9.5 Amortization. Objectives


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1 9.5 Aotization Objectives 1. Calculate the payent to pay off an aotized loan. 2. Constuct an aotization schedule. 3. Find the pesent value of an annuity. 4. Calculate the unpaid balance on a loan. Congatulations! You just bought a new hoe it s lovely and in a good neighbohood. Only 360 oe payents and it s all yous. When you ake such a lage puchase, you usually have to take out a loan that you epay in onthly payents. The pocess of paying off a loan (plus inteest) by aking a seies of egula, equal payents is called aotization, and such a loan is called an aotized loan. If you wee to ake such a puchase, one of the fist questions you ight ask is, What ae y onthly payents? Of couse, the lende can answe this question, but you ay Copyight 2010 Peason Education, Inc.
2 9.5 y Aotization 431 find it inteesting to lean the atheatics involved with paying off a otgage so that you can answe that question youself. KEY POINT Paying off a loan with egula payents is called aotization. Aotization Assue that you have puchased a new ca and afte you down payent, you boowed $10,000 fo a bank to pay fo the ca. Also assue that you have ageed to pay off this loan by aking equal onthly payents fo 4 yeas. Let s look at this tansaction fo two points of view: Banke s point of view: Instead of thinking about you payents, the banke ight think of this tansaction as a futue value poble in which she is aking a $10,000 loan to you now and copounding the inteest onthly fo 4 yeas. At the end of 4 yeas, she expects to be paid the full aount due. Recall fo Section 9.2 that this futue value is A = Pa1 + b n. You point of view: Fo the tie being, you could also ignoe the question of onthly payents and choose to pay the banke in full with one payent at the end of 4 yeas. In ode to have this oney available, you could ake onthly payents into a sinking fund to have the aount A available in 4 yeas. As you saw in Section 9.4, the foula fo doing this is a1 + n b  1 A = R. Thus, to find you onthly payent, we will set the aount the banke expects to eceive equal to the aount that you will save in the sinking fund and then solve fo R. FORMULA FOR FINDING PAYMENTS ON AN AMORTIZED LOAN Assue that you boow an aount P, which you will epay by taking out an aotized loan. You will ake peiodic payents pe yea fo n total payents and the annual inteest ate is. Then, you can find you payent by solving fo R in the equation P a1 + b n a1 + = R ± b n *  1. Do not let this equation intiidate you. You have done the calculation on the left side any ties in Section 9.2 and the coputation on the ight side in Section 9.4. Once you find these two nubes, you do a siple division to solve fo R, as you will see in Exaple 1. EXAMPLE 1 Deteining the Payents on an Aotized Loan Assue that you have taken out an aotized loan fo $10,000 to buy a new ca. The yealy inteest ate is 18% and you have ageed to pay off the loan in 4 yeas. What is you onthly payent? *Cetainly we could do the necessay algeba to solve this equation fo R. Then we could use this new foula fo solving pobles to find the onthly payents fo aotized loans. We chose not to do this because ou philosophy is to iniize the nube of foulas that you have to eoize to solve the pobles in this chapte. We will ound payents on a loan up to the next cent. Copyight 2010 Peason Education, Inc.
