Survey of Math: Chapter 22: Consumer Finance Borrowing Page 1

Transcription

1 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 1 APR and EAR Borrowng s savng looked at from a dfferent perspectve. The dea of smple nterest and compound nterest stll apply. A new term s the annual percentage rate (APR), whch s the rate of nterest per compoundng perod tmes the number of compoundng perods per year. Recall: s the rate of nterest per compoundng perod. We have calculated = r/m, where r s the annual rate of nterest, and n s the number of compoundng perods per year. Therefore, APR = m = r. Note that n general r mght not be an annual rate of nterest, whch s why we have the APR termnology. The APR s not the effectve annual rate (EAR), as the followng shows. Recall the EAR s the amount of smple nterest that would produce the same accumulated amount at the end of the year, whch we called APY for savngs. Example Calculate the APR and EAR for a credt card on whch nterest per year s 18% and requres monthly payments. Ignore late payment charges and other fees. Soluton ( EAR = 1 + m) r m ( = ) = = 19.56% The APR s 18% (by defnton). The APR underestmates the true cost of borrowng, snce t does not take nto account compoundng. The EAR s the true cost of borrowng. Credt card companes are requred to report the APR, whch s fne snce as long as each company reports the same thng consumers can compare companes on a level playng feld, and quckly understand the dfferences between each account. Conventonal Loans In a conventonal loan, each payment pays towards the current nterest that would be due over the lfe of the loan and also repays part of the prncpal. The payments are expressed n terms of an amortzaton table, whch shows how much of each payment s gong towards nterest and how much towards payng off the prncpal. Example You borrow $100,000 at 8% per year for a 30 year loan for a house whch wll be pad off n (equal) monthly nstalments. How much s your monthly payment? Let s check ths out ntally usng one of the many onlne resources: We should fnd that the monthly payment s d = $ To fgure out how ths monthly payment s calculated, we use both the compound nterest formula and the savngs formula.

2 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 2 Thought One: Compound Interest Ths stuaton can be thought of as borrowng the entre $100,000 mmedately, 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, 000( /12) 360 = $1, 093, The amount you wll have saved (whch s the amount that you wll have to repay) n 30 years wll be $1,093, Thought Two: Savngs Savng $d each month (that s what we want to fnd!) for 30 years means that you wll save: (1 + ) n A = d ( /12) 360 = d 0.08/ = d = d( ) We want these two amounts to be exactly equal. $1, 093, = d d = $ The monthly payments should be $ What we have done s called amortze the loan. Part of each monthly payment goes towards reducng the prncpal, and part goes toward reducng the nterest that the loan would accumulate over the lfe of the loan. Note that the mllon dollars tself s not what s beng pad for the home, snce the home s beng pad off over the course of tme and the prncpal s beng reduced as tme goes on. The cost of the house s $ = $264, , where $164, s due to nterest. There s a small correcton made at the end due to the roundng that has been done. The Amortzaton Formula Leavng thngs n general n the example above, we see that: (1 + ) P (1 + ) n n = d where P s the prncpal, s the nterest rate per compoundng perod, and n s the number of compoundng perods for whch you are takng out the loan.

3 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 3 A lttle algebra can be used to rewrte ths as the Amortzaton Formula: 1 (1 + ) or r m 1 (1 + r m ) mt The amortzaton formula s used to determne the monthly payments d on a conventonal loan. Constructng The Amortzaton Table Frst payment: $733.76: Interest for frst month on the prncpal s P = P r/m = $100, /12 = $ What s left goes towards reducng the prncpal: $ $ = $ At the end of the frst month, the prncpal s $100, 000 $67.09 = $ Second Payment: $733.76: Interest for second month on the prncpal s P = P r/m = $ /12 = $ What s left goes towards reducng the prncpal: $ $ = $ At the end of the frst month, the prncpal s $ $67.54 = $ Ths contnues untl the loan s pad off. The amount you are payng per month towards prncpal ncreases, and the amount 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 amortzaton table. Verfyng the dtech.com Ad The ad we saw n class sad the monthly payments would ncrease by 11% f the APR changed by 1%. Let s verfy ths. Use the amortzaton formula: 1 (1 + ) P = $200, 000 APR = 6% = r n = 360 (360 months = 30 years) = r/12 = 0.06/12 = d s the monthly payment 1 (1 + ) d = $200, (1.005) 360 d = $200, d = $

