Math of Finance Math 111: College Algebra Academic Systems

Transcription

1 Math of Fiace Math 111: College Algebra Academic Systems Writte By Bria Hoga Mathematics Istructor Highlie Commuity College Edited ad Revised by Dusty Wilso Mathematics Istructor Highlie Commuity College October 2004 Cotets Cotets... 1 Itroductio... 2 Sequeces... 3 Arithmetic Sequeces ad Series... 5 Geometric Sequeces ad Series... Future Value of Ivestmets ad Compoudig Your Moey Periodic Compoudig Cotiuous Compoudig Effective Aual Iterest Rate Auities Payig off a Loa (Amortizatio) Pre-Qualifyig for a Home Mortgage Appedices Appedix 1: Summary of Formulas Appedix 2: Miscellaeous Problems Appedix 3: Selected Solutios/Hits Page 1 of 51

2 Itroductio This hadout will itroduce some cocepts related to the expoetial growth of moey. This is particularly relevat to studets with busiess related majors as well as the average participat i today s world of persoal fiace ivolvig loas, savigs, ivestmets ad auities. Sice this is a mathematics course, the mathematical priciples uderlyig the cocepts will be preseted. The authors of this material caot claim to be busiess/fiace specialists, so the problems preseted are ot as complex as you would fid out i the real world. Sice this is a itroductory treatmet, oly a selectio of cocepts will be covered. If you are majorig i subject matter that uses this type of material sigificatly, I m sure your later courses will go much more i depth. That said I will try to itroduce you to a few of the basics i persoal fiace that I do feel comfortable speakig to. Let s begi with a check-up. I will develop formulas that cover basic savigs plas ad repaymet of loas. Though challegig, uderstadig how these formulas are reached might be required to do some of the more advaced homework problems. You will be required to memorize/kow the formulas - at least to the extet of beig able to match each formula with its ame. Essetially, the first three sectios cover the material i Academic Systems Lesso The followig sectios use this mathematical foudatio to build a uderstadig of compoudig, auities, ad loas. Complete solutios to a few of the problems are give i the appedix at the back, as (0.1) Persoal Fiace: How healthy are you fiacially? Where do you stad fiacially? Compare your debts to your earigs ad fid out. The debt-to-icome is oe of the most straightforward ways to determie your fiacial health. a.) Collect credit billig statemets to get a good estimate of what you owe each moth. b.) Make a outlie of bills (car loa, studet loas, mortgage, credit card, ad ret) ad the amout you pay each moth. Do ot i- c.) clude taxes ad utilities. Calculate your mothly icome, before taxes, icludig additioal icome (if applicable) for allowaces, ivestmets, or child support. d.) Divide your paymets from (1.) by your mothly icome from (3.). If your icome is $2000 per moth ad you make loa paymets of $700, your debt-to-icome ratio is 35% ($700/$2000 = 0.35). What does it mea? 36% or less This is where you wat to be. May leders take this ito cosideratio whe you apply for a loa. 37% to 42% Start payig your debts dow. You may be headed for fiacial difficulties. 43% to 49% This is a high debt-to-icome ratio. You eed to take immediate actio to take care of debts. Above 50% You should seek professioal help to help reduce your debts. From a ewsletter distributed by the Boeig Employees Credit Uio well as some hits other their solutios. Aswers will be give to may of the odd umbered problems. Be sure you show your use of the appropriate formulas that lead to your solutios. Page 2 of 51

3 Sectio 1 Sequeces To uderstad the cocepts to be preseted later, I eed to itroduce you to the cocept of a sequece. (This may be review--but bear with me as I try to make this material stad o its ow.) Defiitio: A sequece is a fuctio f whose domai (iputs) is {1, 2, 3, 4,...}. The outputs f(1), f(2), f(3),... are the terms of the sequece, with f(1) the first term, f(2) the secod term, etc. Example (1.1): If you were told that f(x) = 3x-5 was to represet a sequece, you would immediately kow that x was oly to take o values of 1, 2, 3,... etc. The terms of this sequece would be -2 (by lettig x = 1), 1 (by lettig x = 2 ), 4 (by lettig x = 3), etc. I fact if you calculated a few more terms for higher values of x, you would fid the terms of this sequece are as follows: -2, 1, 4, 7, 10, 13, 16, 19,... Sice f is a sequece you would ot let x =.5, or x = -7.2, or ay other values you might ormally cosider if f was a fuctio whose domai was all real umbers. NOTATION: Subscripts are most ofte used i describig the terms of a sequece rather tha stadard fuctio otatio. f 1, f 2, f 3, etc. istead of f(1), f(2), f(3), etc. Usually letters at the begiig of the alphabet are used to ame a sequece fuctio, rather tha f, g, or h. Also the letter is commoly used for the domai variable rather tha x. Thus istead of f(x) = 2x, it would be commo to see the sequece described by a = 2. The use of this alterate otatio should alert you that the fuctio is a sequece. There are coutless examples of sequeces - just thik up ay fuctio that will accept the atural umbers 1, 2, 3... as iputs! The outputs are usually displayed i order, separated by commas. It s the outputs that are ormally cosidered to be the sequece. Example (1.2): Let a = 3-1. The first 6 terms of this sequece are: 2, 5, 8, 11, 14, ad 17. The 100 th term is 299. (just let = 100 ) Note that what you see are the outputs of the fuctio ad the order i which you see the outputs tell you what the iputs were (i.e. 1st umber idicates = 1; secod umber meas = 2 ; third umber idicates = 3; etc.). Example (1.3): Let 2 b = +. The first 6 terms of this sequece are 2, 6,, 20, 30, 42. The 80 th term is (Check me out usig your calculator with = 80 ). While there are may classes of sequeces, our primary iterest (i algebra) lies i arithmetic ad geometric sequeces. Page 3 of 51

4 The Problem Set I fidig sums ad terms, show that you re usig formulas rather tha just simply doig all the work o your calculator. Of course, a calculator double-check is a fu way to check to see if your theory is o the mark. 1. Fid the values of the first 6 terms of these sequeces: (a) a = (b) b = ( + 2) (c) c = (d) 1 1 ( 1) + f = 2 2. For each of the followig sequeces, (i) give the ext 3 terms of the sequece ad (ii) give a fuctio defiitio of the sequece. (a) The sequece a starts as 1, 4, 9, 16, 25,... (b) The sequece f starts as 1/2, 2/3, 3/4, 4/5,... (c) The sequece d starts as 3, 8, 13, 18, 23,.... Page 4 of 51

5 Sectio 2 Arithmetic Sequeces ad Series Oe special type of sequece is the arithmetic sequece. A few examples iclude: Example (2.1) 1, 4, 7, 10, Example (2.3),, 1,, Example (2.2) 10, 8, 6, 4, Example (2.4) -9, -5, -1, 3, What do these sequeces have i commo? Well, i each case the sequece is obtaied by addig a fixed costat to the previous term (sometimes the costat is egative as i (2.2)). What should we call this costat? Cosider what happes whe we fid the differece betwee subsequet terms say i example (2.1). 4 1 = = = 3 Sice subtractio of subsequet terms always results i the same umber, we call this umber the commo differece. If you kow the commo differece d ad oe elemet, say a 1, the: 1st term is: 2d term is: 3rd term is: th term is: a = a + d a = a a = a + d a = a + d Workig from the last lie, we ca subtract a 1 from both sides of the equatio a = a 1 + d ad discover that d = a a 1. Oce agai justifyig d s title as the commo differece. Example (2.5): Fid the 10 th term i the arithmetic sequece where a 1 = 3 ad d = 2. 3 Solutio: We start with a 1 ad add d to fid the subsequet elemet: a 2 = 3+ 2 = 1. To fid a we repeat the same process: a 3 = 1+ 2 = 1. Cotiuig i the same maer, we fid the sequece -3, -1, 1, 3, 5, 7, 9, 11, 13, 15 ad so a 10 = 15 That was t too bad, but imagie the difficulty i fidig a 100 ad a 1000 must be a better way. Let s cosider a patter. 1 i this maer. There Page 5 of 51

