Chapter 4

Managing YourMoneyManaging your personal finances is a complex taskin the modern world. If you are like most Ameri-cans, you already have a bank account and at leastone credit card. You may also have student loans, ahome mortgage, and various investment plans. Inthis chapter, we discuss key issues in personal finan-cial management, including budgeting, savings,loans, taxes, and investments. We also explore howthe government manages its money, which affectsall of us.

A fool and his money aresoon parted.

—English proverb

215

UNIT 4ATaking Control of Your Finances: You cannotachieve financial success unless you knowhow to make wise decisions about yourmoney. We discuss the basics of personalbudgeting.

UNIT 4BThe Power of Compounding: We explore theway in which you can increase your savingsthrough the mathematics of compound interest.

UNIT 4CSavings Plans and Investments: We calcu-late the future value of savings plans in whichyou make monthly deposits and study invest-ments in stocks and bonds.

UNIT 4DLoan Payments, Credit Cards, and Mort-gages: We calculate monthly payments andexplore loan issues.

UNIT 4EIncome Taxes: We explore the mathematicsof income taxes and a few of the hot politicalissues that surround them.

UNIT 4FUnderstanding the Federal Budget:Everyone’s personal finances are ultimatelytied to government finances. We examine thefederal budget process and related politicalissues.

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216 CHAPTER 4 Managing Your Money

UNIT 4A Taking Control of Your Finances

Money isn’t everything, but it certainly has a great influence on our lives. Most peo-ple would like to have more money, and there’s no doubt that more money allows youto do things that simply aren’t possible with less. However, when it comes to personalhappiness, studies show that the amount of money you have is less important thanhaving your personal finances under control. People who lose control of theirfinances tend to suffer from financial stress, which in turn leads to higher divorcerates and other difficulties in personal relationships, higher rates of depression, and avariety of other ailments. In contrast, people who manage their money well are morelikely to say they are happy, even when they are not particularly wealthy. So if youwant to attain happiness—along with any financial goals you might have—the firststep is to make sure you understand your personal finances enough to keep them wellunder control.

Take Control

If you’re reading this book, chances are that you are in college somewhere. In thatcase, you are almost certainly facing financial challenges that you’ve never had to dealwith before. If you are a recent high school graduate, this may be the first time thatyou are fully responsible for your own financial well-being. If you are coming back toschool after many years in the work force or as a parent, you now have the challengeof juggling the cost of college with all the other financial challenges of daily life.

The key to success in meeting these financial challenges is to make sure you alwayscontrol them, rather than letting them control you. And the first step in gaining con-trol is to make sure you keep track of your finances. Unless you happen to be amongthe superrich, keeping track of your finances probably isn’t that difficult, but itrequires diligence. For example, you should always know your bank account balance,so that you never have to worry about bouncing a check or having your debit cardrejected. Similarly, you should know what you are spending on your credit card—andif it’s going to be possible for you to pay off the card at the end of the month or if yourspending will dig you deeper into debt. And, of course, you should spend moneywisely and at a level that you can afford.

There are lots of books and Web sites designed to help you control your finances,but in the end they all come back to the same basic idea: You need to know how muchmoney you have and how much money you spend, and then find a way to live withinyour means. If you can do that, as summarized in the following box, you have a goodchance at financial success and happiness.

Money can’t buy melove . . .

—THE BEATLES

By the WayCollege may be costingyou a lot now, but statis-tically it’s worth it: Theaverage college gradu-ate earns nearly $20,000per year more than aperson who graduatesonly from high school,which adds up to nearly$1 million in extraincome over the courseof a career. Of course,this is only an average:Students who takeharder classes and getbetter grades tend toget higher-paying jobsand earn even morethan those who take aneasier route throughschool.

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4A Taking Control of Your Finances 217

CONTROLLING YOUR FINANCES

• Know your bank balance. You should never bounce a check or have your debitcard rejected.

• Know what you spend; in particular, keep track of your debit and credit cardspending.

• Don’t buy on impulse. Think first; then buy only if you are sure the purchasemakes sense for you.

• Make a budget, and don’t overspend it.

❉EXAMPLE 1 Latte MoneyCalvin isn’t rich, but he gets by, and he loves sitting down for a latte at the college cof-fee shop on a busy day. With tax and tip, he usually spends $5 on his large latte. Hegets at least one every day (on average), and about every three days he has a secondone. He figures it’s not such a big indulgence. Is it?

SOLUTION One a day means 365 per year. A second one every third day adds aboutmore (rounding down). That means lattes a year. At

$5 apiece, this comes to

Calvin’s coffee habit is costing him more than $2400 per year. That might not bemuch if he’s financially well off. But it’s more than two months of rent for an averagecollege student; it’s enough to allow him to take a friend out for a $100 dinner twice amonth; and it’s enough so that if he saved it, with interest he could easily build a sav-ings balance of more than $25,000 over the next ten years. Now try Exercises 23–30.

❉EXAMPLE 2 Credit Card InterestCassidy has recently begun to keep her spending under better control, but she stillcan’t fully pay off her credit card. She’s maintaining an average monthly balance ofabout $1100, and her card charges a 24% annual interest rate, which it bills at a rateof 2% per month. How much is she spending on credit card interest?

SOLUTION Her average monthly interest is 2% of the $1100 average balance,which is

Multiplying by the 12 months in a year gives her annual interest payment:

Interest alone is costing Cassidy more than $260 per year—a significant amount forsomeone living on a tight budget. Clearly, she’d be a lot better off if she could find away to pay off that credit card balance quickly and end those interest payments.

Now try Exercises 31–34. ➽

12 3 $22 5 $264

0.02 3 $1100 5 $22

➽

486 3 $5 5 $2430

365 1 121 5 486365>3 5 121

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Master Budget BasicsAs you can see from Examples 1 and 2, one of the keys to deciding what you canafford is knowing your personal budget. Making a budget means keeping track of howmuch money you have coming in and how much you have going out and then decid-ing what adjustments you need to make. The following box summarizes the four basicsteps in making a budget.

A FOUR-STEP BUDGET

1. List all your monthly income. Be sure to include a prorated amount—that is,what it averages out to per month—for any income you do not receive monthly(such as once-per-year payments).

2. List all your monthly expenses. Be sure to include a prorated amount forexpenses that don’t recur monthly, such as expenses for tuition, books, vacations,and holiday gifts.

3. Subtract your total expenses from your total income to determine your netmonthly cash flow.

4. Make adjustments as needed.

For most people, the most difficult part of the budget process is making sure youdon’t leave anything out of your list of monthly expenses. A good technique is to keepcareful track of your expenses for a few months. For example, carry a small note padwith you, and write down everything you spend. And don’t forget to prorate youroccasional expenses, or else you may severely underestimate your average monthlycosts.

Once you’ve made your lists for steps 1 and 2, the third step is just arithmetic: Sub-tracting your monthly expenses from your monthly income gives you your overallmonthly cash flow. If your cash flow is positive, you will have money left over at theend of each month, which you can use for savings. If your cash flow is negative, youhave a problem: You’ll need to find a way to balance it out, either by earning more orspending less or in some cases deciding it’s worthwhile to get a loan.

❉EXAMPLE 3 College ExpensesIn addition to your monthly expenses, you have the following college expenses thatyou pay twice a year: $3500 for your tuition each semester, $750 in student fees eachsemester, and $500 for textbooks each semester. How should you handle theseexpenses in computing your monthly budget?

SOLUTION Since you pay these expenses twice a year, the total amount you pay overa whole year is

To prorate this total expense on a monthly basis, we divide it by 12:

$9500 4 12 < $792

2 3 A$3500 1 $750 1 $500 B 5 $9500

By the Way

The cost of a collegeeducation is significantlymore than what stu-dents actually pay intuition and fees. Onaverage, tuition andfees cover about two-thirds of the total cost atprivate colleges anduniversities, one-third ofthe cost at public four-year institutions, and 20%of the cost at two-yearpublic colleges. The restis covered by taxpayers,alumni donations,grants, and other rev-enue sources.

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Your average monthly college expense for tuition, fees, and textbooks comes to justunder $800, so you should put $800 per month into your expense list.

Now try Exercises 35–40.

❉EXAMPLE 4 College Student BudgetBrianna is creating a budget. The expenses she pays monthly are $700 for rent, $120for gas for her car, $140 for health insurance, $75 for auto insurance, $25 for renters’insurance, $110 for her cell phone, $100 for utilities, about $300 for groceries, andabout $250 for entertainment, including eating out. In addition, over the entire yearshe spends $12,000 for college expenses, about $1000 on gifts for family and friends,about $1500 for vacations at spring and winter break, about $800 on clothes, and$600 in gifts to charity. Her income consists of a monthly, after-tax paycheck of about$1600 and a $3000 scholarship that she received at the beginning of the school year.Find her total monthly cash flow.

SOLUTION Step 1 in creating her budget is to come up with her total monthlyincome. Her $3000 scholarship means an average of per month ona prorated basis. Adding this to her $1600 monthly paycheck makes her total income$1850.

Step 2 is to look at her monthly expenses. Those paid monthly come to Her annual

expenses come to dividingthis sum by the 12 months in a year gives on a prorated monthlybasis. Thus, her total monthly expenditures are

Step 3 is to find her cash flow by subtracting her expenses from her income:

Her monthly cash flow is about The fact that this amount is negative meansshe is spending about $1300 per month—or about per year—more than she is taking in. Unless she can find a way to earn more or spend less, shewill have to cover this excess expenditure either by drawing on past savings (her ownor her family’s) or by going into debt. Now try Exercises 41–44. ➽

$1300 3 12 5 $15,6002$1300.

5 2$1295 5 $1850 2 $3145

monthly cash flow 5 monthly income 2 monthly expenses

$1820 1 $1325 5 $3145.$15,900>12 5 $1325

$12,000 1 $1000 1 $1500 1 $800 1 $600 5 $15,900;$120 1 $140 1 $75 1 $25 1 $110 1 $100 1 $300 1 $250 5 $1820.

$700 1

$3000>12 5 $250

➽


Time out to thinkLook carefully at the list of expenses for Brianna in Example 4. Do you have any cat-egories of expenses that are not covered on her list? If so, what?

Adjust Your BudgetIf you’re like most people, a careful analysis of your budget will prove very surpris-ing. For example, many people find that they are spending a lot more in certain cate-gories than they had imagined, and that the items they thought were causing theirbiggest difficulties are small compared to other items. Once you evaluate your current

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Food

Housing

Clothing and services

Transportation

Health care

Entertainment

Donations to charity

Personal insurance, pensions

0 10 20 30Percent

Percentage of Spending by Category and Age Group

Under 3535 to 6465 and older

FIGURE 4.1 Average spending patterns by age group. Technical note: Thedata show spending per “consumer unit,” which is defined tobe either a single person or a family sharing a household.Source: U.S. Department of Labor Statistics.

budget, you’ll almost certainly want to make adjustments to improve your cash flowfor the future.

There are no set rules for adjusting your budget, so you’ll need to use your criticalthinking skills to come up with a plan that makes sense for you. If your finances arecomplicated—for example, if you are a returning college student who is juggling a joband family while attending school—you might benefit from consulting a financialadvisor or reading a few books about financial planning.

You might also find it helpful to evaluate your own spending against averagespending patterns. For example, if you are spending a higher percentage of yourmoney on entertainment than the average person, you might want to consider findinglower-cost entertainment options. Figure 4.1 summarizes the average spending pat-terns for people of different ages in the United States.

By the WaySpending patterns haveshifted a great deal overtime. At the beginning ofthe twentieth century,the average Americanfamily spent 43% of itsincome on food and23% on housing. Today,food accounts for only13% of the average fam-ily’s spending, whilehousing takes 33%.Notice that thecombined percentagefor food and housinghas declined from 66%to 46% over the pastcentury, implying thatfamilies now spend sig-nificantly higher per-centages of income onother items, includingleisure activities.

❉EXAMPLE 5 Affordable Rent?You’ve worked up a budget and find that you have $1500 per month available for allyour personal expenses combined. According to the spending averages in Figure 4.1,how much should you be spending on rent?

SOLUTION Figure 4.1 shows that the percentage of spending for housing varies verylittle across age groups; it is close to or 33%, across the board. Based on thisaverage and your available budget, your rent would be about 33% of $1500, or $500per month. That’s low compared to rents for apartments in most college towns, whichmeans you face a choice: Either you can put a higher proportion of your incometoward rent—in which case you’ll have less left over for other types of expendituresthan the average person—or you can seek a way of keeping rent down, such as findinga roommate. Now try Exercises 45–50. ➽

1>3,

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Look at the Long TermFiguring out your monthly budget is a crucial step in taking control of your personalfinances, but it is only the beginning. Once you have understood your budget, youneed to start looking at longer-term issues. There are far too many issues to be listedhere, and many of them depend on your personal circumstances and choices. But thegeneral principle is always the same: Before making any major expenditure or invest-ment, be sure you figure out how it will affect your finances over the long term.

❉EXAMPLE 6 Cost of a CarJorge commutes both to his job and to school, driving a total of about 250 miles perweek. His current car is fully paid off, but it’s getting old. He is spending about $1800per year on it for repairs, and it gets only about 18 miles per gallon. He’s thinkingabout buying a new hybrid that will cost $25,000 but that should be maintenance-freeaside from oil changes over the next five years, and it gets 54 miles per gallon. Shouldhe do it?

SOLUTION To figure out whether the new car expense makes sense, Jorge needs toconsider many factors. Let’s start with gas. His 250 miles per week of driving meansabout miles per year of driving. In his current carthat gets 18 miles per gallon, this means he needs about 720 gallons of gas:

If we assume that gas costs $3 per gallon, this comes to per year.Notice that the 54-miles-per-gallon gas mileage for the new car is three times the 18-miles-per-gallon mileage for his current car, so gasoline cost for the new car would beonly as much, or about $720. Thus, he’d save each yearon gas. He would also save the $1800 per year that he’s currently spending on repairs,making his total annual savings about

Over five years, Jorge’s total savings on gasoline and repairs would come to aboutAlthough this is still short of the $25,000 he would

spend on the new car, the savings are starting to look pretty good, and they will getbetter if he keeps the new car for more than five years or if he can sell it for a decentprice at the end of five years. On the other hand, if he has to take out a loan to buy thenew car, his interest payments will add an extra expense; insurance for the new carmay cost more as well. What would you do in this situation? Now try Exercises 51–56.

❉EXAMPLE 7 Is a College Class Worth Its Cost?Across all institutions, the average cost of a three-credit college class is approximately$1500. Suppose that, between class time, commute time, and study time, the averageclass requires about 10 hours per week of your time. Assuming that you could havehad a job paying $10 per hour, what is the net cost of the class compared to working?Is it a worthwhile expense?

➽

$3240>yr 3 5 yr 5 $16,200.

$1440 1 $1800 5 $3240.

$2160 2 $720 5 $14401>3

720 3 $3 5 $2160

13,000 mi

18 migal

< 720 gal

250 mi>wk 3 52 wk>yr 5 13,000

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Time out to thinkFollowing up on Example 7, suppose that you are having difficulty in a particularclass, but know you could raise your grade by cutting back on your work hours toallow more time for studying. How would you decide whether you should do this?Explain.

SOLUTION A typical college semester lasts 14 weeks, so your “lost” work wages forthe time you spend on the class come to

Adding this to the $1500 that the class itself costs gives your total net cost of takingthe class rather than working: $2900. Whether this expense is worthwhile is subjec-tive, but remember that the average college graduate earns nearly $1 million moreover a career than a high school graduate. And also remember that, on average, stu-dents who do better in college also do better in terms of their career earnings.


14 wk 310 hr

wk3

$10hr

5 $1400

Base Financial Goals on Solid UnderstandingThese days, it’s rare for a financial decision to have a clear “best” answer for everyone.Instead, your decisions will depend on your current circumstances, your goals for thefuture, and some unavoidable uncertainty. The key to your future financial success isto approach all your financial decisions with a clear understanding of the availablechoices.

In the rest of this chapter, we’ll study several crucial topics in finance, helping youto build the understanding you’ll need to reach your financial goals. To prepare your-self for this study, it’s worth taking a few moments to think about the impact that eachof these topics will have on your financial life. In particular:

• Achieving your financial goals will almost certainly require that you build up sav-ings over time. Although it may be difficult to save while you are still in college,ultimately you will need to find a way to make your budget allow for savings andthen understand how savings work and how to choose appropriate savings plans;these are the topics of Units 4B and 4C.

• You will probably need to borrow money at various points in your life. You mayalready have credit cards, or you may be taking out student loans to help pay forcollege. In the future, you may need loans for large purchases, such as a car or ahome. Because borrowing is very expensive, it’s critical that you understand thebasic mathematics of loans so that you can make wise choices; this is the topic ofUnit 4D.

• Whether we like it or not, many of the financial decisions we make have conse-quences on our taxes. Sometimes, these tax consequences can be large enough toinfluence our decisions. For example, the fact that interest on house payments istax deductible while rent is not may influence your decision to rent or buy. Whileno one can expect to understand tax law fully, it’s important to have at least a basicunderstanding of how taxes are computed and how they can affect your financialdecisions; this is the topic of Unit 4E.

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• Finally, we do not live in isolation, and our personal finances are inevitably inter-twined with the government’s finances. For example, when politicians allow thegovernment to run deficits now, it means that future politicians will have to collectmore tax dollars from you or your children. We’ll devote Unit 4F to discussingthe federal budget and what it may mean for you in the future.

QUICK QUIZChoose the best answer to each of the following questions.Explain your reasoning with one or more complete sentences.

1. By evaluating your monthly budget, you can learn how to

a. keep your personal spending under control.

b. make better investments.

c. earn more money.

2. The two things you must keep track of in order to under-stand your budget are

a. your income and your spending.

b. your wages and your bank interest.

c. your wages and your credit card debt.

3. A negative monthly cash flow means that

a. your investments are losing value.

b. you are spending more money than you are taking in.

c. you are taking in more money than you are spending.

4. When you are making your monthly budget, what shouldyou do with your once-a-year expenses for December holi-day gifts?

a. Ignore them.

b. Include them only in your calculation for December’sbudget.

c. Divide them by 12 and include them as a monthlyexpense.

5. For the average person, the single biggest category ofexpense is

a. food. b. housing. c. entertainment.

6. According to Figure 4.1, which of the following expensestends to increase the most as a person ages?

a. housing b. transportation c. health care

7. Which of the following is necessary if you want to makemonthly contributions to savings?

a. You must have a positive monthly cash flow.

b. You must be spending less than 20% of your income onfood and clothing.

c. You must not owe money on any loans.

8. Trey smokes about packs of cigarettes per day and paysabout $3.50 per pack. His monthly spending on cigarettes isclosest to

a. $50. b. $100. c. $150.

9. Kira drinks about 6 cans of soda each day, generally buyingthem from vending machines at an average price of $1.25.Her annual spending on soda is closest to

a. $500. b. $1500. c. $3000.

10. You drive an average of 400 miles per week in a car thatgets 18 miles per gallon. With gasoline priced at $3 pergallon, approximately how much would you save each yearon gas if you instead had a car that got 50 miles per gallon?

a. $500 b. $2200 c. $4500

REVIEW QUESTIONS11. Why is it so important to understand your personal

finances? What types of problems are more commonamong people who do not feel they have their financesunder control?

