A fool ad his moey are soo parted. Eglish proverb Maagig Your Moey Maagig your persoal fiaces is a complex task i the moder world. If you are like most Americas, you already have a bak accout ad at least oe credit card. You may also have studet loas, a home mortgage, ad various ivestmet plas. I this chapter, we discuss key issues i persoal fiacial maagemet, icludig budgetig, savigs, loas, taxes, ad ivestmets. We also explore how the govermet maages its moey, which affects all of us. UNIT 4A Takig Cotrol of Your Fiaces: You caot achieve fiacial success uless you kow how to make wise decisios about your moey. We discuss the basics of persoal budgetig. UNIT 4B The Power of Compoudig: We explore the way i which you ca icrease your savigs through the mathematics of compoud iterest. UNIT 4C Savigs Plas ad Ivestmets: We calculate the future value of savigs plas i which you make mothly deposits ad study ivestmets i stocks ad bods. UNIT 4D Loa Paymets, Credit Cards, ad Mortgages: We calculate mothly paymets ad explore loa issues. UNIT 4E Icome Taxes: We explore the mathematics of icome taxes ad a few of the hot political issues that surroud them. UNIT 4F Uderstadig the Federal Budget: Everyoe s persoal fiaces are ultimately tied to govermet fiaces. We examie the federal budget process ad related political issues. 215
216 CHAPTER 4 Maagig Your Moey UNIT 4A Takig Cotrol of Your Fiaces Moey ca t buy me love... THE BEATLES Moey is t everythig, but it certaily has a great ifluece o our lives. Most people would like to have more moey, ad there s o doubt that more moey allows you to do thigs that simply are t possible with less. However, whe it comes to persoal happiess, studies show that the amout of moey you have is less importat tha havig your persoal fiaces uder cotrol. People who lose cotrol of their fiaces ted to suffer from fiacial stress, which i tur leads to higher divorce rates ad other difficulties i persoal relatioships, higher rates of depressio, ad a variety of other ailmets. I cotrast, people who maage their moey well are more likely to say they are happy, eve whe they are ot particularly wealthy. So if you wat to attai happiess alog with ay fiacial goals you might have the first step is to make sure you uderstad your persoal fiaces eough to keep them well uder cotrol. By the Way College may be costig you a lot ow, but statistically it s worth it: The average college graduate ears early $20,000 per year more tha a perso who graduates oly from high school, which adds up to early $1 millio i extra icome over the course of a career. Of course, this is oly a average: Studets who take harder classes ad get better grades ted to get higher-payig jobs ad ear eve more tha those who take a easier route through school. Take Cotrol If you re readig this book, chaces are that you are i college somewhere. I that case, you are almost certaily facig fiacial challeges that you ve ever had to deal with before. If you are a recet high school graduate, this may be the first time that you are fully resposible for your ow fiacial well-beig. If you are comig back to school after may years i the work force or as a paret, you ow have the challege of jugglig the cost of college with all the other fiacial challeges of daily life. The key to success i meetig these fiacial challeges is to make sure you always cotrol them, rather tha lettig them cotrol you. Ad the first step i gaiig cotrol is to make sure you keep track of your fiaces. Uless you happe to be amog the superrich, keepig track of your fiaces probably is t that difficult, but it requires diligece. For example, you should always kow your bak accout balace, so that you ever have to worry about boucig a check or havig your debit card rejected. Similarly, you should kow what you are spedig o your credit card ad if it s goig to be possible for you to pay off the card at the ed of the moth or if your spedig will dig you deeper ito debt. Ad, of course, you should sped moey wisely ad at a level that you ca afford. There are lots of books ad Web sites desiged to help you cotrol your fiaces, but i the ed they all come back to the same basic idea: You eed to kow how much moey you have ad how much moey you sped, ad the fid a way to live withi your meas. If you ca do that, as summarized i the followig box, you have a good chace at fiacial success ad happiess.
4A Takig Cotrol of Your Fiaces 217 CONTROLLING YOUR FINANCES Kow your bak balace. You should ever bouce a check or have your debit card rejected. Kow what you sped; i particular, keep track of your debit ad credit card spedig. Do t buy o impulse. Thik first; the buy oly if you are sure the purchase makes sese for you. Make a budget, ad do t oversped it. EXAMPLE 1 Latte Moey Calvi is t rich, but he gets by, ad he loves sittig dow for a latte at the college coffee shop o a busy day. With tax ad tip, he usually speds $5 o his large latte. He gets at least oe every day (o average), ad about every three days he has a secod oe. He figures it s ot such a big idulgece. Is it? SOLUTION Oe a day meas 365 per year. A secod oe every third day adds about 365>3 5 121 more (roudig dow). That meas 365 1 121 5 486 lattes a year. At $5 apiece, this comes to Calvi s coffee habit is costig him more tha $2400 per year. That might ot be much if he s fiacially well off. But it s more tha two moths of ret for a average college studet; it s eough to allow him to take a fried out for a $100 dier twice a moth; ad it s eough so that if he saved it, with iterest he could easily build a savigs balace of more tha $25,000 over the ext te years. Now try Exercises 23 30. EXAMPLE 2 Credit Card Iterest Cassidy has recetly begu to keep her spedig uder better cotrol, but she still ca t fully pay off her credit card. She s maitaiig a average mothly balace of about $1100, ad her card charges a 24% aual iterest rate, which it bills at a rate of 2% per moth. How much is she spedig o credit card iterest? SOLUTION which is 486 3 $5 5 $2430 Her average mothly iterest is 2% of the $1100 average balace, 0.02 3 $1100 5 $22 Multiplyig by the 12 moths i a year gives her aual iterest paymet: 12 3 $22 5 $264 Iterest aloe is costig Cassidy more tha $260 per year a sigificat amout for someoe livig o a tight budget. Clearly, she d be a lot better off if she could fid a way to pay off that credit card balace quickly ad ed those iterest paymets. Now try Exercises 31 34.