6 a = a + 0d 1 1 a = a + d = a + 1d a = a + d = ( a + d) + d = a + 2d a = a + d = ( a + 2 d) + d = a + 3d a = a + d = [ a + ( 2) d] + d 1 1 = a + ( 1 1) d 1 = a + ( 2 1) d 1 = a + ( 3 1) d 1 = a + ( 4 1) d 1 = a + ( 1) d 1 So, we ow have the followig three formulas to use whe workig with arithmetic sequeces: a = a + d (2.6) 1 d a a 1 = (2.7) a = a1 + ( 1) d (2.8) Example (2.9): Fid the 37 th term of the arithmetic sequece 4, 1, -2, -5, Solutio: Step 1: Fid a 1 ad d. Notice that the first term is a 1 = 4. Calculate d usig formula (2.7) (or observe) that d = 3. d is egative because the terms are decreasig. Step 2: Use formula (2.8) to fid a 37 = 4 + (37 1)( 3) = ( 3) = 104 Example (2.10): Fid the th term of the arithmetic sequece 4, 1, -2, -5, th Solutio: Aother way to fid the term i a arithmetic sequece is to graph the poits (, a ). These poits fall o a lie whose slope is d ad whose y-itercept is a1 d The graph shows the poits alog with the lie that itersects them. Choosig two poits, we fid that 3 1, 4, ad substitute these values ito the formula d =. Now select a poit, say ( ) y = m x + b (the slope-itercept form of a lie). From this, we have 4 = 3(1) + b b = 7 Solvig for a1 i the formula b = a1 d, we have that a1 = b + d which i our case meas that a 1 = 7 + ( 3) = 4. Thus, the th term is a 4 ( 1)( 3) = +. Page 6 of 51

7 Example (2.11): Fid the 53 rd term of the arithmetic sequece where a 13 = 17 ad a 24 = 49 Solutio: Step 1: Fid a 1 ad d. To do this, we must costruct a system of two liear equatios with two ukows. We fid these equatios by substitutig the kow values ito equatio (2.8). For = 13: 17 = a1 + d ad for = 24: 49 = a1 + 23d. We ca solve this system usig a umber of methods. Here we will use the elimiatio method (also called additio). a1 + d = 17 a1 + 23d = 49 (2.) Subtract the secod lie from the first i (2.) resultig i the equatio: 11d = 22 (2.13) Divide both sides of (2.13) by -11 to determie that d = 2. Substitute this result back ito lie 1 of (2.) to form the equatio: a + (2) = 17 1 a + 24 = 17 1 a = 7 1 (2.14) Step 2: Use formula (2.8) to fid a 53 = 7 + (53 1)(2) = (2) = 97. So far, we have cosidered sequeces or lists of umbers. Now, we will focus our attetio o addig up the terms i a sequece. We ofte call the sum of a sequece a series. Specifically, let us fid the sum of the arithmetic series. However, o expositio o the arithmetic series would be complete without the followig etertaiig story about the great mathematicia Carl Gauss Carl Gauss ( ): A Historical Note Carl Friedrich Gauss, who was bor i 1777 i Brauschweig, Germay, the so of a masory forema, was a master at exposig ususpected coectios. He was a mathematical prodigy, ad i his old age he liked to tell stories of his childhood triumphs. At the age of te, he was a show-off i arithmetic class at St Catherie elemetary school, "a squalid relic of the Middle Ages... ru by a virile brute, oe Butter, whose idea of teachig the hudred or so boys i his charge was to thrash them ito such a state of stupidity that they forgot their ow ames." Oe day, as Butter paced the room, ratta cae i had, he asked the boys to fid the sum of all the whole umbers from 1 to 100. The studet who solved the problem first was supposed to go Page 7 of 51

8 ad lay his slate o Butter's desk; the ext to solve it would lay his slate o top of the first slate; ad so o. Butter thought the problem would preoccupy the class, but after a few secods Gauss rushed up, tossed his slate o the desk, ad retured to his seat. Butter eyed him scorfully, as Gauss sat there quietly for the ext hour while his classmates completed their calculatios. As Butter tured over the slates, he saw oe wrog aswer after aother, ad his cae grew warm from costat use. Fially he came to Gauss's slate, o which was writte a sigle umber: 5,050, with o supportig arithmetic. Astoished, Butter asked Gauss how he did it. Whe Gauss explaied it to him, the teacher realized that this was the most importat evet i his life ad from the o worked with Gauss always, plyig him with textbooks, for which Gauss was grateful all his life. So, how did Gauss solve the problem? Well, we will ever kow exactly what wet o i his mid. However, perhaps he solved it by usig the followig logic. Example (2.15): Fid the sum of the series Solutio: Cosider the series Reversig the terms does ot chage sum Now we will add (2.16) ad (2.17) together (2.16) (2.17) (2.18) How may terms of 101 are we addig? That is right, 100 terms are beig added. So our ew sum is 100(101). However, this sum is t idetical to the origial because it cotais (2.16) twice. So, we divide our result by 2 to get the fial aswer. More geerally, 100(101) 100(1 + = 100) = (101) = 5050 (2.19) 2 So, what was Gauss trick? Well, we presume that Gauss added the first ad last terms, multiplied the sum by the total umber of terms, ad divided this result by two. Very cool! Page 8 of 51

9 Theorem: The Sum of a Arithmetic Series. S ( a + a ) 2 1 = (2.20) We ll use the otatio S to idicate the sum of terms of the arithmetic sequece. Example (2.21): Fid the sum of the series Solutio: This is a arithmetic series, so all we eed is the first term, the last term, ad the total umber of terms i the series. We are give a 1 = 17 ad a = 1513 ; oly requires work to fid. By ispectio, we ca see that d = 11. Usig formula (2.8), with as the oly ukow, we have that: 1513 = 17 + ( 1)(11) 1513 = = = 11 = 137 But ow that we have, the sum is withi our grasp usig formula (2.20). S 137 ( )(137) = = Example (2.22): Add the 3 rd term through the 103 rd term i the series where a = Solutio: Sometimes we are at a loss as to how to proceed. At such times, lookig for a patter ca be advatageous. To fid the patter, let s start pluggig i values for. This time, we will begi with = 3. Doig this we arrive at the series: (2.23) Wait a secod! That looks just like Gauss problem. So, we could use the trick used i example (2.15) of reversig the sum or we could simply observe that the first term is 1, the last is 201, ad there are... how may terms? How may itegers are there betwee 3 ad 103 icludig the ed values? You should be able to covice yourself there are 101 terms, so = 101. Oce agai, we apply formula (2.20), ad are left with: ( )(101) S = = Page 9 of 51

10 You may have oticed that we did ot use S101 or S 103, but used S i the precedig example. This was to avoid subscript cofusio. Remember, the mathematical laguage is desiged for clarity. Gauss could solve the problem without subscripts ad so ca we. Ofte times the subscripts hide the elegace of the solutio. Page 10 of 51