12. List four crucial things you should do if you hope to keepyour finances under control, and describe how you canachieve each one.

13. What is a budget? Describe the four-step process of figur-ing out your monthly budget.

14. What is cash flow? Briefly describe your options if youhave a negative monthly cash flow, and contrast them withyour options if you have a positive monthly cash flow.

1 12

EXERCISES 4A

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15. Summarize how average spending patterns change withage. How can comparing your own spending to averagespending patterns help you evaluate your budget?

16. What items should you include when calculating howmuch it is costing you to attend college? How can youdecide whether this is a worthwhile expense?

DOES IT MAKE SENSE?Decide whether each of the following statements makes sense(or is clearly true) or does not make sense (or is clearly false).Explain your reasoning.

17. When I figured out my monthly budget, I included onlymy rent and my spending on gasoline, because nothingelse could possibly add up to much.

18. My monthly cash flow was which explained whymy credit card debt kept rising.

19. My vacation travel cost a total of $1800, which I enteredinto my monthly budget as $150 per month.

20. Emma and Emily are good friends who do everythingtogether, spending the same amounts on eating out, enter-tainment, and other leisure activities. Yet Emma has a neg-ative monthly cash flow while Emily’s is positive, becauseEmily has more income.

21. Brandon discovered that his daily routine of buying a sliceof pizza and a soda at lunch was costing him more than$15,000 per year.

22. I bought the cheapest health insurance I could find,because that’s sure to be the best option for my long-termfinancial success.

BASIC SKILLS & CONCEPTSExtravagant Spending? In Exercises 23–30, compute the totalcost per year of the first set of expenses. Then complete the sen-tence: On an annual basis, the first set of expenses is ____ % ofthe second set of expenses.

23. Natasha buys five $1 lottery tickets every week and spends$120 per month on food.

24. Jeremy buys the New York Times from the newsstand for $1a day (skipping Sundays) and spends $20 per week on gaso-line for his car.

25. Suzanne’s cell phone bill is $85 per month, and she spends$200 per year on student health insurance.

26. Marcus spends an average of $4 per day on iTunes; his rentis $350 per month.

2$150,

27. Sheryl buys a $9 pack of cigarettes each week and spends$30 a month on dry cleaning.

28. Ted goes to a club or concert every two weeks at an averageticket price of $60; he spends $500 a year on car insurance.

29. Vern drinks three 6-packs of beer each week at a cost of $7each and spends $700 per year on his textbooks.

30. Sandy fills the gas tank on her car an average of once everytwo weeks at a cost of $35 per tank; her cable TV/Internetcosts $60 per month.

Interest Payments. Find the annual interest payments in thesituations described in Exercises 31–34. Assume that you pay the interest monthly, at a rate of exactly the annual inter-est rate.

31. You maintain an average balance of $650 on your creditcard, which carries an 18% annual interest rate.

32. Brooke’s credit card has an annual interest rate of 21% onher unpaid balance, which averages $900.

33. Vic bought a new plasma TV for $2200. He made a downpayment of $300 and then financed the balance throughthe store. Unfortunately, he was unable to make the firstmonthly payments and now pays 3% interest per monthon the balance (while he watches his TV).

34. Deanna owes a clothing store $700, but until she makes apayment, she pays 9% interest per month.

Prorating Expenses. In Exercises 35–40, prorate the givenexpenses to find the monthly cost.

35. Sara pays $4500 for tuition and fees for each of two semes-ters, plus an additional $300 for textbooks each semester.

36. Jake enrolls for 15 credit-hours for each of two semestersat a cost of $550 per credit-hour (tuition and fees). In addi-tion, textbooks cost $400 per semester.

37. Moriah takes courses on a quarter system. Three times ayear, she takes 15 credits at a tuition rate of $280 percredit; her fees are $190 per quarter, and her dorm roomcosts $2300 per quarter.

38. Juan pays $500 per month in rent, a semiannual car insur-ance premium of $800, and an annual health club member-ship fee of $900.

39. Nguyen makes an annual contribution of $200 to his localfood bank and pays a life insurance premium of $400 twicea year.

40. Randy spends an average of $25 per week on gasoline and$45 every three months on the daily newspaper.

1>12


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Net Cash Flow. In Exercises 41–44, the expenses and incomeof an individual are given in table form. In each case, find thenet monthly cash flow (it could be positive or negative). Assumesalaries and wages are after taxes. When you need to convertbetween weeks and months, assume that 1 month 4 weeks.5

43.

42.Income Expenses

Part-time job: Rent:

Student loan: Groceries:

Scholarship: Tuition and fees:

Health insurance:

Entertainment:

Phone: $65>month

$200>month

$40>month

$7500>year$8000>year

$70>week$7000>year

$600>month$1200>month

44.

Income Expenses

Salary: Rent:

Groceries:

Utilities:

Health insurance: $360 semiannually

Car insurance: $400 semiannually

Gasoline:

Miscellaneous:

Phone: $85>month

$400>month

$25>week

$125>month

$90>week


Budget Allocation. Use Figure 4.1 to determine whether thespending patterns described in Exercises 45–50 are at, above, orbelow the national average. Assume all salaries and wages areafter taxes.

45. A single 30-year-old woman with a monthly salary of$3200 spends $900 per month on rent.

46. A couple under the age of 30 has a combined householdincome of $3500 per month and spends $400 per monthon entertainment.

47. A single 42-year-old man with a monthly salary of $3600spends $200 per month on health care.

48. A 32-year-old couple with a combined household income of$45,500 per year spends $700 per month on transportation.

49. A retired (over 65 years old) couple with a fixed monthlysalary of $4200 spends $600 per month on health care.

50. A family with a 45-year-old wage earner has an annualhousehold income of $48,000 and spends $1500 per monthon housing.

Making Decisions. Exercises 51–56 present two options.Determine which option is less expensive. Are there other fac-tors that might affect your decision?

51. You currently drive 250 miles per week in a car that gets21 miles per gallon of gas. You are considering buying anew fuel-efficient car for $16,000 (after trade-in on yourcurrent car) that gets 45 miles per gallon. Insurance premi-ums for the new and old car are $800 and $400 per year,respectively. You anticipate spending $1500 per year onrepairs for the old car and having no repairs on the newcar. Assume gas costs $3.50 per gallon. Over a five-yearperiod, is it less expensive to keep your old car or buy thenew car?

41.

Income Expenses

Salary: House payments:

Groceries:

Pottery sales: Household expenses:

Health insurance:

Car insurance: $500 semiannually

Savings plan:

Donations:

Miscellaneous: $800>month

$600>year

$200>month

$150>month$200>month$450>month

$150>week$32,000>year$700>month

Income Expenses

Part-time job: Rent:

College fund from Groceries: grandparents: Tuition and fees: $3000Scholarship: twice a year

Incidentals: $100>week$5000>year

$400>month$50>week


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52. You currently drive 300 miles per week in a car that gets15 miles per gallon of gas. You are considering buying anew fuel-efficient car for $12,000 (after trade-in on yourcurrent car) that gets 50 miles per gallon. Insurance premi-ums for the new and old car are $800 and $600 per year,respectively. You anticipate spending $1200 per year onrepairs for the old car and having no repairs on the new car.Assume gas costs $3.50 per gallon. Over a five-year period,is it less expensive to keep your old car or buy the new car?

53. You must decide whether to buy a new car for $22,000 orlease the same car over a three-year period. Under theterms of the lease, you make a down payment of $1000 andhave monthly payments of $250. At the end of three years,the leased car has a residual value (the amount you pay ifyou choose to buy the car at the end of the lease period) of$10,000. Assume you sell the new car at the end of threeyears at the same residual value. Is it less expensive to buyor to lease?

54. You must decide whether to buy a new car for $22,000 orlease the same car over a four-year period. Under theterms of the lease, you make a down payment of $1000 andhave monthly payments of $300. At the end of four years,the leased car has a residual value (the amount you pay ifyou choose to buy the car at the end of the lease period) of$8000. Assume you sell the new car at the end of four yearsat the same residual value. Is it less expensive to buy or tolease?

55. You have a choice between going to an in-state collegewhere you would pay $4000 per year for tuition and anout-of-state college where the tuition is $6500 per year.The cost of living is much higher at the in-state college,where you can expect to pay $700 per month in rent, com-pared to $450 per month at the other college. Assuming allother factors are equal, which is the less expensive choiceon an annual (12-month) basis?

56. If you stay in your home town, you can go to ConcordCollege at a reduced tuition of $3000 per year and pay$800 per month in rent. Or you can leave home, go toVersalia College on a $10,000 scholarship (per year), pay$16,000 per year in tuition, and pay $350 per month to livein a dormitory. You will pay $2000 per year to travel backand forth from Versalia College. Assuming all other factorsare equal, which is the less expensive choice on an annual(12-month) basis?

You Could Be Doing Something Else. Exercises 57–58 pres-ent two options. Determine which option is better financially.Are there other factors that might affect your decision?

57. You could take a 15-week, three-credit college course,which requires 10 hours per week of your time and costs$500 per credit-hour in tuition. Or during those hours you

could have a job paying $10 per hour. What is the net costof the class compared to working? Given that the averagecollege graduate earns nearly $20,000 per year more than ahigh school graduate, is it a worthwhile expense?

58. You could have a part-time job (20 hours per week) thatpays $15 per hour, or you could have a full-time job (40hours per week) that pays $12 per hour. Because of theextra free time, you will spend $150 per week more onentertainment with the part-time job than with the full-time job. After accounting for the extra entertainment, howmuch more is your cash flow with the full-time job thanwith the part-time job? Neglect taxes and other expenses.

FURTHER APPLICATIONS59. Laundry Upgrade. Suppose that you currently own a

clothes dryer that costs $25 per month to operate. A newefficient dryer costs $620 and has an estimated operatingcost of $15 per month. How long will it take for the newdryer to pay for itself?

60. Break-Even Point. You currently drive 300 miles perweek in a car that gets 18 miles per gallon of gas. A newfuel-efficient car costs $15,000 (after trade-in on your cur-rent car) and gets 45 miles per gallon. Insurance premiumsfor the new and old car are $800 and $500 per year, respec-tively. You anticipate spending $1500 per year on repairsfor the old car and having no repairs on the new car.Assuming that gas remains at $3.50 per gallon, estimatethe number of years after which the costs of owning thenew and old cars are equal. Hint: You might make a tableshowing the accumulated annual expenses for each car foreach year.

61. Insurance Deductibles. Many insurance policies carry adeductible provision that states how much of a claim youmust pay out of pocket before the insurance company paysthe remaining expenses. For example, if you file a claim for$350 on a policy with a $200 deductible, you pay $200 andthe insurance company pays $150. In the following cases,determine how much you would pay with and without theinsurance policy.

a. You have a car insurance policy with a $500 deductibleprovision (per claim) for collisions. During a two-yearperiod, you file claims for $450 and $925. The annualpremium for the policy is $550.

b. You have a car insurance policy with a $200 deductibleprovision (per claim) for collisions. During a two-yearperiod, you file claims for $450 and $1200. The annualpremium for the policy is $650.

c. You have a car insurance policy with a $1000 deductibleprovision (per claim) for collisions. During a two-year

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period, you file claims for $200 and $1500. The annualpremium for the policy is $300.

d. Explain why lower insurance premiums go with higherdeductibles.

62. Car Leases. Consider the following three lease optionsfor a new car. Determine which lease is least expensive,assuming you buy the car when the lease expires. Theresidual is the amount you pay if you choose to buy the carwhen the lease expires. Discuss other factors that mightaffect your decision.

• Plan A: $1000 down payment, $400 per month for twoyears, residual

• Plan B: $500 down payment, $250 per month for threeyears, residual

• Plan C: $0 down payment, $175 per month for fouryears, residual

63. Health Costs. Assume that you have a (relatively simple)health insurance plan with the following provisions:

• Office visits require a co-payment of $25.

• Emergency room visits have a $200 deductible (you paythe first $200).

• Surgical operations have a $1000 deductible (you paythe first $1000).

• You pay a monthly premium of $350.

During a one-year period, your family has the followingexpenses.

value 5 $8000

value 5 $9500

value 5 $10,000

a. Determine your health-care expenses for the year withthe insurance policy.

b. Determine your health-care expenses for the year if youdid not have the insurance policy.

64. Health-Care Choices. You have a choice of two healthinsurance policies with the following terms.

Total costExpense (before insurance)

Feb. 18: Office visit $100

Mar. 26: Emergency room $580

Apr. 23: Office visit $100

May 14: Surgery $6500

July 1: Office visit $100

Sept. 23: Emergency room $840

Suppose that during a one-year period your family has thefollowing expenses.

a. Determine your annual health-care expenses if you havePlan A.

b. Determine your annual health-care expenses if you havePlan B.

c. Would having no health insurance be better than eitherPlan A or Plan B?

Exercises 65–68 ask you to evaluate your own personal finances.(Note to instructors: If these problems are assigned to be turnedin, you should allow students to fictionalize their answers so thatthey are not being asked to reveal personal financial data.)

65. Daily Expenditures. Keep a list of everything you spendmoney on during one entire day. Categorize each expendi-ture, and then make a table with one column for the cate-gories and one column for the expenditures. Add a thirdcolumn in which you compute how much you’d spend in ayear if you spent the same amount every day.

Plan A Plan B

Office visits require a Office visits require a co-payment of $25. co-payment of $25.

Emergency room visits Emergency room visits have a $500 deductible have a $200 deductible (you pay the first $500). (you pay the first $200).

Surgical operations have Surgical operations havea $5000 deductible a $1500 deductible (you pay the first $5000). (you pay the first $1500).

You pay a monthly You pay a monthly premium of $300. premium of $700.

Total costExpense (before insurance)

Jan. 23: Emergency room $400

Feb. 14: Office visit $100

Apr. 13: Surgery $1400

June 14: Surgery $7500

July 1: Office visit $100

Sept. 23: Emergency room $1200

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66. Weekly Expenditures. Repeat Problem 65, but this timemake the list for a full week of spending rather than justone day.

67. Prorated Expenditures. Make a list of all the majorexpenses you have each year that you do not pay on amonthly basis, such as college expenses, holiday expenses,and vacation expenses. For each item, estimate the amountyou spend in a year, and then determine the proratedamount that you should use when you determine yourmonthly budget.

68. Monthly Cash Flow. Create your complete monthlybudget, listing all sources of income and all expenditures,and use it to determine your net monthly cash flow. Besure to include small but frequent expenditures and pro-rated amounts for large expenditures. Explain any assump-tions you make in creating your budget. When the budgetis complete, write a paragraph or two explaining what youlearned about your own spending patterns and whatadjustments you may need to make to your budget.

WEB PROJECTSFind useful links for Web Projects on the text Web site:www.aw.com/bennett-briggs

69. Personal Budgets. Many Web sites provide personalbudget advice and worksheets. Visit several of these sitesand choose one to help you organize your budget for atleast three months. Is the site effective in helping you plan

your finances? Discuss how the site led to insights that youwould not have had otherwise.

70. U.S. Spending Patterns. Find the complete (two-page)paper from which Figure 4.1 was taken (Spending Patternsby Age, U.S. Department of Labor Statistics). Write a sum-mary of the conclusions of the paper and discuss whetheryour personal finances fit the patterns described in thepaper.

IN THE NEWS71. Personal Bankruptcies. The rate of personal bankrupt-

cies has been increasing for several years. Find at leastthree news articles on the subject, document the increasein bankruptcies, and explain the primary reasons for theincrease.

72. Consumer Debt. Find data on the increase in consumer(credit card) debt in the United States. Based on yourreading, do you think consumer debt is (a) a crisis, (b) asignificant occurrence but nothing to worry about, or (c) agood thing? Justify your conclusion.

73. U.S. Savings Rate. When it comes to saving disposableincome, Americans have a remarkably low savings rate.Find sources that compare the savings rates of Asian andEuropean countries to that of the United States. Discussyour observations and put your own savings habits on thescale.

UNIT 4B The Power of Compounding

On July 18, 1461, King Edward IV of England borrowed the modern equivalent of$384 from New College of Oxford. The King soon paid back $160, but never repaidthe remaining $224. The debt was forgotten for 535 years. Upon its rediscovery in1996, a New College administrator wrote to the Queen of England asking for repay-ment, with interest. Assuming an interest rate of 4% per year, he calculated that thecollege was owed $290 billion.

This example illustrates what is sometimes called the “power of compounding”:the remarkable way that money grows when interest continues to accumulate yearafter year. In the New College case, there is no clear record of a promise to repay thedebt with interest, and even if there were, the Queen might not feel obliged to pay adebt that had been forgotten for more than 500 years. But anyone can take advantageof compound interest simply by opening a savings account. With patience, the resultsmay be truly astonishing.

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Simple versus Compound InterestImagine that you deposit $1000 in Honest John’s Money Holding Service, whichpromises to pay 5% interest each year. At the end of the first year, Honest John’ssends you a check for

You also get $50 at the end of the second and third years. Over the 3 years, youreceive total interest of

Your original $1000 has grown in value to $1150. Honest John’s method of paymentrepresents simple interest, in which interest is paid only on your actual investment,or principal.

Now, suppose that you place the $1000 in a bank account that pays the same 5%interest once a year. But instead of paying you the interest directly, the bank adds theinterest to your account. At the end of the first year, the bank deposits $50 interestinto your account, raising your balance to $1050. At the end of the second year, thebank again pays you 5% interest. This time, however, the 5% interest is paid on thebalance of $1050, so it amounts to

Adding this $52.50 raises your balance to

This is the new balance on which your 5% interest is computed at the end of thethird year. So your third interest payment is

Therefore, your balance at the end of the third year is

Despite identical interest rates, you end up with $7.63 more if you use the bankinstead of Honest John’s. The difference comes about because the bank pays youinterest on the interest as well as on the original principal. This type of interest paymentis called compound interest.

$1102.50 1 $55.13 5 $1157.63

5% 3 $1102.50 5 0.05 3 $1102.50 5 $55.13

$1050 1 $52.50 5 $1102.50

5% 3 $1050 5 0.05 3 $1050 5 $52.50

3 3 $50 5 $150

5% 3 $1000 5 0.05 3 $1000 5 $50

4B The Power of Compounding 229

By the WayThe New Collegeadministrator did notseriously believe that theQueen would pay$290 billion. However, hesuggested a compro-mise of assuming a 2%per year interest rate, inwhich case the collegewas owed only $8.9 mil-lion. This, he said, wouldbe enough to pay for amodernization projectat the College. TheQueen has not yet paid.

DEFINITIONS

The principal in financial formulas is the balance upon which interest is paid.

Simple interest is interest paid only on the original principal, and not on anyinterest added at later dates.

Compound interest is interest paid both on the original principal and on allinterest that has been added to the original principal.

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❉EXAMPLE 1 Savings BondWhile banks almost always pay compound interest, bonds usually pay simple interest.Suppose you invest $1000 in a savings bond that pays simple interest of 10% per year.How much total interest will you receive in 5 years? If the bond paid compound inter-est, would you receive more or less total interest? Explain.

SOLUTION With simple interest, every year you receive the same interest paymentof Thus, you receive a total of $500 in interest over 5 years.With compound interest, you receive more than $500 in interest because the interesteach year is calculated on your growing balance rather than on your original princi-pal. For example, because your first interest payment of $100 raises your balance to$1100, your next compound interest payment is which ismore than the simple interest payment of $100. For the same interest rate, compoundinterest always raises your balance faster than simple interest.

