The Present Value of an Annuity

Transcription

1 Module 4.4 Page 492 of 944. Module 4.4: The Preset Value of a Auty Here we wll lear about a very mportat formula: the preset value of a auty. Ths formula s used wheever there s a seres of detcal paymets whether gog towards payg o a debt, supportg someoe (cludg yourself) durg retremet, payg o a house, compesatg someoe for jury, or ay other reaso. Before we beg to use ths very powerful formula, we wll t derve two ways. The frst represets a mathematca s vew, ad the secod a ecoomst s vew. Hopefully, oe vew or the other wll match your learg style. Alteratvely, you ca just memorze the formula, so log as you kow how to use t. How may tmes have we heard a televso commercal say 12 easy equal mothly paymets for owershp of some sazzy product? A auty s actually ths sequece of equally spaced paymets of equal amouts. Ths paymet method apples to a eormous umber of facal stuatos. Our goal the ext few boxes s to derve ad expla the followg: PV = c 1 (1 + ) whch wll be used to solve a whole rage of problems, from car loas to home mortgages ad all sorts of other facal arragemets. Do you remember the formula for a geometrc seres that stops early? If ot, the go reread t o Page 479, before readg the rest of ths box. Let us say that Chuck has a mortgage, whose terest compouds mothly at 6%. He s sedg checks at the value of $ 2000 o the frst of the moth, every moth for 30 years. O the day that ths arragemet s cotracted, how much s ths sequece of paymets worth, from the bak s perspectve? Ths wll be the mortgage amout, dollars, that Fred should expect to receve from the bak upo sgg the cotract. Let us assume that the cotract s sged oe moth before the frst paymet, as s stadard practce. I ths case the the frst paymet occurs after oe moth, the secod after two moths, as well as the thrd after three moths. The th paymet occurs after moths. Usg the Tme Value of Moey formula (see Page 452), we would the value the frst paymet at 2000/(1 + ) 1, the secod paymet at 2000/(1 + ) 2, ad so forth. Ths gves us: PV = 2000 (1 + ) (1 + ) (1 + ) (1 + ) 4 (1 + ) 359 (1 + ) 360 where 360 = s the umber of paymets (more precsely, the total umber of moths 30 years). Yet, what s? Sce r =0.06 ad m = 12, the =0.06/12 = Substtutg that we get: PV = 2000 (1.005) (1.005) (1.005) (1.005) 4 (1.005) 359 (1.005) 360 Does ths seres look famlar? It should, because we calculated t o Page 479. Its value s $ 333,583, ad s how much the bak wll gve Fred for ths mortgage agreemet.

2 Module 4.4 Page 493 of 944. Now suppose I have a three-year car loa wth mothly paymets. My mothly paymets wll last for 3 years, whch comes to 3 12 = 36 paymets. The frst paymet wll be oe moth after purchase, followed by the secod paymet two moths after, the three moths after, ad so o utl thrty sx moths have passed. Usg the Tme Value of Moey formula (see Page 452) I ca compute the value of each paymet. As wth the prevous example, the th paymet occurs after moths. If s the prevalg rate of terest (wrtte mothly) ad c s the dollar value of the check I sed the bak each moth, the the preset value of the th paymet s PV = FV (1 + ) = c (1 + ) The total value of all 36 paymets would be ther sum: PV = c (1 + ) 1 + c (1 + ) 2 + c (1 + ) 3 + c (1 + ) 4 + c (1 + ) c (1 + ) 35 + c (1 + ) 36 At ths pot please pause ad take a momet to covce yourself that ths s a geometrc seres, stoppg early, wth frst etry c/(1+) ad last term c/(1+) 36, ad commo rato 1/(1 + ). Oly oce you uderstad how ths seres s composed should you cotue to the ext box, where we wll calculate the value. Okay, so the sum of the above geometrc seres (stoppg early) s: S = a c rz 1 c r I the prevous box, we computed that the frst etry s a = c/(1 + ). We also computed that the last etry s z = c/(1 + ) 36 ad that the commo rato s c r =1/(1 + ). We should plus these ow, ad we obta S = c c (1+) 36 Now we wll multply the umerator ad deomator by (1 + ) ad obta: c c (1) (1+) S = 36 (1 + ) 1 h 1 c 1 (1+) S = (1 + ) S = c

