000. You plan is to put $200 each onth into an odinay annuity that pays an annual inteest ate of 8%. In how any yeas will you be able to etie? SOLUTION: We can use the futue value foula fo an odinay annuity to solve this poble: a1 + n b - 1 A = R Hee, A is the futue value of $1,000,000, is the onthly inteest ate of 12 L , and R is 200. We ust find n, the nube of onths fo which you will be aking deposits. Theefoe, we ust solve fo n in the following equation: ( ) 1,000, A R n We begin by ultiplying both sides of the equation by : 6,670 = 200[( ) n - 1] Copyight 2010 Peason Education, Inc.

7 428 CHAPTER 9 y Consue Matheatics Quiz Youself 17 If you put $150 pe onth into an odinay annuity that pays an annual inteest ate of 9%, how long will it take fo the annuity to have a value of,000? Next, we divide both sides of the equation by 200 and then add 1 to both sides of the equation: = ( ) n. Then we take the log of both sides: log = log( ) n. Now we use the exponent popety of the log function to siplify the equation: log = n log We divide by log and use a calculato to find n: log n = = L532. log This tells how any onths it will take you to save $1,000,000. Dividing 532 by 12 gives 532 us 12 = yeas until you etieent. Now ty Execises 29 to 34. ] 17 Execises 9.4 Looking Back* These execises follow the geneal outline of the topics pesented in this section and will give you a good oveview of the ateial that you have just studied. 1. To calculate the futue value of the annuity in Exaple 2, we used the expession a1 + n b - 1 R. What is the eaning of? What is the eaning of? What is R? 2. What iplications do you see of the situation pesented in the Highlight egading the Social Secuity tust fund? Shapening You Skills In Execises 3 and 4, siplify each algebaic expession, as in Exaple x 7 + x x 2 + x x 8 + x x 2 + x Execises 5 and 6 ae based on the vacation account exaple at the beginning of this section. Assue that, beginning in Januay, you ake payents at the end of each onth into an account paying the specified yealy inteest. Inteest is copounded onthly. How uch will you have available fo you vacation by the specified date? 5. Monthly payent, ; yealy inteest ate, 6%; August 1 6. Monthly payent, $200; yealy inteest ate, 3%; May 1 In Execises 7 and 16, find the value of each odinay annuity at the end of the indicated tie peiod. The payent R, fequency of deposits (which is the sae as the fequency of copounding), annual inteest ate, and the tie t ae given. 7. Aount, $200; onthly; 3%; 8 yeas 8. Aount, $450; onthly; 2.4%; 10 yeas 9. Aount, $400; onthly; 9%; 4 yeas 10. Aount, $350; onthly; 10%; 10 yeas 11. Aount, $600; onthly; 9.5%; 8 yeas 12. Aount, $500; onthly; 7.5%; 12 yeas 13. Aount, $500; quately; 8%; 5 yeas 14. Aount, $750; quately; 9%; 3 yeas 15. Aount, $280; quately; 3.6%; 6 yeas 16. Aount, $250; quately; 4.8%; 18 yeas In Execises 17 20, find the onthly payent R needed to have a sinking fund accuulate the futue value A. The yealy inteest ate and the tie t is given. Inteest is copounded onthly. Round you answe up to the next cent. 17. A = $2,000; = 6%; t = A = $10,000; = 12%; t = A = $5,000; = 7.5%; t = A = $8,000; = 4.5%; t = 3 Solve each equation fo x x = x = x = x = x x = = x x = = In Execises 29 34, use the foula fo finding the futue value of an odinay annuity, a1 + n b - 1 A = R, *Befoe doing these execises, you ay find it useful to eview the note How to Succeed at Matheatics on page xix. Copyight 2010 Peason Education, Inc.