Now try Exercises 41– 44.

The Compound Interest FormulaLet’s return to King Edward’s debt to the New College. We can calculate the amountowed to the College by pretending that the $224 he borrowed was deposited into aninterest-bearing account for 535 years. Let’s assume, as did the New College adminis-trator, that the interest rate was 4% per year. For each year, we can calculate the inter-est and the new balance with interest. The first three columns of Table 4.1 show thesecalculations for 4 years.

➽

10% 3 $1100 5 $110,

10% 3 $1000 5 $100.

TABLE 4.1 Calculating Compound Interest

After N Years Interest Balance Or Equivalently

1 year

2 years

3 years

4 years $224 3 A1.04 B 4 5 $262.05$251.97 1 $10.08 5 $262.054% 3 $251.97 5 $10.08

$224 3 A1.04 B 3 5 $251.97$242.28 1 $9.69 5 $251.974% 3 $242.28 5 $9.69

$224 3 A1.04 B 2 5 $242.28$232.96 1 $9.32 5 $242.284% 3 $232.96 5 $9.32

$224 3 1.04 5 $232.96$224 1 $8.96 5 $232.964% 3 $224 5 $8.96

To find the total balance, we could continue the calculations to 535 years. Fortu-nately, there’s a much easier way. The 4% annual interest rate means that each end-of-year balance is 104% of, or 1.04 times, the previous year’s balance. Thus, as shownin the last column of Table 4.1, we can get each balance as follows:

• The balance at the end of 1 year is the original principal times 1.04:

• The balance at the end of 2 years is the 1-year balance times 1.04:

• The balance at the end of 3 years is the 2-year balance times 1.04:

$224 3 1.04 3 1.04 3 1.04 5 $224 3 A1.04 B 3 5 $251.97

$224 3 1.04 3 1.04 5 $224 3 A1.04 B 2 5 $242.28

$224 3 1.04 5 $232.96

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Continuing the pattern, we find that the balance after N years is the original principaltimes 1.04 raised to the Nth power. For example, the balance after years is

We can generalize this result by looking carefully at the last equation above.Notice that the $224 is the original principal that we began with. The 1.04 is 1 plusthe interest rate of 4%, or 0.04. The exponent 10 is the number of times that theinterest has been compounded. Let’s write the equation again, adding these identifiersand turning it around to put the result on the left:

d number of compounding periods

When interest is compounded just once a year, as it is in this case, the interest rateis called the annual percentage rate, or APR. The number of compounding periodsis then simply the number of years, which we call Y, over which the principal earnsinterest. We therefore obtain the following general formula for interest compoundedonce a year.

$331.57(''')'''*

accumulated balance, A 5 $224

('')''*

original principal, P

3 A1.04 B 10('')''*

11 interest rate

$224 3 A1.04 B 10 5 $331.57

N 5 10

In the New College case, the annual interest rate is and interestis paid over a total of 535 years. Thus, the accumulated balance after yearswould be

As the administrator claimed, a 4% interest rate means the Queen owes about$290 billion.

❉EXAMPLE 2 Simple and Compound InterestYou invest $100 in two accounts that each pay an interest rate of 10% per year. How-ever, one account pays simple interest and one account pays compound interest. Makea table that shows the growth of each account over a 5-year period. Use the com-pound interest formula to verify the result in the table for the compound interest case.

SOLUTION The simple interest is the same absolute amount each year. The com-pound interest grows from year to year, because it is paid on the accumulated interestas well as on the starting balance. Table 4.2 summarizes the calculations.

5 $224 3 1,296,691,085 < $2.9 3 1011 5 $290 billion A 5 P 3 A1 1 APR B Y 5 $224 3 A1 1 0.04 B 535 5 $224 3 A1.04 B 535

Y 5 535APR 5 4% 5 0.04,

where

Be sure to note that the annual interest rate, APR, should always be expressed as adecimal rather than as a percentage.

Y 5 number of years APR 5 annual percentage rate Aas a decimal B

P 5 starting principal A 5 accumulated balance after Y years

A 5 P 3 A1 1 APR B Y

Technical NoteFor the more generalcase in which theinterest rate is notnecessarily set on anannual (APR) basis,you may see thecompound interestformula written

where i is the interestrate and N is the totalnumber of com-pounding periods.

A 5 P 3 A1 1 i B N

USING YOURC A L C U L A T O R

Most calculators have a key forraising to powers, labeled or . For example, cal-culate by

pressing 1.04 535

or 1.04 535.�

�y x

1.04535

�

y x

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TABLE 4.2 Calculations for Example 2

SIMPLE INTEREST ACCOUNT COMPOUND INTEREST ACCOUNT

End of year Interest paid Interest paid

1

2

3

4

5 $146.41 1 $14.64 5 $161.0510% 3 $146.41 5 $14.64$140 1 $10 5 $15010% 3 $100 5 $10

$133.10 1 $13.31 5 $146.4110% 3 $133.10 5 $13.31$130 1 $10 5 $14010% 3 $100 5 $10

$121 1 $12.10 5 $133.1010% 3 $121 5 $12.10$120 1 $10 5 $13010% 3 $100 5 $10

$110 1 $11 5 $12110% 3 $110 5 $11$110 1 $10 5 $12010% 3 $100 5 $10

$100 1 $10 5 $11010% 3 $100 5 $10$100 1 $10 5 $11010% 3 $100 5 $10

5 new balance5 new balanceOld balance 1 interestOld balance 1 interest


To verify the final entry in the table with the compound interest formula, we use astarting principal and an annual interest rate with inter-est paid for years. The accumulated balance A is

This result agrees with the one in the table. Overall, the account paying compoundinterest builds to $161.05 while the simple interest account reaches only $150, eventhough both pay at the same 10% rate. This is a significant difference, especiallywhen you consider that the difference will continue to grow with time. Clearly, com-pound interest is much better for an investor than simple interest at the same rate.


Compound Interest as Exponential GrowthThe New College case demonstrates the remarkable way in whichmoney can grow with compound interest. Figure 4.2 shows how thevalue of the New College debt rises during the first 100 years,assuming a starting value of $224 and an interest rate of 4% peryear. Note that while the value rises slowly at first, it rapidly acceler-ates, so in later years the value grows by much more each year thanit did during earlier years.

This rapid growth is a hallmark of what we generally callexponential growth. You can see how exponential growth gets itsname by looking again at the general compound interest formula:

Because the principal P and the interest rate APR have fixed val-ues for any particular compound interest calculation, the growthof the accumulated value A depends only on Y (the number oftimes interest has been paid), which appears in the exponent of thecalculation.


➽

5 $161.05 A 5 P 3 A1 1 APR B Y 5 $100 3 A1 1 0.1 B 5 5 $100 3 1.15 5 $100 3 1.6105

Y 5 5APR 5 10% 5 0.1P 5 $100

Years

Acc

umul

ated

val

ue

$5000

$224

$10,000

$15,000

0 20 40 60 80 100

FIGURE 4.2 The value of the debt in the NewCollege case during the first 100 years, at an inter-est rate of 4% per year. Note that the value risesmuch more rapidly in later years than in earlieryears—a hallmark of exponential growth.

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Exponential growth is one of the most important topics in mathematics, withapplications that include population growth, resource depletion, and radioactivity.We will study exponential growth in much more detail in Chapter 8. In this chapter,we focus only on its applications in finance.

❉EXAMPLE 3 New College Debt at 2%If the interest rate is 2%, calculate the amount due to New College using

a. simple interest b. compound interest

SOLUTION

a. At a rate of 2%, the simple interest due each year is

Over 535 years, the total interest due would be

Adding this to the original loan principal of $224 gives the payoff amount of

b. To find the amount due with compound interest, we set the annual interestrate to and the number of years to Then weuse the formula for compound interest paid once a year:

The amount due with compound interest isabout $8.94 million—far higher than theamount due with simple interest. You shouldnote the remarkable effects of small changes inthe compound interest rate. Here, we found thata 2% compound interest rate leads to $8.94 mil-lion after 535 years. Earlier, we found that a 4%interest rate for the same 535 years leads to$290 billion—which is more than 30,000 timesas large as $8.94 million. Figure 4.3 contrasts thevalues of the New College debt during the first100 years at interest rates of 2% and 4%. Notethat the rates don’t make much difference forthe first few years, but over time the higher ratebecomes far more valuable.


< $8.94 3 106

5 $224 3 39,911 5 $224 3 A1.02 B 535

A 5 P 3 A1 1 APR B Y 5 $224 3 A1 1 0.02 B 535

Y 5 535.APR 5 2% 5 0.02

$224 1 $2396.80 5 $2620.80

535 3 $4.48 5 $2396.80

2% 3 $224 5 0.02 3 $224 5 $4.48

By the WayFinancial planners oftencall the principal, P, thepresent value (PV) ofthe money and theaccumulated amount,A, the future value (FV).This terminology is alsoused on many financialcalculators and soft-ware packages.

APR � 4%

APR � 2%

Years

Acc

umul

ated

val

ue

$5000

$224

$10,000

$15,000

0 20 40 60 80 100

FIGURE 4.3 This figure contrasts the debt in theNew College case during the first 100 years atinterest rates of 2% and 4%.

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❉EXAMPLE 4 Mattress InvestmentsYour grandfather put $100 under his mattress 50 years ago. If he had instead investedit in a bank account paying 3.5% interest compounded yearly (roughly the averageU.S. rate of inflation during that period), how much would it be worth now?

SOLUTION The starting principal is The annual percentage rate isThe number of years is So the accumulated balance is

Invested at a rate of 3.5%, the $100 would be worth over $550 today. Unfortunately,the $100 was put under a mattress, so it still has a face value of only $100.


Compound Interest Paid More Than Once a YearSuppose you deposit $1000 into a bank that pays compound interest at an annual per-centage rate of If the interest is paid all at once at the end of a year, you’llreceive interest of

Thus, your year-end balance will be Now, assume instead that the bank pays the interest quarterly, or four times a year

(that is, once every 3 months). The quarterly interest rate is one-fourth of the annualinterest rate:

Table 4.3 shows how quarterly compounding affects the $1000 starting principalduring the first year.

quarterly interest rate 5APR

45

8%4

5 2% 5 0.02

$1000 1 $80 5 $1080.

8% 3 $1000 5 0.08 3 $1000 5 $80

APR 5 8%.

➽

5 $558.49 5 $100 3 A1.035 B 50

A 5 P 3 A1 1 APR B Y 5 $100 3 A1 1 0.035 B 50

Y 5 50.APR 5 3.5% 5 0.035.P 5 $100.

Time out to thinkSuppose the interest rate for the New College debt were 3%. Without calculating,do you think the value after 535 years would be halfway between the values at 2%and 4% or closer to one or the other of these values? Now, check your guess by cal-culating the value at 3%. What happens at an interest rate of 6%? Briefly discusswhy small changes in the interest rate can lead to large changes in the accumu-lated value.

TABLE 4.3 Quarterly Interest Payments

After N Quarters Interest Paid New Balance

1st quarter (3 months)

2nd quarter (6 months)

3rd quarter (9 months)

4th quarter (1 full year) $1061.21 1 $21.22 5 $1082.432% 3 $1061.21 5 $21.22

$1040.40 1 $20.81 5 $1061.212% 3 $1040.40 5 $20.81

$1020 1 $20.40 5 $1040.402% 3 $1020 5 $20.40

$1000 1 $20 5 $10202% 3 $1000 5 $20

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Note that the year-end balance with quarterly compounding ($1082.43) is greaterthan the year-end balance with interest paid all at once ($1080). That is, when interestis compounded more than once a year, the balance increases by more than the APR in1 year.

We can find the same results with the compound interest formula. Remember thatthe basic form of the compound interest formula is

where A is the accumulated balance and P is the original principal. In our currentcase, the starting principal is the quarterly payments have an interest rateof and in one year the interest is paid four times. Thus, the accumu-lated balance at the end of one year is

We see that if interest is paid quarterly, the interest rate at each payment is Generalizing, if interest is paid n times per year, the interest rate at each payment is

The total number of times that interest is paid after Y years is nY. We there-fore find the following formula for interest paid more than once each year. APR>n.

APR>4.

A 5 P 3 A1 1 interest rate Bnumber ofcompoundings

5 $1000 3 A1 1 0.02 B 4 5 $1082.43

APR>4 5 0.02,P 5 $1000,

A 5 P 3 A1 1 interest rate Bnumber ofcompoundings

COMPOUND INTEREST FORMULA FOR INTEREST PAID n TIMES PER YEAR

where

Y 5 number of years n 5 number of compounding periods per year

APR 5 annual percentage rate Aas a decimal B P 5 starting principal A 5 accumulated balance after Y years

A 5 P a1 1APR

nbAnYB

❉EXAMPLE 5 Monthly Compounding at 3%You deposit $5000 in a bank account that pays an APR of 3% and compounds interestmonthly. How much money will you have after 5 years? Compare this amount to theamount you’d have if interest were paid only once each year.

Note that Y is not necessarily an integer; for example, a calculation for three and ahalf years would have Y 5 3.5.

Time out to thinkConfirm that substituting into the formula for interest paid n times per yeargives you the formula for interest paid once a year. Explain why this should be true.

n 5 1

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SOLUTION The starting principal is and the interest rate is Monthly compounding means that interest is paid times a year, and we areconsidering a period of years. We put these values into the compound interestformula to find the accumulated balance, A.

For interest paid only once each year, we find the balance after 5 years by using theformula for compound interest paid once a year:

After 5 years, monthly compounding gives you a balance of $5808.08 while annualcompounding gives you a balance of $5796.37. That is, monthly compounding earns

more, even though the APR is the same in both cases.Now try Exercises 53–60. ➽

$5808.08 2 $5796.37 5 $11.71

5 $5796.37 5 $5000 3 A1.03 B 5

A 5 P 3 A1 1 APR B Y 5 $5000 3 A1 1 0.03 B 5

5 $5808.08 5 $5000 3 A1.0025 B 60

A 5 P 3 a1 1APR

nbAnY B

5 $5000 3 a1 10.0312

bA1235B

Y 5 5n 5 12

APR 5 0.03.P 5 $5000

The Compound Interest Formula (for Interest Paid More Than Once per Year)

You can do compound interest calculations on any calculator that has a or key for raising to powers. Here’s a five-step procedure that will work on most sci-entific calculators, along with an example in which and (monthly compounding).With a graphing calcula-tor, you may be able to do the calculation more directly by using the parentheses keys. Some business calculators have built-in functions for calculating compoundinterest in a single step. Note: It is very important that you not round any answers until you have completed all the calculations.

n 5 12Y 5 10 years,APR 5 8% 5 0.08,P 5 $1000,

�y x


IN GENERAL EXAMPLE DISPLAY

STARTING FORMULA: ____

STEP 1. Multiply factors in exponent. n Y 12 10 120.

STEP 2. Store product in memory

(or write down). 120.

STEP 3. Add terms 1 and 1 APR n 1 0.08 12 1.0066666667

STEP 4. Raise result to power in memory. 2.219640235

STEP 5. Multiply result by P. P $1000 2219.640235

With the calculation complete, you can round to the nearest cent, writing the answer as $2219.64. Finally, because it’s easy to push the wrong buttons by accident, you should always check the calculation

(preferably twice) and make sure your answer makes sense.

��

�Recally x�Recally x

��APR>n.

StoreStore

��

$1000 3 a1 10.08

12bA12310B

A 5 P 3 a1 1APR

nbAnY B

Benn.8206.04.pgs 12/15/06 7:51 AM Page 236


Annual Percentage Yield (APY)We’ve seen that in one year, money grows by more than the APR when interest iscompounded more than once a year. For example, we found that with quarterly com-pounding and an 8% APR, a $1000 principal increases to $1082.43 in one year. Thisrepresents a relative increase of 8.243%:

This relative increase over one year is called the annual percentage yield (APY).Note that the APY of 8.243% is greater than the APR of 8%.

relative increase 5absolute increasestarting principal

5$82.43$1000

5 0.08243 5 8.243%

By the WayBanks usually list boththe annual percentagerate (APR) and theannual percentageyield (APY). The APY iswhat your money reallyearns and is the moreimportant number whenyou are comparinginterest rates. Banks arerequired by law to statethe APY on interest-bearing accounts. TheAPY is sometimes calledthe effective yield, orsimply the yield.

Technical NoteMost banks divide theAPR by 360, ratherthan 365, when cal-culating the interestrate and APY for dailycompounding. Thus,the results found heremay not agreeexactly with actualbank results.

DEFINITION

The annual percentage yield (APY) is the actual percentage by which a balanceincreases in one year. It is equal to the APR if interest is compounded annually. Itis greater than the APR if interest is compounded more than once a year. TheAPY does not depend on the starting principal.

❉EXAMPLE 6 More Compounding Means a Higher YieldYou deposit $1000 into an account with Find the annual percentage yieldwith monthly compounding and with daily compounding.

SOLUTION The easiest way to find the annual percentage yield is by finding the bal-ance at the end of one year. We have Formonthly compounding, we set Thus, at the end of one year, the accumulatedbalance with monthly compounding is

Your balance increases by $83.00, so the annual percentage yield is

With monthly compounding, the annual percentage yield is 8.3%.Daily compounding means that interest is paid times per year. At the end

of one year, your accumulated balance with daily compounding is

Your balance increases by $83.28, so the annual percentage yield is

APY 5 relative increase in 1 year 5$83.28$1000

5 0.08328 5 8.328%

5 $1000 3 A1.000219178 B 365 5 $1083.28

A 5 P 3 a1 1APR

nbAnY B

5 $1000 3 a1 10.08365

bA36531B

n 5 365

APY 5 relative increase in 1 year 5$83.00$1000

5 0.083 5 8.3%

5 $1000 3 A1.006666667 B 12 5 $1083.00

A 5 P 3 a1 1APR

nbAnY B

5 $1000 3 a1 10.0812

bA1231B

n 5 12.Y 5 1 year.APR 5 8% 5 0.08,P 5 $1000,

APR 5 8%.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 237


With an APR of 8% and daily compounding, the annual percentage yield is 8.328%,slightly higher than the APY for monthly compounding. The same APY would havebeen found using any starting principal. Now try Exercises 61–64.

Continuous CompoundingSuppose that interest were compounded more often than daily—say, every second orevery trillionth of a second. How would this affect the annual percentage yield?

Let’s examine what we’ve found so far for If interest is compoundedannually (once a year), the annual yield is simply With quarterlycompounding, we found With monthly compounding, we found

With daily compounding, we found Clearly, morefrequent compounding means a higher APY (for a given APR).

However, notice that the change gets smaller as the frequency of compoundingincreases. For example, changing from annual compounding to quarterlycompounding increases the APY quite a bit, from 8% to 8.243%. In contrast,going from monthly to daily compounding increases the APYonly slightly, from 8.300% to 8.328%. You probably won’t be surprised to learn thatthe APY can’t get much bigger than it already is for daily compounding.

Table 4.4 shows the APY for various compounding periods and Figure 4.4 is agraph of the results. As expected, the annual yield does not grow indefinitely. Instead,

An 5 365 BAn 5 12 BAn 5 4 B

An 5 1 B

APY 5 8.328%.APY 5 8.300%.APY 5 8.243%.

APY 5 APR 5 8%.APR 5 8%.