11 The Problem Set I fidig sums ad terms, show that you re usig formulas rather tha just simply doig all the work o your calculator. Of course, a calculator double-check is a fu way to check to see if your theory is o the mark. 1. Fid the forty-ith term, a49 of the arithmetic sequece 7, 4, 1, 2. Fid the first term, a 1, ad the commo differece, d, of the arithmetic sequece whose third term, a 3, is 16 ad whose fifteeth term, a 15, is For the arithmetic sequece 18, 13, 8, 3, a. Fid the commo differece d. b. Fid the twety-fourth term a 24. c. Fid the sum, S 36, of the first 36 terms. 4. Fid the first term, a 1, ad commo differece, d, of the arithmetic sequece whose sum of the first terms, S, is 246 ad whose twelfth term, a, is For the arithmetic sequece 3, 7/2, 4, 9/2, a. Fid the commo differece d. b. Fid the eleveth term, a 11. c. Fid the sum, S 50, of the first 50 terms. 6. A ball rollig dow a iclied plae moves 8 feet the first secod. I each secod thereafter it moves 16 feet more tha i the precedig secod. a. How far will the ball move durig the teth secod? b. How far will it have moved durig the first 10 secods? 7. Fid the sum of the first 10,000 terms of the arithmetic sequece whose teth term, a 10, is -11 ad whose ieteeth term, a 19, is -71. Page 11 of 51

12 Sectio 3 Geometric Sequeces ad Series Example (3.1): Let c = 3(2 ). The first 6 terms of this sequece are: 6,, 24, 48, 96, 192. The 60 th term is 3,458,764,513,820,540,928. WOW! (I used a TI-92 - you ca t get this exact value o a ormal calculator. A ormal scietific calculator will chage to scietific otatio ad give you x which is good eough for govermet work!) Example (3.2): Let a = 2(3-1 ). The first 5 terms are 2, 6, 18, 54, ad 162. The 60 th term is 28,260,772,183,477,469,009,529,622,134. (Wat to do that by had? Agai, a ormal calculator would give you x ) You might otice that examples above are expoetial fuctios of the type: x y = c b, b > 0 ad b 0 where c. (See Lesso.1 i Academic Systems.) Expoetial fuctios give us what are called geometric sequeces. Defiitio: A geometric sequece is of the form: a 1 = a1r. Note that whe = 1, a 1 = a 1 (r 1-1 ) = a 1 (r 0 )= a 1 (1) = a 1. The value of r is called the commo ratio. (If you look at example (3.2) above, the ratio of a term compared to the previous term is always 3.) The ratio r of a geometric sequece is a for ay two cosecutive terms a ad a -1. Thus a 1 a = r a 1. Hece multiply a term by r to get the ext term. Sice we are repeatedly multiplyig by r, each term ivolves powers of r, as i our defiitio of a geometric sequece above. From a practical poit of view, to costruct a geometric sequece start with a umber (called a 1 ), the multiply by r, the multiply by r agai, multiply by r, etc. To costruct the sequece i example (3.2), start with 2, multiply by 3 to get 6, multiply by 3 to get 18, the by 3 to get 54, etc. Example (3.3): Costruct a geometric sequece whose first term is 4 ad has a commo ratio of 5 ad the give the formula for this sequece. Solutio: Start with the value of 4, the cotiue to multiply by the commo ratio of 5. If you do this you will get 4, 20, 100, 500, 2500, 500, etc. The formula for this sequece is a = 4(5-1 ) You should check out that this formula gives you the terms of the sequece by tryig = 1, 2, 3, 4,... Page of 51

13 Example (3.4): Give a formula for the geometric sequece: 2, 1, 1/2, 1/4, 1/8,... Solutio: The first term is 2, so a 1 = 2. We eed to determie r. Ca you see what umber is beig used as a multiplier? Sometimes it s very obvious. If ot, take ay term ad divide by the previous term. (like 1/2 1 or 1/8 1/4) For the sequece, this always gives you a aswer of 1/2. So, r = ½ ad a = 2 (1/2) -1 Example (3.5): Iflatio over the past 10 years has bee about 3.4% per year. This meas the price of somethig will icrease by about 3.4% per year. If a loaf of bread i 1990 was $0.95, about what would that loaf of bread be worth i 2002? Solutio: If P is the price at ay year, the P +.034P is the ew price the ext year. This is equal to 1.034P. I other words, we just simply multiply the old price by to get the ext ew price. Thus the ext year, we have 1.034(1.034P) = P, etc. We geerate a geometric sequece P, 1.034P, P, P,..., P with a ratio of If P = $0.95, the after years, $0.95(1.034) $1.42 will be the sellig price of the loaf of bread. Now, let s see if we ca fid a formula that gives the sum of a fiite geometric sequece. Let s look at a few examples to see if we might guess at what the formula might be =??? (a = 1 (2) -1, = 1, 2, 3, 4, 5, a 1 = 1, ad r = 2 ) If you add this up with your calculator you get 31. Hmmm... that s simply 1 less tha the ext term. Let s try a few more terms: If you add this up with your calculator, you will get You might otice that this is simply 1 less tha the ext term i the origial sequece. But before we jump to ay coclusios, maybe we d better try a more complicated example (a 1 = 2 ad r = 4). This sequece adds to 682. Now, the ext term would be Whoops - our sum is NOT the ext term mius 1. You might otice that the sum of 682 is almost 1/3 of I fact, if we subtract 2 from 486 ad divide by 3 we get the right aswer for the sum. Page 13 of 51

14 Hmmm. You might check that for this sequece, if you re addig the 1 st terms, just take what would be the ext term, subtract 2, (Hmmm - 2 is the first term) the divide by 3 (Hmmm agai... 3 is oe less tha the ratio). This seems to give you the right aswer for the sum - try it! Do you thik you see a patter? If you thik you do, try to fid the sum of the first 25 terms of the sequece: a = 4(3) -1 OK, check your guess with the followig theorem: Theorem: The Sum of a Geometric Series. S = a + a r + a r + a r a r a1r a1 = r 1 ext term first term = ratio 1 We ll use the otatio S to idicate the sum of terms of the geometric sequece. I everyday laguage, to add up terms of a geometric sequece, simply take the value of what would be the ext term, subtract the first term, the divide by 1 less tha the ratio. Note that this works for each of the examples give above. See your Academic Systems text for a more formal 1 proof i the subsectio titled The Derivatio of the Formula (1 a r S ) = toward the ed of 1 r Lesso Example (3.6): Fid by this algorithm Solutio: The ext term would be 15 ad r = 3. So S 5 = = (Check this aswer by actually addig up the series with your calculator). Example (3.7): Fid the sum of the first te terms of a = 3(2-1 ). Solutio: S = = = Check by writig out the first te terms, the addig them up o your calculator. Example (3.8): Fid the sum of 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32. Page 14 of 51