218 CHAPTER 4 Maagig Your Moey Master Budget Basics As you ca see from Examples 1 ad 2, oe of the keys to decidig what you ca afford is kowig your persoal budget. Makig a budget meas keepig track of how much moey you have comig i ad how much you have goig out ad the decidig what adjustmets you eed to make. The followig box summarizes the four basic steps i makig a budget. A FOUR-STEP BUDGET 1. List all your mothly icome. Be sure to iclude a prorated amout that is, what it averages out to per moth for ay icome you do ot receive mothly (such as oce-per-year paymets). 2. List all your mothly expeses. Be sure to iclude a prorated amout for expeses that do t recur mothly, such as expeses for tuitio, books, vacatios, ad holiday gifts. 3. Subtract your total expeses from your total icome to determie your et mothly cash flow. 4. Make adjustmets as eeded. By the Way The cost of a college educatio is sigificatly more tha what studets actually pay i tuitio ad fees. O average, tuitio ad fees cover about twothirds of the total cost at private colleges ad uiversities, oe-third of the cost at public fouryear istitutios, ad 20% of the cost at two-year public colleges. The rest is covered by taxpayers, alumi doatios, grats, ad other reveue sources. For most people, the most difficult part of the budget process is makig sure you do t leave aythig out of your list of mothly expeses. A good techique is to keep careful track of your expeses for a few moths. For example, carry a small ote pad with you, ad write dow everythig you sped. Ad do t forget to prorate your occasioal expeses, or else you may severely uderestimate your average mothly costs. Oce you ve made your lists for steps 1 ad 2, the third step is just arithmetic: Subtractig your mothly expeses from your mothly icome gives you your overall mothly cash flow. If your cash flow is positive, you will have moey left over at the ed of each moth, which you ca use for savigs. If your cash flow is egative, you have a problem: You ll eed to fid a way to balace it out, either by earig more or spedig less or i some cases decidig it s worthwhile to get a loa. EXAMPLE 3 College Expeses I additio to your mothly expeses, you have the followig college expeses that you pay twice a year: $3500 for your tuitio each semester, $750 i studet fees each semester, ad $500 for textbooks each semester. How should you hadle these expeses i computig your mothly budget? SOLUTION Sice you pay these expeses twice a year, the total amout you pay over a whole year is 2 3 A$3500 1 $750 1 $500B 5 $9500 To prorate this total expese o a mothly basis, we divide it by 12: $9500 4 12 < $792
4A Takig Cotrol of Your Fiaces 219 Your average mothly college expese for tuitio, fees, ad textbooks comes to just uder $800, so you should put $800 per moth ito your expese list. Now try Exercises 35 40. EXAMPLE 4 College Studet Budget Briaa is creatig a budget. The expeses she pays mothly are $700 for ret, $120 for gas for her car, $140 for health isurace, $75 for auto isurace, $25 for reters isurace, $110 for her cell phoe, $100 for utilities, about $300 for groceries, ad about $250 for etertaimet, icludig eatig out. I additio, over the etire year she speds $12,000 for college expeses, about $1000 o gifts for family ad frieds, about $1500 for vacatios at sprig ad witer break, about $800 o clothes, ad $600 i gifts to charity. Her icome cosists of a mothly, after-tax paycheck of about $1600 ad a $3000 scholarship that she received at the begiig of the school year. Fid her total mothly cash flow. SOLUTION Step 1 i creatig her budget is to come up with her total mothly icome. Her $3000 scholarship meas a average of $3000>12 5 $250 per moth o a prorated basis. Addig this to her $1600 mothly paycheck makes her total icome $1850. Step 2 is to look at her mothly expeses. Those paid mothly come to $700 1 $120 1 $140 1 $75 1 $25 1 $110 1 $100 1 $300 1 $250 5 $1820. Her aual expeses come to $12,000 1 $1000 1 $1500 1 $800 1 $600 5 $15,900; dividig this sum by the 12 moths i a year gives $15,900>12 5 $1325 o a prorated mothly basis. Thus, her total mothly expeditures are $1820 1 $1325 5 $3145. Step 3 is to fid her cash flow by subtractig her expeses from her icome: mothly cash flow 5 mothly icome 2 mothly expeses 5 $1850 2 $3145 52$1295 Her mothly cash flow is about 2$1300. The fact that this amout is egative meas she is spedig about $1300 per moth or about $1300 3 12 5 $15,600 per year more tha she is takig i. Uless she ca fid a way to ear more or sped less, she will have to cover this excess expediture either by drawig o past savigs (her ow or her family s) or by goig ito debt. Now try Exercises 41 44. Time out to thik Look carefully at the list of expeses for Briaa i Example 4. Do you have ay categories of expeses that are ot covered o her list? If so, what? Adjust Your Budget If you re like most people, a careful aalysis of your budget will prove very surprisig. For example, may people fid that they are spedig a lot more i certai categories tha they had imagied, ad that the items they thought were causig their biggest difficulties are small compared to other items. Oce you evaluate your curret
220 CHAPTER 4 Maagig Your Moey budget, you ll almost certaily wat to make adjustmets to improve your cash flow for the future. There are o set rules for adjustig your budget, so you ll eed to use your critical thikig skills to come up with a pla that makes sese for you. If your fiaces are complicated for example, if you are a returig college studet who is jugglig a job ad family while attedig school you might beefit from cosultig a fiacial advisor or readig a few books about fiacial plaig. You might also fid it helpful to evaluate your ow spedig agaist average spedig patters. For example, if you are spedig a higher percetage of your moey o etertaimet tha the average perso, you might wat to cosider fidig lower-cost etertaimet optios. Figure 4.1 summarizes the average spedig patters for people of differet ages i the Uited States. Percetage of Spedig by Category ad Age Group By the Way Spedig patters have shifted a great deal over time. At the begiig of the twetieth cetury, the average America family spet 43% of its icome o food ad 23% o housig. Today, food accouts for oly 13% of the average family s spedig, while housig takes 33%. Notice that the combied percetage for food ad housig has declied from 66% to 46% over the past cetury, implyig that families ow sped sigificatly higher percetages of icome o other items, icludig leisure activities. Food Housig Clothig ad services Trasportatio Health care Etertaimet Doatios to charity Persoal isurace, pesios Uder 35 35 to 64 65 ad older 0 10 20 30 Percet FIGURE 4.1 Average spedig patters by age group. Techical ote: The data show spedig per cosumer uit, which is defied to be either a sigle perso or a family sharig a household. Source: U.S. Departmet of Labor Statistics. EXAMPLE 5 Affordable Ret? You ve worked up a budget ad fid that you have $1500 per moth available for all your persoal expeses combied. Accordig to the spedig averages i Figure 4.1, how much should you be spedig o ret? SOLUTION Figure 4.1 shows that the percetage of spedig for housig varies very little across age groups; it is close to 1>3, or 33%, across the board. Based o this average ad your available budget, your ret would be about 33% of $1500, or $500 per moth. That s low compared to rets for apartmets i most college tows, which meas you face a choice: Either you ca put a higher proportio of your icome toward ret i which case you ll have less left over for other types of expeditures tha the average perso or you ca seek a way of keepig ret dow, such as fidig a roommate. Now try Exercises 45 50.