➽

8.4

8.3

8.2

8.1

8.0

0 2412 36 6048 72 9684 108 120Compoundings per year

Ann

ual y

ield

8.3287068

As the numberof compoundingsper year increases…

…the APY gets closer andcloser to the APY forcontinuous compounding.

FIGURE 4.4 The annual percentage yield (APY) for depends on thenumber of times interest is compounded per year.

APR 5 8%

TABLE 4.4 Annual Yield (APY) for with Various Numbers of Compounding Periods ( )

n APY n APY

1 1000

4 10,000

12 1,000,000

365 10,000,000

500 1,000,000,000 8.328 706 8%8.328 013 5%

8.328 706 7%8.327 757 2%

8.328 706 4%8.299 950 7%

8.328 672 1%8.243 216 0%

8.328 360 1%8.000 000 0%

nAPR 5 8%

Benn.8206.04.pgs 12/15/06 7:51 AM Page 238


it approaches a limit that is very close to the APY of 8.3287068% found for In other words, even if we could compound infinitely many times per year, the

annual yield would not go much above 8.3287068%. Compounding infinitely manytimes per year is called continuous compounding. It represents the best possiblecompounding for a particular APR. With continuous compounding, the compoundinterest formula takes the following special form.

1 billion.n 5

HISTORICAL NOTE

Like the number thatarises so frequently inmathematics, the num-ber e is one of the uni-versal constants ofmathematics. It appearsin countless applica-tions, most importantlyto describe exponentialgrowth and decayprocesses. The notatione was proposed by theSwiss mathematicianLeonhard Euler in 1727.Like the number e isnot only an irrationalnumber, but also a tran-scendental number.

p,

p

COMPOUND INTEREST FORMULA FOR CONTINUOUS COMPOUNDING

where

The number e is a special irrational number with a value of You cancompute e to a power with the key on your calculator.ex

e < 2.71828.

Y 5 number of years APR 5 annual percentage rate Aas a decimal B

P 5 starting principal A 5 accumulated balance after Y years

A 5 P 3 eAAPR3Y B

Time out to thinkLook for the key on your calculator. Use it to enter and thereby verify thate < 2.71828.

e1ex

❉EXAMPLE 7 Continuous CompoundingYou deposit $100 in an account with an APR of 8% and continuous compounding.How much will you have after 10 years?

SOLUTION We have and of continuouscompounding. The accumulated balance after 10 years is

Your balance will be $222.55 after 10 years. Note: Be sure you can get the aboveanswer by using the on your calculator. Now try Exercises 65–70.

Planning Ahead with Compound InterestSuppose you have a new baby and want to make sure that you’ll have $100,000 for hisor her college education. Assuming your baby will start college in 18 years, how muchmoney should you deposit now?

If we know the interest rate, this problem is simply a “backwards” compound inter-est problem. We start with the amount A needed after 18 years and then calculate thenecessary starting principal, P. The following two examples illustrate the calculations.

➽ex

5 $222.55 5 $100 3 e0.8

A 5 P 3 eAAPR3Y B5 $100 3 eA0.08310B

Y 5 10 yearsAPR 5 8% 5 0.08,P 5 $100,

Benn.8206.04.pgs 12/15/06 7:51 AM Page 239


A Brief Review

Three Basic Rules of AlgebraIn prior units, we’ve already encountered severalinstances in which we needed to solve an equation byadding, subtracting, multiplying, or dividing both sidesby the same quantity. These operations will be useful inthis unit and the rest of the book, so it is important toreview the basic rules.

Three Basic RulesThe following three rules can always be used:

1. We can add or subtract the same quantity on bothsides of an equation.

2. We can multiply or divide both sides of an equationby the same quantity, as long as we do not multiply ordivide by zero.

3. We can interchange the left and right sides of anequation. That is, if it is also true that

Adding and SubtractingThe following examples show how adding to or sub-

tracting from both sides can help solve equations thatinvolve unknowns.

Example: Solve the equation for x.Solution: We isolate x by adding 9 to both sides:

Example: Solve the equation for y.Solution: Because we have y on the left side and 2y onthe right, we can isolate y on the right by subtracting yfrom both sides:

We interchange the left and right sides, writing theanswer as

Example: Solve the equation for p.Solution: We isolate p by subtracting 4q from both sideswhile also adding 2 to both sides:

4q 2 15 5 p T

8q 2 17 2 4q 1 2 5 p 1 4q 2 2 2 4q 1 2

8q 2 17 5 p 1 4q 2 2

y 5 6.

y 1 6 2 y 5 2y 2 y S 6 5 y

y 1 6 5 2yx 2 9 1 9 5 3 1 9 S x 5 12

x 2 9 5 3

y 5 x.x 5 y,

We interchange the left and right sides, writing theanswer more simply as Note that theequation is solved for p, but we cannot state a numericalvalue for p until we know the value of q.

Multiplying and DividingWhen we cannot isolate a variable by addition or sub-

traction alone, we may need to multiply or divide bothsides of an equation by the same quantity.Example: Solve the equation for x.Solution: We isolate x by dividing both sides by 4:

Example: Solve the equation for z.

Solution: First, we isolate the term containing z byadding 2 to both sides:

Now we multiply both sides by

Example: Solve the equation for s.Solution: We isolate the term containing s by subtracting5 from both sides:

Next we divide both sides by 3 to isolate s. To write thefinal answer more simply, we also switch the left andright sides after dividing.

Remember that you should always check that yoursolution satisfies the original equation.


7w 2 53

53s3 S s 5

7w 2 53

7w 2 5 5 3s 1 5 2 5 S 7w 2 5 5 3s

7w 5 3s 1 5

3z4

343

5 124

3431

S z 5 16

43 :

3z4

2 2 1 2 5 10 1 2 S 3z4

5 12

3z4

2 2 5 10

4x4

5244 S x 5 6

4x 5 24

p 5 4q 2 15.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 240


❉EXAMPLE 8 College Fund at 5%Suppose you could make a single deposit in an investment with an interest rate of

compounded annually, and leave it there for the next 18 years. Howmuch would you have to deposit now to realize $100,000 after 18 years?

SOLUTION We know the interest rate the number of years of com-pounding and the amount desired after 18 years We wantto find the starting principal, P, that must be deposited now. We therefore solve thecompound interest formula (for interest paid once a year) for P, by dividing both sidesby

Now we substitute the given values for A, APR, and Y. The original starting princi-pal is

Depositing $41,552.07 now will yield the desired $100,000 in 18 years—assumingthat the 5% APR doesn’t change and that you make no withdrawals or additionaldeposits. Now try Exercises 71–74.

❉EXAMPLE 9 College Fund at 7%, Compounded MonthlyRepeat Example 8, but with an interest rate of and monthly compound-ing. Compare the results.

SOLUTION This time we must solve for P in the compound interest formula forinterest paid more than once a year.

We substitute the given interest rate the number of years and the balance after 18 years With monthly compounding, we have

The required starting principal is

With a 7% APR and monthly compounding, you can reach $100,000 in 18 years bydepositing about $28,469.43 today. This is over $13,000 less than you must deposit toreach the same goal with an interest rate of 5% (compounded annually).


P 5A

a1 1APR

nbAnY B 5

$100,000

a1 10.0712

bA12318B 5$100,000

A1.0058333333 B 216 5 $28,469.43

n 5 12.AA 5 $100,000 B .

AY 5 18 B ,AAPR 5 0.07 B ,

(11111)1111*

compound interest formula(interest paid n times per year)

P 5A

a1 1APR

nbnY

divide both sides

then interchangeby a11

APRn

bAnY B

left and right sides

>A 5 P 3 a1 1APR

nbAnY B

APR 5 7%

➽

P 5A

A1 1 APR B Y 5$100,000

A1 1 0.05 B 18 5$100,000A1.05 B 18 5 $41,552.07

(11111)1111*

compound interest formula(interest paid once a year)

P 5A

A1 1 APR B Ydivide both sides

then interchangeby A11APRBY;

left and right sides

>A 5 P 3 A1 1 APR B Y

A1 1 APR B Y.

AA 5 $100,000 B .AY 5 18 B ,AAPR 5 0.05 B ,

APR 5 5%,

By the WayThe process of findingthe principal (presentvalue) that must bedeposited today to yieldsome particular futureamount is calleddiscounting by financialplanners.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 241



1. Consider two bank accounts, one earning simple interestand one earning compound interest. If both start with thesame initial deposit (and you make no other deposits orwithdrawals) and earn the same annual interest rate, aftertwo years the account with simple interest will have

a. a greater balance than the account with compoundinterest.

b. a smaller balance than the account with compoundinterest.

c. the same balance as the account with compoundinterest.

2. An account with interest compounded annually and anAPR of 5% increases in value each year by a factor of

a. 1.05. b. 1.5. c. 1.005.

3. After five years, an account with interest compoundedannually and an APR of 6.6% increases in value by a factor of

a. b. c.

4. An account with an APR of 4% and quarterly compound-ing increases in value every three months by

a. 1%. b. c. 4%.

5. With the same deposit, APR, and length of time, anaccount with monthly compounding yields a

a. greater balance than an account with daily compounding.

b. smaller balance than an account with quarterly compounding.

c. greater balance than an account with annual compounding.

1>4%.

1.0665.5 3 1.066.1.665.

6. The annual percentage yield (APY) of an account is always

a. less than the APR.

b. at least as great as the APR.

c. the same as the APR.

7. Consider two bank accounts earning compound interest,one with an APR of 10% and the other with an APR of5%, both with the same initial deposit (and no furtherdeposits or withdrawals). After twenty years, how muchmore interest will the account with haveearned than the account with

a. less than twice as much

b. exactly twice as much

c. more than twice as much

8. If you deposit $500 in an account with an APR of 6% andcontinuous compounding, the balance after two years is

a. b.

c.

9. Suppose you use the compound interest formula to calcu-late how much you must deposit into a college fund todayif you want it to grow in value to $20,000 in ten years.Your calculated amount will be the actual amount after tenyears only if

a. the average APR remains as you assumed throughoutthe ten years.

b. the account has continuous compounding.

c. the account earns simple interest rather than compoundinterest.

10. A bank account with compound interest exhibits what we call

a. linear growth. b. compound growth.

c. exponential growth.

$500 3 A1 1 0.06 B 2.

$500 3 e2.$500 3 e0.12.

APR 5 5%?APR 5 10%

Time out to thinkAside from long-term government bonds, it is extremely difficult to find investmentswith a constant interest rate for 18 years.Nevertheless,financial planners often makesuch assumptions when exploring investment options. Explain why such calculationscan be useful, despite the fact that you can’t be sure of a steady interest rate.

EXERCISES 4B

Benn.8206.04.pgs 12/15/06 7:51 AM Page 242


REVIEW QUESTIONS11. What is the difference between simple interest and com-

pound interest? Why do you end up with more moneywith compound interest?

12. Explain how New College could claim that a debt of $224from 535 years ago is worth $290 billion today. How doesthis show the “power of compounding”?

13. Explain why the term appears in the compoundinterest formula for interest paid n times a year.

14. State the compound interest formula for interest paid oncea year. Define APR and Y.

15. State the compound interest formula for interest paidmore than once a year.

16. What is an annual percentage yield (APY)? Explain why,for a given APR, the APY is higher if the interest is com-pounded more frequently.

17. What is continuous compounding? How does the APY forcontinuous compounding compare to the APY for, say,daily compounding? Explain the use of the formula forcontinuous compounding.

18. Give an example of a situation in which you might want tosolve the compound interest formula to find the principalP that must be invested now to yield a particular amount Ain the future.


19. Simple Bank was offering simple interest at 4.5% per year,which was clearly a better deal than the 4.5% compoundinterest rate at Complex Bank.

20. Both banks were paying the same annual percentage rate(APR), but one had a higher annual percentage yield thanthe other (APY).

21. The bank that pays the highest annual percentage rate(APR) is always the best deal.

22. No bank could afford to pay interest every trillionth of asecond because, with compounding, it’d soon owe every-one infinite dollars.

23. My bank paid an annual interest rate (APR) of 5.0%, but atthe end of the year my account balance had grown by 5.1%.

APR>n

24. If you deposit $10,000 in an investment account today, itcan double in value to $20,000 in just a couple decadeseven at a relatively low interest rate (say, 4%).

BASIC SKILLS & CONCEPTSAlgebra Review. Exercises 25–40 use skills covered in the BriefReview on p. 240. Solve the equations for the unknown quantity.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

Simple Interest. In Exercises 41–44, calculate the amount ofmoney you’ll have at the end of the indicated period of time.

41. You invest $1000 in an account that pays simple interest of5% for 10 years.



44. You invest $5000 in an account that pays simple interest of6.5% for 20 years.

Simple vs. Compound Interest. Exercises 45–46 describe twosimilar, but not identical, investment accounts. Make a table thatshows the performance of both accounts for 5 years. The tableshould list the amount of interest earned each year and the bal-ance in each account. Compare the balances after 5 years.

45. Yancy invests $5000 in an account that earns simple inter-est at an annual rate of 5% per year. Samantha invests$5000 in a savings account with annual compounding at arate of 5% per year.

46. Trevor invests $1000 in an account that earns simple inter-est at an annual rate of 6% per year. Kendra invests $1000in a savings account with annual compounding at a rate of6% per year.

Compound Interest. In Exercises 47–52, use the compoundinterest formula to determine the accumulated balance after thestated period. Assume that interest is compounded annually.

47. $3000 is invested at an APR of 3% for 10 years.

2x>3 1 4 5 2xt>4 1 5 5 25

5w 2 5 5 3w 2 256q 2 20 5 60 1 4q

3n 2 16 5 533a 1 4 5 6 1 4a

5 2 4s 5 6s 2 53x 2 4 5 2x 1 6

1 2 6y 5 135z 2 1 5 19

4y 1 2 5 183p 5 12

2x 5 8z 2 10 5 6

y 1 4 5 7x 2 3 5 9

Benn.8206.04.pgs 9/29/07 9:48 AM Page 243


48. $10,000 is invested at an APR of 5% for 20 years.

49. $40,000 is invested at an APR of 7% for 25 years.



52. $40,000 is invested at an APR of 8.5% for 30 years.

Compounding More Than Once per Year. In Exercises 53–60,use the compound interest formula for compounding more thanonce per year to determine the accumulated balance after thestated period.

53. A $4000 deposit at an APR of 3.5% with monthly com-pounding for 10 years

54. A $2000 deposit at an APR of 3% with daily compoundingfor 5 years

55. A $15,000 deposit at an APR of 5.6% with quarterly com-pounding for 20 years

56. A $10,000 deposit at an APR of 2.75% with monthly com-pounding for 5 years

57. A $2000 deposit at an APR of 7% with monthly com-pounding for 15 years

58. A $3000 deposit at an APR of 5% with daily compoundingfor 10 years

59. A $25,000 deposit at an APR of 6.2% with quarterly com-pounding for 30 years

60. A $15,000 deposit at an APR of 7.8% with monthly com-pounding for 15 years

Annual Percentage Yield (APY). Find the annual percentageyield (APY) for the banks described in Exercises 61–64.

61. A bank offers an APR of 3.5% compounded daily.

62. A bank offers an APR of 4.5% compounded monthly.

63. A bank offers an APR of 4.25% compounded monthly.

64. A bank offers an APR of 2.25% compounded quarterly.

Continuous Compounding. In Exercises 65–70, use the com-pound interest formula for continuous compounding to deter-mine the accumulated balance after 1 year, 5 years, and 20 years.Also find the APY for each account.

65. A $3000 deposit in an account with an APR of 4%


67. A $10,000 deposit in an account with an APR of 8%

68. A $3000 deposit in an account with an APR of 7.5%

69. A $2500 deposit in an account with an APR of 6.5%


Planning Ahead with Compounding. For Exercises 71–74,suppose you start saving today for a $20,000 down payment thatyou plan to make on a house in 10 years. Assume that you makeno deposits into the account after your initial deposit. For eachaccount described, how much would you have to deposit now toreach your $20,000 goal in 10 years? Round answers to the near-est dollar.

71. An account with annual compounding and an APR of 5%

72. An account with quarterly compounding and an APR of 4.5%

73. An account with monthly compounding and an APR of 6%

74. An account with daily compounding and an APR of 4%

College Fund. You want to have a $100,000 college fund in18 years. How much will you have to deposit now under each ofthe scenarios in Exercises 75–78? Assume that you make nodeposits into the account after the initial deposit. Round answersto the nearest dollar.

75. An APR of 4%, compounded daily

76. An APR of 5.5%, compounded daily

77. An APR of 9%, compounded monthly

78. An APR of 3.5% compounded monthly

FURTHER APPLICATIONSSmall Rate Differences. Exercises 79–80 describe two similarinvestment accounts. In each case, compare the balances after10 years and after 30 years. Briefly discuss the effects of thesmall difference in interest rates.

79. Chang invests $500 in a savings account that earns 3.5%compounded annually. Kio invests $500 in a different sav-ings account that earns 3.75% compounded annually.

80. José invests $1500 in a savings account that earns 5.6%compounded annually. Marta invests $1500 in a differentsavings account that earns 5.7% compounded annually.

81. Comparing Annual Yields. Consider an account with anAPR of 6.6%. Find the APY with quarterly compounding,monthly compounding, and daily compounding. Commenton how changing the compounding period affects theannual yield.

82. Comparing Annual Yields. Consider an account with anAPR of 5%. Find the APY with quarterly compounding,monthly compounding, and daily compounding. Commenton how changing the compounding period affects theannual yield.

Benn.8206.04.pgs 9/29/07 9:48 AM Page 244


83. Rates of Compounding. Compare the accumulated bal-ance in two accounts that both start with an initial depositof $1000. Both accounts have an APR of 5.5%, but oneaccount compounds interest annually while the otheraccount compounds interest daily. Make a table that showsthe interest earned each year and the accumulated balancein both accounts for the first 10 years. Compare the bal-ance in the accounts, in percentage terms, after 10 years.Round all figures to the nearest dollar.

84. Understanding Annual Percentage Yield (APY).

a. Explain why APR and APY are the same with annualcompounding.

b. Explain why APR and APY are different with daily com-pounding.

c. Does APY depend on the starting principal, P? Why orwhy not?

d. How does APY depend on the number of compoundingsduring a year, n? Explain.

85. Comparing Investment Plans. Bernard deposits $1600in a savings account that compounds interest annually at anAPR of 4%. Carla deposits $1400 in a savings account thatcompounds interest daily at an APR of 5%. Who will havethe higher accumulated balance after 5 years and after20 years? Explain.

86. Comparing Investment Plans. Brian invests $1600 in anaccount with annual compounding and an APR of 5.5%.Celeste invests $1400 in an account with continuous com-pounding and an APR of 5.2%. Determine who has thehigher accumulated balance after 5 years and after20 years. Discuss the effect of the APR and the compound-ing period.

87. Retirement Fund. You want to accumulate $75,000 foryour retirement in 35 years. You have two choices. Plan Ais an account with annual compounding and an APR of5%. Plan B is an account with continuous compoundingand an APR of 4.5%. How much of an investment doeseach plan require to reach your goal?