15 Solutio: r = 1/2, a 1 = 1, ad a = 1(1/2) -1 Math of Fiace S = = = = Remember, start with the ext term, subtract the first term, ad the divide by 1 less tha the ratio. Example (3.9): You start workig at a job for $800 per moth. However, as you gai experiece, the boss assumes you are worth more (because you work faster ad more efficietly) ad gives you a raise of 10% for the ext moth. (i.e. You d receive $880 for the ext moth). Assume a raise of 10% takes place each moth. How much have you eared i 1 year of work? Solutio: Your pay for each moth is as follows: Moth 1: $800 Moth 2: $ (.10)=800(1.10) (otice we just multiply the previous moth s balace by 1.10) Moth 3: 800(1.10)(1.10)=800(1.10) 2 Moth 4: 800(1.10) 2 (1.10) = 800(1.10) 3 We may cotiue i this maer all the while oticig that the expoet is 1 less tha the moth util we get to Moth : 800(1.10) 11 What we wat is (1.10) + 800(1.10) (1.10) 11 S ext term first term 800(1.10) = = = $17, ratio As you ca see, that 10% raise at the ed of each moth really adds up. Cotrast the total with just (800) = $9600 whe o raise was give. Example (3.10): A geometric sequece has a first term of 3.25 ad a ratio of 2. Fid which term is equal to Solutio: We eed to solve = 3.25(2) -1 for. This is a job for logarithms! First divide by 3.25 to get = 2-1. The take the log of both sides to get l l(16384) = (-1) l(2) ( ) l ( 2) = 1 14 = 1 = 15. Page 15 of 51

16 So the 15 th term is the oe that equals You may have oticed that the atural logarithm l was used i example (3.10) istead of log base 2 or the commo log. This purely a matter of preferece ay base would suffice. However, most calculators oly have the commo ad atural logs ad some (like the TI-92) are limited to the atural log. I ay case, you ca use the chage of base formula: l( a) log( a) log c( a) log b( a) = = = (3.11) l( b) log( b) log ( b) For a more thorough review of logarithms, see Lesso.2 i the Academic Systems text. c Page 16 of 51

17 The Problem Set I fidig sums ad terms, show that you re usig formulas rather tha just simply doig all the work o your calculator. Whe workig with moey, roud off the fial aswer to the earest pey. Of course, a calculator double-check is a fu way to check to see if your theory is o the mark. 1. Write out the first 6 terms of a geometric sequece whose first term is 5 ad the ratio is 3. Give the formula for this sequece. Use your formula to fid the 17th term. Fid the sum of the first terms usig the formula for the sum of a geometric sequece. 2. Write out the first 6 terms of a geometric sequece whose first term is 9 ad the ratio is 1/2. Give the formula for this sequece. Use this formula to fid the 11th term. Fid the sum of the first 8 terms usig the formula for the sum of a geometric sequece. 3. Give the formula for the geometric sequece whose first term is 7 ad ratio is -1/2. Fid the sum of the first 20 terms of this sequece. Give a aswer to the full accuracy of your calculator ad also try to give a exact aswer as a fractio. 4. If iflatio causes the value of a house to icrease about 8% per year, what would a house that is worth $0,000 today, cost i 9 years? Give a formula for the cost of the house years from ow. What is the ratio r for this geometric sequece? 5. Write the first 6 terms of a geometric sequece whose first term is -2 ad whose ratio is - 2. Fid the sum of the first 8 terms. Fid the sum of the first 29 terms. 6. A famous problem goes as follows: Bob tells his boss he will work for peies every day durig the moth of April. O April 1st he will work for 1 cet. The ext day he will work for 2 cets, the ext day for 4 cets, the ext day for 8 cets, etc. The boss, beig greedy (but who slept through busiess math, readily agrees). How much does Bob ear o the last day of April? What is the total of all of Bob s wages for the moth of April? (Express your fial aswers i dollar format - i.e. if aswer was 867 cets, express aswer as $8.67). 7. A $5,000 loa take out o the first of March is beig repaid to a fried (o iterest is beig charged) by makig mothly paymets of % of the upaid balace. You will make your paymets o the first day of the followig moths. Write several terms of a sequece called B that idicates the upaid balace at the begiig of each moth (right after you make the paymet). Start your sequece with B 1 = (The balace after the first paymet.) I got this by takig % of 5000 ad figurig out what was left. Is this a geometric sequece? If so, what is the ratio? What is the upaid balace after 1 year? (Write the formula for the upaid balace B i terms of, the umber of moths you made a paymet.) Usig logarithms, fid the umber of moths it will take before the upaid balace reaches $10; reaches $1. Page 17 of 51

18 Sectio 4 Future Value of Ivestmets ad Compoudig Your Moey You probably were exposed to the cocept of simple iterest i your begiig algebra classes. I that case, you simply took the amout of moey you were savig (called the priciple) ad multiplied it by the yearly iterest rate ad the umber of years. The iterest you were earig did ot become part of the priciple ad hece the iterest did ot ear further iterest. Thus, if you had $200 at 6% for years, the iterest eared would be $200(.06)() = $144. Thus after years, you would have $200 + $144 = $344. Periodic Compoudig However, whe moey is compouded, you calculate the iterest for a short period of time ad add that to the priciple. This forms a ew priciple that ears more iterest tha i the previous period. If P is the origial priciple, the New Priciple = P + Pi = P(1 + i), where i is the iterest rate per compoudig period. Note, we just take the priciple (what ever it may be at the time, ad multiply it by (1 + i) to get the ew priciple at the ext stage of compoudig. If you cotiue to multiply by (1 + i) over ad over, you get the formula give by Periodic Compoudig (4.1) where A = amout accumulated (Future Value) ad m = total umber compoudig periods A = P(1 + i) m ad i = iterest rate per compoudig period ad P = the origial pricipal (Preset Value) Importat Note: i versus r If = the umber of times moey is compouded per year, ad r is the yearly (omial) iterest rate, the r i = For example, if your credit card s yearly iterest rate is 18%, the mothly iterest rate would be 1.5%. (4.2) Types of Iterest Baks traditioally pay iterest compouded mothly or daily. However, there are other places where you ca ear iterest that is compouded over much loger periods. Treasury bods pay iterest every six moths ad most compaies pay divideds o a quarterly basis. So, what is a divided? A divided is whe a compay gives some of its earigs back to the shareholder. Ulike iterest from the bak which is determied (roughly) by the Federal Reserve, divideds are determied by a compay s board of directors. The catch the yield (divided/share price) is subject to the share price. This meas that the yield is chagig alog with the share price. Page 18 of 51

19 (4.3) A Case Study i Divideds: Geeral Electric. Take Geeral Electric (GE) as a example of a divided payig stock. GE has paid a divided each quarter for over oe hudred years. I additio, GE's divideds have bee raised for 27 cosecutive years. Does this mea that the iterest rate a ivestor ears i GE has icreased each year for loger tha may of us have bee alive? (You ca aswer this questio). So, what is GE s curret yield ( iterest )? O August 29, 2003, shares of GE sold for $29.40 each. GE curretly pays a aual divided of $0.76 per share. So, the yield is $0.76/$29.40 = or 2.59%. By way of compariso, my bak pays about 1% iterest. Ofte the words quarterly, mothly, daily are used to idicate how ofte we compoud moey. Compoudig quarterly meas 4 times per year; compoudig daily meas 365 times per year (some books might cosider = 360, rather tha makes oly a differece of a few peies). Importat Note: May sources give the geeral compoudig formula as r A = P1 + where is the umber of compoudig periods per year ad r is the omial (yearly) iterest rate. This is equivalet to formula (4.1), sice m = t would be the total umber of compoudig periods i t years ad r/ would be i, the iterest rate per period. Let me illustrate the previous commets by doig a compoud iterest problem, usig the basic cocept of compoudig, the comparig it with usig the compoud iterest formula. Example (4.4): Suppose $8,000 is compouded quarterly for two years. Suppose the yearly iterest rate is 6%. t Solutio: Compoudig quarterly, meas that the iterest rate is 6% = 1.5%, that is = 4 ad 4 i = Sice we are compoudig 4 times per year for two years, we will have a total of 8 compoudig periods (i.e., m = 8 ). So ow, let s step by step show the ew priciple for each successive compoudig period. Period 1: (.015) = $8,0 Period 2: 80+80(.015) =$8, Period 3: (.015) = $8, Period 4: (.015) = $ Full accuracy is importat! Period 5: (.015) = $8, Period 6: (.015) = $8, Period 7: (.015) = $8, Period 8: (.015) = $9, Page 19 of 51