4A Takig Cotrol of Your Fiaces 221 Look at the Log Term Figurig out your mothly budget is a crucial step i takig cotrol of your persoal fiaces, but it is oly the begiig. Oce you have uderstood your budget, you eed to start lookig at loger-term issues. There are far too may issues to be listed here, ad may of them deped o your persoal circumstaces ad choices. But the geeral priciple is always the same: Before makig ay major expediture or ivestmet, be sure you figure out how it will affect your fiaces over the log term. EXAMPLE 6 Cost of a Car Jorge commutes both to his job ad to school, drivig a total of about 250 miles per week. His curret car is fully paid off, but it s gettig old. He is spedig about $1800 per year o it for repairs, ad it gets oly about 18 miles per gallo. He s thikig about buyig a ew hybrid that will cost $25,000 but that should be maiteace-free aside from oil chages over the ext five years, ad it gets 54 miles per gallo. Should he do it? SOLUTION To figure out whether the ew car expese makes sese, Jorge eeds to cosider may factors. Let s start with gas. His 250 miles per week of drivig meas about 250 mi>wk 3 52 wk>yr 5 13,000 miles per year of drivig. I his curret car that gets 18 miles per gallo, this meas he eeds about 720 gallos of gas: 13,000 mi 18 mi gal < 720 gal If we assume that gas costs $3 per gallo, this comes to 720 3 $3 5 $2160 per year. Notice that the 54-miles-per-gallo gas mileage for the ew car is three times the 18- miles-per-gallo mileage for his curret car, so gasolie cost for the ew car would be oly 1>3 as much, or about $720. Thus, he d save $2160 2 $720 5 $1440 each year o gas. He would also save the $1800 per year that he s curretly spedig o repairs, makig his total aual savigs about $1440 1 $1800 5 $3240. Over five years, Jorge s total savigs o gasolie ad repairs would come to about $3240>yr 3 5 yr 5 $16,200. Although this is still short of the $25,000 he would sped o the ew car, the savigs are startig to look pretty good, ad they will get better if he keeps the ew car for more tha five years or if he ca sell it for a decet price at the ed of five years. O the other had, if he has to take out a loa to buy the ew car, his iterest paymets will add a extra expese; isurace for the ew car may cost more as well. What would you do i this situatio? Now try Exercises 51 56. EXAMPLE 7 Is a College Class Worth Its Cost? Across all istitutios, the average cost of a three-credit college class is approximately $1500. Suppose that, betwee class time, commute time, ad study time, the average class requires about 10 hours per week of your time. Assumig that you could have had a job payig $10 per hour, what is the et cost of the class compared to workig? Is it a worthwhile expese?
222 CHAPTER 4 Maagig Your Moey SOLUTION A typical college semester lasts 14 weeks, so your lost work wages for the time you sped o the class come to 14 wk 3 10 hr wk 3 $10 hr 5 $1400 Addig this to the $1500 that the class itself costs gives your total et cost of takig the class rather tha workig: $2900. Whether this expese is worthwhile is subjective, but remember that the average college graduate ears early $1 millio more over a career tha a high school graduate. Ad also remember that, o average, studets who do better i college also do better i terms of their career earigs. Now try Exercises 57 58. Time out to thik Followig up o Example 7, suppose that you are havig difficulty i a particular class, but kow you could raise your grade by cuttig back o your work hours to allow more time for studyig. How would you decide whether you should do this? Explai. Base Fiacial Goals o Solid Uderstadig These days, it s rare for a fiacial decisio to have a clear best aswer for everyoe. Istead, your decisios will deped o your curret circumstaces, your goals for the future, ad some uavoidable ucertaity. The key to your future fiacial success is to approach all your fiacial decisios with a clear uderstadig of the available choices. I the rest of this chapter, we ll study several crucial topics i fiace, helpig you to build the uderstadig you ll eed to reach your fiacial goals. To prepare yourself for this study, it s worth takig a few momets to thik about the impact that each of these topics will have o your fiacial life. I particular: Achievig your fiacial goals will almost certaily require that you build up savigs over time. Although it may be difficult to save while you are still i college, ultimately you will eed to fid a way to make your budget allow for savigs ad the uderstad how savigs work ad how to choose appropriate savigs plas; these are the topics of Uits 4B ad 4C. You will probably eed to borrow moey at various poits i your life. You may already have credit cards, or you may be takig out studet loas to help pay for college. I the future, you may eed loas for large purchases, such as a car or a home. Because borrowig is very expesive, it s critical that you uderstad the basic mathematics of loas so that you ca make wise choices; this is the topic of Uit 4D. Whether we like it or ot, may of the fiacial decisios we make have cosequeces o our taxes. Sometimes, these tax cosequeces ca be large eough to ifluece our decisios. For example, the fact that iterest o house paymets is tax deductible while ret is ot may ifluece your decisio to ret or buy. While o oe ca expect to uderstad tax law fully, it s importat to have at least a basic uderstadig of how taxes are computed ad how they ca affect your fiacial decisios; this is the topic of Uit 4E.
4A Takig Cotrol of Your Fiaces 223 Fially, we do ot live i isolatio, ad our persoal fiaces are ievitably itertwied with the govermet s fiaces. For example, whe politicias allow the govermet to ru deficits ow, it meas that future politicias will have to collect more tax dollars from you or your childre. We ll devote Uit 4F to discussig the federal budget ad what it may mea for you i the future. EXERCISES 4A QUICK QUIZ Choose the best aswer to each of the followig questios. Explai your reasoig with oe or more complete seteces. 1. By evaluatig your mothly budget, you ca lear how to a. keep your persoal spedig uder cotrol. b. make better ivestmets. c. ear more moey. 2. The two thigs you must keep track of i order to uderstad your budget are a. your icome ad your spedig. b. your wages ad your bak iterest. c. your wages ad your credit card debt. 3. A egative mothly cash flow meas that a. your ivestmets are losig value. b. you are spedig more moey tha you are takig i. c. you are takig i more moey tha you are spedig. 4. Whe you are makig your mothly budget, what should you do with your oce-a-year expeses for December holiday gifts? a. Igore them. b. Iclude them oly i your calculatio for December s budget. c. Divide them by 12 ad iclude them as a mothly expese. 5. For the average perso, the sigle biggest category of expese is a. food. b. housig. c. etertaimet. 6. Accordig to Figure 4.1, which of the followig expeses teds to icrease the most as a perso ages? a. housig b. trasportatio c. health care 7. Which of the followig is ecessary if you wat to make mothly cotributios to savigs? a. You must have a positive mothly cash flow. b. You must be spedig less tha 20% of your icome o food ad clothig. c. You must ot owe moey o ay loas. 1 8. Trey smokes about 1 2 packs of cigarettes per day ad pays about $3.50 per pack. His mothly spedig o cigarettes is closest to a. $50. b. $100. c. $150. 9. Kira driks about 6 cas of soda each day, geerally buyig them from vedig machies at a average price of $1.25. Her aual spedig o soda is closest to a. $500. b. $1500. c. $3000. 10. You drive a average of 400 miles per week i a car that gets 18 miles per gallo. With gasolie priced at $3 per gallo, approximately how much would you save each year o gas if you istead had a car that got 50 miles per gallo? a. $500 b. $2200 c. $4500 REVIEW QUESTIONS 11. Why is it so importat to uderstad your persoal fiaces? What types of problems are more commo amog people who do ot feel they have their fiaces uder cotrol? 12. List four crucial thigs you should do if you hope to keep your fiaces uder cotrol, ad describe how you ca achieve each oe. 13. What is a budget? Describe the four-step process of figurig out your mothly budget. 14. What is cash flow? Briefly describe your optios if you have a egative mothly cash flow, ad cotrast them with your optios if you have a positive mothly cash flow.