88. Your Bank Account. Find the current APR, the com-pounding period, and the claimed APY for your personalsavings account or pick a rate from a nearby bank if youdon’t have an account.

a. Calculate the APY on your account. Does your calcula-tion agree with the APY claimed by the bank? Explain.

b. Suppose you receive a gift of $10,000 and place it inyour account. If the interest rate never changes, howmuch will you have in 10 years?

c. Suppose you find another account that offers interest atan APR that is 2 percentage points higher than yours,with the same compounding period. For the $10,000deposit, how much will you have after 10 years? Brieflydiscuss how this result compares to the result from part b.

Finding Time Periods. Use a calculator and possibly sometrial and error to answer Exercises 89–91.

89. How long will it take your money to triple at an APR of8% compounded annually?

90. How long will it take your money to grow by 50% at anAPR of 7% compounded annually?

91. You deposit $1000 in an account that pays an APR of 7%compounded annually. How long will it take for your bal-ance to reach $100,000?

92. Continuous Compounding. Explore continuous com-pounding by answering the following questions.

a. For an APR of 12%, make a table similar to Table 4.4 inwhich you display the APY for 4, 12, 365, 500,1000.

b. Find the APY for continuous compounding at an APRof 12%.

c. Show the results of parts a and b on a graph similar toFigure 4.3.

d. In words, compare the APY with continuous compound-ing to the APY with other types of compounding.

e. You deposit $500 in an account with an APR of 12%.With continuous compounding, how much money willyou have at the end of 1 year? at the end of 5 years?

93. A Savings Plan. Suppose that on January 1 you deposit$500 into an account that earns interest annually at a rateof 6%. For the next four years, on January 1 you deposit$500 into the same account at the same interest rate (fivedeposits total). How much money will be in the account atthe end of the fifth year? Assume that interest is com-pounded on December 31 of each year. Hint: Note thateach deposit earns interest for a different length of time.


94. Compound Interest Calculators. Although you knowhow to calculate balances with the compound interest for-mula, the Web has many compound interest calculators.Find such a calculator on the Web. Experiment with vari-ous APRs, initial deposits, and compounding periods todetermine if the Web calculator is accurate. Note and

n 5 1,

Benn.8206.04.pgs 12/15/06 7:51 AM Page 245


discuss any terms that are new or different from those youencountered in this unit.

95. Money Stretcher. The Money Stretcher is a Web-basedtutorial on compound interest. Read this short article andcomment on its accuracy, given what you have learned inthis unit.

96. Rate Comparisons. Find a Web site that compares inter-est rates available for ordinary savings accounts at differentbanks. What is the range of rates currently being offered?What is the best deal? How does your own bank accountcompare?

IN THE NEWS97. Bank Advertisement. Find two bank advertisements that

refer to compound interest rates. Explain the terms in eachadvertisement. Which bank offers the better deal? Explain.

98. Power of Compounding. In an advertisement or articleabout an investment, find a description of how money hasgrown (or will grow) over a period of many years. What isthe annual yield listed? How does the value of the accountchange?

UNIT 4C Savings Plans and Investments

Suppose you want to save money for retirement, for your child’s college expenses, orfor some other reason. You could deposit a lump sum of money today and let it growthrough the power of compound interest. But what if you don’t have a large lump sumto start such an account?

For most people, a more realistic way to save is by depositing smaller amounts on aregular basis. For example, you might put $50 a month into savings. Such long-termsavings plans are so popular that many have special names—and some even get spe-cial tax treatment (see Unit 4E). Popular types of savings plans include IndividualRetirement Accounts (IRAs), 401(k) plans, Keogh plans, and employee pension plans.

The Savings Plan FormulaWe can study savings plans with a simple example. Suppose you deposit $100 intoyour savings plan at the end of each month. Further suppose that your plan paysinterest monthly at an annual rate of or 1% per month.

• You begin with $0 in the account. At the end of month 1, you make the firstdeposit of $100.

• At the end of month 2, you receive the monthly interest on the $100 already inthe account, which is In addition, you make your monthlydeposit of $100. Thus, your balance at the end of month 2 is

• At the end of month 3, you receive 1% interest on the $201 already in theaccount, or Adding your monthly deposit of $100, you havea balance at the end of month 3 of

Table 4.5 continues these calculations through 6 months.

$201.00(')'*

prior balance

1 $2.01(')'*

interest

1 $100(')'*

new deposit

5 $303.01

1% 3 $201 5 $2.01.

$100(')'*

prior balance

1 $1.00(')'*

interest

1 $100(')'*

new deposit

5 $201.00

1% 3 $100 5 $1.

APR 5 12%,

By the WayFinancial planners callany series of equal, reg-ular payments anannuity. Thus, savingsplans are a type ofannuity, as are loans thatyou pay with equalmonthly payments.

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4C Savings Plans and Investments 247

TABLE 4.5 Savings Plan Calculations

Prior Interest on End-of-Month New End of . . . Balance Prior Balance Deposit Balance

Month 1 $0 $0 $100 $100

Month 2 $100 $100 $201

Month 3 $201 $100 $303.01

Month 4 $303.01 $100 $406.04

Month 5 $406.04 $100 $510.10

Month 6 $510.10 $100 $615.20

Note: The last column shows the new balance at the end of each month, whichis the sum of the prior balance, the interest, and the end-of-month deposit.

1% 3 $510.10 5 $5.10

1% 3 $406.04 5 $4.06

1% 3 $303.01 5 $3.03

1% 3 $201 5 $2.01

1% 3 $100 5 $1

❉EXAMPLE 1 Using the Savings Plan FormulaUse the savings plan formula to calculate the balance after 6 months for an APR of12% and monthly payments of $100.

SOLUTION We have monthly payments of annual interest rate ofbecause the payments are made monthly, and because

6 months is a half year. Using the savings plan formula, we can find the balance after6 months:

Note that this answer agrees with Table 4.5. Now try Exercises 45–48. ➽

5 $100 33 A1.01 B 6 2 1 4

0.015 $615.20

A 5 PMT 3

c a1 1APR

nbAnY B

2 1 daAPR

nb

5 $100 3

c a1 10.1212

bA12312B

2 1 da0.12

12b

Y 512n 5 12APR 5 0.12,

PMT 5 $100,

By the WayA savings plan in whichpayments are made atthe end of each monthis called an ordinaryannuity. A plan in whichpayments are made atthe beginning of eachperiod is called anannuity due. In bothcases, the accumulatedamount, A, at somefuture date is called thefuture value of the annu-ity. The formulas in thisunit apply only to ordi-nary annuities.

SAVINGS PLAN FORMULA (REGULAR PAYMENTS)

where

Y 5 number of years n 5 number of payment periods per year

APR 5 annual percentage rate Aas a decimal B PMT 5 regular payment Adeposit B amount

A 5 accumulated savings plan balance

A 5 PMT 3

c a1 1APR

nbAnY B

2 1 daAPR

nb

In principle, we could extend this table indefinitely—but it would take a lot ofwork! Fortunately, there’s a much easier way: the savings plan formula. Technical Note

This formula assumesthe same paymentand compoundingperiods. For example,if payments aremade monthly, inter-est also is calculatedand paid monthly. Ifthe compoundingperiod is differentfrom the paymentperiod, replace theterm by theeffective yield foreach paymentperiod.

APR>n

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thinking about . . .

Derivation of the Savings Plan Formula

We can derive the savings plan formula by looking at theexample in Table 4.5 in a different way. Instead of calcu-lating the balance at the end of each month (as inTable 4.5), let’s calculate the value of each individualpayment (deposit) and its interest at the end of month 6.

The first $100 payment was made at the end of month1. Therefore, by the end of 6 months, it has collectedinterest for months (at the end of months 2,3, 4, 5, and 6). We can find its value at the end of month6 with the compound interest formula (Unit 4B). Thepayment amount is and the monthly inter-est rate is Thus, after the collection of 5 inter-est payments, its value is

Similarly, the second $100 payment has collected inter-est for months, so its value at the end ofmonth 6 is

PMT 3 A1 1 i B 4 5 $100 3 1.014

6 2 2 5 4

PMT 3 A1 1 i B 5 5 $100 3 1.015

i 5 0.01.PMT 5 $100

6 2 1 5 5

The first two columns of the following table continuethe calculations for the remaining months. The last col-umn shows how the compound interest formula appliesin general to each individual payment. The accumulatedbalance A after months is the sum of the values ofthe individual payments. Note that the second columnsum agrees with the result found in Table 4.5.

N 5 6

End-of-month Value after Value generalized payment month 6 for N months

123456 $100 PMT

Total $615.20 (sum of terms (accumulated above)balance, A)

PMT 3 A1 1 i B 1$100 3 1.01$100 3 1.012

($100 3 1.013PMT 3 A1 1 i BN22$100 3 1.014PMT 3 A1 1 i BN21$100 3 1.015

❉EXAMPLE 2 Retirement PlanAt age 30, Michelle starts an IRA to save for retirement. She deposits $100 at the endof each month. If she can count on an APR of 6%, how much will she have when sheretires 35 years later at age 65? Compare the IRA’s value to her total deposits over thistime period.

SOLUTION We use the savings plan formula for payments of an inter-est rate of and for monthly deposits. The balance after years is

A 5 PMT 3

c a1 1APR

nbAnYB

2 1 daAPR

nb

5 $100 3

c a1 10.0612

bA12335B2 1 d

a0.0612

b

Y 5 35n 5 12APR 5 0.06,PMT 5 $100,

Benn.8206.04.pgs 12/15/06 7:51 AM Page 248

Because 35 years is 420 months the total amount of her depositsover 35 years is

She will deposit a total of $42,000 over 35 years. However, thanks to compounding,her IRA will have a balance of more than $142,000—more than three times the amountof her contributions. Now try Exercises 49 –52. ➽

420 months 3$100

month5 $42,000

A35 3 12 5 420B , 5 $142,471.03

5 $100 33 A1.005 B 420 2 1 4

0.005


Equation 1:

5 PMT A1 1 i BN 2 PMT A A1 1 i B 2 A 5 2PMT 1 PMT A1 1 i BN

2A 5 PMT 1 PMT A1 1 i B 1 1 c 1 PMT A1 1 i BN21 A A1 1 i B 5 PMT A1 1 i B 1 1 c 1 PMT A1 1 i BN21 1 PMT A1 1 i BN

We do this by rewriting the term on the left as

Thus, the equation becomes

(The last step above comes from factoring out PMTfrom both terms on the right.) Now, we divide bothsides by i to get the savings plan formula:

To put the savings plan formula in the form given inthe text, we simply substitute for the inter-est rate per period and for the total number ofpayments (where n is the number of payments per yearand Y is the number of years).

N 5 nYi 5 APR>n

A 5 PMT 33 A1 1 i B N 2 1 4

i

5 PMT 3 3 A1 1 i B N 2 1 4 Ai 5 PMT A1 1 i B N 2 PMT

A A1 1 i B(')'*

5A 1 Ai

2 A 5 A 1 Ai 2 A 5 Ai

Because the last column contains only general formu-las, we can use the sum of its terms as the accumulatedbalance A for any savings plan after N months:

(We’ve used to indicate a continuing pattern.)We could use this formula for A, but we can simplify

it further with the algebra shown in Equation 1 below.As shown, we multiply both sides by and thensubtract the original equation from the result. Note howall but two terms cancel on the right. The last line inEquation 1 contains the formula we seek, but we need tosolve it for A.

A1 1 i B 1,

“c”

1 PMT 3 A1 1 i B N21 1 c

1 PMT 3 A1 1 i B 1 A 5 PMT

Benn.8206.04.pgs 12/15/06 7:51 AM Page 249


By the WayThe lump sum depositthat would give you thesame end result as regu-lar payments into a sav-ings plan is called thepresent value of the sav-ings plan.


The Savings Plan Formula

There are many ways to do the savings plan formula on your calculator.On a graphing calculator, you may be able to do the calculations directly if you use the parentheseskeys. Some business calculators have built-in functions that allow you to make savings plan calculations in a single step. However, the following procedure will work onmost scientific calculators.The example uses numbers from Example 2 (pp.248–249): payments (monthly payments), and

years. It is very important that you not round any number until the end of the calculation.Y 5 35APR 5 6% 5 0.06,PMT 5 $100,n 5 12

Planning Ahead with Savings PlansMost people start savings plans with a particular goal in mind, such as saving enoughfor retirement or enough to buy a new car in a couple of years. For planning ahead,the important question is this: Given a financial goal (the total amount, A, desiredafter a certain number of years), what regular payments are needed to reach the goal?The following two examples show how the calculations work.

❉EXAMPLE 3 College Savings Plan at 7%You want to build a $100,000 college fund in 18 years by making regular, end-of-month deposits. Assuming an APR of 7%, calculate how much you should depositmonthly. How much of the final value comes from actual deposits and how muchfrom interest?

SOLUTION The goal is to accumulate over years. The interestrate is and monthly payments mean The goal is to calculate then 5 12.APR 5 0.07

Y 5 18A 5 $100,000


STARTING FORMULA: ——

STEP 1. Multiply factors in exponent. n Y 12 35 420.

STEP 2. Store product in memory (or write down). 420.

STEP 3. Add terms 1 and 1 APR n 1 0.06 12 1.005

STEP 4. Raise result to power in memory. 8.123551494

STEP 5. Subtract 1 from result. 1 1 7.123551494

STEP 6. Store result in memory (or write down). 7.123551494

STEP 7. Compute denominator, then take its reciprocal. APR n 0.06 12 200.

STEP 8. Multiply by result in memory and payment. PMT 100 142,471.0299

With the calculation complete, you can round to the nearest cent, writing the answer as $142,471.03. Be sure to check the calculation.

��Recall��Recall�

1 x/��1 x/��

StoreStore

��


��APR>n.

StoreStore

��

$100 3 ≥a1 1

0.06

12bA12335B

2 1

0.06

12

¥A 5 PMT 3 ≥a1 1

APR

nbAnY B

2 1

APR

n

¥

Benn.8206.04.pgs 12/15/06 7:51 AM Page 250


required monthly payments, PMT. We therefore need to solve the savings plan for-mula for PMT. The savings plan formula is

To isolate PMT, we multiply both sides by and divide both sides by

You should confirm the following result:

Now we substitute the given values for A, APR, n, and Y.

Assuming the APR remains 7%, monthly payments of $232.17 will give you $100,000after 18 years. During that time, you deposit a total of

Just over half of the $100,000 comes from your actual deposits; the rest is the result ofcompound interest. Now try Exercises 53–56. ➽

18 yr 312 mo

yr3

$232.17mo

5 $50,148.72

5 $232.17

5$100,000 3 0.0058333333 A1.005833333B 216 2 1 4

PMT 5

A 3APR

n

c a1 1APR

nbAnY B

2 1 d5

$100,000 30.0712

c a1 10.0712

bA12318B2 1 d

PMT 5

A 3APR

n

c a1 1APR

nbAnY B

2 1 d

c a1 1APR

nbAnY B

2 1 d .aAPR

nb

A 5 PMT 3

c a1 1APR

nbAnY B

2 1 daAPR

nb


On most calculators, it is easiest tocalculate the denominator firstand then take its reciprocal andmultiply by the other terms.

Time out to thinkSuppose you want a $100,000 college fund in 18 years and you are counting on anAPR of 7%. In Example 3 above, we found that you could reach your goal withmonthly deposits of about $232. In Unit 4B (Example 9), we found that you couldreach the same goal with a lump sum deposit of $28,469.43 today. Discuss the cir-cumstances under which you might choose either the lump sum or the savings plan.

❉EXAMPLE 4 A Comfortable RetirementYou would like to retire 25 years from now, and you would like to have a retirementfund from which you can draw an income of $50,000 per year—forever! How can youdo it? Assume a constant APR of 9%.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 251


SOLUTION You can achieve your goal by building a retirement fund that is largeenough to earn $50,000 per year from interest alone. In that case, you can withdraw theinterest for your living expenses while leaving the principal untouched (for yourheirs!). The principal will then continue to earn the same $50,000 interest year afteryear (assuming there is no change in interest rates).

What balance do you need to earn $50,000 annually from interest? Since we areassuming an APR of 9%, the $50,000 must be of the total balance. That is,

Dividing both sides by 0.09, we find

In other words, with a 9% APR, a balance of about $556,000 allows you to withdraw$50,000 per year without ever reducing the principal.

Let’s assume you will try to accumulate this balance of by makingregular, monthly deposits into a savings plan. We have (formonthly deposits), and years. As in Example 3, we calculate the requiredmonthly deposits by using the savings plan formula solved for PMT.

If you deposit about $500 per month over the next 25 years, you will achieve yourretirement goal—assuming you can count on a 9% APR (which is high by historicalstandards). Although saving $500 per month may seem like a lot, it can be easier thanit sounds thanks to special tax treatment for retirement plans (see Unit 4E).


Total and Annual ReturnIn the examples so far, we’ve assumed that you get a constant interest rate for a longperiod of time. In reality, interest rates usually vary over time. Consider a case inwhich you invest a starting principal of $1000 and it grows to $1500 in 5 years.Although the interest rate may have varied during the 5 years, we can still describe thechange in both total and annual terms.

Your total return is the relative change in the investment value over the 5-yearperiod (see Unit 3A for a discussion of relative change):

The total return on this investment is 50% over 5 years.

total return 5new value 2 starting principal

starting principal5

$1500 2 $1000$1000

5 0.5

➽

5 $495.93

5$556,000 3 0.00753 A1.0075B 300 2 1 4

PMT 5

A 3APR

n

c a1 1APR

nbAnY B

2 1 d5

$556,000 30.0912

c a1 10.0912

bA12325B2 1 d

Y 5 25n 5 12APR 5 0.09,

A 5 $556,000

total balance 5$50,000

0.095 $555,556

$50,000 5 0.09 3 Atotal balance B9% 5 0.09

By the WayAn account that pro-vides a permanentsource of income with-out reducing its principalis called an endowment.Many charitable foun-dations are endow-ments. They spend eachyear’s interest (or a por-tion of the interest) oncharitable activities,while leaving the princi-pal untouched to earninterest again in futureyears. Of course, thevalue of a particular dollar amount tends todecline with time,because inflationreduces the value of adollar.

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Your annual return is the average annual rate at which your money grew over the5 years. That is, it is the constant annual percentage yield (APY) that would give thesame result in 5 years. In this case, the annual return is about 8.5%. We can see whyby using the compound interest formula for interest paid once a year. We set theinterest rate (APR) to the annual yield of and the number ofyears to The compound interest formula confirms that a starting principal

grows to about in 5 years:

In this case, we “guessed” the APY, then confirmed our guess with the compoundinterest formula. We can calculate the annual return more directly by solving thecompound interest formula above for APY. The algebra is shown in the Brief Reviewbelow. The result is summarized in the following box.

A 5 P 3 A1 1 APYB Y 5 $1000 3 A1 1 0.085B 5 5 $1503.66

A 5 $1500P 5 $1000Y 5 5.

APY 5 8.5% 5 0.085

TOTAL AND ANNUAL RETURN

Consider an investment that grows from an original principal P to a later accumu-lated balance A.

The total return is the relative change in the investment value:

The annual return is the annual percentage yield (APY) that would give the sameoverall growth. The formula is

where Y is the investment period in years.

annual return 5 aAPbA1>Y B

2 1

total return 5AA 2 PB

P

A Brief Review

Algebra with Powers and RootsAs we have seen, powers and roots are often used infinancial calculations. A review of these operations maybe helpful.