3 Module 4.4 Page 494 of 944. A sequece of paymets equally spaced throughout tme ad of equal value s called a auty. I the prevous box we derved the formula for a auty wth 36 paymets, but keep md that ay umber of paymets s acceptable. If there are m paymets per year for t years, the = mt s the umber of total paymets. The value of each paymet s c. If the terest rate s r, thelet = r/m as usual. The preset value of ths sequece of paymets, the, s calculated by PV = c 1 (1 + ) ad ths s a formula that we re gog to get some serous mleage out of. We just saw the classc mathematca s dervato of the preset value formula for a auty. The paymets (after adjustmet by the Tme Value of Moey Formula) form a geometrc seres that stops early, so we aturally use the formula for geometrc seres stoppg early to fd the sum. There s really othg else to t. The ecoomst s dervato of the preset value of a auty s a bt d eret, but brgs you to the same aswer. It ca be descrbed as a seres of observatos. We wll beg wth those observatos the ext box. We wll ow approach the dervato of the PV formula from the perspectve of a ecoomst or facer. Before we beg you should probably revew the cocept of a perpetutyforbore, gve o Page 472. Let s say (very hypothetcally) that you have a perpetuty-due wth the bak, ad that the bak has a perpetuty-due wth you. That meas that you gve the bak a check for c dollars o the frst of every moth, ad they also preset you wth a check for c dollars o the frst of every moth. Remember, wth a perpetuty, the paymets ever ed. I ths case, the whole thg s a complete wash. The paymets cacel out ad othg happes facally. Now let s say that the bak forbears ts perpetuty by three years. Ths meas you wll have a proper perpetuty-due gvg the bak a check for the frst three years o the frst of the moth, every moth, utl the ed of tme but the bak delays by 37 moths ad you get a check from them startg oly o the moth after the three year aversary moth 37. Ths wll be followed by a check from them moth 38, moth 39, moth 40, ad so o. The prevous bullet ca be thought of two stages: Stage Oe s the frst 36 moths, ad Stage Two s everythg afterward. I Stage Two the stuato s detcal to the frst bullet ths lst, so facally othg s happeg. I Stage Oe, however, you re gvg the bak a check o the frst of the moth, moth after moth, for three years. Stage Oe represets a stadard loa wth 3 12 = 36 equal mothly paymets. It s a regular schedule of detcal ad equally spaced paymets, ad therefore a auty. What s the preset value of ths arragemet? The value of the perpetuty-due that you are payg the bak wth your mothly checks (all of value c) would be PV = c/. The value of the perpetuty-forbore (wth = 37) that you are gettg from the bak s c PV = (1 + ) 37 1 = c (1 + ) 36 whch s based o the formula from Page 472. Now we ca complete ths dervato the ext box.

4 Module 4.4 Page 495 of 944. Cotug wth the prevous box, we kow that the etre arragemet s value must be the perpetuty-due mus the perpetuty forbore. The calculato s t so bad: c c (1 + ) 36 = c(1 + )36 c (1 + ) 36 (1 + ) 36 = c(1 + )36 c (1 + ) 36 = c (1 + ) 36 1 (1 + ) 36 c (1 + ) 36 1 = (1 + ) 36 c (1 + ) 36 1 = (1 + ) 36 (1 + ) 36 c 1 = 1 (1 + ) 36 c = 1 (1 + ) (1 + ) = c We ve derved ths mportat formula ow, wth two very d eret pots of vew. I hope that you uderstood at least oe or the other dervato. If you uderstood both, the that s pheomeally good ews. However, f you uderstood ether, you have two choces. You ca just memorze the formula f you lke but that s accdet proe ad also wo t assst you usg the formula correctly. Alteratvely, you could reread oe of the dervatos. Now that we have derved our formula, let s use t! Perhaps I ca a ord a car paymet of $ 300 a moth. If I wat a fve-year car loa, what should the value of my loa be? Let the terest rate be 4.80%. We beg wth the formula PV = c 1 (1 + ) ( ) = 300 = = 15, where =0.004 because = r/m =0.048/12. So I should get a car loa for $ 15, What f the car loa from the prevous example s ot eough? What paymet c, all other factors beg the same, s requred order to get a $ 20,000 loa? 1 (1 + ) PV = c ( ) 20, 000 = c , 000 = c , 000 = c( ) = c Well that was t so bad, but t would be ce f there were a shortcut!

5 Module 4.4 Page 496 of 944. # What f I wated a $ 40,000 car? Or a $ 10,000 car? What s the mothly paymet each case? [Aswer: $ ad $ ] Note, that s exactly double ad half the prevous example, respectvely. These computatos are ot horrbly log, but they are probably too much for the average bak employee to perform o the spot. Istead, we eed a a shortcut. As you saw the prevous boxes, f you double the PV the you double the paymet c; f you halve the PV the you halve the paymet c. We ca coclude that the PV ad paymet are drectly proportoal. That eables us to use the cost per thousad techque for autes. (We frst saw that techque o Page 344.) Frst, use PV = 1000 to fd c. The f someoe wats a $ 40,000 car, just multply the cost per thousad by 40 to fd ther paymet. Lkewse, f someoe wats a $ 30,000 car, multply the cost per thousad by 30 to fd ther paymet. If you have may calculatos to do, ths ca be a huge tme-saver. Suppose you work for a car dealershp that s o erg 2.5% facg compouded mothly, wth o dow-paymet. What s the cost-per-thousad for a 3-year car loa? Frst we must fd. That s 0.025/12 = The we use 36 1 (1 + ) PV = c 36 1 ( ) 1000 = c = c = c = = c We lear that the cost-per-thousad s $ Ths would be oe of those tmes where you do ot roud moey to the earest pey, because you d be destroyg a good deal of accuracy f you dd that. These problems ca be a bt loger tha what you re used to. The trck to dog them successfully s to allow your pecl to share the burde wth your bra. Frst, wrte out the formula full. The t s much easer to plug the correct values. Next, proceed to slowly move toward the correct aswer by calculato ad algebrac mapulato. Not oly does ths reduce the chaces of a error beg made, but also t creases the partal credt you ca obta. You ve show the structor that you kow both the formula ad also where to plug the data.