224 CHAPTER 4 Maagig Your Moey 15. Summarize how average spedig patters chage with age. How ca comparig your ow spedig to average spedig patters help you evaluate your budget? 16. What items should you iclude whe calculatig how much it is costig you to atted college? How ca you decide whether this is a worthwhile expese? DOES IT MAKE SENSE? Decide whether each of the followig statemets makes sese (or is clearly true) or does ot make sese (or is clearly false). Explai your reasoig. 17. Whe I figured out my mothly budget, I icluded oly my ret ad my spedig o gasolie, because othig else could possibly add up to much. 18. My mothly cash flow was 2$150, which explaied why my credit card debt kept risig. 19. My vacatio travel cost a total of $1800, which I etered ito my mothly budget as $150 per moth. 20. Emma ad Emily are good frieds who do everythig together, spedig the same amouts o eatig out, etertaimet, ad other leisure activities. Yet Emma has a egative mothly cash flow while Emily s is positive, because Emily has more icome. 21. Brado discovered that his daily routie of buyig a slice of pizza ad a soda at luch was costig him more tha $15,000 per year. 22. I bought the cheapest health isurace I could fid, because that s sure to be the best optio for my log-term fiacial success. BASIC SKILLS & CONCEPTS Extravagat Spedig? I Exercises 23 30, compute the total cost per year of the first set of expeses. The complete the setece: O a aual basis, the first set of expeses is % of the secod set of expeses. 23. Natasha buys five $1 lottery tickets every week ad speds $120 per moth o food. 24. Jeremy buys the New York Times from the ewsstad for $1 a day (skippig Sudays) ad speds $20 per week o gasolie for his car. 25. Suzae s cell phoe bill is $85 per moth, ad she speds $200 per year o studet health isurace. 26. Marcus speds a average of $4 per day o itues; his ret is $350 per moth. 27. Sheryl buys a $9 pack of cigarettes each week ad speds $30 a moth o dry cleaig. 28. Ted goes to a club or cocert every two weeks at a average ticket price of $60; he speds $500 a year o car isurace. 29. Ver driks three 6-packs of beer each week at a cost of $7 each ad speds $700 per year o his textbooks. 30. Sady fills the gas tak o her car a average of oce every two weeks at a cost of $35 per tak; her cable TV/Iteret costs $60 per moth. Iterest Paymets. Fid the aual iterest paymets i the situatios described i Exercises 31 34. Assume that you pay the iterest mothly, at a rate of exactly 1>12 the aual iterest rate. 31. You maitai a average balace of $650 o your credit card, which carries a 18% aual iterest rate. 32. Brooke s credit card has a aual iterest rate of 21% o her upaid balace, which averages $900. 33. Vic bought a ew plasma TV for $2200. He made a dow paymet of $300 ad the fiaced the balace through the store. Ufortuately, he was uable to make the first mothly paymets ad ow pays 3% iterest per moth o the balace (while he watches his TV). 34. Deaa owes a clothig store $700, but util she makes a paymet, she pays 9% iterest per moth. Proratig Expeses. I Exercises 35 40, prorate the give expeses to fid the mothly cost. 35. Sara pays $4500 for tuitio ad fees for each of two semesters, plus a additioal $300 for textbooks each semester. 36. Jake erolls for 15 credit-hours for each of two semesters at a cost of $550 per credit-hour (tuitio ad fees). I additio, textbooks cost $400 per semester. 37. Moriah takes courses o a quarter system. Three times a year, she takes 15 credits at a tuitio rate of $280 per credit; her fees are $190 per quarter, ad her dorm room costs $2300 per quarter. 38. Jua pays $500 per moth i ret, a semiaual car isurace premium of $800, ad a aual health club membership fee of $900. 39. Nguye makes a aual cotributio of $200 to his local food bak ad pays a life isurace premium of $400 twice a year. 40. Rady speds a average of $25 per week o gasolie ad $45 every three moths o the daily ewspaper.
4A Takig Cotrol of Your Fiaces 225 Net Cash Flow. I Exercises 41 44, the expeses ad icome of a idividual are give i table form. I each case, fid the et mothly cash flow (it could be positive or egative). Assume salaries ad wages are after taxes. Whe you eed to covert betwee weeks ad moths, assume that 1 moth 5 4 weeks. 41. 42. Icome Expeses Part-time job: $600>moth Ret: $450>moth College fud from Groceries: $50>week gradparets: $400>moth Tuitio ad fees: $3000 Scholarship: $5000>year twice a year Icidetals: $100>week Icome Part-time job: $1200>moth Studet loa: $7000>year Scholarship: $8000>year Expeses Ret: $600>moth Groceries: $70>week Tuitio ad fees: $7500>year Health isurace: $40>moth Etertaimet: $200>moth Phoe: $65>moth 44. Icome Salary: $32,000>year Pottery sales: $200>moth Expeses House paymets: $700>moth Groceries: $150>week Household expeses: $450>moth Health isurace: $150>moth Car isurace: $500 semiaually Savigs pla: $200>moth Doatios: $600>year Miscellaeous: $800>moth Budget Allocatio. Use Figure 4.1 to determie whether the spedig patters described i Exercises 45 50 are at, above, or below the atioal average. Assume all salaries ad wages are after taxes. 45. A sigle 30-year-old woma with a mothly salary of $3200 speds $900 per moth o ret. 46. A couple uder the age of 30 has a combied household icome of $3500 per moth ad speds $400 per moth o etertaimet. 47. A sigle 42-year-old ma with a mothly salary of $3600 speds $200 per moth o health care. 48. A 32-year-old couple with a combied household icome of $45,500 per year speds $700 per moth o trasportatio. 49. A retired (over 65 years old) couple with a fixed mothly salary of $4200 speds $600 per moth o health care. 50. A family with a 45-year-old wage earer has a aual household icome of $48,000 ad speds $1500 per moth o housig. 43. Icome Salary: $2300>moth Expeses Ret: $800>moth Groceries: $90>week Utilities: $125>moth Health isurace: $360 semiaually Car isurace: $400 semiaually Gasolie: $25>week Miscellaeous: $400>moth Phoe: $85>moth Makig Decisios. Exercises 51 56 preset two optios. Determie which optio is less expesive. Are there other factors that might affect your decisio? 51. You curretly drive 250 miles per week i a car that gets 21 miles per gallo of gas. You are cosiderig buyig a ew fuel-efficiet car for $16,000 (after trade-i o your curret car) that gets 45 miles per gallo. Isurace premiums for the ew ad old car are $800 ad $400 per year, respectively. You aticipate spedig $1500 per year o repairs for the old car ad havig o repairs o the ew car. Assume gas costs $3.50 per gallo. Over a five-year period, is it less expesive to keep your old car or buy the ew car?