3 432 CHAPTER 9 y Consue Matheatics TI83 calculato confis ou coputations in Exaple 1. Quiz Youself 18 What would you payents be in Exaple 1 if you agee to pay off the loan in 5 yeas? SOLUTION: We will use the peceding equation. The values of the vaiables in this equation ae We ust solve fo R in the equation onthly inteest ate aount of loan P = 10,000 n = 12 * 4 = 48 = 18% 12 = If we calculate the nueical expessions on both sides of this equation as we did in Sections 9.2 and 9.4, we get Theefoe, you onthly payent is Now ty Execises 3 to 10. ] 20, = R( ). R = 20, L $ nube of payents 10,000( ) 48 R ( ) Aotization Schedules Payents that a boowe akes on an aotized loan patly pay off the pincipal and patly pay inteest on the outstanding pincipal. As the pincipal is educed, each successive payent pays oe towad pincipal and less towad inteest. A list showing payentbypayent how uch is going to pincipal and inteest is called an aotization schedule. We illustate such a schedule in Exaple 2. EXAMPLE 2 Constucting an Aotization Schedule To expand you business selling collectibles on the Intenet, you need a loan of $5,000. You banke loans you the oney at a 12% annual inteest ate, which you agee to pay back in thee equal onthly installents of $1, * Constuct an aotization schedule fo this loan. SOLUTION: At the end of the fist onth, you have boowed $5,000 fo 1 onth at a 1% onthly inteest ate. So using the siple inteest foula, the inteest that you owe the bank is P t I $5,000 * 0.01 * 1 = $50. Payent Nube Aount of Payent Inteest Payent Applied to Pincipal Balance $5, $1, $50.00 $1, $3, $1, $33.50 $1, $1, $1, $16.83 $1, $0.03 TABLE 9.4 An aotization schedule. You payent is $1,700.12; theefoe, $50 pays the inteest, and the est, $1, $50 = $1,650.12, is applied to the pincipal. Fo the second onth, you ae now boowing $5,000  $1, = $3, at 1% onthly inteest. We coplete the coputations fo the payents on this loan in Table 9.4. *We used the ethod fo Exaple 1 to calculate the exact payent to be $1, Because we incease this eve so slightly to $1,700.12, afte the thid payent we have ovepaid by $0.03. Copyight 2010 Peason Education, Inc.
4 9.5 y Aotization 433 As expected, we ended with a negative balance because the payent of $1, is a faction of a cent lage than it needs to be. In an actual banking situation, the bank would adjust the final payent so that the final balance is exactly $0.00. ] Exaple 3 illustates how discouaging it can be when you ake you fist payent on a otgage fo a house and ealize how little of you payent goes towad paying the pincipal. EXAMPLE 3 Constucting an Aotization Schedule Assue that you have saved oney fo a down payent on you dea house, but you still need to boow $120,000 fo you bank to coplete the deal. The bank offes you a 30 yea otgage at an annual ate of 7%. The onthly payent is $ Constuct an aotization schedule fo the fist thee payents on this loan. SOLUTION: We copute Table 9.5* as we did Table 9.4 in Exaple 2. Payent Nube Aount of Payent Inteest Payent Applied to Pincipal Balance $120, $ $ $98.37 $119, $ $ $98.94 $119, $ $ $99.52 $119, TABLE 9.5 Making an aotization schedule fo a lengthy otgage. Quiz Youself 19 Copute the fouth line of Table 9.5. You see that fo such a lengthy aotized loan, the ealy payents ae ostly inteest. Fotunately, because the debt is being educed, each onth a little oe of the payent goes towad pincipal and a little less towad inteest. Now ty Execises 11 to 14. ] 19 Between the Nubes Can They Really Do That to You? How would you feel if you took out a $200,000 otgage fo a house, faithfully ade all of you payents on tie, and at the end of 1 yea owed $201,118? Incedibly, this can actually happen if you have an adjustable ate otgage, o ARM. Soe ARMs allow you to ake payents that do not even cove the inteest on the loan, so the aount you owe inceases even though you ake you payents on tie. ARMs can have othe vey seious pobles fo the consue. With an ARM, it is possible to stat with a low inteest ate, say 4%, and with yealy inceases afte seveal yeas you inteest ate could be uch highe. Motgage lendes use an index, often tied to govenent secuities, to HIGHLIGHT deteine how uch to incease you inteest ate. Thee ae any diffeent types of ARMs soe liit the ate incease fo yea to yea, and othes liit the axiu ate that can be chaged. Howeve, even with these liits, you onthly payents in an ARM could incease fo $900 to $1,400 ove a 3yea peiod, causing you geat financial distess. The Consue Handbook on Adjustable Rate Motgages, available fo the Fedeal Reseve Boad, is an excellent guide to ARMs and contains nueous exaples, cautions, and a woksheet to help you ake sensible decisions egading otgages. *If you veify these coputations by hand, you answes ay diffe slightly fo ous due to a diffeence in the way we ae ounding off ou inteediate calculations. Copyight 2010 Peason Education, Inc.