6 Module 4.4 Page 497 of 944. The large fracto 1 (1 + ) keeps comg up our calculatos. Ths s sometmes called the growth factor, butless commoly the compoudg factor or the power of compoudg. (We llusetheterm growth factor. ) You mght wat to revew the cocept of compoudg factor from Page 344. Ths s a mportat value, ad t reflects the mpact of the paymet c. I the prevous example, the fracto evaluated to ad multplyg ths by the mothly paymet of $ yelds = wth the mssg pey just beg roudg-error. It turs out that face people have ther ow symbol for the growth factor because t s so mportat. We wll have o use of the symbol, but you should see t oce so that f you see t later aother class or durg a job tervew you wll ot pac. Face people wll ofte wrte to represet the growth factor of a seres of paymets. If a loa s to be mothly for fve years, ad the terest rate s 7%, the what s the cost per thousad? [Aswer: $ ]... what s the compoudg factor? [Aswer: ] # Suppose my fred loas me some moey, whch I wat to pay back fve moths. The loa s for $ 2000, ad we agree upo a terest rate of 10% (compouded mothly) because he s sellg vestmets to make the loa to me. How much should each paymet be? [Aswer: $ ] What s the compoudg factor? [Aswer: ] # What s the cost-per-thousad? [Aswer: $ ]

7 Module 4.4 Page 498 of 944. A Amortzato Table for a loa ca be a excellet way to uderstad what s gog o sde the loa. It s very easy to make oe. I the frst colum of the table I wll wrte the headg Perod # ad the Startg Value the secod. I the thrd colum I wll put Iterest Accrued ad the fourth colum I wll put Paymet. Fally, the ffth ad last colum s Edg Balace. We wll use the loa from the prevous problem. The startg prcpal s $ 2000, so for the frst perod that goes uder Startg Value. The terest s gog to be tmes the startg value for that perod, whch s /12 = (Notce I m devatg from our ormal roudg polcy of trucato. For ths problem we ll roud to the earest umbers, because a amortzato table roudg errors accrue very fast ad buld o each other.) The paymet s always $ , as we calculated the prevous box. The edg balace for Perod 1 s therefore = For the ext moth, the starg value s the edg balace of the prevous moth, or $ The the terest rate s /12 = The paymet remas $ , ad we have the edg balace as = The remag three moths work just lke the secod moth. The table s gve the ext box. The amortzato table from the prevous example would be tabulated for Row 3, Row 4, ad Row 5 by the same methods as Row 2. After that, the complete table should look lke: Perod # Startg Iterest Paymet Edg Value Accrued Balace 1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ 6.75 $ $ $ $ 3.39 $ $ 0.03 (The extra three cets at the ed s roudg error that we ll address two boxes from ow.) Now try t yourself wth a $ 3000 loa, wth terest 12% compouded mothly, ad durato of sx moths. Use the PV formula to fd the paymet, whch should be $ Oce you have that, costruct the Amortzato Table. The soluto s gve at the ed of the module, o Page 513. # # The extra three cets two boxes ago mght bother some studets. Let s explore t ths way: Cosder the $ 2000 loa at 10% compouded mothly, for whch we just dd a amortzato table. What wll 5 moths at 10% compouded mothly produce for a paymet of $ ? [Aswer: $ ]... for a paymet of $ ? [Aswer: $ ]

8 Module 4.4 Page 499 of 944. Ths roudg dscrepacy almost always happes. The math actually calls for a paymet of c = but there s o way to pay for $ so mathematcal correctess just s ot possble. You ca ether roud up to or dow to I practce, baks wll roud upward to the earest pey, thus you slghtly overpay them, whe you take out a loa. If they rouded dowward, the you would slghtly uderpay them. It s mportat to realze that we are oly talkg about a few cets for small loas, ad eve for home mortgages t s ever as large as a $ 10 d erece. Cosder a mortgage of $ 500,000, at 7% terest for 30 years. Of course, t s compouded mothly, as are all mortgages. What value of c does the math call for? [Aswer: ] What s the PV of a auty at 7% compouded mothly for 30 years ad a paymet of $ ? [Aswer: $ 499, ] # What s the PV of a auty at 7% compouded mothly for 30 years ad a paymet of $ ? [Aswer: $ 500, ] As you ca see from the prevous box, the fact that we caot pay the exact value of c s a matter of $ 1.51 eve for a half-mllo dollar house. Ths ty d erece s ot worth frettg over or spedg more tme o. The purpose of ths box ad the prevous two boxes was to teach you that... roudg error does exst, ad t must be take to accout, but... roudg c upward to the earest pey wll esure that ths s ot a problem for ay busess. Suppose I ca a ord a mothly house paymet of $ The gog rate for mortgages s perhaps 7%. I have $ 35,000 my savgs accout for the dow paymet, whch my bak tells me must be 10% of the value of the house (or more). How much of a house ca I a ord? I would lke a 30-year mortgage. Frst, we should fd ad. Wehave = r/m =0.07/12 = , ad = m t = = 360. Now we ca use our formula: 1 (1 + ) PV = c ( ) PV = PV = PV = 300, Thus I lear that I ll get $ 300, from my mortgage. The maxmum value I ca purchase would therefore be 300, , 000 = 335,