Basics of PowersA number raised to the nth power is that number

multiplied by itself n times (n is called an exponent). Forexample:

21 5 2 22 5 2 3 2 5 4 23 5 2 3 2 3 2 5 8

A number to the zero power is defined to be 1. Forexample:

Negative powers are the reciprocals of the correspon-ding positive powers. For example:

522 5152 5

15 3 5

5125

223 5123 5

12 3 2 3 2

518

20 5 1

Benn.8206.04.pgs 12/15/06 7:51 AM Page 253


Power RulesIn the following rules, x represents a number being

raised to a power, and n and m are exponents. Note thatthese rules work only when all powers involve the samenumber x. (See also A Brief Review on p. 105 [Unit 2B].)

• To multiply powers of the same number, add theexponents:

Example:

• To divide powers of the same number, subtract theexponents:

Example:

• When a power is raised to another power, multiply theexponents:

Example:

Basics of Roots

Finding a root is the reverse of raising a number to apower. Second roots, or square roots, are written with anumber under the root symbol More generally, weindicate an nth root by writing a number under the sym-bol For example:

Roots as Fractional PowersThe nth root of a number is the same as the number

raised to the power. That is,

For example:

1,000,0001>6 5 "6 1,000,000 5 10

641>3 5 "3 64 5 4

"n x 5 x1>n1>n

"6 1,000,000 5 10 because 106 5 1,000,000

"4 16 5 2 because 24 5 2 3 2 3 2 3 2 5 16

"3 27 5 3 because 33 5 3 3 3 3 3 5 27

"4 5 2 because 22 5 2 3 2 5 4

!n .

! .

A22 B 3 5 2233 5 26 5 64

Axn Bm 5 xn3m

53

52 5 5322 5 51 5 5

xn

xm 5 xn2m

23 3 22 5 2312 5 25 5 32

xn 3 xm 5 xn1m

Power and Root AlgebraThe following two rules hold true for working with

equations:

1. We can raise both sides of an equation to the samepower.

2. We can take the same root of both sides of an equa-tion, which is equivalent to raising both sides to thesame fractional power. (Note: This process may pro-duce both positive and negative roots; we consideronly positive roots here.)

Example: Find the positive solution of the equation

Solution: We isolate x by raising both sides to the power:

Therefore, one solution of the equation is Notethat, in the last step, we recognized that

Example: Solve the equation forAPY.

Solution: First, we isolate the term containing APY bydividing both sides of the equation by P:

Next, we raise both sides of the equation to the power, then simplify the right side:

Finally, we isolate APY by subtracting 1 from bothsides:

(In the last step, we interchanged the left and rightsides.) To get the formula in the box on p. 253, wereplace APY by annual return. This replacement is validbecause the annual return is the same as the constantAPY that would lead to an accumulated balance A.


aAPb1>Y

2 1 5 1 1 APY 2 1 S APY 5 aAPb1>Y

2 1

xn>n 5 x1 5 xthe rule Axn Bm 5 xn3m

fromfrom the ruleThis stepThis step

(')'*(''')'''*

5 A1 1 APYB Y31>Y 5 1 1 APY

aAPb1>Y

5 3 A1 1 APYB Y 4 1>Y

1>Y

AP

5 A1 1 APYB Y

A 5 P 3 A1 1 APYB Y

x431>4 5 x1 5 x.x 5 2.

Ax4 B 1>4 5 161>4 S x4314 5 161>4 S x 5 161>4 5 2

1>4

x4 5 16.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 254



Raising a number to the power is the same as taking theYth root.The key for taking a rootwill be labeled something like

or . For example, onmany calculators you would cal-culate by pressing 2.8 4 .�y

x

"4 2.8

x 1y/y

x

1>Y

❉EXAMPLE 5 Mutual Fund GainYou invest $3000 in the Clearwater mutual fund. Over 4 years, your investment growsin value to $8400. What are your total and annual returns for the 4-year period?

SOLUTION You have a starting principal and an accumulated value ofafter years. Thus, your total and annual returns are

and

Your total return is 1.8, or 180%, meaning that the value of your investment after4 years is 1.8 times its original value. Your annual return is 0.294, or 29.4%, meaningthat your investment has grown by an average of 29.4% each year.

You should check your answer for the annual return by using the compound inter-est formula. If you use the annual return as the APY, the compound interest formulashould give you the correct accumulated value. In this case,

This is very close to the correct value of $8400. The slight difference is due to round-ing when we calculated the annual return. Now try Exercises 59–62.

❉EXAMPLE 6 Investment LossYou purchased shares in NewWeb.com for $2000. Three years later, you sold themfor $1100. What were your total return and annual return on this investment?

SOLUTION You had a starting principal and an accumulated value ofafter years. Thus, your total and annual returns were

and

The returns are negative because you lost money on this investment. Your total returnwas or meaning that your investment lost 45% of its original value.Your annual return was or meaning that your investment lost an aver-age of 18% of its value each year. Now try Exercises 63– 66. ➽

218%,20.18,245%,20.45,


2 1 5 a$1100$2000

bA1>3B2 1 5 "3 0.55 2 1 5 20.18


P5

A$1100 2 $2000B$2000

5 20.45

Y 5 3A 5 $1100P 5 $2000

➽

A 5 P 3 A1 1 APYB Y 5 $3000 3 A1 1 0.294B 4 5 $8411.21

5 "4 2.8 2 1 5 0.294 5 29.4%


2 1 5 a$8400$3000

bA1>4B2 1

5 1.8 5 180%


P5

A$8400 2 $3000B$3000

Y 5 4A 5 $8400P 5 $3000

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Investment Considerations: Liquidity, Risk, and ReturnNo matter what type of investment you make, you should evaluate the investment interms of three general considerations.

• How difficult is it to take out your money? An investment from which you canwithdraw money easily, such as an ordinary bank account, is said to be liquid. Theliquidity of an investment like real estate is much lower because real estate can bedifficult to sell.

• Is your investment principal at risk? The safest investments are federally insuredbank accounts and U.S. Treasury bills—there’s virtually no risk of losing the prin-cipal you’ve invested. Stocks and bonds are much riskier because they can drop invalue, in which case you may lose part or all of your principal.

By the WayThe U.S. Treasury issuesbills, notes, and bonds.Treasury bills are essen-tially cash investmentsthat are highly liquidand very safe. Treasurynotes are essentiallybonds with 2- to 10-yearterms. Treasury bondshave 20- to 30-yearterms.

By the WayThere are many othertypes of investmentsbesides the basic three,such as rental proper-ties, precious metals,commodities futures,and derivatives. Theseinvestments are gener-ally more complex andoften higher risk than thebasic three.

THREE BASIC TYPES OF INVESTMENTS

Stock (or equity) gives you a share of ownership in a company. You invest someprincipal amount to purchase the stock, and the only way to get your money out isto sell the stock. Because stock prices change with time, the sale may give youeither a gain or a loss on your original investment.

A bond (or debt) represents a promise of future cash. You buy a bond by payingsome principal amount to the issuing government or corporation. The issuer paysyou simple interest (as opposed to compound interest) and promises to pay backyour principal at some later date.

Cash investments include money you deposit into bank accounts, certificates ofdeposit (CD), and U.S. Treasury bills. Cash investments generally earn interest.

Types of InvestmentsSavings plans can involve many types of investments. By combining what we’ve cov-ered about savings plans with the ideas of total and annual return, we can now studyinvestment options. Most investments fall into one of the three basic categoriesdescribed in the following box.

There are two basic ways to invest in any of these categories. First, you caninvest directly, which means buying individual investments yourself. For example,you can directly purchase individual stocks through a stockbroker and you can buybonds directly from the government. In general, the only costs associated withdirect investments are commissions that you pay to brokers.

Alternatively, you can invest indirectly by purchasing shares in a mutual fund,where a professional fund manager invests your money (and the money of others par-ticipating in the fund). Stock mutual funds invest primarily in stocks, bond mutual fundsinvest primarily in bonds, money market funds invest only in cash, and diversified fundsinvest in a mixture of stocks, bonds, and cash.

Benn.8206.04.pgs 12/15/06 7:51 AM Page 256


• How much return (total or annual) can you expect on your investment? A higherreturn means you earn more money. In general, low-risk investments offer rela-tively low returns, while high-risk investments offer the prospects of higherreturns—along with the possibility of losing your principal.

Historical ReturnsOne of the most difficult tasks of investing is trying to balance risk and return.Although there are no guarantees for the future, historical trends offer at least someguidance. Table 4.6 shows historical average annual returns for several different typesof investments.

Building a Portfolio

Before you bought a new television for a few hundreddollars, you’d probably do a fair amount of research tomake sure that you were getting a good buy. You shouldbe even more diligent when making investments thatmay determine your entire financial future.

The best way to plan your savings is to learn aboutinvestments by reading financial pages of newspapersand some of the many books and magazines devoted tofinance. You may also want to consult a professionalfinancial planner. With this background, you will be pre-pared to create a personal financial portfolio (set ofinvestments) that meets your needs.

Most financial advisors recommend that you create adiversified portfolio—that is, a portfolio with a mixture oflow-risk and high-risk investments. No single mixture is rightfor everyone.Your portfolio should balance risk and returnin a way that is appropriate for your situation. For exam-ple, if you are young and retirement is far in the future,you may be willing to have a relatively risky portfolio that

offers the hope of high returns. In contrast, if you arealready retired, you may want a low-risk portfolio thatpromises a safe and steady stream of income.

However you structure your portfolio, the most impor-tant step in meeting your financial goals is making surethat you save enough money.You can use the tools in thisunit to help you determine what is “enough.” Make a rea-sonable estimate of the annual return you can expectfrom your overall portfolio. Use this annual return as theinterest rate in the savings plan formula, and calculatehow much you must invest each month or each year tomeet your goals (see Examples 3 and 4). Then make sureyou actually put this money in your investment plan. If youneed further motivation, consider this: Every $100 youspend today is gone, but even at a fairly low (by historicalstandards) annual return of 4%, every $100 you investtoday will be worth $148 in 10 years, $219 in 20 years, and$711 in 50 years.

TABLE 4.6 Returns on Different Investment Categories, 1926–2005

Investment Type Average Annual Return* Best Year Worst Year

Small-company stocks 12.6% 142.9% (1933) (1937)

Large-company stocks 10.4% 54.0% (1933) (1931)

Long-term corporate

bonds 5.9% 42.6% (1982) (1969)

Cash (U.S. Treasury bills) 3.7% 14.7% (1981) (1938)

*Includes both increases in price and any dividends or interest.Source: Stocks, Bonds, Bills & Inflation Yearbook™, Ibbotson Associates, Chicago.

20.02%

28.1%

243.3%

258.0%

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A common way of tracking an investment category over time is to use an indexthat describes the average return for some category of investments. The best-knownindex is the Dow Jones Industrial Average (DJIA), which reflects the average stockprices of 30 of the largest and most stable companies. Different investment cate-gories are tracked by different indices. Figure 4.5 shows the historical performanceof the DJIA.

By the Way

The DJIA is the mostfamous financial index,but many others areimportant. Standard andPoor’s 500 (S&P 500)tracks 500 large-company stocks; theRussell 2000 tracks2000 small-companystocks; the NASDAQcomposite tracks100 large-companystocks listed on the NAS-DAQ exchange; theLehman Brothers T-BondIndex tracks the per-formance of U.S. Treasurybonds; and the FederalFunds Index tracks short-term interest rates.

Time out to thinkHow do the data in Table 4.6 confirm that higher returns tend to involve higher risk?Explain.

DJI

A

Years2000 20101990198019701960195019401930192019101900

0

2000

1000

3000

4000

5000

6000

7000

12,000

8000

9000

10,000

11,000

Dow Jones Industrial Averages(year end closings)

13,000

FIGURE 4.5 Historical values of the Dow Jones Industrial Average through 2005.

❉EXAMPLE 7 Historical ReturnsSuppose your great-grandmother invested $1000 at the beginning of 1926 in each ofthe following: small-company stocks, large-company stocks, long-term corporatebonds, and U.S. Treasury bills. Assuming her investments grew at the rates given inTable 4.6, approximately how much was each investment worth at the end of 2005?

SOLUTION We find the value of each of the four investments with the compoundinterest formula, setting the interest rate (APR) to the annual return. In all four cases,the starting principal is and years from the beginning of 1926 tothe end of 2005. Table 4.7 shows the calculations.

Y 5 80P 5 $1000

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Note the enormous difference between the categories. The $1000 investment incash grew to $18,300 in 80 years, while the same investment in small-company stocksgrew to over $13 million! Now try Exercises 67–68. ➽

TABLE 4.7 Calculations for Example 7

Investment Type Annual Return Investment Value:

Small-company stocks

Large-company stocks

Corporate bonds

Treasury bills A 5 $1000 3 A1 1 0.037 B 80 5 $18,3003.7% 5 0.037

A 5 $1000 3 A1 1 0.059 B 80 5 $98,1005.9% 5 0.059

A 5 $1000 3 A1 1 0.104 B 80 5 $2,738,60010.4% 5 0.104

A 5 $1000 3 A1 1 0.126 B 80 5 $13,276,10012.6% 5 0.126

A 5 P 3 11 1 APR2Y

Time out to thinkAlthough stocks have outperformed other investments over the long term,Figure 4.5shows that there have been some periods during which stocks gained little or lostvalue. For example, what happened to typical stock portfolios during 2000–2002?What do you think will happen to the stock market over the next 5 years? the next50 years? Why?

By the WayA corporation is a legalentity created to con-duct a business. Owner-ship is held throughshares of stock. Forexample, owning 1% ofa company’s stockmeans owning 1% of thecompany. Shares ofstock in privately heldcorporations are ownedonly by a limited groupof people. Shares ofstock in publicly heldcorporations are tradedon a public exchange,such as the New YorkStock Exchange or theNASDAQ, where anyonemay buy or sell them.

The Financial PagesIf you are investing money, you can track your investments in the financial pages orthrough many Web sites. Let’s look briefly at what you must know to understandcommonly published data about stocks, bonds, and mutual funds.

StocksIn general, there are two ways to make money on stocks:

• You can make money if you sell a stock for more than you paid for it, in whichcase you have a capital gain on the sale of the stock. Of course, you also can losemoney on a stock (a capital loss) if you sell shares for less than you paid for themor if the company goes into bankruptcy.

• You can make money while you own the stock if the corporation distributes partor all of its profits to stockholders as dividends. Each share of stock is paid thesame dividend, so the amount of money you receive depends on the number ofshares you own. Not all companies distribute profits as dividends. Some reinvestall profits within the corporation.

Daily stock tables provide a wealth of information about stocks, summarized inFigure 4.6. Nevertheless, it pays to get even more information if you are buyingstocks. For example, you can learn a lot by studying a company’s annual report. Manycompanies have Web sites with information for investors. You can also get independ-ent research reports from many investment services (usually for a fee) or by workingwith a stockbroker (to whom you pay commissions when you buy or sell stock).

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❉EXAMPLE 8 Motorola StockSuppose that Figure 4.6 comes from today’s paper.

a. What is the ticker symbol for the Motorola Corporation?b. What was the range of selling prices for Motorola shares yesterday? How

do these prices compare to prices over the past year?c. What was the closing price of Motorola shares yesterday and 2 days ago?d. How many shares of Motorola were traded yesterday?e. Suppose you own 100 shares of Motorola. What total dividend payment

should you expect this year?f. Compare what you can expect to earn from dividends to what you would

earn from a bank account offering a 1.5% annual interest rate.g. Has Motorola made a profit in the past year?

SOLUTION

a. The symbol column shows that Motorola’s ticker symbol is MOT.b. The high and low columns show that, yesterday, Motorola stock traded in

the range from $8.88 to $9.57 per share. The middle of this range is close to$9.20, or about 25%, above the 52-week low of $7.30.

c. The close column shows that Motorola closed at $9.43 per share. The changecolumn shows that the share price rose $0.05 from the previous day. Thus,the closing price 2 days ago was per share.$9.43 2 $0.05 5 $9.38

annual dividendshare price × 100%

52-Week High/Low The highest and lowestprices for the stock overthe past 52 weeks

Percent Yield

The percent yield =(the number in the Div column dividedby the number in the Close column)

Volume (sales) in 100sThe number of shares traded yesterdayin 100s (The actual number of sharestraded is 100 times the number shown.)

StockThe company name,often abbreviated

SymbolA 2- to 5-letter tickersymbol used to identifythe stock

Net ChangeThe change in price fromthe market close two daysago to yesterday's marketclose

CloseThe price at which sharestraded when the stockexchange closed yesterday

High, LowThe highest and lowestprices at which stockswere traded yesterday

Price-to-Earnings Ratio (P/E)The share price divided byearnings per share over thepast year (dd indicates a lossover past year)

DividendThe current annualdividend, if any, in dollarsper share

FIGURE 4.6

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d. The volume column reads 17,149. Because this figure is in hundreds ofshares, the actual number of shares traded was

e. The annual dividend rate is $0.16 per share. If you own 100 shares, yourtotal dividend payment will be (However, dividends areusually paid quarterly, so your actual dividend may be different if the com-pany changes its dividend rate during the year.)

f. The percent yield column shows that dividends alone represent an annualreturn of 1.7%—slightly above the 1.5% interest rate offered by the bank.

g. The entry for Motorola’s price-to-earnings ratio is dd, which means that ithad a loss, not a profit, in the past year. Now try Exercises 69–76.

❉EXAMPLE 9 RatioUsing the data from Figure 4.6, compare Monsanto’s share price to its profit per sharein the past year. How much profit per share did Monsanto earn in the past year? His-torically, stocks trade at an average ratio of about 12–14. Based on this historicalaverage and its current ratio, does Monsanto’s stock price seem cheap or expen-sive right now? Given this ratio, what might explain the current stock price?

SOLUTION Monsanto’s price-to-earnings ratio is 55, which means that its current(closing) stock price is 55 times its earnings (profit) per share in the past year. Thus,its earnings per share over the past year must have been of its current stock price:

Monsanto earned a profit of about per share over the past year. The ratio of55 is far above the historical average at which stocks trade, which makes the stockseem quite expensive on this basis alone. One possible explanation for the high ratio is that investors expect Monsanto’s profits to grow substantially in the nearfuture, since higher earnings would bring the ratio closer to historical norms.


BondsMost bonds are issued with three main characteristics:

• The face value (or par value) of the bond is the price you must pay the issuer tobuy it at the time it is issued.

➽P>E

P>EP>E

P>E39¢

earnings per share 5stock priceP>E ratio

5$21.64

555 $0.393

155

P>EP>E

P>E

P/E

➽

100 3 $0.16 5 $16.

17,149 3 100 5 1,714,900. By the WayHistorically, most gainsfrom stocks have comefrom increases in stockprices, rather than fromdividends. Stocks incompanies that payconsistently high divi-dends are calledincome stocks becausethey provide ongoingincome to stockholders.Stocks in companiesthat reinvest most profitsin hopes of growinglarger are called growthstocks.

1997 Thaves/Reprinted with permission. Newspaper distribution by NEA, Inc.

By the WayA company that needscash can raise it eitherby issuing new shares ofstock or by issuingbonds. Issuing newshares of stock reducesthe ownership fractionrepresented by eachshare and hence candepress the value of theshares. Issuing bondsobligates the companyto pay interest to bond-holders. Companiesmust balance these fac-tors in deciding whetherto raise cash throughbond issues or stockofferings.

