Concepts: simple interest, compound interest, annual percentage yield, compounding continuously, mortgages

Transcription

1 Precalculus: Matheatcs of Fnance Concepts: sple nterest, copound nterest, annual percentage yeld, copoundng contnuously, ortgages Note: These topcs are all dscussed n the text, but I a usng slghtly dfferent language and spendng ore te settng up where the forulas coe fro usng Exaples/tables whch s why these notes are so long). The atheatcal concepts we use to descrbe fnance are also used to descrbe how populatons of organss vary over te, how dsease spreads through a populaton, how ruours spread through a populaton, even the oton of partcles suspended n a flud, as well as any other stuatons. Matheatcs s so beautful because the technques you learn to solve one type of proble typcally can be used to solve other probles! Money deposted n a savngs account n a bank wll earn nterest. The ntal aount you depost s called the prncpal, and the oney whch s earned s called the nterest. How does the oney grow? What wll your balance be after one year? There s a lot that goes nto answerng these questons, snce nterest can be pad n dfferent ways. Growth of Savngs: Sple Interest Sple nterest pays nterest only on the prncpal, not on any nterest whch has accuulated. Sple nterest s rarely used for savng accounts, but t s used for bonds. Exaple You put $98.45 n a savngs account whch pays sple nterest of 6% a onth. How uch oney do you have n the savngs account after 4 onths? Soluton To answer ths queston, we can buld fro what we know. Sple nterest eans we pay nterest only on the ntal aount deposted prncpal), whch was $ The nterest aount wll be 6%= 6/100 = 0.06 of the prncpal, and added to the account balance once a onth. Interest Perod Date Interest Added Accuulated Aount 0 Jan 1 0 $ Feb 1 $ =$5.91 $ $5.91 = $ Mar 1 $ =$5.91 $ $5.91 = $ Apr 1 $ =$5.91 $ $5.91 = $ May 1 $ =$5.91 $ $5.91 = $ Ths table s the for an Excel spreadsheet would take to calculate sple nterest. ntalzaton and t s the second row that contans forulas. Notce the frst row s an We see that the growth s by a constant aount $ =$5.91) every te perod onth n ths case). Ths s the requreent for lnear or arthetc growth. It gets the nae lnear snce the graph of the aount versus the te s a straght lne lnear functon). Sple Interest Forula For sple nterest of r percent pad every te perod wth a prncpal P, we get Page 1 of 9

2 Precalculus: Matheatcs of Fnance Years Accuulated Aount 0 P 1 P ) + P r 2 P + P r) + P r = P + 2P r 3 P + 2P r) + P r = P + 3P r 4 P + 3P r) + P r = P + 4P r t. P + P rt e., for a prncpal of P wth sple nterest of r% pad every te perod, we get an accuulated aount after t years of A = P + P rt = P 1 + rt). The forula gves you another way of calculatng a quantty that could be done usng a spreadsheet style table. Growth of Savngs: Copound Interest Copound nterest pays nterest on the prncpal and the accuulated nterest, not just the prncpal. Exaple You put $98.45 n a savngs account whch pays copound nterest of 6% a onth. How uch oney do you have n the savngs account after 4 onths? Soluton To answer ths queston, we can buld fro what we know. Copound nterest eans we pay nterest on the accuulated aount n the account. The nterest aount wll be 6%= 6/100 = 0.06 of ths aount, and added to the account balance once a onth. Copoundng Perod Date Interest Added Accuulated Aount 0 Jan 1 0 $ Feb 1 $ =$5.91 $ $5.91 = $ Mar 1 $ =$6.26 $ $6.26 = $ Apr 1 $ =$6.64 $ $6.64 = $ May 1 $ =$7.04 $ $7.04 = $ We see that the aount of growth ncreases as te ncreases. The aount of growth s proportonal to the aount present, whch s the requreent for geoetrc growth. Interest Ternology Savngs probles typcally nvolve a bt ore ternology than we ve used so far. The copoundng perod s the te whch elapses before copound nterest s pad. The te when copoundng s done effects the accuulated aount, snce the current aount affects the aount of nterest added, and the current aount wll change f we copound ore frequently. The nonal rate s the stated rate of nterest for a specfed length of te. The nonal rate does not take nto account how nterest s copounded! Page 2 of 9