226 CHAPTER 4 Maagig Your Moey 52. You curretly drive 300 miles per week i a car that gets 15 miles per gallo of gas. You are cosiderig buyig a ew fuel-efficiet car for $12,000 (after trade-i o your curret car) that gets 50 miles per gallo. Isurace premiums for the ew ad old car are $800 ad $600 per year, respectively. You aticipate spedig $1200 per year o repairs for the old car ad havig o repairs o the ew car. Assume gas costs $3.50 per gallo. Over a five-year period, is it less expesive to keep your old car or buy the ew car? 53. You must decide whether to buy a ew car for $22,000 or lease the same car over a three-year period. Uder the terms of the lease, you make a dow paymet of $1000 ad have mothly paymets of $250. At the ed of three years, the leased car has a residual value (the amout you pay if you choose to buy the car at the ed of the lease period) of $10,000. Assume you sell the ew car at the ed of three years at the same residual value. Is it less expesive to buy or to lease? 54. You must decide whether to buy a ew car for $22,000 or lease the same car over a four-year period. Uder the terms of the lease, you make a dow paymet of $1000 ad have mothly paymets of $300. At the ed of four years, the leased car has a residual value (the amout you pay if you choose to buy the car at the ed of the lease period) of $8000. Assume you sell the ew car at the ed of four years at the same residual value. Is it less expesive to buy or to lease? 55. You have a choice betwee goig to a i-state college where you would pay $4000 per year for tuitio ad a out-of-state college where the tuitio is $6500 per year. The cost of livig is much higher at the i-state college, where you ca expect to pay $700 per moth i ret, compared to $450 per moth at the other college. Assumig all other factors are equal, which is the less expesive choice o a aual (12-moth) basis? 56. If you stay i your home tow, you ca go to Cocord College at a reduced tuitio of $3000 per year ad pay $800 per moth i ret. Or you ca leave home, go to Versalia College o a $10,000 scholarship (per year), pay $16,000 per year i tuitio, ad pay $350 per moth to live i a dormitory. You will pay $2000 per year to travel back ad forth from Versalia College. Assumig all other factors are equal, which is the less expesive choice o a aual (12-moth) basis? You Could Be Doig Somethig Else. Exercises 57 58 preset two optios. Determie which optio is better fiacially. Are there other factors that might affect your decisio? 57. You could take a 15-week, three-credit college course, which requires 10 hours per week of your time ad costs $500 per credit-hour i tuitio. Or durig those hours you could have a job payig $10 per hour. What is the et cost of the class compared to workig? Give that the average college graduate ears early $20,000 per year more tha a high school graduate, is it a worthwhile expese? 58. You could have a part-time job (20 hours per week) that pays $15 per hour, or you could have a full-time job (40 hours per week) that pays $12 per hour. Because of the extra free time, you will sped $150 per week more o etertaimet with the part-time job tha with the fulltime job. After accoutig for the extra etertaimet, how much more is your cash flow with the full-time job tha with the part-time job? Neglect taxes ad other expeses. FURTHER APPLICATIONS 59. Laudry Upgrade. Suppose that you curretly ow a clothes dryer that costs $25 per moth to operate. A ew efficiet dryer costs $620 ad has a estimated operatig cost of $15 per moth. How log will it take for the ew dryer to pay for itself? 60. Break-Eve Poit. You curretly drive 300 miles per week i a car that gets 18 miles per gallo of gas. A ew fuel-efficiet car costs $15,000 (after trade-i o your curret car) ad gets 45 miles per gallo. Isurace premiums for the ew ad old car are $800 ad $500 per year, respectively. You aticipate spedig $1500 per year o repairs for the old car ad havig o repairs o the ew car. Assumig that gas remais at $3.50 per gallo, estimate the umber of years after which the costs of owig the ew ad old cars are equal. Hit: You might make a table showig the accumulated aual expeses for each car for each year. 61. Isurace Deductibles. May isurace policies carry a deductible provisio that states how much of a claim you must pay out of pocket before the isurace compay pays the remaiig expeses. For example, if you file a claim for $350 o a policy with a $200 deductible, you pay $200 ad the isurace compay pays $150. I the followig cases, determie how much you would pay with ad without the isurace policy. a. You have a car isurace policy with a $500 deductible provisio (per claim) for collisios. Durig a two-year period, you file claims for $450 ad $925. The aual premium for the policy is $550. b. You have a car isurace policy with a $200 deductible provisio (per claim) for collisios. Durig a two-year period, you file claims for $450 ad $1200. The aual premium for the policy is $650. c. You have a car isurace policy with a $1000 deductible provisio (per claim) for collisios. Durig a two-year
4A Takig Cotrol of Your Fiaces 227 period, you file claims for $200 ad $1500. The aual premium for the policy is $300. d. Explai why lower isurace premiums go with higher deductibles. 62. Car Leases. Cosider the followig three lease optios for a ew car. Determie which lease is least expesive, assumig you buy the car whe the lease expires. The residual is the amout you pay if you choose to buy the car whe the lease expires. Discuss other factors that might affect your decisio. Pla A: $1000 dow paymet, $400 per moth for two years, residual value 5 $10,000 Pla B: $500 dow paymet, $250 per moth for three years, residual value 5 $9500 Pla C: $0 dow paymet, $175 per moth for four years, residual value 5 $8000 63. Health Costs. Assume that you have a (relatively simple) health isurace pla with the followig provisios: Office visits require a co-paymet of $25. Emergecy room visits have a $200 deductible (you pay the first $200). Surgical operatios have a $1000 deductible (you pay the first $1000). You pay a mothly premium of $350. Durig a oe-year period, your family has the followig expeses. Pla A Pla B Office visits require a Office visits require a co-paymet of $25. co-paymet of $25. Emergecy room visits Emergecy room visits have a $500 deductible have a $200 deductible (you pay the first $500). (you pay the first $200). Surgical operatios have Surgical operatios have a $5000 deductible a $1500 deductible (you pay the first $5000). (you pay the first $1500). You pay a mothly You pay a mothly premium of $300. premium of $700. Suppose that durig a oe-year period your family has the followig expeses. Total cost Expese (before isurace) Ja. 23: Emergecy room $400 Feb. 14: Office visit $100 Apr. 13: Surgery $1400 Jue 14: Surgery $7500 July 1: Office visit $100 Sept. 23: Emergecy room $1200 Total cost Expese (before isurace) Feb. 18: Office visit $100 Mar. 26: Emergecy room $580 Apr. 23: Office visit $100 May 14: Surgery $6500 July 1: Office visit $100 Sept. 23: Emergecy room $840 a. Determie your health-care expeses for the year with the isurace policy. b. Determie your health-care expeses for the year if you did ot have the isurace policy. 64. Health-Care Choices. You have a choice of two health isurace policies with the followig terms. a. Determie your aual health-care expeses if you have Pla A. b. Determie your aual health-care expeses if you have Pla B. c. Would havig o health isurace be better tha either Pla A or Pla B? Exercises 65 68 ask you to evaluate your ow persoal fiaces. (Note to istructors: If these problems are assiged to be tured i, you should allow studets to fictioalize their aswers so that they are ot beig asked to reveal persoal fiacial data.) 65. Daily Expeditures. Keep a list of everythig you sped moey o durig oe etire day. Categorize each expediture, ad the make a table with oe colum for the categories ad oe colum for the expeditures. Add a third colum i which you compute how much you d sped i a year if you spet the same amout every day.