20 So, after two years, we have a future value of $9, (Now we ll roud to cets). Usig the formula, A = 8000( ) 8 = $9, (Wow - same aswer!). By cotrast, if your moey was earig simple iterest, you would have had the followig amout of moey: A = (.06)(2) = $8,960. Example (4.5): If we ivest $5,000 by puttig ito a savigs accout payig 6% yearly iterest, compouded mothly, fid the future value of that moey i years. Solutio: P = Sice we re compoudig mothly ( times per year), we have that 0.06 i =. Sice we re compoudig times per year for years, that s a total of 144 compoudig periods. Thus m = 144. So, A = = $10, (rouded to earest pey). Warig: Be sure to use the full accuracy of your calculator. Remember to oly roud off your results i your fial aswer to the earest pey. You should lear to use your calculator well eough so that you basically ever have to re-eter itermediate results by had. Example (4.6): If the future value of a ivestmet after 10 years is to be $5,500 where the moey was compouded quarterly at 7.7% omial iterest, what is the preset value? Solutio: I this case, we are to solve for the origial priciple that was ivested. Quarterly.077 compoudig implies i = (compouded 4 times per year). So, we must solve ,500 = P ad get: 40. (Did you catch that m = 4(10) = 40?) Simply divide both sides by 5500 P = = $2, Example (4.7): If you ivest $2,500, compouded mothly at a aual rate of 5.5%, how much would you have after 15 moths? After 30 years? (Always roud off fial aswer to earest pey). Solutio: Let m = 15. Sice there are compoudig periods per year, we have that 15 Page 20 of i = A = $ = $2, If there are compoudig periods per year, after 30

21 years there d be (30) (which equals 360) compoudig periods, hece m = 360 ad you ll get $, Now that we uderstad periodic compoudig ad future value, let s move o to cotiuous compoudig a special case of periodic compoudig where we cosider to be ifiitely large. Cotiuous Compoudig Now, if we take the formula A = P(1 + i) m ad cosider our compoudig periods per year to be ifiitely umerous (faster tha oce every secod, or oce every micro-secod, etc) the formula evolves ito (See Lesso.1 i your text): Cotiuously Compoudig Where r = omial(yearly) iterest rate ad t = umber of years rt A = Pe e is a irratioal umber e It should be oted that the umber of compoudig periods has o meaig i this case - i oe sese, the umber of compoudig periods is =. For the cotiuously compoudig case, sice there is o such thig as a compoudig period, you may use ay value you wat for t as log it represets years. You do ot have to have a whole umber as a value for t. For example, use t = 2.5 for 2 1/2 years or if the time is 9 moths, use ¾ for the value of t. Example (4.8): If you ivest $2,500, compouded cotiuously at a aual rate of 5.5%, how much would you have after 7.5 years? After 200 moths? Solutio: A = $2500 e 7.5(.055) = $3,776.47; A = $2500 e (200/)(.055) = $6, (the trick is that t must be expressed i years - thus 200 moths = 200/ yrs) Example (4.9): Compoudig more ofte (keepig all other factors the same) always results i a larger future value. Cosequetly, cotiuous compoudig is the ultimate compoudig strategy. Cosider P = 8000, 10 years, 6% rate: Solutio: With icreasig values for, we have: (a) compouded quarterly: A = 8000(1+.06/4) 40 = $ (b) compouded mothly: A = 8000(1+.06/) 0 = $ (c) compouded daily: A = 8000(1+.06/365) 3650 = $ (d) compouded hourly: A = 8000(1+.06/8760) = $ (e) compouded cotiuously: A = 8000 e 10(.06) = $ Now, let s look at a series of examples that illustrate tryig to solve these compoud iterest equatios for various variables, depedig o the kow iformatio. The three examples should demostrate the eed to kow a variety of equatio solvig techiques. Page 21 of 51

22 As you will see, similar questios ca require differet strategies for solvig the resultig equatio! That s why we study various techiques i your geeral algebra classes before we tackle some of the more sophisticated problems. Example (4.10): Suppose you have received a large amout of moey from a uexpected source. You wish to take part of that moey ad ivest it i a log-term CD that pays 4.5% omial iterest, compouded cotiuously. You wat the CD to be worth $70,000 i 15 years for your child s educatio at Staford. How much should you ivest? Solutio: You wish to fid the preset value of a ivestmet that will have a future value of $70,000 i 15 years. To do this is to solve the algebra equatio: 70,000 = P e.045(15) Divide both sides by e.045(15) ad get $70, 000 = P. Usig a calculator, you get P = $35, ( 15) e Example (4.11): Suppose you wish to ivest $5,000 i a accout that will have a future value of $10,000 i 7 years. If you wish to have your ivestmet compouded cotiuously, what omial iterest rate would be required? Solutio: Solve = 5000 e 7r. Ha - we eed logarithms sice the ukow is part of a expoet! Divide by 5000 results i: 7r l 2 2 = e l 2 = 7 r r = = = 9.902% 7 We eed a ivestmet with a retur of 9.9% i order to double our moey i 7 years. Example (4.): Same questio as i previous example, except the ivestmet is compouded mothly. Solutio: Sice r is our omial (yearly) rate, r/ is the iterest rate per period we eed for our compoudig formula. (7) 84 r r Solve = = 1 +. I this case, our ukow is ot i the expoet, so logarithms are ot eeded. All we eed to do is to take the 84 th root of both sides. 84 r 84 r 84 So r ( 2 1) = + = =. Our calculator gives us r =.0994 = 9.94%. As you ca see we d eed a slightly higher ivestmet rate tha i the previous example. Of all compoudig choices, compoudig cotiuously always gives us the highest retur (assumig time ad omial rates are the same!). Prior to workig through the problem set, write dow defiitios for the followig vocabulary words. Page 22 of 51

23 Future Value: Preset Value: Yearly Nomial Rate: Page 23 of 51

24 The Problem Set Geeral Istructios: Show formulas used to solve problems. Display the appropriate umbers i the formulas, so partial credit ca be assiged if the results are t quite right. Whe workig with moey, roud off the fial aswer to the earest pey. 1. If you put a lump sum of $5,000 ito a ivestmet that pays 6%, compouded mothly, how much will you have after 15 years? 2. If the preset value of a $6,000 ivestmet pays a omial iterest rate of 7.5%, compouded cotiuously, what will be the future value of that ivestmet i 20 years? 3. For a preset value of $10,000 ad a aual iterest rate of 8%, compute the future value after 20 years for each of these compoudig strategies: a.) compouded yearly b.) compouded quarterly c.) compouded mothly d.) compouded daily (365 days per year) e.) compouded every hour f.) compouded cotiuously 4. If the future value of a ivestmet i 30 years is $150,000, what was the preset value, assumig the ivestmet was compouded daily at 5.5% yearly iterest. 5. You just iherited a large sum of moey. You pla o usig part of it i risky ivestmets, but you wat to be sure that i case those fail, you have part of your wiigs that will provide for you whe you tur 60 years old i 15 years. How much should you put ito a safe 4.5% ivestmet, compouded mothly, that will give you $600,000 whe you tur 60? After solvig the give problem, aswer the same questio, but use your actual age istead of assumig a age of 45 (uless, of course, you are 45). 6. Populatio growth is ofte cosidered as a compoudig problem, compouded aually. If a populatio of Black Bears is 60,000 i 1995, how may bears do we estimate there will be i the year 2020 if the aual growth rate is 2.4%? 7. $5,000 is ivested i a accout compouded mothly at a omial iterest rate of 6%, for 10 years. At the ed of that time, your moey is pulled from the accout ad is reivested i a accout compouded cotiuously at a rate of 6.5%. The moey is left i that secod accout for aother 10 years. How much will you have after that time? 8. Iflatio is cosidered to be growth with aual compoudig. Suppose the iflatio rate has bee about 3.1% for the past 10 years. A loaf of bread that today costs $2.18, would probably have cost about how much 10 years ago? 9. Exactly 7 years ago, Betty put i $8,000 ito a ivestmet that compouded her moey cotiuously at 7.5% aual iterest rate. She the took out all of her moey from the ivestmet ad used 60% of it as a dow paymet o a car. If she puts the balace i her credit uio that Page 24 of 51