3 Precalculus: Matheatcs of Fnance The effectve rate s the actual percentage rate of ncrease for a length of te whch takes nto account copoundng. It represents the aount of sple nterest that would yeld exactly as uch nterest over that length of te. The effectve annual rate EAR) s the effectve rate gven over a year. For savngs accounts, the EAR s also called the annual percentage yeld APY). Copound Interest Forula For a nonal annual rate r, copounded tes per year, we have = r/ as the nterest rate per copoundng perod. Now let s try to derve a forula for copound nterest. Copoundng Perod Aount 0 P 1 P + P = P 1 + ) 2 P 1 + ) + P 1 + ) = P 1 + ) 2 3 P 1 + ) 2 + P 1 + ) 2 = P 1 + ) 3 4 P 1 + ) 3 + P 1 + ) 3 = P 1 + ) 4 n. P 1 + ) n e., for a prncpal of P wth copound nterest of = r/ pad every copoundng perod, we get an accuulated aount after n = t copoundng perods t s nuber of years, s nuber of copoundng perods per year) of A = P 1 + ) n = P 1 + ) r t. Annual Percentage Yeld APY) By defnton, the APY s the sple nterest rate that earns the sae nterest as the copound nterest after one year t = 1). Copound Interest: A = P 1 + r ) t = P 1 + r ) Sple Interest: A = P 1 + rt) = P 1 + APY) Set these quanttes equal, and solve for APY: P 1 + APY) = P 1 + r ) 1 + APY) = 1 + r ) APY = 1 + r ) Exaple $1000 s deposted at 7.5% per year. Fnd the balance at the end of one year, and two years, f the nterest pad s copounded daly. What s the APY? Soluton The nonal annual rate s r =7.5% = 0.075, when copounded daly, eans we have = 365, so = r/ = 0.075/365 = One year corresponds to n = t = = 365, so after one year we have A = P 1 + ) n = $ ) 365 = $ Two years corresponds to n = t = = 730, so after two years we have A = P 1 + ) n = $ ) 730 = $ APY = 1 + r ) = ) 365 = 7.79%. 365 Page 3 of 9

4 Precalculus: Matheatcs of Fnance A Lt to Copoundng Sketch the graph of the accuulated aount for 10 years f the prncpal s P =$1000 and the annual nterest rate s r = 10% for sple nterest, copound nterest copounded yearly, copound nterest copounded quarterly, and copound nterest copounded daly assue 365 days n a year). To get the values, we can use the forulas we derved. Here s the process for gettng the accuulated aount after 1 year so t = 1 n all forulas); the rest are calculated n a slar fashon usng t = 2, 3, 4,.... Sple nterest after 1 year: A = P 1 + rt) = $ ) = $ after 1 year. Copound nterest copounded yearly = 1, = r/ = 0.10/1 = 0.10, and n = t = 1): A = P 1 + ) n = $ ) 1 = $ after 1 year. Copound nterest copounded quarterly = 4, = r/ = 0.10/4 = 0.025, and n = t = 4): A = P 1 + ) n = $ ) 4 = $ after 1 year. Copound nterest copounded daly = 365, = r/ = 0.10/365 = , and n = t = 365): A = P 1 + ) n = $ ) 365 = $ after 1 year. black: sple nterest. red: copound nterest, copounded yearly. green: copound nterest, copounded quarterly. blue: copound nterest, copounded daly. The curves are all essentally the sae for short tes. There are ore ponts for copoundng quarterly than yearly snce nterest s pad ore often durng the year. There s not uch dfference over 10 years to copoundng quarterly and copoundng daly. Copoundng ore frequently leads to a larger accuulated balance, but there s a lt to ths process. The lt would be f we copounded contnuously. Copoundng Contnuously Consder a prncpal P = $1 and a rate of r=100% whch s copounded over shorter and shorter te perods. We are nterested n how uch the accuulated aount wll be after one year. Copound nterest copounded n tes a year = 1/, and n = t = to get one year, t = 1)): A = P 1 + ) n = 1 + ) 1 after 1 year. Here s a sketch Page 4 of 9

5 Precalculus: Matheatcs of Fnance We see that the accuulated aount s approachng a nuber: 1 + ) l Ths nuber s slar to π = n that t s atheatcally sgnfcant, appears n any stuatons, and s a nonrepeatng nonternatng decal and so we gve t a specal desgnaton not surprsngly snce ths was the defnton gven n Secton 3.1!): e = l ) Ths leads the the contnuous nterest forula, whch s A = P e rt after t years f nterest s copounded contnuously at annual rate r. The contnuous nterest forula s the upper lt on the accuulated aount that can accrue due to copoundng nterest. Revew of nterest forulas prncpal P and annual rate r) Sple nterest: A = P 1 + rt) s the aount after t years. Copound nterest, copounded tes over 1 year for t years: A = P 1 + ) n s the aount where = r/. Contnuously copounded nterest: A = P e rt s the aount after t years. APY = 1 + ) r s the annual percentage yeld. The forulas allow us to answer questons whch would be dffcult to answer usng a table, and also to answer questons quckly wthout a lot of calculaton. However, the tables allow us to answer questons that do not atch the condtons under whch the forulas were derved. Therefore, both forulas and spreadsheet tables are useful n understandng how personal fnance works. Accuulaton: Future Value of Annutes The Savngs Forula) An portant aspect of savng s the dea of accuulaton, whch answers the queston: What sze depost do I have to ake at regular te nterval d to save a certan aount of oney n a certan aount of te? Ths would be portant for savng for retreent, or a down payent on a house, or a car, or a chld s educaton. Obvously, f there was no nterest, you would just break the aount you need to save nto d even peces and depost that aount regularly. Interest akes the proble ore nterestng! Page 5 of 9