5 434 CHAPTER 9 y Consue Matheatics HIGHLIGHT Using a Speadsheet to Make an Aotization Schedule A speadsheet can ceate an aotization schedule in the blink of an eye. The following is a speadsheet that calculates the schedule fo an aotized loan fo $10,000 with 60 onthly payents of $ We fist show the speadsheet displaying the foulas in the cells of the speadsheet. A B C D E F 1 End of Month Payent Inteest Pincipal Balance 2 0 $ $10, $ E2*0.08/12 B3 C3 E2 D3 4 2 $ E3*0.08/12 B4 C4 E3 D4 5 3 $ E4*0.08/12 B5 C5 E4 D5 6 4 $ E5*0.08/12 B6 C6 E5 D6 7 5 $ $ $ Hee is the sae speadsheet when the foulas in the speadsheet ae evaluated. A B C D E F 1 End of Month Payent Inteest Pincipal Balance 2 0 $ $10, $ $66.67 $ $9, $ $65.76 $ $9, $ $64.85 $ $9, $ $63.93 $ $9, $ $ $ To geneate a new schedule fo a otgage, all we have to do is change the foulas in seveal cells and the entie speadsheet will be ecalculated. KEY POINT We use the foula fo finding the size of onthly payents to deteine the pesent value of an annuity. Finding the Pesent Value of an Annuity When buying a ca, you budget deteines the size of the onthly payents you can affod, and that deteines how uch you can pay fo the ca you buy. Assue that you can affod ca payents of $200 pe onth fo 4 yeas and you bank will gant you a ca loan at an annual ate of 12%. We can think of this as a futue value of an annuity poble whee R is 200, is 1%, and n is 48 onths. We know fo Section 9.4 that the futue value of this annuity is A = 200c ( )481 d = $12, This esult does not ean that now you can affod a $12,000 ca! This aount is the futue value of you annuity, not what that aount of oney would be woth in the pesent. Copyight 2010 Peason Education, Inc.
6 9.5 y Aotization 435 DEFINITION If we know the onthly payent, the inteest ate, and the nube of payents, then the aount we can boow is called the pesent value of the annuity. We can find the pesent value of an annuity by setting the expession fo the futue value of an account using copound inteest equal to the expession fo finding the futue value of an annuity and solving fo the pesent value P. FINDING THE PRESENT VALUE OF AN ANNUITY Assue that you ae aking peiodic payents pe yea fo n total payents into an annuity that pays an annual inteest ate of. Also assue that each of you payents is R. Then to find the pesent value of you annuity, solve fo P in the equation P a1 + b n a1 + = R ± b n  1. Again, you have done the coputations on both sides of this equation any ties in Sections 9.2 and 9.4. EXAMPLE 4 Deteining the Pice You Can Affod fo a Ca If you can affod to spend $200 each onth on ca payents and the bank offes you a 4yea ca loan with an annual ate of 12%, what is the pesent value of this annuity? SOLUTION: To solve this poble, we can use the foula fo finding payents on an aotized loan: TI83 sceen confis ou coputations in Exaple 4. Quiz Youself Redo Exaple 4, but now assue that you can affod payents of $250 pe onth. KEY POINT 20 To efinance a loan, we ust know the unpaid balance on the loan. Pa1 + n a1 + n b b  1 = R ±. We know R = 200, = 1% = 0.01, and n = 48 onths. If we substitute these fo the vaiables in equation (1), we get P = 200c d Calculating the nueical expessions on both sides of equation (2) gives us P( ) = 12, Now dividing both sides of this equation by , we find P = c d L $ You ay find this answe supising, but the atheatics of this poble ae clea. If you can only affod payents of $200 pe onth, then you can only affod to finance a ca loan fo about $7,600! Now ty Execises 25 to 30. ] 20 Finding the Unpaid Balance of a Loan Duing ties when inteest ates ae high, people ae foced to boow oney at these high ates if they want to buy a ca o a house on cedit. If inteest ates decline, then it is wise to conside paying off the eaining debt on the fist loan by taking out a second loan at a (1) (2) Copyight 2010 Peason Education, Inc.