25 compouds her moey quarterly at a 5% omial iterest rate, how log before she will have $8,000 agai? 10. Aswer the followig questios regardig the importace of P. a.) If you start with preset value of $1,500, compouded cotiuously at 7%, how log will it take to triple i value? b.) If you start with a preset value of $10, compouded cotiuously at 7%, how log will it take to triple i value? c.) State a coclusio based o these two examples (alog with others made up to make sure you re right - chage oly your startig amout). 11. Suppose you start with a preset value of $800, how log will it take to double i value if you are compoudig the moey mothly at a yearly iterest rate of 6.5%?. $5,000 is put i to accout A ad $6,000 ito aother accout B. The moey i accout A is compouded quarterly at a omial rate of 6.8%, whereas the moey i accout B is compouded daily at a omial rate of 5%. At the ed of 6 years, they are cashed out ad the total is put ito a sigle savigs accout payig 6% compouded cotiuously. 10 years later the savigs is used as a dow paymet buy a vacatio cabi that will cost $4,000. How big of a mortgage will have to be take out to make up the differece. 13. What iterest rate will result i a future value of $8,000, startig with a preset value of $3,500 that was compouded quarterly for 10 years? Page 25 of 51

26 Sectio 5 Effective Aual Iterest Rate Whe baks (or other istitutios) offer may choices i savigs or ivestmets (differet iterest rates, differet compoudig periods) it s cofusig as to what is the best deal. Oe way to help compare these differet optios is to covert these to a aual iterest rate, called the effective aual iterest rate (or the APY; Aual Percetage Yield). Most states require this type of disclosure from fiacial istitutios - the ame may be somewhat differet, but the cocept is the same. (5.1) Historical Note: Iflatio Never uderestimate the power of iflatio. I the U.S., iflatio has fluctuated throughout the years. However, our iflatio has ever rivaled that i pre-hitler Germay ( ) where postage o a letter cost 2 Germa marks o Jauary 2, 1922 ad 100,000,000,000 marks o December 1, It is commo to calculate iterest rates (after covertig them ito percets) to two decimal places. To calculate the effective iterest rate r eff we fid out the effect o a $1 ivestmet for 1 year: (i) Periodic Compoudig: If you re calculatig the future value of $1 for 1 year usig ormal compoudig of periods per year, the future value of $1 would be: (1) r r $11 + = 1 + where is umber of compoudig periods per year. (ii) Cotiuous Compoudig: If compoudig cotiuously, the $1 becomes: $1 e r(1) = e r The effective iterest rate r eff is the rate that would result i the same future value whe you compouded the $1 aually for oe year. Thus the future value of the $1 would be $1(1 + r eff ) = 1 + r eff. r (a) 1+ reff = 1 + Page 26 of 51

27 r r Or reff = = (1 + i) 1, where i = r r (b) 1+ r = e r = e 1 eff eff The Effective Iterest Rate: a.) Periodic Compoudig: r = (1 + i) 1, where r b.) Cotiuous Compoudig: r = e 1 eff eff r i = Example (5.2): Fid the effective aual iterest rate of moey that is compouded cotiuously at 7% iterest. Solutio: r eff = e = 7.25% (2 decimal places). Example (5.3): Fid the APY of moey that is beig compouded mothly at a omial iterest rate of 7%. Solutio: APY 0.07 = = % (2 decimal places). Note that the effective iterest rate is always slightly higher tha the origial iterest rate. 1 This is because the effective iterest rate assumes you are oly compoudig yearly, thus to make up for this lower umber of compoudig periods, the iterest rate has to be made higher i order to make up the differece. Example (5.4): I the previous example, you foud that the 7% rate, compouded mothly had a APY of 7.229%. Now, we will use both to fid the future value of $6,000 over 7 years. Solutio: Two Methods. (a) The future value, usig 7%, compouded mothly is 7() 0.07 $ = $9, (b) If usig the APY, we are compoudig yearly ( = 1): $ $9, Note the amouts are the same. (However, if you use the slightly rouded off value of 7.23% you will be off by about 60 cets). 7 1 Uless, of course, the iterest is compouded aually. Page 27 of 51

28 The above example should covice you there is really othig special about the umber of compoudig periods i ad of itself; has to be tied ito the iterest rate beig charged too. If a bak is oly compoudig yearly, or quarterly, or... it could be just as good of a deal as the bak with cotiuous compoudig provided the iterest rate is right. Page 28 of 51

29 The Problem Set Geeral Istructios: Show formulas used to solve problems. Display the appropriate umbers i the formulas, so partial credit ca be assiged if the results are t quite right. Display iterest rate results to earest hudredth of a percet. Whe workig with moey, roud off the fial aswer to the earest pey. 1. Fid the aual effective rate of a savigs accout that is advertised to be 5.6%, compouded daily. 2. Fid the aual effective rate of a savigs accout that cotiuously compouds your moey at 6.5%. 3. If the aual effective rate of a ivestmet is 7.2%. What is its actual iterest rate if the ivestmet is compouded cotiuously? 4. If the aual effective rate of a ivestmet that s compouded mothly is 7.2%, what is the actual iterest rate? 5. If a credit uio that is compoudig your moey cotiuously, advertises that the effective iterest rate is 7.4%, what oe-time amout do you put ito a accout i that credit uio that will have a future value of $10,500 i years? (There are a couple of ways to approach this problem; however, oe-way is defiitely much easier!) 6. Suppose you have some moey tied up ito two ivestmets: 1/3 of it at 5%, compouded mothly ad 2/3 of it at 6.5% compouded cotiuously. What would be a reasoable umber that represets the aual effective iterest rate of your total ivestmet? (Hit: You will have to create your ow math i the spirit of the defiitio of the effective aual iterest rate). Would you expect your aswer to be closer to oe iterest rate tha the other? Which oe? 7. Lookig back at how we foud the APY (aual effective iterest rate), fid the mothly iterest rate that is equivalet to a cotiuous iterest rate of 7%. 8. What cotiuous iterest rate would be equivalet to a 6% iterest rate, compouded quarterly? Page 29 of 51