9 Module 4.4 Page 500 of 944. Let s recosder the prevous example, f the terest rate were stead oly 5%. How much s the value of a mortgage wth 5% terest, $ 2000 mothly paymets, ad 30 years durato? [Aswer: $ 372, ] # Alteratvely, f there s a house costg $ 335,615 whch I d lke, ad I use $ 35,000 for the dow paymet, ad allocate $ 300,615 for the mortgage, what wll my mothly paymet be? [Aswer: $ ] Now we ca see why lowerg the terest rate ca be a boo to the ecoomy. Whe the Federal Reserve Bak ( The Fed ) lowers the prme rate, the baks ca gve lower terest rates o mortgages. Note, each dvdual bak stll decdes what rate to charge. However, the terest rates are tercoected ways that we should probably ot go to ow. I the prevous checkerboard box, we saw that whe the terest rate fell from 7% to 5%, the future homebuyer could get a mortgage for $ 372,563 stead of for oly $ 300,615. That s a huge d erece! It meas that people ca a ord larger homes, or eve to buld ew homes, ad that wll employ people the costructo dustry. Also, whe people move to larger homes, state, couty, ad tow govermets ca collect more property taxes, causg growth schools, roads, mass trast, prsos, ad bureaucratc jobs to oversee that growth. We also saw that f the homeower decded to stck wth the $ 335,615 house, the stead of payg $ 2000 per moth, the mothly paymet would fall to $ That extra $ per moth s extremely sgfcat. It could be spet o vacatos, etertamet, or a ew car. Ths would help the toursm, ght lfe, or automoble dustry. Lkewse, the moey could also be vested, ad that would (slghtly) rase the value of all stocks as more moey flowed to the market. Wth the prevous box md, we should also cosder the dark sde of lowerg the terest rate. Frst, mage yourself as the seller of that $ 335,615 house. Image that you have several potetal customers who are lookg to sped aroud $ 320k to $ 340k, whe the terest rate s 7%. Suddely, the terest rate s ow 5% ad they all ca a ord aroud $ 360k to $ 380k! Is t ot temptg to try to ask more for your house? After all, f you receve several o ers, you ca beg to egotate ad try to get a better prce. For ths reaso, housg prces ca rse ad durg the housg boom of , they rose out of cotrol to very hgh prces. Ths s called a bubble, ad caused a whole host of problems. We do t have tme to go to that ow, but you ca see how terest rates ca have a huge mpact o the ecoomy. It s worthwhle to make a ote that the terest rate would ever chage from 7% to 5% overght. Usually the cremets are 1/4th of 1%, or 25 bass-pots. However, the example s easer to uderstad wth a larger (though urealstc) chage.

10 Module 4.4 Page 501 of 944. If you get a mortgage from a bak, ad the caot make the paymets, the the bak loses a great deal of moey. Of course, the bak ca foreclose o the mortgage but the they have to seze the house, evct the resdets, fx t up, put the house o the market, fd a buyer, ad close the sale. All that requres a lot of work by hghly pad professoals. Oe of the ways that the bak ca avod ths fasco s to make sure that people do ot purchase more house tha they ca a ord. Two major strateges for that would be to set a mmum dowpaymet sze, ad to ask the future homeower what ther maxmum mothly paymet could be, ad the lmt the homebuyer accordg to these two restrctos. Let s suppose that my bak requres 10% dow. I ca a ord $ 2000 per moth, ad the bak s o erg a 7% mortgage. Furthermore suppose that I ca a ord a dowpaymet of $ 35,000. How much of a house ca I a ord? The dow paymet would be at least 10% of the mortgage, ad s lmted to $ 35,000; lkewse the mortgage would be at most $ 300, Why? Because we calculated the PV of a mortgage of 30 years, 7% terest, ad c = 2000 as a example o Page 499, ad we leared that the preset value of such a sequece of paymets s $ 300, If p s the prce of the house, the 0.1p s the mmum dow paymet (lmted to $ 2000) ad 0.9p s the maxmum mortgage (lmted to $ 300,615.13). So I would wrte 0.1p apple 35, 000 whch becomes p apple 350, 000. Now we ve modeled the dowpaymet, so we should model the PV of the mortgage as well. We would wrte 0.9p apple 300, whch becomes 300, p apple = 334, to take care of the PV of the mortgage. The fal aswer s the strcter of these two requremets. Therefore, I must restrct myself to houses that cost $ 334, or less, to satsfy both requremets. # Repeat the prevous example f I have oly $ 25,000 savgs. All other detals rema the same. How much of a house ca I a ord? What requremet, the dowpaymet or the mothly paymet, s the restrcto? [Aswer: I ths case, the 10% dow-paymet s the lmtato, ad I am restrcted to houses that cost up to $ 250,000.] # Repeat the prevous example f I have $ 30,000 savgs, but the terest rate becomes 9% for mortgages. (That s a bt hgh, but t happeed durg the Reaga era.) All other detals rema the same. How much of a house ca I a ord? What requremet, the dowpaymet or the mothly paymet, s the restrcto? [Aswer: I ths case, the mothly paymet s the restrcto. I am lmted to houses that cost $ 276, or less.]

11 Module 4.4 Page 502 of 944. Well, that was depressg. Let s try some uplftg examples. Let s suppose I budget coservatvely. Image that I buy a smaller/cheaper house tha I ca a ord because I do t wat a oppressve mortgage paymet each moth. I m gog to buy a $ 250,000 house, payg a 10% dow paymet. Let s suppose the terest rate s 6.50%. What s the value of the dow paymet? [Aswer: $ 25,000.] What must the PV of the mortgage be? [Aswer: $ 225,000.] # What s the requred mothly paymet, c? [Aswer: $ ] Now, the ext box, we re gog to explore how I ca save some moey the log ru, by overpayg the bak each moth. Cotug wth our aalyss from the prevous box, ow suppose that I pay the bak double the mothly paymet every moth. You re always allowed to smply overpay. (Some mortgages but very few wll have pre-paymet pealtes; because these are rare, we ll gore them.) Suppose you pay = per moth whch s double the requred amout. Ths s a eormous facal sacrfce, but as we are about to see, a extremely e ectve oe. How log wll the mortgage take to be pad? (I other words, what s the value of whe PV = 225, 000 ad c = ?) 1 ( ) 225, 000 = , = 1 ( ) = 1 ( ) = ( ) log( ) = log ( ) = log( ) = ( ) = Thus, after 104 moths (because 103 wo t be qute eough, beg less tha the that the math foud) you wll be mortgage-free. That s 8 years ad 8 moths far, far shorter tha 30 years! I thk t s worthwhle to reflect upo the power of the facal techque the prevous example. If I get the mortgage at age 28, the I would be makg the $ per moth paymets for 30 years, or utl I am age 58. However, f I pay the $ , the I wll be fshed at age 36 (ad 8 moths.) Ths way, I would be able to ejoy all that extra moey each moth, from age 37 utl age 58. Let s look at how to quatfy how much I am savg.