7 436 CHAPTER 9 y Consue Matheatics lowe inteest ate. This pocedue is called efinancing the loan. To undestand efinancing, we ust be able to copute how uch debt eains on a loan afte a cetain nube of payents have been ade. EXAMPLE 5 Finding the Unpaid Balance on a Loan a) Assue that you take out a 30yea otgage fo $100,000 at an annual inteest ate of 9%. If, afte 10 yeas, inteest ates dop and you want to efinance, how uch eains to be paid on you otgage? b) If you can efinance you otgage fo the eaining 20 yeas at an annual inteest ate of 7.2%, what will you onthly payents be? c) How uch will you save in inteest in 20 yeas by paying the lowe ate? SOLUTION: a) Doing the sae kind of calculations as we did in Exaple 1, we find that the onthly payent is $ You onthly payent was based on the assuption that you would be paying the loan fo 30 yeas. Theefoe, afte 10 yeas, you will not have accuulated enough in you annuity to pay off the banke. This eans that the aount you owe, PA1 + B n, ust be lage than the aount that you have accuulated in you sinking fund, The unpaid balance U on the loan is theefoe It is ipotant to ecognize in equation (3) that only 10 yeas have elapsed, so n = 12 * 10 = 120, not 360. We can now substitute the values P = $100,000, = 0.09/12 = , n = 120, and R = $ in equation (3). Theefoe, you still owe $89, on this otgage. b) Because you have a balance of $89, on you otgage, in effect you ae now taking out a new otgage fo this aount at an annual inteest ate of 7.2% fo the eaining 20 yeas. In this case, P = $89,428.32, n = 12 * 20 = 240, and = 0.072/12 = We now solve fo the onthly payent R in the equation aount boowed fo second loan what you owe U P 1 n R a1 + n b  1 R±. what you have accuulated 1 new onthly inteest ate n yeas 10 yeas 10 yeas U 100,000( ) ( ) $89, ,428.32( ) 240 R ( ) * (3) *The unpaid balance appeas in the balance colun of an aotization table. Copyight 2010 Peason Education, Inc.
8 9.5 y Execises 437 Execises 9.5 If we calculate the nueical expessions on both sides of this equation as we did in Sections 9.2 and 9.4, we get Theefoe, you new onthly payent is 375, = R( ). R = 375, L $ Theefoe, by efinancing, you ae able to educe you onthly otgage payent by $ $ = $ pe onth. c) If you ake onthly payents on the unpaid balance fo 20 yeas at the old inteest ate, you total payents will be 240 * $ = $193, If you ake onthly payents fo 20 yeas at the new inteest ate, you will pay 240 * $ = $168, The diffeence $193, $168, = $24,122.40* is the aount that you save in inteest ove 20 yeas at the educed ate. Now ty Execises 31 to 36. ] Often, when efinancing a otgage you ust pay a efinancing fee. The efinancing fee is often calculated as a pecentage of the balance eaining on the otgage. Suppose in Exaple 5 that you had to pay a 2% efinancing fee. Two pecent of $89, is $1, You would gain this fee back in 18 onths with the educed payents on the loan. In this case, it is clea that afte 18 onths, you would benefit fo the efinancing. 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. What does the left side of the fist equation in Exaple 1 epesent? What does the ight side epesent? 2. Afte eading the Highlight on adjustable ate otgages, nae two possible danges of ARMs. Shapening You Skills Solve the equation Pa1 + a1 + n n b b  1 = R± fo R to find the onthly payent necessay to pay off the loan. You ae given the loan aount, the annual inteest ate, and the length of the loan. 3. Aount, $5,000; ate, 10%; tie, 4 yeas 4. Aount, $6,000; ate, 8%; tie, 3 yeas 5. Aount, $8,000; ate, 7.5%; tie, 4 yeas 6. Aount, $10,000; ate, 8.4%; tie, 4 yeas 7. Aount, $12,500; ate, 8.25%; tie, 4 yeas 8. Aount, $10,500; ate, 9.75%; tie, 4 yeas 9. Aount, $1,900; ate, 8%; tie, 18 onths 10. Aount, $1,050; ate, 6.5%; tie, 15 onths In Execises 11 14, coplete the fist thee lines of an aotization schedule fo each loan. You answe should look like Table The loan descibed in Execise The loan descibed in Execise The loan descibed in Execise The loan descibed in Execise 9 Applying What You ve Leaned 15. Paying off a otgage. Assue that you have taken out a 30yea otgage fo $100,000 at an annual ate of 7%. a. Constuct the fist thee lines of an aotization schedule fo this otgage. b. Assue that you have decided to pay an exta $100 pe onth to pay off the otgage oe quickly. Find the fist thee lines of you payent schedule unde this assuption. c. What is the diffeence in inteest that you will pay on the otgage duing the fouth onth if you pay the exta $100 pe onth vesus paying only the equied payent? *Note that you can find this aount oe quickly by ultiplying the diffeence in otgage payents, $100.51, by 240 onths. 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.