228 CHAPTER 4 Maagig Your Moey 66. Weekly Expeditures. Repeat Problem 65, but this time make the list for a full week of spedig rather tha just oe day. 67. Prorated Expeditures. Make a list of all the major expeses you have each year that you do ot pay o a mothly basis, such as college expeses, holiday expeses, ad vacatio expeses. For each item, estimate the amout you sped i a year, ad the determie the prorated amout that you should use whe you determie your mothly budget. 68. Mothly Cash Flow. Create your complete mothly budget, listig all sources of icome ad all expeditures, ad use it to determie your et mothly cash flow. Be sure to iclude small but frequet expeditures ad prorated amouts for large expeditures. Explai ay assumptios you make i creatig your budget. Whe the budget is complete, write a paragraph or two explaiig what you leared about your ow spedig patters ad what adjustmets you may eed to make to your budget. WEB PROJECTS Fid useful liks for Web Projects o the text Web site: www.aw.com/beett-briggs 69. Persoal Budgets. May Web sites provide persoal budget advice ad worksheets. Visit several of these sites ad choose oe to help you orgaize your budget for at least three moths. Is the site effective i helpig you pla your fiaces? Discuss how the site led to isights that you would ot have had otherwise. 70. U.S. Spedig Patters. Fid the complete (two-page) paper from which Figure 4.1 was take (Spedig Patters by Age, U.S. Departmet of Labor Statistics). Write a summary of the coclusios of the paper ad discuss whether your persoal fiaces fit the patters described i the paper. IN THE NEWS 71. Persoal Bakruptcies. The rate of persoal bakruptcies has bee icreasig for several years. Fid at least three ews articles o the subject, documet the icrease i bakruptcies, ad explai the primary reasos for the icrease. 72. Cosumer Debt. Fid data o the icrease i cosumer (credit card) debt i the Uited States. Based o your readig, do you thik cosumer debt is (a) a crisis, (b) a sigificat occurrece but othig to worry about, or (c) a good thig? Justify your coclusio. 73. U.S. Savigs Rate. Whe it comes to savig disposable icome, Americas have a remarkably low savigs rate. Fid sources that compare the savigs rates of Asia ad Europea coutries to that of the Uited States. Discuss your observatios ad put your ow savigs habits o the scale. UNIT 4B The Power of Compoudig O July 18, 1461, Kig Edward IV of Eglad borrowed the moder equivalet of $384 from New College of Oxford. The Kig soo paid back $160, but ever repaid the remaiig $224. The debt was forgotte for 535 years. Upo its rediscovery i 1996, a New College admiistrator wrote to the Quee of Eglad askig for repaymet, with iterest. Assumig a iterest rate of 4% per year, he calculated that the college was owed $290 billio. This example illustrates what is sometimes called the power of compoudig : the remarkable way that moey grows whe iterest cotiues to accumulate year after year. I the New College case, there is o clear record of a promise to repay the debt with iterest, ad eve if there were, the Quee might ot feel obliged to pay a debt that had bee forgotte for more tha 500 years. But ayoe ca take advatage of compoud iterest simply by opeig a savigs accout. With patiece, the results may be truly astoishig.
4B The Power of Compoudig 229 Simple versus Compoud Iterest Imagie that you deposit $1000 i Hoest Joh s Moey Holdig Service, which promises to pay 5% iterest each year. At the ed of the first year, Hoest Joh s seds you a check for 5% 3 $1000 5 0.05 3 $1000 5 $50 You also get $50 at the ed of the secod ad third years. Over the 3 years, you receive total iterest of 3 3 $50 5 $150 Your origial $1000 has grow i value to $1150. Hoest Joh s method of paymet represets simple iterest, i which iterest is paid oly o your actual ivestmet, or pricipal. Now, suppose that you place the $1000 i a bak accout that pays the same 5% iterest oce a year. But istead of payig you the iterest directly, the bak adds the iterest to your accout. At the ed of the first year, the bak deposits $50 iterest ito your accout, raisig your balace to $1050. At the ed of the secod year, the bak agai pays you 5% iterest. This time, however, the 5% iterest is paid o the balace of $1050, so it amouts to By the Way The New College admiistrator did ot seriously believe that the Quee would pay $290 billio. However, he suggested a compromise of assumig a 2% per year iterest rate, i which case the college was owed oly $8.9 millio. This, he said, would be eough to pay for a moderizatio project at the College. The Quee has ot yet paid. Addig this $52.50 raises your balace to 5% 3 $1050 5 0.05 3 $1050 5 $52.50 $1050 1 $52.50 5 $1102.50 This is the ew balace o which your 5% iterest is computed at the ed of the third year. So your third iterest paymet is 5% 3 $1102.50 5 0.05 3 $1102.50 5 $55.13 Therefore, your balace at the ed of the third year is $1102.50 1 $55.13 5 $1157.63 Despite idetical iterest rates, you ed up with $7.63 more if you use the bak istead of Hoest Joh s. The differece comes about because the bak pays you iterest o the iterest as well as o the origial pricipal. This type of iterest paymet is called compoud iterest. DEFINITIONS The pricipal i fiacial formulas is the balace upo which iterest is paid. Simple iterest is iterest paid oly o the origial pricipal, ad ot o ay iterest added at later dates. Compoud iterest is iterest paid both o the origial pricipal ad o all iterest that has bee added to the origial pricipal.