12 Module 4.4 Page 503 of 944. The total pad to the bak for the mortgage would be the sum of all the values of c. Ithe prevous box, the orgal mortgage, there were = 360 paymets of $ , whch s a total of = 511, I retur for ths, you were pad $ 225,000 to buy the house wth. The questo s, where dd the 511, , 000 = 286, ed up gog? That s the total terest pad. I retur for beg let $ 225,000, you have gve the bak more tha a quarter of a mllo dollars. Ths s why baks lke to ssue mortgages! Ths s how they make moey. I the secod case, there were 104 paymets. The last paymet would be fractoal, ad we ll lear how to tabulate that o Page??. A reasoably close model s to take the 104th paymet as equal to the other 103 paymets. That comes to ad the total terest pad ths case s = 295, , , 000 = 70, Now we ca see how much we are savg, by aalyzg the mathematcs of the prevous box. If we follow the orgal pla, we are payg $ 286,974 terest. However, f we pay double the requred paymet per moth, are gvg payg $ 70, terest. Ths meas that we would save 286, , = 216, by payg double the ormal paymet each moth. That s a huge quatty of moey! The mortgage from two boxes ago was realstc, but payg double the paymet volutarly s extraordary ad ucommo. Suppose, more realstcally, that you pay 10% more tha the requred paymet. What s the paymet? [Aswer: The paymets are $ ] How may terms does the mortgage take to be pad o? [Aswer: = moths whch meas 280 moths or 23 years ad 4 moths.] # What s the total terest pad? [Aswer: $ 213, ] We wll ow repeat the aalyss of the prevous box, but supposg that you pay 20% more. What s the mothly paymet? [Aswer: $ ] How may terms does the mortgage take to be pad o? [Aswer: moths, whch s really 232 moths or 19 years ad 4 moths.] # What s the total terest pad? [Aswer: $ 170, ]

13 Module 4.4 Page 504 of 944. Re-examg the prevous box, take a momet to reflect o how great a deal ths overpaymet of 10% turs out to be. Let s compare t to the orgal loa. Istead of 30 years, you wll make 19 years ad 4 moths of paymets. Istead of gvg $ 511,974 terest to the bakers, you wll be gvg them $ 170, terest. The best way to check a problem of the type we just saw s to otce that we are clamg that the loa wll be pad o betwee the 231 st ad 232 d paymets f you pay $ Therefore, to be really precse, you ca calculate the PV of a auty wth the gve terest rate of 6.5% ad the paymet of $ You should calculate t twce, oce wth = 231 ad oce wth = 232. If you are a rush o a exam, calculatg oly oe of these wll probably su ce. PV = c 1 PV = c 1 (1 + ) (1 + ) ( ) = = =224, ( ) = = =225, As you ca see, they straddle the correct value of $ 225,000, as desred. To check the other checkerboard, where we overpad by 10%, we just chage the paymet to $ ad the umber of perods to = 279 ad = 280. The we have PV = c 1 PV = c 1 (1 + ) (1 + ) ( ) = ( ) = = =224, = =225, Oce aga, they straddle the correct value of $ 225,000, as desred. The total terest pad, as before, s the sum of all the mothly paymets mus the value of the loa. Do ot clude the dow paymet. The face charge s the total terest pad plus ay fees. However, the fees o mortgages ad other loas are ofte extremely complex, so we wll save ths detal for future coursework.

14 Module 4.4 Page 505 of 944. Let s suppose a paymet pla for a home-etertamet system valued at $ 2995 s for 12 equal mothly paymets of $ What s the total of all the paymets? [Aswer: $ ] What s the total terest pad? [Aswer: $ ] Note: Sce there are o fees, ths s also the face charge. # The total of all the paymets represets what percetage markup of the orgal prce? [Aswer: A markup of 8.16%.] I the prevous box, we are makg mothly paymets of $ , whch seems reasoable, ad the markup s oly 8.16%. Thus, we mght be coed to belevg that ths s ot a terrbly bad loa. Careful aalyss wll ow prove otherwse. As you ca see, the schedule of paymets s fxed, ad all the paymets are equal. Whle 12 moths s much shorter tha the 30 years of a mortgage, mathematcally the formulas of a auty stll apply. (Ths s because all the paymets are the same, there s a fxed umber of them, ad they are equally spaced.) I am ow gog to clam that the terest rate s 14.73%. You ll lear to calculate that yourself very shortly (o Page??), but for ow you should verfy my clam. We are payg mothly, so m = 12 ad thus = r/m =0.1473/12 = Next, c = , ad so we are ready to plug to our formula: PV = c 1 (1 + ) whch s close eough ( ) = = = A Pause for Reflecto... I the prevous example, we leared two mportat lessos. Whle the terest rate appeared o a tutve level to be low, we see that fact the rate s 14.73%, whch s rather hgh but ot outrageously hgh. We leared a techcal face lesso: computg the percetage markup requred to chage the value of the tem to the sum of the paymets, whle mathematcally vald, tells you lttle about the terest rate of the loa. However, we also saw a broader lesso that has appeared before: compoud terest s couter-tutve; do ot trust your tuto. Calculato, o the other had, yelds relable aswers.