6 Precalculus: Matheatcs of Fnance Exaple You begn savng for retreent at age 35 by payng $100 a onth nto an account payng 6% annual nterest copounded onthly. How uch wll you have n savngs by the te you are 65? Soluton: The easest way to thnk of ths s backwards, startng by what happens at age 65. For nterest copounded onthly at an annual rate of 6%, we have = r/ = 0.06/12 = The last depost you ake wll be $100, and earn no nterest or nterest for 0 onths). $100 The penultate depost wll be $100, and wll earn nterest for 1 onth: $ ) 1. The second last depost wll be $100, and wll earn nterest for 2 onth: $ ) 2. Ths process contnues, rght up untl the frst depost s ade. In = 30 years, you wll ake = 360 onthly deposts. The aount you save s A = $100 + $ ) 1 + $ ) $ ) 359 = $ ) ) ) 359. We stop at 359 snce we started at 0, not 1. Ths s a geoetrc seres, wth x = 1 + ) and n = 360. Therefore, we can wrte A = $ ) ) ) 359 = $ ) 360 = $ ) The aount we wll save by the age of 65 s 1 + ) ) 360 A = $100 = $100 = $ Only $36, 000 of ths s due to the deposts. The rest s nterest. That should blow your nd. 1 + ) 360. The Savngs Forula Future Value of Annuty): Based on the above exaple, we see that for a unfor depost d per copoundng perod and an nterest rate of per perod, the aount A accuulated after n perods s gven by: 1 + ) n A = d 1 + r = d )t r/ Mortgages: Aortzaton forula Present Value of an Annuty) Conventonal Loans In a conventonal loan, each payent pays towards the current nterest that would be due over the lfe of the loan and also repays part of the prncpal. The payents are expressed n ters of an aortzaton table, whch shows how uch of each payent s gong towards nterest and how uch towards payng off the prncpal. Exaple You borrow $100,000 at 8% per year for a 30 year loan for a house whch wll be pad off n equal) onthly nstalents. How uch s your onthly payent? Let s check ths out ntally usng one of the any onlne resources: We should fnd that the onthly payent s d = $ To fgure out how ths onthly payent s calculated, we use both the copound nterest forula and the savngs forula. Thought One: Copound Interest Ths stuaton can be thought of as borrowng the entre $100,000 edately, and then puttng t n a savngs account where t wll earn nterest for 30 years untl t the nterest and prncpal) has to be repad. A = P 1 + ) n = $100, /12) 360 = $1, 093, Page 6 of 9

7 Precalculus: Matheatcs of Fnance The aount you wll have saved whch s the aount that you wll have to repay) n 30 years wll be $1,093, Thought Two: Savngs Savng $d each onth that s what we want to fnd!) for 30 years eans that you wll save: 1 + ) n A = d /12) 360 = d 0.08/ = d = d ) We want these two aounts to be exactly equal. $1, 093, = d d = $ The onthly payents should be $ What we have done s called aortze the loan. Part of each onthly payent goes towards reducng the prncpal, and part goes toward reducng the nterest that the loan would accuulate over the lfe of the loan. Note that the llon dollars tself s not what s beng pad for the hoe, snce the hoe s beng pad off over the course of te and the prncpal s beng reduced as te goes on. The cost of the house s $ = $264, , where $164, s due to nterest. There s a sall correcton ade at the end due to the roundng that has been done. The Aortzaton Forula Leavng thngs n general n the exaple above, we see that: 1 + ) P 1 + ) n n = d where P s the prncpal, s the nterest rate per copoundng perod, and n s the nuber of copoundng perods for whch you are takng out the loan. A lttle algebra can be used to rewrte ths as the Aortzaton Forula: r 1 + ) n or 1 + r ) t The aortzaton forula s used to deterne the onthly payents d on a conventonal loan. A bt of algebra rearranges ths forula nto the Present Value of an Annuty forula. Constructng The Aortzaton Table Frst payent: $733.76: Interest for frst onth on the prncpal s P = P r/ = $100, /12 = $ What s left goes towards reducng the prncpal: $ $ = $ At the end of the frst onth, the prncpal s $100, 000 $67.09 = $ Second Payent: $733.76: Interest for second onth on the prncpal s P = P r/ = $ /12 = $ What s left goes towards reducng the prncpal: $ $ = $ Page 7 of 9