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• The coupon rate of the bond is the simple interest rate that the issuer promises topay. For example, a coupon rate of 8% on a bond with a face value of $1000means that the issuer will pay you interest of each year.

• The maturity date of the bond is the date on which the issuer promises to repaythe face value of the bond.

Bonds would be simple if that were the end of the story. However, bonds can alsobe bought and sold after they are issued, in what is called the secondary bond market.For example, suppose you own a bond with a $1000 face value and a coupon rate of8%. Further suppose that new bonds with the same level of risk and same time tomaturity are issued with a coupon rate of 9%. In that case, no one would pay $1000for your bond because the new bonds offer a higher interest rate. However, you maybe able to sell your bond at a discount—that is, for less than its face value. In contrast,suppose that new bonds are issued with a coupon rate of 7%. In that case, buyers willprefer your 8% bond to the new bonds and therefore may pay a premium for yourbond—a price greater than its face value.

Consider a case in which you buy a bond with a face value of $1000 and a couponrate of 8% for only $800. The bond issuer will still pay simple interest of 8% of$1000, or $80 per year. However, because you paid only $800 for the bond, yourreturn for each year is

More generally, the current yield of a bond is defined as the amount of interest itpays each year divided by the bond’s current price (not its face value).

amount you earnamount you paid

5$80$800

5 0.1 5 10%

8% 3 $1000 5 $80

A bond selling at a discount from its face value has a current yield that is higherthan its coupon rate. The reverse is also true: A bond selling at a premium over itsface value has a current yield that is lower than its coupon rate. Thus, we have the rulethat bond prices and yields move in opposite directions.

Bond prices are usually quoted in points, which means percentage of face value. Mostbonds have a face value of $1000. Thus, for example, a bond that closes at 102 points isselling for

❉EXAMPLE 10 Bond InterestThe closing price of a U.S. Treasury bond with a face value of $1000 is quoted as105.97 points, for a current yield of 3.7%. If you buy this bond, how much annualinterest will you receive?

102% 3 $1000 5 $1020.

CURRENT YIELD OF A BOND

current yield 5annual interest payment

current price of bond

By the WayBonds are graded interms of risk by inde-pendent rating services.Bonds with a AAA ratinghave the lowest risk andbonds with a D ratinghave the highest risk. U.S.Treasury notes andbonds are not ratedbecause they are con-sidered to be as close torisk-free as is possible.

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SOLUTION The 105.97 points means the bond is selling for 105.97% of its facevalue or

This is the current price of the bond. We are also given its current yield of 3.7%, sowe can solve the current yield formula to find the annual interest payment:

Substituting the price and yield, we find

The annual interest payments on this bond are $39.21. Now try Exercises 83–90.

Mutual FundsWhen you buy shares in a mutual fund, the fund manager takes care of the day-to-daydecisions about when to buy and sell individual stocks or bonds. Thus, in comparingmutual funds, the most important factors are the fees charged for investing and meas-ures of how well the manager is doing with the fund’s money. Figure 4.7 shows a sam-ple mutual fund table. The table makes it easy to compare the past performance offunds. Of course, as stated in every mutual fund prospectus, past performance is noguarantee of future results.

Most mutual fund tables do not show the fees charged. For that, you must call orcheck the Web site of the company offering the mutual fund. Because fees are gener-ally withdrawn automatically from your mutual fund account, they can have a bigimpact on your long-term gains. For example, if you invest $100 in a fund thatcharges a 5% annual fee, only $95 is actually invested. Over many years, this can sig-nificantly reduce your total return.

❉EXAMPLE 11 Mutual Fund GrowthSuppose that Figure 4.7 represents a table from today’s paper. If you invested $500 inthe Calvert Income fund 3 years ago, what is your investment worth now? (Assumeyou reinvested all dividends and gains, and do not count fees.)

SOLUTION Figure 4.7 shows that the annual return for the past 3 years was 10.3%,or 0.103. Therefore, we can use this as the APR in the compound interest formula,with a term of years and principal of

Your $500 investment is now worth about $671. Now try Exercises 91–92. ➽

A 5 P 3 A1 1 APRB Y 5 $500A1 1 0.103B 3 5 $670.96

P 5 $500.Y 5 3

➽

annual interest 5 3.7% 3 $1059.70 5 0.037 3 $1059.70 5 $39.21

annual interest 5 current yield 3 current price

multiply both sides by current priceT

current yield 5annual interestcurrent price

105.97% 3 $1000 5 $1059.70

By the WayMutual funds collect feesin two ways.Some fundscharge a commission,orload, when you buy orsell shares.Funds that donot charge commissionsare called no-loadfunds.Nearly all fundscharge an annual fee,which is usually a per-centage of your invest-ment’s value. In general,fees are higher for fundsthat require moreresearch on the part ofthe fund manager.

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1-year % ReturnThe total return for the past1-year period

YTD % ReturnThe total return for theyear-to-date (since Jan. 1)

Weekly % ReturnThe total return for the week,including capital gains fromsales and any dividends

3-year % ReturnThe annual return over thepast 3 years, calculated byassuming that any dividendsand gains are reinvestedinto that fund

RatingA system for comparing fund performance, with 1 as the worst and 5 as the best (The first number is performance compared to a broad group of similar funds,such as all stock funds, and the second number is performance compared only tofunds of the same type.)

NAVThe net asset value of thefund's shares—that is, theamount that each fund shareis currently worth

Fund FamilyA group of funds fromthe same company

TypeAn abbreviation describingthe type of investments thefund manages (The New YorkTimes categorizes funds intoabout 50 different types andincludes an index eachSunday.)

Fund NameThe name of an individualmutual fund

FIGURE 4.7


1. In the savings plan formula, assuming all other variablesare constant, the accumulated balance in the savingsaccount

a. increases as n increases.

b. increases as APR decreases.

c. decreases as Y increases.

2. In the savings plan formula, assuming all other variablesare constant, the accumulated balance in the savingsaccount

a. decreases as n increases.

b. decreases as PMT increases.

c. increases as Y increases.

3. The total return on a five-year investment is

a. the value of the investment after five years.

b. the difference between the final and initial values of theinvestment.

c. the relative change in the value of the investment.

4. The annual return on a five-year investment is

a. the average of the amounts that you earned in each ofthe five years.

b. the annual percentage yield that gives the same increasein the value of the investment.

c. the amount you earned in the best of the five years.

EXERCISES 4C

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5. Suppose you deposited $100 per month into a savings planfor ten years, and at the end of that period your balancewas $22,200. The amount you earned in interest was

a. $10,200. b. $20,200.

c. impossible to compute without knowing the APR.

6. The best investment would be characterized by which ofthe following choices?

a. low risk, high liquidity, and high return

b. high risk, low liquidity, and high return

c. low risk, high liquidity, and low return

7. Arrange in increasing order by historical annual return:small-company stocks (C), large-company stocks (L), cor-porate bonds (B), and U.S. Treasury bills (T).

a. BTCL b. CLBT c. TBLC

8. Excalibur’s ratio of 75 tells you that

a. its current share price is 75 times its earnings per shareover the past year.

b. its current share price is 75 times the total value of thecompany if it were sold.

c. it offers an annual dividend that is of its currentshare price.

9. The price you pay for a bond with a face value of $5000selling at 103 points is

a. $5300. b. $5150. c. $5103.

10. The one-year return on a mutual fund

a. must be greater than the three-year return.

b. must be less than the three-year return.

c. could be greater than or less than the three-year return.

REVIEW QUESTIONS11. What is a savings plan? Explain the savings plan formula.

12. Give an example of a situation in which you might want tosolve the savings plan formula to find the payments, PMT,required to achieve some goal.

13. Distinguish between the total return and the annual returnon an investment. How do you calculate the annual return?Give an example.

14. Briefly describe the three basic types of investments:stocks, bonds, and cash. How can you invest in these typesdirectly? How can you invest in them indirectly through amutual fund?

1>75

P>E

15. Explain what we mean by an investment’s liquidity, risk,and return. How are risk and return usually related?

16. Contrast the historical returns for different types of invest-ments. How do financial indices, such as the DJIA, helpkeep track of historical returns?

17. Define the face value, coupon rate, and maturity date of abond. What does it mean to buy a bond at a premium? at adiscount? How can you calculate the current yield of abond?

18. Briefly describe the meaning of each column in a typicalfinancial table for stocks and mutual funds.


19. If interest rates stay at 4% APR and I continue to make mymonthly $25 deposits into my retirement plan, I should beable to retire in 30 years with a comfortable income.

20. My financial advisor showed me that I could reach myretirement goal with deposits of $200 per month and anaverage annual return of 7%. But I don’t want to depositthat much of my paycheck, so I’m going to reach the samegoal by getting an average annual return of 15% instead.

21. I’m putting all my savings into stocks because stocksalways outperform other types of investment over thelong term.

22. I’m hoping to withdraw money to buy my first housesoon, so I need to put it into an investment that is fairlyliquid.

23. I bought a fund advertised on the Web that says it uses asecret investment strategy to get an annual return twicethat of stocks, with no risk at all.

24. I’m already retired, so I need low-risk investments. That’swhy I put most of my money in U.S. Treasury bills, notes,and bonds.

BASIC SKILLS & CONCEPTSReview of Powers and Roots. Exercises 25–36 use skills cov-ered in the Brief Review on pp. 253–254. Evaluate the expres-sions and express the answer in simplest terms.

25. 26.

27. 28.

29. 30. 811>2161>232243

3423

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31. 32.

33. 34.

35. 36.

Solving with Powers and Roots. Solve the equations in Exer-cises 37–44 for the unknown.

37. 38.

39. 40.

41. 42.

43. 44.

Savings Plan Formula. In Exercises 45–48, calculate the bal-ance under the given assumptions.

45. Find the savings plan balance after 9 months with an APRof 12% and monthly payments of $200.

46. Find the savings plan balance after 1 year with an APR of12% and monthly payments of $100.



Investment Plans. Use the savings plan formula in Exer-cises 49–52.

49. You set up an IRA (individual retirement account) with anAPR of 5% at age 25. At the end of each month, youdeposit $75 in the account. How much will the IRA con-tain when you retire at age 65? Compare that amount tothe total deposits made over the time period.

50. A friend creates an IRA with an APR of 6.25%. She startsthe IRA at age 25 and deposits $50 per month. How muchwill her IRA contain when she retires at age 65? Comparethat amount to the total deposits made over the timeperiod.

51. You put $300 per month in an investment plan that paysan APR of 7%. How much money will you have after18 years? Compare this amount to the total deposits madeover the time period.

52. You put $200 per month in an investment plan that paysan APR of 4.5%. How much money will you have after18 years? Compare this amount to the total deposits madeover the time period.

v3 1 4 5 68u9 5 512

w2 1 2 5 27A t>3 B 2 5 16

p1>3 5 3Ax 2 4 B 2 5 36

y3 5 27x2 5 25

33 1 23251>2 4 2521>262 3 62234 4 32

23 3 256421>3 Investment Planning. Use the savings plan formula in Exer-cises 53–56.

53. You intend to create a college fund for your baby. If youcan get an APR of 7.5% and want the fund to have a valueof $75,000 after 18 years, how much should you depositmonthly?

54. At age 35 you start saving for retirement. If your invest-ment plan pays an APR of 6% and you want to have$2 million when you retire in 30 years, how much shouldyou deposit monthly?

55. You want to purchase a new car in 3 years and expect thecar to cost $15,000. Your bank offers a plan with a guaran-teed interest rate of if you make regularmonthly deposits. How much should you deposit eachmonth to end up with $15,000 in 3 years?

56. At age 20 when you graduate, you start saving for retire-ment. If your investment plan pays an APR of 8% and youwant to have $5 million when you retire in 45 years, howmuch should you deposit monthly?

57. Comfortable Retirement. Suppose you are 30 years oldand would like to retire at age 60. Furthermore, you wouldlike to have a retirement fund from which you can draw anincome of $100,000 per year—forever! How can you do it?Assume a constant APR of 6%.

58. Very Comfortable Retirement. Suppose you are25 years old and would like to retire at age 65. Further-more, you would like to have a retirement fund from whichyou can draw an income of $200,000 per year—forever!How can you do it? Assume a constant APR of 6%.

Total and Annual Returns. In Exercises 59–66, compute thetotal and annual returns on the described investment.

59. Five years after buying 100 shares of XYZ stock for $60per share, you sell the stock for $9400.

60. You pay $8000 for a municipal bond. When it maturesafter 20 years, you receive $12,500.

61. Twenty years after purchasing shares in a mutual fund for$6500, you sell them for $11,300.

62. Three years after buying 200 shares of XYZ stock for $25per share, you sell the stock for $8500.

63. Three years after paying $3500 for shares in a startup com-pany, you sell the shares for $2000 (at a loss).

64. Five years after paying $5000 for shares in a new company,you sell the shares for $3000 (at a loss).

65. Ten years after purchasing shares in a mutual fund for$7500, you sell them for $12,600.

APR 5 5.5%

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66. Ten years after purchasing shares in a mutual fund for$10,000, you sell them for $2200 (at a loss).

67. Historical Returns. Suppose your great-uncle invested$500 at the beginning of 1940 in each of the following:small-company stocks, large-company stocks, long-termcorporate bonds, and U.S. Treasury bills. Assuming hisinvestments grew at the long-term average annual returnsin Table 4.6, approximately how much will each invest-ment be worth at the end of 2010?

68. Best and Worst Years. Suppose you invest $2000 in eachof the following: small-company stocks, large-companystocks, long-term corporate bonds, and U.S. Treasury bills.Using the returns shown in Table 4.6, calculate how muchyour investments would be worth a year later if it was thebest of years? How much would your investments be wortha year later if it was the worst of years?

Reading Stock Tables. Use the data in Figure 4.6 to answerthe questions in Exercises 69–76. Assume the data come fromtoday’s newspaper.

69. Of the four stocks shown in Figure 4.6, which one had thebiggest gain in price yesterday? State the company’s nameand symbol and the amount it gained per share. Based onthe closing price and the gain, what was its closing pricetwo days ago?

70. Of the four stocks shown in Figure 4.6, which one had thebiggest decline in price yesterday? State the company’sname and symbol and the amount it lost per share. Basedon the closing price and the loss, what was its closing pricetwo days ago?

71. Of the four stocks shown in Figure 4.6, which one is cur-rently trading at prices nearest to its highest price over thepast year? Explain.

72. Of the four stocks shown in Figure 4.6, which one is cur-rently trading at prices nearest to its lowest price over thepast year? Explain.

73. Suppose you own 1000 shares of Monsanto. What totaldividend payment can you expect this year?

74. Suppose you own 100 shares of each of the four stocksshown in Figure 4.6. Which one will pay you the highestdividend, in absolute dollars? How much will your divi-dend payment be?

75. Suppose your primary investment goal is to receive incomefrom dividends. Assuming the stock prices and dividends inFigure 4.6 continue to hold, which of the four stockswould be the best investment for you? Explain.

76. Suppose your primary investment goal is to receive incomefrom dividends. Which stock(s) in Figure 4.6 would itmake no sense for you to invest in? Explain.

Price-to-Earning Ratio. For each stock listed in Exercises77–82, answer the following questions:

a. Did the company earn a profit in the past year? If so, howdoes its share price compare to the profit per share that itearned in the past year?

b. How much profit per share did the company earn in thepast year?

c. Based on the fact that stocks historically trade at an averageratio of about 12–14, does the stock price seem cheap,

about right, or expensive right now? If it seems cheap orexpensive, what might explain the current stock price?

77. McDonald’s, assuming Figure 4.6 comes from today’s newspaper

78. McDonald’s, based on yesterday’s actual closing stock price(from a newspaper or Web site)

79. Motorola, assuming Figure 4.6 comes from today’s newspaper

80. Motorola, based on yesterday’s actual closing stock price(from a newspaper or Web site)

81. Mueller Industries, assuming Figure 4.6 comes fromtoday’s newspaper

82. Mueller Industries, based on yesterday’s actual closingstock price (from a newspaper or Web site)

Bond Yields. In Exercises 83–86, calculate the current yield onthe described bond.

83. A $1000 Treasury bond with a coupon rate of 2.0% thathas a market value of $950



86. A $10,000 Treasury bond with a coupon rate of 3.0% thathas a market value of $9500

Bond Interest. In Exercises 87–90, calculate the annual inter-est that you will receive on the described bond.

87. A $1000 Treasury bond with a current yield of 3.9% that isquoted at 105 points

88. A $1000 Treasury bond with a current yield of 1.5% that isquoted at 98 points

P>E

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89. A $1000 Treasury bond with a current yield of 6.2% that isquoted at 114.3 points

90. A $10,000 Treasury bond with a current yield of 3.6% thatis quoted at 102.5 points

91. Mutual Fund Growth. Assume that Figure 4.7 comesfrom today’s paper. Suppose you invested $500 in theCalvert Social Investment Bond fund (SocInvBdA) threeyears ago and reinvested all dividends and gains. What isyour investment worth now?

92. Mutual Fund Growth. Assume that Figure 4.7 comesfrom today’s paper. Suppose you invested $500 in theCalvert Social Investment Equity fund (SocInvEqA) threeyears ago and reinvested all dividends and gains. What isyour investment worth now?

FURTHER APPLICATIONSWho Comes Out Ahead? Exercises 93–96 each describe twosavings plans. Compare the balances in the two plans after10 years. Who deposits more money in each case? Who comesout ahead in each case? Comment on any lessons about savingsplans that you find in the results. (Assume that, for each plan,the payment and compounding periods are the same, so the sav-ings plan formula is valid.)

93. Yolanda deposits $200 per month in an account with anAPR of 5%, while Zach deposits $2400 at the end of eachyear in an account with an APR of 5%.

94. Polly deposits $50 per month in an account with an APRof 6%, while Quint deposits $40 per month in an accountwith an APR of 6.5%.

95. Juan deposits $400 per month in an account with an APRof 6%, while Maria deposits $5000 at the end of each yearin an account with an APR of 6.5%.

96. George deposits $40 per month in an account with an APRof 7%, while Harvey deposits $150 per quarter in anaccount with an APR of 7.5%.

Comparing Investment Plans. Suppose you want to accumu-late $50,000 for your child’s college fund within the next 15 years.Explain fully whether the investment plans in Exercises 97–100will allow you to reach your goal.

97. You deposit $50 per month into an account with an APR of 7%.

98. You deposit $75 per month into an account with an APR of 7%.

99. You deposit $100 per month into an account with an APRof 6%.

100. You deposit $200 per month into an account with an APRof 5%.

101. Total Return on Stock. Suppose you bought XYZ stock1 year ago for $5.80 per share and sell it at $8.25. You alsopay a commission of $0.25 per share on your sale. What isthe total return on your investment?

102. Total Return on Stock. Suppose you bought XYZ stock1 year ago for $46.00 per share and sell it at $8.25. Youalso pay a commission of $0.25 per share on your sale.What is the total return on your investment?

103. Death and the Maven (A True Story). In December1995, 101-year-old Anne Scheiber died and left $22 mil-lion to Yeshiva University. This fortune was accumulatedthrough shrewd and patient investment of a $5000 nestegg over the course of 50 years. In turning $5000 into$22 million, what were her total and annual returns? Howdid her annual return compare to the average annualreturn for large-company stocks (see Table 4.6)?