15 Module 4.4 Page 506 of 944. I the home-etertamet system example above, we saw how the percetage markup told us a uderestmate of the terest rate. You mght be woderg f ths s always the case. It turs out that sometmes the percetage markup s a overestmate. Cosder the orgal mortgage that we aalyzed o Page 502. The paymets were each $ , mothly for 30 years retur for $ 225,000 to buy the house wth. (Note: we exclude the tal dow paymet of $ 25,000.) What s the sum of all the paymets? [Aswer: $ 511,974.] What s the markup? # [Aswer: A markup of % would tur $ 225,000 to $ 511,974.] As you ca see, % s much larger tha 6.5%, whch s the actual terest rate of the loa. Because calculatg a percetage markup s a mathematcal operato that s both easy ad frequetly used, busess studets are very famlar wth t. However, the percetage markup a auty problem does ot tell you the terest rate. Sadly, may busess studets thk that t does. We saw that oe case, the percetage markup was a drastc uderestmate of the true terest rate. We saw the other case, that the percetage markup was a pheomeal overestmate of the true terest rate. The bottom le s that the percetage markup s ot a useful tool aalyzg a auty. There s a car that you wat sellg for $ 8995, ad you put a tal $ 1000 dow paymet o t. You obta a car loa for 9% compouded mothly. (Ths s ot a great rate, actually, but t ca happe f your credt ratg s poor.) What s the mothly paymet o the $ 7995 loa f you get a 3-year loa? [Aswer: $ ] How about a 4-year loa? [Aswer: $ ] # Ad a 5-year loa? [Aswer: $ ]

16 Module 4.4 Page 507 of 944. The followg problem was suggested by Joel Casser, a chldhood fred of me who works at a car dealershp. We re gog to look at a car loa ad make the problem as realstc as possble. You wsh to purchase a car, whose total cost s $ 29,995 plus 7.5% sales tax. The requred dow paymet s 20%. You re tradg a car worth $ Thus you eed to pay to whch we apply the sales tax ad get 29, = 22, , 495(1.075) = 24, You have three optos. The frst opto s 0% facg for three years. The secod opto s 2.99% facg for 3 years, but wth a $ 5000 cash-back cetve. The thrd opto s a $ 6000 rebate, but 3.25% facg. All three optos are compouded mothly. Your moey-market accout s curretly yeldg 3.50%, compouded daly. (Use a 360-day year.) Sce the amout left s $ 24, the a 20% dow paymet wll be 24, = ad the loa tself wll have a prcpal of 24, = 19, Now we ll vestgate all three optos. The 0% facg case s easy. Ths s probably the frst tme you ve see 0% facg ths book, so do t feel bad f you dd t kow what to do. We ll exame ths stuato a bt more o Page 545. As you ca see, zero-percet facg s just a log way of sayg that the paymets are a equal dvso of the cost of the tem, wth o terest pad at all. The shortcut for calculatg ths s to realze that there are 36 equal mothly paymets, so the paymets wll be 19, /36 = The 2.99% case s just a preset-value calculato. We kow that the rate r = s compouded mothly, so m = 12 ad the we have = r/m =0.0299/12 = The preset value s the amout to be pad mus the cash-back cetve, or 19, = 14, Sce the loa s for 3 years, there wll be = 36 paymets. The we have 36 1 (1 + ) 1 ( ) 14, = c = c = c We the ca calculate c = 14, / = That s very sgfcatly cheaper! = ( )c The thrd opto s more terestg. We have r = 3.25% therefore = /12 = The preset value s $ 19, The we have 36 1 (1 + ) 1 ( ) 19, = c = c = c = ( )c ad the we ca calculate c = 19, / = That s the most expesve paymet yet. However, that does t mea that t s the worst opto. Wat! What about the $ 6000 rebate? Ths s moey that s maled to you after the fact you smply get a check for ths amout. Perhaps you ll use t to replesh your checkg accout after makg the dow paymet, or perhaps you ll place t a savgs accout, case you re dager of mssg a few paymets. Alteratvely, you mght have some other plas for the moey. If you have o use for t, the you should take to cosderato that you could vest t ad ear terest. I ay case, ths s a huma cosderato that mathematcal face caot address.

17 Module 4.4 Page 508 of 944. Fally, the real way to settle the ssue s to see how much s pad for the car each case. I the frst case, I have paymets of $ for 36 moths or a total of = 19, I the secod case, I have paymets of $ for 36 moths or a total of = 15, , much better. I the thrd case, I have paymets of $ for 36 moths or a total of = 20, , whch looks hgh, but oce we subtract the $ 6000 rebate we get $ 14, Ths turs out to actually be the best opto. Suppose a state wth 5% sales tax, you are tradg a $ 5000 car for a $ 18,500 car. You ca have 6% facg wth a $ 4000 cash-back cetve, or you ca have 0% facg. The loa s for three years ad s compouded mothly. If you choose 0% facg, what s the value of your loa? [Aswer: $ 14,175.] If you choose 6% facg, what s the value of your loa? [Aswer: $ 10,175.] If you choose 0% facg, what s the mothly paymet? [Aswer: $ ] If you choose 6% facg, what s the mothly paymet? [Aswer: $ ] # What s the total that you pay for the car the 0% case? [Aswer: $ 14,175.] What s the total that you pay for the car the 6% case? [Aswer: $ 11, ] Whch s the better deal? [Aswer: the cash-back cetve, clearly.] Whle problems volvg the preset value of a auty formula are usually about debts, they also ca go the reverse drecto. The key s that there must be a regular ad fxed schedule of equal paymets. Pesos ad retremet beefts that go for a fxed legth of tme are excellet examples. Whle ot pleasat to thk about, Accdetal Death & Dsmembermet Isurace s a mportat part of may employer s beefts packages, partcularly f the work that the employee s dog s ay way dagerous. If you become completely dsabled ad uable to work, the most geerc plas wll gve you a bweekly paymet equal to your former wages utl you tur 65-years old. A mportat questo s, how much s that worth?