9 438 CHAPTER 9 y Consue Matheatics 16. Paying off a otgage. Repeat Execise 15, but assue that the otgage is a 20yea otgage fo $80,000 and the annual ate is 8%. In Execises 17 20, a) the onthly payent fo each aotized loan and b) the total inteest paid on the loan. Assue that all inteest ates ae annual ates. 17. Paying off a boat. Wilfedo bought a new boat fo $13,500. He paid $2,000 fo the down payent and financed the est fo 4 yeas at an inteest ate of 7.2%. 18. Paying off a ca. Beatice bought a new ca fo $14,800. She eceived $3,500 as a tadein on he old ca and took out a 4 yea loan at 8.4% to pay the est. 19. Paying off a consue debt. Fanklin s new skis cost $350. Afte his down payent of $75, he financed the eainde at 18% fo 5 onths. 20. Paying off a consue debt. Richad s used otocycle cost $3,500. He paid $1,100 down and financed the est at 8.5% fo 2 yeas. In Execises 21 24, assue that all otgages ae 30yea, adjustable ate otgages. In each situation, use the additional infoation given to calculate the onthly payent on the otgage a) in yea one and b) in yea five.* 21. P = $200,000; beginning inteest ate, 4%; ate inceases 2% pe yea 22. P = $180,000; beginning inteest ate, 3.5%; ate inceases 7% by yea five 23. P = $220,000; beginning inteest ate, 4.4%; ate inceases 2%, then 2%, then 1%, then 1.8% 24. P = $160,000; beginning inteest ate, 3.6%; ate inceases 2%, then 1.6%, then 1.8%, then 2% In Execises 25 30, find the pesent value of each annuity. Assue that all ates ae annual ates. 25. The value of a lottey pize. Macus has won a $1,000,000 state lottey. He can take his pize as eithe 20 yealy payents of $50,000 o a lup su of $425,000. Which is the bette option? Assue an inteest ate of 10%. 26. The value of a lottey pize. Belinda has won a $3,400,000 lottey. She can take he pize as eithe 20 yealy payents of $170,000 o a lup su of $1,500,000. Which is the bette option? Assue an inteest ate of 10%. 27. Pesent value of a ca. If Addison can affod ca payents of $350 pe onth fo 4 yeas, what is the pice of a ca that she can affod now? Assue an inteest ate of 10.8%. 28. Pesent value of a ca. If Pete can affod ca payents of $250 pe onth fo 5 yeas, what is the pice of a ca that he can affod now? Assue an inteest ate of 9.6%. 29. Planning fo etieent. Shane has a etieent plan with an insuance copany. He can choose to be paid eithe $350 pe onth fo 20 yeas, o he can eceive a lup su of $40,000. Which is the bette option? Assue an inteest ate of 9%. 30. Planning fo etieent. Nico has a etieent plan with an investent copany. She can choose to be paid eithe $400 pe onth fo 10 yeas, o she can eceive a lup su of $30,000. Which is the bette option? Assue an inteest ate of 9%. In Execises 31 36, find the unpaid balance on each loan. 31. Paying off a loan ealy. You have taken an aotized loan at 8.5% fo 5 yeas to pay off you new ca, which cost $12,000. Afte 3 yeas, you decide to pay off the loan. 32. Paying off a loan ealy. In ode to pay fo new scuba equipent, you took an aotized loan fo $1,800 at 11%, which you ageed to epay in 3 yeas. Afte 18 payents, you decide to pay off the loan. 33. Paying off a otgage ealy. The MacGuffs took out a 30yea otgage fo $120,000 on thei vacation hoe at an annual inteest ate of 7%. They decide to efinance the otgage afte 8 yeas. 34. Paying off a otgage ealy. In ode to odenize thei estauant, the Buccos took out a 25yea otgage fo $135,000 at an annual inteest ate of 6%. They decide to efinance the otgage afte 10 yeas. 35. Paying off a loan ealy. Gaett