230 CHAPTER 4 Maagig Your Moey EXAMPLE 1 Savigs Bod While baks almost always pay compoud iterest, bods usually pay simple iterest. Suppose you ivest $1000 i a savigs bod that pays simple iterest of 10% per year. How much total iterest will you receive i 5 years? If the bod paid compoud iterest, would you receive more or less total iterest? Explai. SOLUTION With simple iterest, every year you receive the same iterest paymet of 10% 3 $1000 5 $100. Thus, you receive a total of $500 i iterest over 5 years. With compoud iterest, you receive more tha $500 i iterest because the iterest each year is calculated o your growig balace rather tha o your origial pricipal. For example, because your first iterest paymet of $100 raises your balace to $1100, your ext compoud iterest paymet is 10% 3 $1100 5 $110, which is more tha the simple iterest paymet of $100. For the same iterest rate, compoud iterest always raises your balace faster tha simple iterest. Now try Exercises 41 44. The Compoud Iterest Formula Let s retur to Kig Edward s debt to the New College. We ca calculate the amout owed to the College by pretedig that the $224 he borrowed was deposited ito a iterest-bearig accout for 535 years. Let s assume, as did the New College admiistrator, that the iterest rate was 4% per year. For each year, we ca calculate the iterest ad the ew balace with iterest. The first three colums of Table 4.1 show these calculatios for 4 years. TABLE 4.1 Calculatig Compoud Iterest After N Years Iterest Balace Or Equivaletly 1 year 4% 3 $224 5 $8.96 $224 1 $8.96 5 $232.96 $224 3 1.04 5 $232.96 2 years 4% 3 $232.96 5 $9.32 $232.96 1 $9.32 5 $242.28 $224 3 A1.04B 2 5 $242.28 3 years 4% 3 $242.28 5 $9.69 $242.28 1 $9.69 5 $251.97 $224 3 A1.04B 3 5 $251.97 4 years 4% 3 $251.97 5 $10.08 $251.97 1 $10.08 5 $262.05 $224 3 A1.04B 4 5 $262.05 To fid the total balace, we could cotiue the calculatios to 535 years. Fortuately, there s a much easier way. The 4% aual iterest rate meas that each edof-year balace is 104% of, or 1.04 times, the previous year s balace. Thus, as show i the last colum of Table 4.1, we ca get each balace as follows: The balace at the ed of 1 year is the origial pricipal times 1.04: $224 3 1.04 5 $232.96 The balace at the ed of 2 years is the 1-year balace times 1.04: $224 3 1.04 3 1.04 5 $224 3 A1.04B 2 5 $242.28 The balace at the ed of 3 years is the 2-year balace times 1.04: $224 3 1.04 3 1.04 3 1.04 5 $224 3 A1.04B 3 5 $251.97
4B The Power of Compoudig 231 Cotiuig the patter, we fid that the balace after N years is the origial pricipal times 1.04 raised to the Nth power. For example, the balace after N 5 10 years is $224 3 A1.04B 10 5 $331.57 We ca geeralize this result by lookig carefully at the last equatio above. Notice that the $224 is the origial pricipal that we bega with. The 1.04 is 1 plus the iterest rate of 4%, or 0.04. The expoet 10 is the umber of times that the iterest has bee compouded. Let s write the equatio agai, addig these idetifiers ad turig it aroud to put the result o the left: $331.57 (''')'''* accumulated balace, A 5 $224 ('')''* origial pricipal, P d umber of compoudig periods ('')''* 11 iterest rate 3 A1.04B 10 Whe iterest is compouded just oce a year, as it is i this case, the iterest rate is called the aual percetage rate, or APR. The umber of compoudig periods is the simply the umber of years, which we call Y, over which the pricipal ears iterest. We therefore obtai the followig geeral formula for iterest compouded oce a year. where A 5 P 3 A1 1 APRB Y A 5 accumulated balace after Y years P 5 startig pricipal APR 5 aual percetage rate Aas a decimalb Y 5 umber of years Be sure to ote that the aual iterest rate, APR, should always be expressed as a decimal rather tha as a percetage. I the New College case, the aual iterest rate is APR 5 4% 5 0.04, ad iterest is paid over a total of 535 years. Thus, the accumulated balace after Y 5 535 years would be A 5 P 3 A1 1 APRB Y 5 $224 3 A1 1 0.04B 535 5 $224 3 A1.04B 535 5 $224 3 1,296,691,085 < $2.9 3 10 11 5 $290 billio As the admiistrator claimed, a 4% iterest rate meas the Quee owes about $290 billio. Techical Note For the more geeral case i which the iterest rate is ot ecessarily set o a aual (APR) basis, you may see the compoud iterest formula writte A 5 P 3 A1 1 i B N where i is the iterest rate ad N is the total umber of compoudig periods. USING YOUR CALCULATOR Most calculators have a key for raisig to powers, labeled y x or. For example, calculate 1.04 535 by pressig 1.04 y x 535 or 1.04 535. EXAMPLE 2 Simple ad Compoud Iterest You ivest $100 i two accouts that each pay a iterest rate of 10% per year. However, oe accout pays simple iterest ad oe accout pays compoud iterest. Make a table that shows the growth of each accout over a 5-year period. Use the compoud iterest formula to verify the result i the table for the compoud iterest case. SOLUTION The simple iterest is the same absolute amout each year. The compoud iterest grows from year to year, because it is paid o the accumulated iterest as well as o the startig balace. Table 4.2 summarizes the calculatios.