8 9.4 y Execises 429 to solve fo n. You ae given A, R, and. Assue that payents ae ade onthly and that the inteest ate is an annual ate. 29. A = $10,000; R = 200; = 9% 30. A = $12,000; R = 400; = 8% 31. A = $5,000; R = 150; = 6% 32. A = $8,000; R = 400; = 5% 33. A = $6,000; R = 250; = 7.5% 34. A = $7,500; R = 100; = 8.5% Applying What You ve Leaned In Execises 35 40, assue that the copounding is being done onthly. 35. Saving fo a scoote. Matt is saving to buy a new Vespa. If he deposits $75 at the end of each onth in an account that pays an annual inteest ate of 6.5%, how uch will he have saved in 30 onths? 36. Saving fo a tip. Angelina wants to save fo an Afican safai. She is putting $200 each onth in an odinay annuity that pays an annual inteest ate of 9%. If she akes payents fo 2 yeas, how uch will she have saved fo he tip? 42. Saving to stat a business. Victoy is aking onthly payents into an annuity that pays 0.8% onthly inteest to save enough fo a down payent to stat he own business. If she wants to save $10,000 in 5 yeas, what should he onthly payents be? 43. Saving fo consue goods. Sanda Lee is aking onthly payents into an annuity. She wants to have $600 in the fund to buy a new convection ange in 6 onths, and the account pays 8.2% annual inteest. What ae he onthly payents to the account? 44. Saving fo consue goods. Lennox is aking onthly payents into an annuity. He wishes to have $1,150 in 10 onths to buy execise equipent. His account pays 9% annual inteest. What ae his onthly payents to the account? Tax-defeed annuities wok like this: If, fo exaple, you plan to set aside $400 pe onth fo you etieent in 30 yeas in a taxdefeed plan, the $400 is not taxed now, so all of the $400 is invested each onth. In a nondefeed plan, the $400 is fist taxed and then the eainde is invested. So, if you tax backet is 25%, afte you pay taxes, you would have only 75% of the $400 to invest each onth. Howeve, in the tax-defeed plan, all of you oney is taxed when you withdaw the oney. In the nondefeed plan, only the inteest that you have eaned is taxed. In Execises 45 50, we give the aount you ae setting aside each onth, you cuent tax ate, the nube of yeas you will contibute to the annuity, and you tax ate when you begin withdawing fo the annuity. Answe the following questions fo each situation: a) Find the value of the tax-defeed and the nondefeed accounts. b) Calculate the inteest that was eaned in both accounts. This will be the value of the account inus the payents you ade. c) If you withdaw all oney fo each account and pay the elevant taxes, which account is bette and by how uch? 37. Saving fo a ca. Kisty Joe deposits $150 each onth in an odinay annuity to save fo a new ca. If the annuity pays a onthly inteest ate of 0.85%, how uch will she be able to save in 3 yeas? 38. Saving fo etieent. Cohutta is saving fo his etieent in 10 yeas by putting $500 each onth into an odinay annuity. If the annuity has an annual inteest ate of 9.35%, how uch will he have when he eties? 39. Saving fo a vacation hoe. Wendy has set up an odinay annuity to save fo a etieent hoe in Floida in 15 yeas. If he onthly payents ae $400 and the annuity has an annual inteest ate of 6.5%, what will be the value of the annuity when she eties? 40. Saving fo etieent. Thiep has set up an odinay annuity to save fo his etieent in 20 yeas. If his onthly payents ae $350 and the annuity has an annual inteest ate of 7.5%, what will be the value of the annuity when he eties? 41. Saving fo a condoiniu. Kanye wants to save $14,000 in 8 yeas by aking onthly payents into an odinay annuity fo a down payent on a condoiniu at the shoe. If the annuity pays 0.7% onthly inteest, what will his onthly payent be? Monthly Payent Nube of Yeas Annual Inteest Rate Cuent Tax Rate Futue Tax Rate 45. $ % 25% 18% 46. $ % 25% 15% 47. $ % 30% 30% 48. $ % 25% 25% 49. $ % 25% 30% 50. $ % 18% 25% In Execises 51 54, assue that onthly deposits ae being placed in an odinay annuity and inteest is copounded onthly. 51. Saving fo a fie tuck. The Reliance Voluntee Fie Copany wants to take advantage of a state poga to save oney to puchase a new fie tuck. The tuck will cost $400,000, and ebes of the finance coittee estiate that with counity and state contibutions, they can save $5,000 pe onth in an account paying 10.8% annual inteest. How long will it take to save fo the tuck? 52. Saving fo new equipent. BioCon, a bioengineeing copany, ust eplace its wate teatent equipent within 2 yeas. The Copyight 2010 Peason Education, Inc.

9 430 CHAPTER 9 y Consue Matheatics new equipent will cost $80,000, and the copany will tansfe $3,800 pe onth into a special account that pays 9.2% inteest. How long will it take to save fo this new equipent? 53. Saving fo a condoiniu. Kisten wants to save $30,000 fo a down payent on a condoiniu at a ski esot. She feels that she can save $550 pe onth in an account that has a 7.8% annual inteest ate. How long will it take to acquie he down payent? 54. Saving fo a business. Leo needs to save $25,000 fo a down payent to stat a photo estoation business. He intends to save $300 pe onth in an account that pays an annual inteest ate of 6%. How long will it take fo hi to save fo the down payent? Counicating Matheatics 55. What popety of the log function would you use to solve an equation of the fo y = a x fo x? 56. What is siila about the poble of finding the futue value of an annuity and finding the payents to ake into a sinking fund? How ae the two pobles elated? Using Technology to Investigate Matheatics 57. See you instucto fo tutoials fo the TI-83 to do financial calculations. Use you calculato to epoduce soe of the exaples in this section.* 58. You can find any inteactive annuity calculatos on the Intenet. Soe ipleent the calculations that you have leaned in this section and othes ipleent othe types of annuity pobles. Find soe inteesting annuity calculatos, expeient with the, and epot on you findings. Fo Exta Cedit 59. Saving fo etieent. Reconside Execise 38. Suppose that Cohutta had stated his annuity 10 yeas ealie. What would his onthly payents have been to accuulate the sae futue value in his annuity as you found in Execise 38? 60. Saving fo etieent. Reconside Execise 40. Suppose that Thiep had stated his annuity 10 yeas ealie. What would his onthly payents have been to accuulate the sae futue value in his annuity as you found in Execise 40? 61. Saving fo etieent. Calos began to save fo his etieent at age 25, and fo 10 yeas he put $200 pe onth into an odinay annuity at an annual inteest ate of 6%. Afte the 10 yeas, he could no longe ake payents, so he placed the value of the annuity into anothe account that paid 6% annual inteest copounded onthly. He left the oney in this account fo 30 yeas until he was eady to etie. How uch did he have fo etieent? 62. Saving fo etieent. If Calos had waited until age 45 to think about etieent and then decided to put oney into an odinay annuity fo 20 yeas, what would his onthly payents have to be to accuulate the sae aount fo etieent as you found in Execise 61? (We assue that the inteest ate fo the annuity is the sae.) 63. Saving fo etieent. Exaine you solutions to Execises 61 and 62. What do you notice? Explain why this is so. 64. Copaing annuities. The diffeence between an annuity due and an odinay annuity is that with an annuity due, the payent is ade at the beginning of the onth athe than at the end of the onth. This eans that each payent geneates one oe onth of inteest than with an odinay annuity. a. How does this change the foula fo finding the futue value of an annuity? b. Use this foula to find the value of the annuity in Exaple 2, assuing that the annuity is an annuity due. Copyight 2010 Peason Education, Inc.

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