took out an aotized loan fo $8,000 fo 2 yeas at 8% to finish culinay school. Afte 12 payents, he decided to pay off the loan. 36. Paying off a loan ealy. Sheila boowed $14,000 to invest in he floal shop. She took out an aotized loan at 6% fo 5 yeas. Afte aking payents fo 1 yea, she decided to pay off the loan. *To keep these execises siple, duing yea five, use the sae values fo P and n as you used in yea one. Technically, to get a oe exact answe, you should take into account that by yea five, soe of the pincipal would have been paid off and also that only 26 yeas of payents eain. Ignoing these does not affect the spiit of the execise because duing the fist few yeas, ost of you otgage payents ae going towads inteest. Copyight 2010 Peason Education, Inc.
10 9.6 y Looking Deepe: Annual Pecentage Rate 439 In Execises 37 42, you ae given the aount of an aotized loan P, the annual inteest ate, the nube of payents of the loan n, and the onthly payent R. Afte the specified nube of onths, the boowe decides to efinance at the new inteest ate fo the eaining length of the loan. a) What is the new onthly payent? b) How uch does the boowe save on inteest? 37. P = $10,000; = 8%; n = 48; R = $244.13; efinance afte 24 onths at an inteest ate of 6.5%. 38. P = $20,000; = 9%; n = 48; R = $497.71; efinance afte 12 onths at an inteest ate of 7%. 39. P = $100,000; = 8%; n = 240; R = $836.45; efinance afte 60 onths at an inteest ate of 7%. 40. P = $100,000; = 9.5%; n = 360; R = $840.86; efinance afte 120 onths at an inteest ate of 7.5%. 41. P = $40,000; = 8%; n = 48; R = $976.52; efinance afte 24 onths at an inteest ate of 7.5%. 42. P = $50,000; = 10%; n = 60; R = $1,062.36; efinance afte 24 onths at an inteest ate of 9.5%. Counicating Matheatics 43. In the foula fo finding the payents on an aotized loan, whee have you seen the expessions on the left and ight sides of the equation befoe? 44. What does Exaple 5 show you with egad to the benefit of efinancing a loan? 45. What do the wods aotize and otality have in coon? In light of the TheeWay Pinciple, how does this help you undestand the wod aotization? 46. Why is the inteest payent on the otgage deceasing in Table 9.4? Using Technology to Investigate Matheatics 47. Ask you instucto fo tutoials and speadsheets to pefo the aotization coputations that we did in this section. Duplicate soe of the esults that we obtained in ou exaples.* 48. Find otgage calculatos on the Intenet and use the to epoduce soe of the exaples that we explained in this section. Repot on you findings. Fo Exta Cedit 49. In deciding to efinance at a lowe inteest ate, how does it affect you payents if you decide to efinance ealy in the loan peiod vesus late in the loan peiod? Fo exaple, fo a 60onth loan, would the new payents be lage, salle, o the sae if you efinance afte 12 onths instead of 36 onths? Make up nueical exaples to answe this question. Explain you answe. 50. Soe otgage ageeents allow the boowe to ake payents that ae lage than what is equied. Because this exta oney goes towad educing pincipal, inceasing you payents ay allow you to pay off the otgage any yeas ealie, thus saving a geat aount of inteest. Assue you take out a 30yea aotized loan at 8% fo $100,000 and you onthly payents will be $ Suppose that instead of aking the specified payent, you incease it by $100 to $ How uch do you save on inteest ove the life of the loan if you ake the lage payent? Copyight 2010 Peason Education, Inc.
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