30 Sectio 6 Auities Okay, let s cosider a more sophisticated savigs strategy to achieve a future value at some later time. Sceario: Usually, most people do t have a big chuk of cash to set aside as savigs (or a ivestmet) at oe time. Most of us might expect to achieve our savigs goal by puttig a smaller amout of moey ito the savigs or ivestmet o a regular basis over a period of time. Such a fiacial pla is called a auity. Note: We refer to the sceario where a fixed sum of moey is paid (or distributed) o a regular basis (such as from a retiremet pla) a auity. (6.1) Iterest Rates As with iflatio, iterest rates have chaged dramatically over time. The graph give shows iterest rates from Jauary, 1983 through July, While the problems i this text assume a costat iterest rate, the world is ot always that straightforward. As a cosumer, you must be ready to shop aroud for the best rates ad/or wait util you ca fid the rate you desire. Data courtesy of There are may possible variatios of the basic sceario ad some of those variatios would complicate matters i tryig to come up with a simple formula. Hopefully, this class will provide you eough backgroud that you might be able to uderstad a uusual situatio. We will provide you a example of the most commo use of this savigs sceario. Savig a fixed amout of moey P at each compoudig period: Suppose a amout P is put ito savigs. This will ear iterest durig the compoudig period ad the a paymet of P is made agai. This is repeated over ad over. Give a formula that gives the future value at the ed of t years. Let r = the aual iterest rate ad = the umber of compoudig periods per year. Formig the equatio: Let s look at the patter of moey formed by a sequece of paymets. r Let i =, the iterest rate per period. We start with P ad our first paymet accumulates iterest for a period, so we have P P(1 + i) the make the secod paymet P P(1 + i) + P Page 30 of 51

31 this accumulates iterest for a period [ P(1 + i) + P](1 + i) the we make aother paymet 2 [ P(1 + i) + P](1 + i) + P = P(1 + i) + P(1 + i) + P After the ext paymet, we have 3 2 P(1 + i) + P(1 + i) + P(1 + i) + P Ad so o... What is the patter? Just keep multiplyig the previous amout accumulated by 1+ i the add aother paymet P. Notice that i the last step, we ve made 4 paymets, but the largest expoet is 3 (oe less tha the umber of paymets!). So, after m paymets, the largest expoet i our expressio above is m 1. So our amout accumulated is: (1 ) m m P i P(1 i).... P(1 i) P(1 i) P (6.2) Reversig the terms i the expressio, we have: P P i P i P i P i 2 m-2 m-1 + (1 + ) + (1 + ) (1 + ) + (1 + ) (6.3) We see that expressio (6.3) is a sum of a geometric sequece whose first term is P ad whose ratio is 1+ i. If you recall our formula from the previous sectio, we subtract the first term from what would be the ext term after the last term, ad the divide by 1 less tha the ratio. This would give us: m P(1 + i) P 1+ i 1 which ca be simplified to (1 + i) m 1 P i We will call this the Future Value, FV, of this auity. Thus, we have the formula The Future Value of a Auity: FV (1 + i) m 1 = P i Example (6.4): Suppose I save $100 per moth toward my daughter s college educatio, compouded mothly, at aual rate of 5.5%. At the ed of 18 years, whe my daughter is ready to go to college, how much moey will there be i the savigs accout? Solutio: i =, 18( ) m =, P = 100. Puttig these ito the formula gives us Page 31 of 51

32 FV (18) = 100 = $36, Are you impressed? If we paid $100 per moth for 18 years, we made 216 paymets for a total of $21,600. We almost doubled the moey by makig those relatively small paymets over that time. Try some what-ifs to see how you ca chage the amout at the ed. For example, if the paymet is icreased to $5, you will have $45, Note: Keep i mid, m is the umber of times you are addig a paymet to the savigs accout, ad i is the iterest rate durig each period of time. We are assumig the moey is beig compouded at the same rate at which you are addig moey. The usual situatio is mothly deposits ad mothly compoudig. Example (6.5): Suppose Bob s compay made mothly paymets toward his retiremet fud for 20 years. The paymets were $250 ad the fud eared iterest at a rate of 5.6%. Bob the left the compay ad was allowed to roll-over his moey i his retiremet fud ito a ormal savigs accout. That savigs accout paid iterest cotiuously at a rate of 6.2%. Bob chose ot to make ay additioal paymets ito the savigs accout. 10 years after Bob left the compay; Bob retired ad started livig off his retiremet savigs. How much moey was i accout at the time Bob retired? Solutio: We first will use our FV formula as i the last example, the that result will be the pricipal for the ormal compoud iterest formula. FV ( ) = $250 = $110, The A 0.062(10) = 110,190 e = $204, Sceario: We wish to establish a auity for which we make periodic paymets (oe paymet for each compoudig period) i order to have a future value FV. Fid the paymet P that will give us that desired FV over a time period of t years. To do the task i the sceario above, just take our auity formula ad simply solve for P. ( i) m 1+ 1 m FV i FV = P FV i = P ( 1+ i) 1 = P m i 1+ 1 ( i) This ca be rewritte as: P = FV i ( i) m 1+ 1 Note: This is sometimes called a Sikig Fud. Page 32 of 51

33 Example (6.6): A fud of $25,000 eeds to be available to a compay i 8 years. What mothly paymets eed to be made to have this moey available at that time if the moey is compouded mothly at a rate of 6.5%? Solutio: P = $25000 = $ ( 8) Prior to workig through the problem set, write dow the defiitio for the vocabulary words auity ad sikig fud. Auity: Sikig Fud: Page 33 of 51

34 The Problem Set Geeral Istructios: Show formulas used to solve problems. Display the appropriate umbers i the formulas, so partial credit ca be assiged if the results are t quite right. Assume all iterest rates are yearly rates, eve if ot explicitly stated. Whe workig with moey, roud off the fial aswer to the earest pey. 1. What will be the future value of a auity if $7,000 is deposited semiaually at a yearly rate of 7%, compouded every 6 moths for 15 years? 2. You are savig towards a house. Towards this ed, you put away $600 per moth ito a ivestmet that is compouded mothly at 9%. How much will you have after 6 years? 3. You eed $54,000 i 10 years for a vacatio cabi. How much do you eed to put aside mothly ito a accout that is compouded mothly, at a aual rate of 7.5%? 4. You pla to pay $150 per moth ito your IRA (Idividual Retiremet Accout) for 29 years. The IRA will be a accout that guaratees a 7.8% omial iterest rate, compouded mothly. What will you have i your accout at the ed of that time? Note: There are restrictios o how much you ca deposit i a IRA (or Roth IRA). As of 2003, most people ca deposit at most $3,000 each year ito their IRA. There is a clause that allows people over the age of 50 to deposit $3,500 i a year. Also, you caot make a deposit larger tha your gross icome for the year. 5. Jae paid $200 per moth ito a IRA accout for 20 years that paid 5.5%, compouded mothly. At the ed of that time, she rolled-over the moey that was i her IRA ito aother accout that eared iterest cotiuously at a rate of 6%. She simply left the moey, without addig ay additioal amout, for aother 10 years. How much moey did she have at the ed of the 30 years? 6. You estimate that you will eed to replace your car i 5 more years. The car you wat will probably cost $25,000. If you thik you will get $3,000 for your preset car, how much will you eed to set aside i order to be able to pay cash for your car? Assume your savigs accout will pay 5.5%, compouded mothly. 7. For some IRA s you oly make aual cotributios. If you ivest $3,000 aually i a ivestmet that pays %, compouded aually, for 10 years, how much will you have? 8. See if you ca make a formula that will calculate the amout you will have after 10 years if you put $500 per moth ito a accout that pays cotiuous iterest of 6%. (This will require you to modify the origial auity formula. The origial formula always assumes the paymet period is the same as the compoudig period. Try oe of two approaches: (1) remember that 1- moth is 1/ of a year ad go back ad see how the origial formula was developed. Or (2) look at the problems at the ed of the previous sectio ad fid the mothly iterest rate that is equivalet to the cotiuous iterest rate.) Page 34 of 51