8 Precalculus: Matheatcs of Fnance At the end of the frst onth, the prncpal s $ $67.54 = $ Ths contnues untl the loan s pad off. The aount you are payng per onth towards prncpal ncreases, and the aount you are payng towards nterest decreases. Note: If you look at FAQ #7 on the webste you wll see the need for creatng your own spreadsheet for the aortzaton table. Verfyng the dtech.co Ad The dtech ad sad the onthly payents would ncrease by 11% f the APR changed by 1%. Let s verfy ths. Use the aortzaton forula: 1 + ) n P = $200, 000 APR = 6% = r n = onths = 30 years) = r/12 = 0.06/12 = d s the onthly payent 1 + ) n d = $200, ) 360 d = $200, d = $ Redo the calculaton, wth APR = r =7%: = r/12 = 0.07/12 = ) n d = $200, ) 360 d = $200, d = $ So the percentage ncrease s gven by $ $ $ = %. Interest Only Loans The ad for Qucken Loans appears to be an nterest only loan. In an nterest only loan, you reduce the onthly payents by not payng anythng towards prncpal hence the nae). Of course, ths reduces your onthly payent, but at the prce of buldng equty. Untl you start payng towards the prncpal, you are not buldng any equty n your hoe through payents towards prncpal any equty you are buldng s based on the fact that hoe prces are ncreasng at a rate greater than nflaton. That ay not always be the case, of course. After an ntal te perod could be 10 years, but t wll vary) you wll have substantally hgher onthly payents snce you need to start repayng prncpal, and the te for the repayent s decreased fro 30 year to 20 years. Interest only loans are not a good dea for people whose ncoe s unlkely to change over te. They are only a good dea f you antcpate a sgnfcant jup n your ncoe that wll allow you to pay the hgher payents that are requred later, or f you plan on flppng the house quckly before the hgher payents kck n. Let s say we have a loan structured n the followng way. P = $100,000 wth an APR of r = 8% for 30 years, the frst 10 of whch are nterest only. Page 8 of 9

9 Precalculus: Matheatcs of Fnance For the frst 10 years, you only pay nterest, P = P r/ = $100, /12 = $ Durng ths te, you have pad nothng towards the prncpal, so after 10 years you need an aortzaton loan for 20 years for the full purchase prce: n = = 240 onths = r/ = 0.08/12 = ) n d = $100, ) 240 d = $100, d = $ Total Pad for Interest Only = $ $ = $280, 753. If we dd ths usng a tradtonal 30 year fxed rate ortgage, we would fnd: n = = 360 onths = r/ = 0.08/12 = ) n d = $100, ) 360 d = $100, d = $ Total Pad for Tradtonal 30 year ortgage = $ = $264, 154. The upshot of all ths s that after 30 years, you pay $16, 600 ore nterest for the nterest only loan than f you had used a tradtonal 30 year aortzaton. So n the long run, an nterest only loan only akes sense f your ncoe s gong ncrease draatcally n cong years, so the ncreased onthly payents aren t a burden, you can ake substantal payents to the prncpal later on whch wll reduce the aount of nterest you ultately pay), or you are gong to sell the house before you pay t off. These condtons are not usually satsfed by an average consuer. An average consuer should typcally look for fxed rate ortgages and f they have extra oney pay ahead on the prncpal. The Qucken Ad If we try to verfy the nubers n the Qucken Loans ad, we wll fnd they are wrong. The onthly payent for a tradtonal 30 year aortzaton of P = $150, 000 at an APR of 7.5% s d = $ they got t rght on ther webste, although they dd choose to round to $1049 nstead of $1048). The nterest only payent I calculate for ths loan s P r/n = $150, /12 = $937.50, whch does not agree wth the $745 n the TV ad or $703 on ther webste. We would need to look a lttle ore closely at how ths loan s structured to fully understand t. Adjustable Rate Mortgage ARM) ARMs have nterest rates that change, and f they change n the wrong drecton you ght fnd that you cannot ake the onthly payents! ARMs are generally a bad dea for an average consuer. Check out ths artcle fro 2004, BEFORE the fnancal crss: x.ht. Ths s what Qucken Loans says about nterest only loans today: FYI, y house lost $4000 n value fro 2009 to Page 9 of 9