18 Module 4.4 Page 509 of 944. Suppose a worker makes $ 46,000 per year, ad s jured at age 28. Let us assume that the prevalg rate s 7%, ad sce he s pad bweekly we wll make the prevalg rate bweekly as well. To keep the math smple, let us assume he s jured o hs 28th brthday. He the has = 37 years of beefts comg to hm. Next, hs bweekly wage s 46, 000/26 = The umber of paymets would be = 962. The, because m = 26 we have = r/m =0.07/26 = Fally, the preset value s 1 (1 + ) PV = c ( ) = = = 607, You mght thk that the prevous box s a bt of a odd problem, but whe the workma gets jured, the surace compay eeds to kow ths value for two reasos. Frst, they wll move that sum of moey from ther operatg fud to ther auty fud (whch s operatg at 7%, we are told) to make the paymets avalable for the worker. Ths way the auty fud does ot have too much or too lttle moey t. Secod, the surace compay wll sed a adjustor to fd out who s resposble (the techcal term s lable ) ad sue that perso for damages. Aother mportat pot s that we dd ot adjust for flato the prevous problem. Most of these polces do ot, but a few wll, ad we wll lear how to calculate that o Page 568. Lastly, t s terestg to ote that these calculatos are farly sestve to the prevalg rate, whch we wll ow explore. Cosder the prevous example, but wth a worker who makes half as much ($ 23,000). [Aswer: $ 303, ] wth a worker who makes twce as much ($ 92,000). [Aswer: $ 1,215, ] # what s the relatoshp betwee the auty depost sum ad the worker s wages? [Aswer: a lear relatoshp, because double the wages doubles the sum, ad half the wages halves the sum.] Naturally, f your aswer s o the pees problems lke ths, where the aswer s the hudreds of thousads of dollars, or eve the mllos, the do ot worry. A aswer correct to sx sgfcat fgures s excellet.

19 Module 4.4 Page 510 of 944. Cotue wth the aalyss of the prevous example, but wth a worker who makes $ 46,000, but the prevalg rate s 6%. [Aswer: $ 683, ] wth a worker who makes $ 46,000, but the prevalg rate s 8%. [Aswer: $ 545, ] wth a worker who makes $ 46,000, but the prevalg rate s 3.5%. [Aswer: $ 953, ] # wth a worker who makes $ 46,000, but the prevalg rate s 14%. [Aswer: $ 326, ] 5. Does the same relatoshp exst wth the prevalg rate? (Ht, look at # 3 ad # 4.) [Aswer: No, ot eve close.] I a preset-value auty problem, double the PV meas double the paymet, ad half the PV meas half the paymet. Ths meas that the paymet ad the PV are drectly proportoal. That s also why t was justfable to use the cost-per-thousad methodology o Page 496. A Pause for Reflecto... Expla your ow words why the depost the prevous chessboard box s larger whe the terest rate s lower, ad s smaller whe the terest s hgher. Let us mage that you have wo the lottery. You are gve the choce of $ 26 mllo dollars pad over 20 years by a aual paymet of $ 1.3 mllo, or a lump-sum paymet of $ 19 mllo. Whch should you choose? Your fred who s a vestor says that 6.5% s the gog rate for secure log-term vestmets, ad so you decde to use ths as the prevalg rate. The way to solve ths problem s to calculate the preset value of the 20 paymets of $ 1.3 mllo. Sce the paymets are aual, we compoud aually ad the m = 1 ad = r/m =6.5%/1 =6.5%. We have PV = c 1 (1 + ) 20 1 ( ) =1, 300, 000 = 14, 324, Therefore the lump-sum of $ 19 mllo s a much better opto ths case.

20 Module 4.4 Page 511 of 944. Now suppose that you take the 19 mllo a lump sum, ad wth t you purchase a 20- year auty cotract that pays 20 equal paymets each year, for 20 years. The rate s 6.5% compouded aually. How much should you expect per year? [Aswer: $ 1,724, per year.] # The oly remag topc that I d lke to share wth you ths module has to do wth extremely log-term loas. Realstcally, these loas are somethg you are ulkely to ecouter your professoal lfe. However, I thk that the qury wll make kow to you the deep relatoshp betwee the PV of a auty, ad the PV of a perpetuty. Furthermore, there s a slght foreshadowg of calculus, ad the qury wll coclude wth wth a eat trck for checkg your work PV problems. Therefore, whle most structors would skp ths last lttle bt of the module, I recommed that you read t ayway. Suppose Percval s lookg to buy a house worth $ 700,000, ad I put 5% dow (or $ 35,000) ad take out a mortgage for the remag $ 665,000. Let the rate be 6% compouded mothly. What s the mothly paymet for: A 10-year loa? [Aswer: $ per moth.] A 15-year loa? [Aswer: $ per moth.] A 20-year loa? [Aswer: $ per moth.] A 30-year loa? [Aswer: $ per moth.] # A 40-year loa? [Aswer: $ per moth.] A 999-year loa? [Aswer: $ per moth.] Warg, the last aswer (of the last box) wll vary depedg o roudg errors teral to your calculator. What I lsted above was calculated usg a computer-algebra package ad 1000 dgts of precso. Now fd the total terest pad by Percval each of the above cases. For 10 years? [Aswer: $ 220, ] For 15 years? [Aswer: $ 345, ] For 20 years? [Aswer: $ 478, ] For 30 years? [Aswer: $ 770, ] # For 40 years? [Aswer: $ 1,091, ] For 999 years? [Aswer: $ 39,195, ]