35 Sectio 7 Payig off a Loa (Amortizatio) Sceario: A item is purchased at a price L, ad it is goig to be paid off by istallmet paymets over a period of t years. Of course the people holdig the loa expect to be paid iterest o the upaid balace. Assume that the loa is paid off i m paymets (a paymet made each period) ad that the aual iterest rate is r, compouded times per year. Formig the Equatio: As with the auity sceario i the previous sectio, let s look at the amout of moey left after each paymet is made. Let PMT be the paymet made each period. Remember, the balace of our loa icreases due to iterest i each compoudig period. Let r i =, the iterest rate durig each compoudig period. We start with a loa of L L after the first paymet, the upaid pricipal (1 + i) L - PMT the upaid pricipal after the secod paymet multiply by (1 + i) ad subtract PMT to fid the upaid priciple after the third paymet Ad so o... (1 + i)[(1 + i) L - PMT ]- PMT 2 L(1 + i) - PMT (1 + i) - PMT L(1 i) - PMT (1 i) - PMT (1 i) - PMT Note the patter: There were 3 paymets ad the largest expoet i the expressio is also 3, so... At the ed of m paymets, whe the loa is paid off completely, we would have (1 ) m m L i - PMT (1 i) PMT (1 i) - PMT ad this should equal $0. Ha, aother equatio to solve! L + i PMT + i PMT + i PMT = m m-1 (1 ) - (1 ) (1 ) - 0 L + i PMT + i + + PMT + i + PMT = m m-1 (1 ) -[ (1 )... (1 ) ] 0 m m-1 Cotiuig o from L(1 + i) -[ PMT (1 + i) PMT (1 + i) + PMT ] = 0, we add the expressio i the brackets to both sides of the equatio. L + i = PMT + i + + PMT + i + PMT m m-1 (1 ) [ (1 )... (1 ) ] Page 35 of 51

36 Now, otice the right side of the equatio is the sum of a geometric sequece. By our formula for summig geometric sequeces, we get: m m m PMT (1 + i) PMT (1 + i) 1 L( 1+ i) = = PMT (1 + i) 1 i Now, let s solve for PMT by dividig ad get a formula for PMT: The Paymet: If a loa $L is to be repaid over m periods at a periodic iterest rate of i, the the m i(1 + i) i paymet required each period is: PMT = L = L m m (1 + i) 1 1 (1 + i) The last form of this formula is what is usually give i texts. You get this from the expressio i frot of it by dividig top ad bottom of the fractio by ( 1 i) x b. m. x + ad rememberig that 1 b is Example (7.1): Suppose that the Joes have a mortgage of $4,000 o their ew house. The loa is at 7.8% ad the paymets are made mothly over 30 years. What are their paymets? Assume iterest is also computed mothly Solutio: PMT = $4000 (30) = $ (7.2) Iterest Rates. Not oly do iterest rates chage daily, but each day you have may differet optios. I this example, I assume that we buy a house for $200,000 with a $40,000 dow paymet. That is, we take out a loa for $160,000. Each of these five choices is a 30-year fixed rate mortgage. The differece is i the iterest rate. The rates vary a full 0.5% based o the umber of discout poits you pay. A discout poit is a additioal closig cost (or credit) equal to 1% of the loa amout. So, i this case, 1 poit = $1,600. Rates from Boeig Employees Credit Uio (BECU) o August 29, Page 36 of 51

37 Example (7.3): You ve borrowed $4,000 from a fried. He agrees to charge iterest of 5%, compouded quarterly, ad you re to make 4 paymets per year for 5 years. How much is your quarterly paymet? How much did you pay i iterest for your loa? 0.05 Solutio: PMT = $ (5) = $ You made 20 paymets of $ which will total $ You will otice this is more tha the amout you borrowed. This extra, of course, is the iterest you paid your fried! You paid $ $4000 = $ i iterest. (7.4) Closig Costs. Whe you get a loa to make a purchase, you are buyig somethig that you caot afford (or are ot willig to pay for). The start-up fees you pay for the privilege of makig purchases that you caot afford are called closig costs. O a home mortgage, closig costs must be paid prior to you takig possessio of the house. This meas that you eed moey for a dow paymet AND moey for closig costs i order to buy a house. I the ote o Iterest Rates (7.2), you probably oticed that the closig costs varied from about $2,600 to over $6,000 depedig o the umber of poits you paid. We will focus i o the fourth case where you pay o poits ad the closig costs were $52. $ % $ = Whe plaig to purchase a home, you ca estimate that you will have to pay closig costs equal to about 3% the value of the loa. Example (7.5): You wish to keep your mothly paymets for a ew car over a 4-year period dow to $250 per moth. The iterest rate will be 8.8%. How expesive of a car ca you afford? Solutio: This meas we will have to solve our formula for L. Multiply both sides of your formula by the deomiator ad you get PMT (1- (1 + i) ) = L( i). Now divide by i. Lettig -m PMT = 250, i =, ad = 4() = 48 moths, we get ( ) L = = $10, You ca afford a car at this. If the car costs more tha that, you ll have to come up with a dow paymet of some sort. Page 37 of 51

38 (7.6) Type of Loas There are may types of home mortgages. The two most commo are fixed rate mortgages ad adjustable rate mortgages. O the same day, drastically differet rates are offered depedig o the type of loa selected. 2 Samples from BECU are give Example (7.7): Suppose Bill saved $500, 000 for his retiremet. The retiremet fud is locked i a ivestmet payig 7% per year, compouded mothly. However, Bill ca make mothly paymets to himself from that fud. How large of a mothly paymet will use up Bill s retiremet fud i 30 years? Solutio: O first glace, this seems like a differet problem tha our first example. But it really is t - we ca preted that the retiremet fud is simply a loa that we wish to pay off i 30 years. That dar retiremet fud keeps growig due to iterest, just like our loa o that house, but we are makig paymets to ourselves ( rather tha to a mortgage compay) to use up that fud i a certai amout of time. So: 0.07 $500,000 PMT = ( 30) = $3, per moth Over the 30 years, Bill paid himself 360($3,326.51) = $1,197,544.49! You gotta love that iterest!!! (7.8) Fixed Rate Mortgages versus Adjustable Rate Mortgages The Pros ad Cos from the BECU website 30-Year Fixed Rate 5-Year Adjustable Rate Best choice if: You pla o stayig i the home log-term. You thik iterest rates will icrease. You do t expect your icome to icrease sigificatly over the comig years. You eed to qualify for the largest loa possible. Advatages: Fixed rate of iterest. Level pricipal ad iterest paymets for the full term of the loa. No risk that chagig market coditios will icrease your mothly paymets Disadvatages: You ed up payig more i iterest charges over the life of the loa. Beefits are ot realized util after the 10th year. Best choice if you wat: The stability of a fixed mothly paymet for first five years of loa. To keep your paymets low. To maximize the amout of loa you qualify for. Advatages: Iitial fixed iterest rate for 5 full years. The rate adjusts periodically thereafter. Allows for higher loa amout qualificatio ad ehaced buyig power. Disadvatages: It s riskier if you do t expect your icome to icrease over the iitial five-year period to cover the chage i mothly paymet. Iterest rate ca rise above the curret fixed rates over time. 2 Iformatio o ARMs may be foud at uder the most recet Math 111 class. Page 38 of 51