232 CHAPTER 4 Maagig Your Moey TABLE 4.2 Calculatios for Example 2 SIMPLE INTEREST ACCOUNT COMPOUND INTEREST ACCOUNT Ed of Old balace 1 iterest Old balace 1 iterest year Iterest paid 5 ew balace Iterest paid 5 ew balace 1 10% 3 $100 5 $10 $100 1 $10 5 $110 10% 3 $100 5 $10 $100 1 $10 5 $110 2 10% 3 $100 5 $10 $110 1 $10 5 $120 10% 3 $110 5 $11 $110 1 $11 5 $121 3 10% 3 $100 5 $10 $120 1 $10 5 $130 10% 3 $121 5 $12.10 $121 1 $12.10 5 $133.10 4 10% 3 $100 5 $10 $130 1 $10 5 $140 10% 3 $133.10 5 $13.31 $133.10 1 $13.31 5 $146.41 5 10% 3 $100 5 $10 $140 1 $10 5 $150 10% 3 $146.41 5 $14.64 $146.41 1 $14.64 5 $161.05 To verify the fial etry i the table with the compoud iterest formula, we use a startig pricipal P 5 $100 ad a aual iterest rate APR 5 10% 5 0.1 with iterest paid for Y 5 5 years. The accumulated balace A is A 5 P 3 A1 1 APRB Y 5 $100 3 A1 1 0.1B 5 5 $100 3 1.1 5 5 $100 3 1.6105 5 $161.05 This result agrees with the oe i the table. Overall, the accout payig compoud iterest builds to $161.05 while the simple iterest accout reaches oly $150, eve though both pay at the same 10% rate. This is a sigificat differece, especially whe you cosider that the differece will cotiue to grow with time. Clearly, compoud iterest is much better for a ivestor tha simple iterest at the same rate. Now try Exercises 45 46. Accumulated value $15,000 $10,000 $5000 $224 0 20 40 60 80 100 Years FIGURE 4.2 The value of the debt i the New College case durig the first 100 years, at a iterest rate of 4% per year. Note that the value rises much more rapidly i later years tha i earlier years a hallmark of expoetial growth. Compoud Iterest as Expoetial Growth The New College case demostrates the remarkable way i which moey ca grow with compoud iterest. Figure 4.2 shows how the value of the New College debt rises durig the first 100 years, assumig a startig value of $224 ad a iterest rate of 4% per year. Note that while the value rises slowly at first, it rapidly accelerates, so i later years the value grows by much more each year tha it did durig earlier years. This rapid growth is a hallmark of what we geerally call expoetial growth. You ca see how expoetial growth gets its ame by lookig agai at the geeral compoud iterest formula: A 5 P 3 A1 1 APRB Y Because the pricipal P ad the iterest rate APR have fixed values for ay particular compoud iterest calculatio, the growth of the accumulated value A depeds oly o Y (the umber of times iterest has bee paid), which appears i the expoet of the calculatio.
4B The Power of Compoudig 233 Expoetial growth is oe of the most importat topics i mathematics, with applicatios that iclude populatio growth, resource depletio, ad radioactivity. We will study expoetial growth i much more detail i Chapter 8. I this chapter, we focus oly o its applicatios i fiace. EXAMPLE 3 New College Debt at 2% If the iterest rate is 2%, calculate the amout due to New College usig a. simple iterest b. compoud iterest SOLUTION a. At a rate of 2%, the simple iterest due each year is 2% 3 $224 5 0.02 3 $224 5 $4.48 Over 535 years, the total iterest due would be By the Way Fiacial plaers ofte call the pricipal, P, the preset value (PV) of the moey ad the accumulated amout, A, the future value (FV). This termiology is also used o may fiacial calculators ad software packages. 535 3 $4.48 5 $2396.80 Addig this to the origial loa pricipal of $224 gives the payoff amout of $224 1 $2396.80 5 $2620.80 b. To fid the amout due with compoud iterest, we set the aual iterest rate to APR 5 2% 5 0.02 ad the umber of years to Y 5 535. The we use the formula for compoud iterest paid oce a year: A 5 P 3 A1 1 APRB Y 5 $224 3 A1 1 0.02B 535 5 $224 3 A1.02B 535 5 $224 3 39,911 < $8.94 3 10 6 The amout due with compoud iterest is about $8.94 millio far higher tha the amout due with simple iterest. You should ote the remarkable effects of small chages i the compoud iterest rate. Here, we foud that a 2% compoud iterest rate leads to $8.94 millio after 535 years. Earlier, we foud that a 4% iterest rate for the same 535 years leads to $290 billio which is more tha 30,000 times as large as $8.94 millio. Figure 4.3 cotrasts the values of the New College debt durig the first 100 years at iterest rates of 2% ad 4%. Note that the rates do t make much differece for the first few years, but over time the higher rate becomes far more valuable. Now try Exercises 47 48. Accumulated value $15,000 $10,000 $5000 APR 4% Years APR 2% $224 0 20 40 60 80 100 FIGURE 4.3 This figure cotrasts the debt i the New College case durig the first 100 years at iterest rates of 2% ad 4%.
234 CHAPTER 4 Maagig Your Moey EXAMPLE 4 Mattress Ivestmets Your gradfather put $100 uder his mattress 50 years ago. If he had istead ivested it i a bak accout payig 3.5% iterest compouded yearly (roughly the average U.S. rate of iflatio durig that period), how much would it be worth ow? SOLUTION The startig pricipal is P 5 $100. The aual percetage rate is APR 5 3.5% 5 0.035. The umber of years is Y 5 50. So the accumulated balace is Ivested at a rate of 3.5%, the $100 would be worth over $550 today. Ufortuately, the $100 was put uder a mattress, so it still has a face value of oly $100. Now try Exercises 49 52. Compoud Iterest Paid More Tha Oce a Year TABLE 4.3 Time out to thik Suppose the iterest rate for the New College debt were 3%. Without calculatig, do you thik the value after 535 years would be halfway betwee the values at 2% ad 4% or closer to oe or the other of these values? Now, check your guess by calculatig the value at 3%. What happes at a iterest rate of 6%? Briefly discuss why small chages i the iterest rate ca lead to large chages i the accumulated value. A 5 P 3 A1 1 APRB Y 5 $100 3 A1 1 0.035B 50 Suppose you deposit $1000 ito a bak that pays compoud iterest at a aual percetage rate of APR 5 8%. If the iterest is paid all at oce at the ed of a year, you ll receive iterest of 8% 3 $1000 5 0.08 3 $1000 5 $80 Thus, your year-ed balace will be $1000 1 $80 5 $1080. Now, assume istead that the bak pays the iterest quarterly, or four times a year (that is, oce every 3 moths). The quarterly iterest rate is oe-fourth of the aual iterest rate: quarterly iterest rate 5 APR 4 5 $100 3 A1.035B 50 5 $558.49 5 8% 4 5 2% 5 0.02 Table 4.3 shows how quarterly compoudig affects the $1000 startig pricipal durig the first year. Quarterly Iterest Paymets After N Quarters Iterest Paid New Balace 1st quarter (3 moths) 2d quarter (6 moths) 3rd quarter (9 moths) 2% 3 $1000 5 $20 2% 3 $1020 5 $20.40 2% 3 $1040.40 5 $20.81 $1000 1 $20 5 $1020 $1020 1 $20.40 5 $1040.40 $1040.40 1 $20.81 5 $1061.21 4th quarter (1 full year) 2% 3 $1061.21 5 $21.22 $1061.21 1 $21.22 5 $1082.43