21 Module 4.4 Page 512 of 944. What s shockg the prevous box s how lttle the paymet falls by gog from 30 years to 40 years. It s a rather small d erece, less tha 10%, but the loa s askg Percval to pay for far loger. Thus, 40 year mortgages are a bt stupd; ths s reflected the total terest pad, whch s far more. To be super-clear, the 40-year loa would cost $ less each moth, but there are a addtoal te years of paymets that s 120 addtoal paymets! As a result, the total amout that Percval would pay s ot lower but hgher. I fact, t s $ 320,958 hgher. Cotug wth the prevous box, what s also cute our aalyss of Percval s optos s how lttle the 40-year loa ad the 999-year loa d er the mothly paymet. The mothly paymet for 999-years was oly $ cheaper tha the mothly paymet for 40-years. Moreover, whle the paymets are essetally the same, otce how the total terest pad by Percval the 999-year case s vastly more tha the 40-year case. Whle a 999-year cotract mght soud absurd to a Amerca, Ballol College of Oxford Uversty has sged a 999-year cotract wth the Epscopala Docese of Oxford for St. Cross Church, to covert t to a ceter for ts ow archves. Ballol College was fouded 1263, ad so beg a roughly 750-year old college, t s o more rratoal for them to sg a 999-year lease tha year old perso to sg a 30-year mortgage cotract. After all, observe that = = A secod example s the St. Margaret s Isttute, whch leased a buldg for 999 years from St. Joh s College of Oxford for 400,000 pouds sterlg However, St. Joh s College s ot a medeval orgazato; t was fouded Yet a thrd example s St. Cross College of Oxford Uversty, whch sged a 999-year lease for a buldg, ad ths college s extremely youg, beg fouded All of these cotracts were sged the tme frame. Now that we ve looked at a 999-year lease let us cosder what would happe f we let = 1. After all, 999-years s = 11, 988 moths whch s much loger tha a huma lfespa (at least at ths pot hstory who kows what the future wll brg). It seems as though 999 years s a good model for fty.

22 Module 4.4 Page 513 of 944. Cotug wth the prevous box, we frst realze that a fte-durato mortgage would have a sequece of paymets of equal value gog o forever. However, we have already studed a facal strumet that does ths: the perpetuty! Furthermore, recallg that r =0.06 ad thus =0.005, a perpetuty costg $ 665,000 would have (see Page 467) as a paymet V = a/ 665, 000 = a/0.005 (665, 000)(0.005) = = a Stll cotug wth the prevous box,otce how close the perpetuty ($ ) s compared to the 999-year loa ($ ): they are detcal to the pey. Of course, some roudg error mght be volved, so I decded to set Maple to use 10,000 dgts of precso, ad calculated to obta ( ) 665, , whch s a d erece of approxmately 1 qutlloth of a dollar, or a teth of a quadrlloth of a cet. So we see that 999-years or = 11, 988 s a really, really good approxmato for fty. We have argued for, but ot qute prove, that (1 + ) lm c1 = c!1 Ths ca be sad more smply: as goes to 1, the PV of the auty approaches the PV of a perpetuty. The aalyss of the prevous box ca actually be used to check your work ay PV problem provded the durato s log. For ay somewhat log auty, the PV of the auty should be a bt less tha the PV of a perpetuty. A good gudele for somewhat log s 30 or more years. Let s see ths acto: The frst example ths module has $ 2000 paymets, oce a moth, for 30 years. The r was 6%, so = A perpetuty-due at that value would be 2000/0.005 = 400, 000. The PV was $ 333,583, whch s slghtly less. Of course, for the 999-year mortgage above, the two PVs were much closer. The soluto to the chessboard box o the amortzato table from Page 498 s gve below: Perod Startg Iterest Paymet Edg 1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ 5.13 $ $ 0.03

23 Module 4.4 Page 514 of 944. We have leared the followg sklls ths module: To derve the PV formula as the d erece of two perpetutes. To derve the PV formula as a geometrc seres stoppg early. To use the PV formula to calculate parameters of loas wth multple paymets. To fd the cost-per-thousad, or compoudg factor, of a auty. To calculate the compoudg factor, whch s also called the growth factor or power of compoudg, ad whch s deoted by a specal symbol. To costruct a amortzato table for a auty. To measure the mpact of a dow-paymet requremet o the maxmum sze of a loa. To calculate how log t wll take to pay o a auty-style loa, both accordg to schedule ad wth overpaymets. To check the type of problem the prevous bullet by pluggg the two earest values ad seeg f they straddle the desred value. To calculate the total terest pad ad the face charge of a loa. To observe that the PV vares learly wth the paymet, but very sestvely to. To calculate the tradeo betwee lump-sum lottery paymets ad auty-style paymets. To see how the paymet vares wth the durato of the loa, ad to see the futlty of 40-year mortgages compared wth 30-year oes. To calculate the tradeo betwee a cash rebate ad 0% facg, or a cash-back cetve ad 0% facg, whe buyg a car. To see that perpetutes have a PV equal to the lmt of the PV of a auty, whe the umber of paymets goes to fty. Furthermore, how to use that to check oe s work problems wth very log loas. (It s such a easy ad e ectve way to check your work!) As well as the vocabulary terms: Accdetal Death & Dsmembermet Isurace, auty, auty forbore, compoudg factor, face charge, growth factor, power of compoudg, zero-percet facg. Comg Soo!