4 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 4 Redo the calculaton, wth APR = r =7%: = r/12 = 0.07/12 = (1 + ) d = $200, ( ) 360 d = $200, d = $ So the percentage ncrease s gven by $ $ $ = %. Home Equty Formula The amount needed for a mortgage payment are generally larger than the amount needed to amortze a loan snce the buyer must also pay taxes and nsurance on ther monthly mortgage. Read the example of page 716 for an example of a medan prced home for a famly wth medan ncome. For what follows, we wll gnore these detals, as well as any ncrease n value of the house. The downpayment smply reduces the sze of the loan that s requred, so we can gnore t for now. The downpayment wll smply add to the equty you have n your house. We know that when a home loan s amortzed, you pay towards the nterest and the prncpal wth each payment. Your early payments are heavly weghted towards the nterest, and very lttle goes towards reducng the prncpal. Equty refers to the amount of money you have pad towards payng off the prncpal n your home. If you have used a spreadsheet to construct an amortzaton table, then you already have the equty t s the cumulatve prncpal. Alternately, you can calculate equty at any gven tme usng formulas n the followng manner. Example You purchase a home for $89, 000 wth an annual nterest rate of 6.375% and a 30 year mortgage. How much equty do you have n the house after 5 years? Frst, we need to know how much the monthly payment s for ths house. We can use the amortzaton formula to fgure ths out, usng = r/m = /12 = : 1 (1 + ) d = $200, ( ) 360 d = $200, d = $ (1 + ) (n) Now, for the equty. The Amortzaton Formula can be wrtten as P = d.

5 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 5 The prncpal at tme 0 s the entre $89,000, and the prncpal after 360 payments should be $0: 1 (1 + ) (360 0) P = d 1 ( /12) (360) = $ ( /12) = $89, (1 + ) ( ) P = d 1 = $ ( /12) = $0 The above equatons, whch only change n the exponent, motvates the followng. Now we can use the amortzaton formula agan, ths tme to fgure out what the prncpal s after 5 years or 5 12 = 60 months. 1 (1 + ) (360 60) P = d 1 ( /12) (360 60) = $ ( /12) = $83, Ykes! We have only pad $89, $83, = $ towards the prncpal, whch s the equty we have bult up after 5 years. The equty bulds slowly ntally, and then grows faster near the end of the mortgage perod. Here s a plot of how equty vares over tme for ths example: Note that we got ths plot by usng some algebra and the formulas, but t s very easy to get the equty from an Amortzaton Table when we use a spreadsheet, as t s smply the column wth cumulatve prncpal.

6 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 6 Interest Only Loans The ad we saw for Qucken Loans appears to be an nterest only loan. In an nterest only loan, you reduce the monthly payments by not payng anythng towards prncpal (hence the name). Of course, ths reduces your monthly payment, but at the prce of buldng equty. Untl you start payng towards the prncpal, you are not buldng any equty n your home through payments towards prncpal any equty you are buldng s based on the fact that home prces are ncreasng at a rate greater than nflaton. That may not always be the case, of course. After an ntal tme perod (could be 10 years, but t wll vary) you wll have substantally hgher monthly payments snce you need to start repayng prncpal, and the tme for the repayment s decreased from 30 year to 20 years. Interest only loans are not a good dea for people whose ncome s unlkely to change over tme. They are only a good dea f you antcpate a sgnfcant jump n your ncome that wll allow you to pay the hgher payments that are requred later, or f you plan on flppng the house quckly before the hgher payments 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. For the frst 10 years, you only pay nterest, P = P r/m = $100, /12 = $ Durng ths tme, you have pad nothng towards the prncpal, so after 10 years you need an amortzaton loan for 20 years for the full purchase prce: n = = 240 months = r/m = 0.08/12 = (1 + ) d = $100, ( ) 240 d = $100, d = $ Total Pad for Interest Only = $ $ = $280, 753. If we dd ths usng a tradtonal 30 year fxed rate mortgage, we would fnd: n = = 360 months = r/m = 0.08/12 = (1 + ) d = $100, ( ) 360 d = $100, d = $ Total Pad for Tradtonal 30 year mortgage = $ = $264, 154.

7 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 7 The upshot of all ths s that after 30 years, you pay $16, 600 more nterest for the nterest only loan than f you had used a tradtonal 30 year amortzaton. So n the long run, an nterest only loan only makes sense f your ncome s gong ncrease dramatcally n comng years, so the ncreased monthly payments aren t a burden, you can make substantal payments to the prncpal later on (whch wll reduce the amount of nterest you ultmately pay), or you are gong to sell the house before you pay t off. These condtons are not usually satsfed by an average consumer. An average consumer should typcally look for fxed rate mortgages and f they have extra money pay ahead on the prncpal. The Qucken Ad If we try to verfy the numbers n the Qucken Loans ad, we wll fnd they are wrong. The monthly payment for a tradtonal 30 year amortzaton 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 payment 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 more 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 mght fnd that you cannot make the monthly payments! ARMs are generally a bad dea for an average consumer. Check out ths artcle from 2004, BEFORE the fnancal crss: x.htm. Ths s what Qucken Loans says about nterest only loans today: FYI, my house lost $4000 n value from 2009 to 2010.