104. Personal Savings Plan. Describe something for whichyou would like to save money right now. How much doyou need to save? How long do you have to save it? Basedon these needs, calculate how much you should depositeach month in a savings plan to meet your goal. For theinterest rate, use the highest rate currently available atlocal banks.

105. Get Started Early! Mitch and Bill are the same age.When Mitch is 25 years old, he begins depositing $1000per year into a savings account. He makes deposits for10 years, at which point he is forced to stop makingdeposits. However, he leaves his money in the account forthe next 40 years (where it continues to earn interest). Billdoesn’t start saving until he is 35 years old, but for the next40 years he makes annual deposits of $1000. Assume thatboth accounts earn interest at an annual rate of 7% andinterest in both accounts is compounded once a year.

a. How much money does Mitch have in his account at age 75?

b. How much money does Bill have in his account at age 75?

c. Compare the amounts of money that Mitch and Billdeposit into their accounts.

d. Write a paragraph summarizing your conclusions aboutthis parable.

WEB PROJECTSFind useful links for Web Projects on the text Web site:www.aw.com/bennett-briggs106. Investment Tracking. Choose three stocks, three bonds,

and three mutual funds that you think would make goodinvestments. Imagine that you invest $100 in each of these

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4D Loan Payments, Credit Cards, and Mortgages 269

nine investments. Use the Web to track the value of yourinvestment portfolio over the next 5 weeks. Based on theportfolio value at the end, find your return for the 5-weekperiod. Which investments fared the best, and which didmost poorly?

107. Dow Jones Industrial Average. The Dow Jones Com-pany has an extensive Web site that includes its history andfunctions, as well as information on the Dow Jones Indus-trial Average (DJIA) and links to the companies that makeup the DJIA. Visit the Web site and choose a specific topicrelated to the DJIA (for example, the history of the DJIA,the original companies in the DJIA, the best and worstdays for the DJIA, how the DJIA is computed). Using theWeb site and any other resources, write a two-page paperon your topic.

108. Company Research. Go to the Web site of a specificcompany (links to the 30 DJIA companies are on the DowJones Web site) and carry out research on that company asif you were a prospective investor. You should consider thefollowing questions: How has the company performedover the last year? 5 years? 10 years? Does the companyoffer dividends? How do you interpret its ratio?Overall, do you think the company is a good investment?Why or why not?

109. Financial Web Sites. Visit one of the many financial newsand advising Web sites. Describe the services offered by

P>E

the Web site. Explain whether, as an active or prospectiveinvestor, you find the Web site useful.

110. Other Averages. Investigate one of several other stockaverages, such as Standard and Poor’s or the Russell 2500.How do these averages differ from the Dow Jones IndustrialAverage? What services do they offer on their Web pages?

111. Online Brokers. It is possible to buy and sell stocks onthe Internet through online brokers. Visit the Web sites ofat least two online brokers. How do their services differ?Compare the commissions charged by the brokers.

IN THE NEWS112. Advertised Investment. Find an advertisement for an

investment plan. Describe some of the cited benefits of theplan. Using what you learned in this unit, identify at leastone possible drawback of the plan.

113. Financial Pages. Choose a major newspaper and study itsfinancial pages. Can you identify all the investment datadescribed in this unit? If not, what data are missing? If so,what additional financial data are offered? Explain how toread the pages.

114. Personal Investment Options. Does your employeroffer you the option of enrolling in a savings or retirementplan? If so, describe the available options and discuss theadvantages and disadvantages of each.

UNIT 4D Loan Payments, Credit Cards, and Mortgages

Do you have a credit card? Do you have a loan for your car? Do you have studentloans? Do you own a house? Chances are that you owe money for at least one of thesepurposes. If so, you not only have to pay back the money you borrowed but also haveto pay interest on the money that you owe. In this unit, we will begin by studying thebasic ideas of loans and then apply these ideas to common loans, including creditcards and mortgages.

Loan BasicsSuppose you borrow $1200 at an annual interest rate of or 1% permonth. At the end of the first month, you owe interest in the amount of

1% 3 $1200 5 $12

APR 5 12%,

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If you paid only this $12 in interest, you’d still owe $1200. That is, the total amountof the loan, called the loan principal, would still be $1200. In that case, you’d owe thesame $12 in interest the next month. In fact, if you paid only the interest from onemonth to the next, the loan would never be paid off and you’d have to pay $12 permonth forever.

If you hope to make progress in paying off the loan, you need to pay part of theprincipal as well as interest. For example, suppose that you paid $200 toward yourloan principal each month, plus the current interest. At the end of the first month,you’d pay $200 toward principal plus $12 for the 1% interest you owe, making a totalpayment of $212. Because you’ve paid $200 toward principal, your new loan principalwould be

At the end of the second month, you’d again pay $200 toward principal and 1%interest. But this time the interest is on the $1000 that you still owe. Thus, your inter-est payment would be making your total payment $210. Table 4.8shows how the calculations continue until the loan is paid off after 6 months.

1% 3 $1000 5 $10,

$1200 2 $200 5 $1000.

LOAN BASICS

For any loan, the principal is the amount of money owed at any particular time.Interest is charged on the loan principal. To pay off a loan, you must gradually paydown the principal. Thus, in general, every payment should include all the inter-est you owe plus some amount that goes toward paying off the principal.

Installment LoansFor the case illustrated in Table 4.8, your total payment decreases from month tomonth because of the declining amount of interest that you owe. There’s nothinginherently wrong with this method of paying off a loan, but most people prefer to paythe same total amount each month because it makes planning a budget easier. A loanthat you pay off with equal regular payments is called an installment loan (oramortized loan).

TABLE 4.8 Payments and Principal for a $1200 Loan with Principal Paid Off at a Constant $200/Month

Payment Prior Interest on Toward Total New

End of . . . Principal Prior Principal Principal Payment Principal

Month 1 $1200 $200 $212 $1000

Month 2 $1000 $200 $210 $800

Month 3 $800 $200 $208 $600

Month 4 $600 $200 $206 $400

Month 5 $400 $200 $204 $200

Month 6 $200 $200 $202 $0 1% 3 $200 5 $2

1% 3 $400 5 $4

1% 3 $600 5 $6

1% 3 $800 5 $8

1% 3 $1000 5 $10

1% 3 $1200 5 $12

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Suppose you wanted to pay off your $1200 loan with 6 equal monthly payments.How much should you pay each month? Because the payments in Table 4.8 varybetween $202 and $212, it’s clear that the equal monthly payments must lie some-where in this range. The exact amount is not obvious, but we can calculate it with theloan payment formula.


By the WayAbout two-thirds of allcollege students takeout student loans, andat the time of gradua-tion these students owean average debt ofabout $20,000.

LOAN PAYMENT FORMULA (INSTALLMENT LOANS)

where

Y 5 loan term in years n 5 number of payment periods per year

APR 5 annual percentage rate P 5 starting loan principal Aamount borrowed B

PMT 5 regular payment amount

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d

In our current example, the starting loan principal is the annual inter-est rate is the loan term is year (6 months), and monthly pay-ments mean The loan payment formula gives

The monthly payments would be $207.06, which, as we expected, is between $202and $212.

Because the loan principal is gradually paid down with the installment payments,the interest due each month must also decline gradually. Thus, because the paymentsremain the same, the amount paid toward principal each month gradually rises. Wetherefore have the general relationship between principal and interest summarized inthe following box.

5 $207.06

5$12

1 2 0.942045235

5$1200 3 A0.01 B

31 2 A1 1 0.01 B26 4

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$1200 3 a0.1212

bc1 2 a1 1

0.1212

bA21231>2B d

n 5 12.Y 5

12APR 5 12%,

P 5 $1200,

PRINCIPAL AND INTEREST FOR INSTALLMENT LOANS

The portions of installment loan payments going toward principal and towardinterest vary as the loan is paid down. Early in the loan term, the portion goingtoward interest is relatively high and the portion going toward principal is rela-tively low. As the term proceeds, the portion going toward interest graduallydecreases and the portion going toward principal gradually increases.

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Technical NoteBecause we assumethe compoundingperiod is the same asthe payment periodand because weround payments tothe nearest cent, thecalculated paymentsmay differ slightly fromactual payments.

❉EXAMPLE 1 Student LoanSuppose you have student loans totaling $7500 when you graduate from college. Theinterest rate is and the loan term is 10 years. What are your monthly pay-ments? How much will you pay over the lifetime of the loan? What is the total inter-est you will pay on the loan?

SOLUTION The starting loan principal is the interest rate isthe loan term is years, and for monthly payments. We

use the loan payment formula to find the monthly payments:

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$7500 3 a0.0912

bc1 2 a1 1

0.0912

bA212310B d

n 5 12Y 5 10APR 5 0.09,P 5 $7500,

APR 5 9%


The Loan Payment Formula

As with other formulas in this chapter, there are many ways to do loan calculations on your calculator. Graphing or business calculators may make the calculations easier.Here is a procedure that will work on most scientific calculators.The example uses (monthly payments), and (6 months). It is important that you not round any numbers until the last step.

Y 512 yearn 5 12APR 5 12%,P 5 $1200,


STARTING FORMULA: ——

Step 1. Multiply factors in exponent. n * Y 12 1 2

Step 2. Store product in memory (or write down).

Step 3. Add denominator terms 1 and 1 APR n 1 0.12 12 1.01

Step 4. Raise result to power in memory. 0.942045235

Step 5. Subtract result from 1 by making result negative and adding 1. 1 1 0.057954765

Step 6. Denominator is now complete; take its reciprocal. 17.25483667

Step 7. Multiply result by factors P and P APR n 1200 0.12 12 207.0580401

With the calculation complete, you can round to the nearest cent, writing the answer as $207.06. Be sure to check the

calculation.

*The key is used on scientific calculators to change the sign of a number.�/�

��APR>n.

1 x/1 x/

��/��/�


��APR>n.

26.StoreStore

26.��/��/�

$1200 3 a0.12

12b

c 1 2 a1 10.12

12bA21231>2B d

PMT 5

P 3 aAPR

nb

c 1 2 a1 1APR

nbA2nY B d

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Your monthly payments are $95.01. Over the 10-year term, your total paymentswill be

Of this amount, $7500 pays off the principal. The rest, or represents interest payments. Now try Exercises 23–34.

❉EXAMPLE 2 Principal and Interest PaymentsFor the loan in Example 1, calculate the portions of your payments that go to princi-pal and to interest during the first 3 months.

SOLUTION The monthly interest rate is For a $7500starting loan principal, the interest due at the end of the first month is

Your monthly payment (calculated in Example 1) is $95.01. We’ve found that theinterest due is $56.25, so the rest, or goes to principal.Thus, after your first payment, your new loan principal is

Table 4.9 continues the same calculations for months 2 and 3. Note that, as expected,the interest payment gradually decreases and the payment toward principal graduallyincreases. But also note that, for these first 3 months of a 10-year loan, more than halfof each payment goes toward interest. We could continue this table through the life ofthe loan, but it’s generally easier to use software that finds principal and interest pay-ments with built-in functions.

$7500 2 $38.76 5 $7461.24

$95.01 2 $56.25 5 $38.76,

0.0075 3 $7500 5 $56.25

APR>12 5 0.09>12 5 0.0075.

➽$11,401 2 $7500 5 $3901,

10 yr 3 12 moyr

3$95.01

mo5 $11,401.20

5 $95.01

5$56.25

31 2 0.407937305 4

5$7500 3 A0.0075 B31 2 A1.0075 B2120 4

By the WayA table of principal andinterest payments overthe life of a loan is calledan amortization sched-ule. Most banks will pro-vide an amortizationschedule for any loanyou are considering.

TABLE 4.9 Interest and Principal Portions of Payments on a $7500 Loan (10-year term, )

Payment Toward End of . . . Principal New Principal

Month 1

Month 2

Month 3 $7422.19 2 $39.34 5 $7382.85$95.01 2 $55.67 5 $39.340.0075 3 $7422.19 5 $55.67

$7461.24 2 $39.05 5 $7422.19$95.01 2 $55.96 5 $39.050.0075 3 $7461.24 5 $55.96

$7500 2 $38.76 5 $7461.24$95.01 2 $56.25 5 $38.760.0075 3 $7500 5 $56.25

0.0075 3 BalanceInterest 5

APR 5 9%


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Choices of Rate and TermYou’ll usually have several choices of interest rate and loan term when seeking aloan. For example, a bank might offer a 3-year car loan at 8%, a 4-year loan at 9%,and a 5-year loan at 10%. You’ll pay less total interest with the shortest-term, lowest-rate loan, but this loan will have the highest monthly payments. Thus, you’ll have toevaluate your choices and make the decision that is best for your personal situation.

❉EXAMPLE 3 Choice of Auto LoansYou need a $6000 loan to buy a used car. Your bank offers a 3-year loan at 8%, a 4-yearloan at 9%, and a 5-year loan at 10%. Calculate your monthly payments and total inter-est over the loan term with each option.

SOLUTION Let’s begin with the 3-year loan at 8%. The starting loan principal isthe interest rate is the loan term is years, and

for monthly payments. Your monthly payments would be

Three years is 36 months, so your payments would total Of this total, $6000 pays off your principal, so the total interest is the remaining$768.72.

For the 4-year loan at 9%, we repeat the calculations with and 4 years:

Your total payments over 4 years, or 48 months, would be After we subtract the $6000 that goes to principal, the total interest is $1166.88.

48 3 $149.31 5 $7166.88.

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nYB d5

$6000 3 a0.0912

bc1 2 a1 1

0.0912

bA21234B d5 $149.31

Y 5APR 5 0.09

36 3 $188.02 5 $6768.72.

5 $188.02

5$40

31 2 A1.006666667 B236 4

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$6000 3 a0.0812

bc1 2 a1 1

0.0812

bA21233B d

n 5 12Y 5 3APR 5 0.08,P 5 $6000,

Time out to thinkIn a case such as the student loan in Examples 1 and 2, many people are surprisedto find that more than half of their early loan payments goes to interest when theannual interest rate is only 9%. By referring to Table 4.9, explain why this is the case.How will the payments toward principal and interest compare toward the end ofthe loan?

By the WayYou should alwayswatch out for financialscams, especially whenborrowing money. Keepin mind what is some-times called the first ruleof finance: If it soundstoo good to be true, itprobably is!

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thinking about . . .

Derivation of the Loan Payment Formula

Suppose you borrow a principal P for a loan term of Nmonths at a monthly interest rate i. In most real cases,you would make monthly payments on this loan. How-ever, suppose the lender did not want monthly pay-ments, but instead wanted you to pay back the principalwith compound interest in a lump sum at the end of theloan term. We can find this lump sum amount with thecompound interest formula:

In financial terms, this lump sum amount, A, is calledthe future value of your loan. (The present value is theoriginal loan principal, P.) From the lender’s point ofview, allowing you to spread your payments out overtime should not affect this future value. Thus, in theend, your monthly payments should represent the samefuture value, A. We already have a formula for determin-ing the future value with monthly payments—it is thegeneral form of the savings plan formula from Unit 4C:

We now have two different expressions for A, so weset them equal:

To find the loan payment formula, we need to solvethis equation for PMT. We first multiply both sides bythe reciprocal of the fraction on the left:

5 P 3 A1 1 i B N 3i

3 A1 1 i B N 4 2 1

PMT 33 A1 1 i B N 2 1 4

i3

i3 A1 1 i B N 2 1 4

PMT 33 A1 1 i B N 2 1 4

i5 P 3 A1 1 i B N

A 5 PMT 33 A1 1 i B N 2 1 4

i

A 5 P 3 A1 1 i B N

Simplified, this becomes

Next, we divide both the numerator and the denomi-nator of the fraction on the right by

The numerator simplifies easily to To simplifythe denominator, note that

Substituting the simplified terms for the numeratorand the denominator, we find the loan payment formula:

To put the loan payment formula in the form given inthe text, we substitute for the interest rateper period and for the total number of pay-ments (where n is the number of payments per year andY is the number of years).

N 5 nYi 5 APR>n

PMT 5P 3 i

1 2 A1 1 i B 2N

5 1 2 A1 1 i B 2N

('')''*

apply rule1xN

5 x2N

3 A1 1 i B N 2 1 4

A1 1 i B N 5A1 1 i B N

A1 1 i B N 21

A1 1 i B N

P 3 i.

PMT 5

P 3 A1 1 i B N 3 iA1 1 i B N

3 A1 1 i B N 2 1 4A1 1 i B N

A1 1 i B N:

PMT 5P 3 A1 1 i B N 3 i3 A1 1 i B N 2 1 4

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For the 5-year loan, we set and years:

Your total payments over 5 years, or 60 months, would be After we subtract the $6000 that goes to principal, the total interest is $1648.80. Aswe expected, the monthly payments are lower with the longer-term loans, but thetotal interest is higher. Now try Exercises 37–38. ➽

60 3 $127.48 5 $7648.80.

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nYB d5

$6000 3 a0.112

bc1 2 a1 1

0.112

bA21235B d5 $127.48

Y 5 5APR 5 0.1

Credit CardsCredit card loans differ from installment loans in that you are not required to pay offyour balance in any set period of time. Instead, you are required to make only a mini-mum monthly payment that generally covers all the interest but very little principal.As a result, it takes a very long time to pay off your credit card loan if you make onlythe minimum payments. If you wish to pay off your loan in a particular amount oftime, you should use the loan payment formula to calculate the necessary payments.

A word of caution: Most credit cards have very high interest rates compared toother types of loans. As a result, it is easy to get into financial trouble if you getoverextended with credit cards. The trouble is particularly bad if you miss your pay-ments. In that case, you will probably be charged a late fee that is added to your prin-cipal, thereby increasing the amount of interest due the next month. With the interestcharges operating like compound interest in reverse, failure to pay on time can put aperson into an ever-deepening financial hole.

❉EXAMPLE 4 Credit Card DebtYou have a credit card balance of $2300 with an annual interest rate of 21%. Youdecide to pay off your balance over 1 year. How much will you need to pay eachmonth? Assume you make no further credit card purchases.

SOLUTION Your starting loan principal is the interest rate is and monthly payments mean Because you want to pay off the loan in

1 year, we set The required payments are

You must pay $214.16 per month to pay off the balance in 1 year.Now try Exercises 39–42. ➽

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$2300 3 a0.2112

bc1 2 a1 1

0.2112

bA21231B d5 $214.16

Y 5 1.n 5 12.0.21,

APR 5P 5 $2300,

Time out to thinkConsider your own current financial situation. If you needed a $6000 car loan, whichoption from Example 3 would you choose? Why?

By the WayAbout three-fourths ofAmerican householdshave at least one creditcard, and their averagecredit card balance isabout $8000. The aver-age credit card interestrate is about 17%—farhigher than the interestrate on most other con-sumer loans.

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Time out to thinkContinuing Example 4, suppose you can get a personal loan at a bank at anannual interest rate of 10%. Should you take this loan and use it to pay off your$2300 credit card debt? Why or why not?

❉EXAMPLE 5 A Deepening HolePaul has gotten into credit card trouble. He has a balance of $9500 and just lost hisjob. His credit card company charges interest of compounded daily.Suppose the credit card company allows him to suspend his payments until he finds anew job—but continues to charge interest. If it takes him a year to find a new job,how much will he owe when he starts his new job?

SOLUTION Because Paul is not making payments during the year, this is not a loanpayment problem. Instead, it is a compound interest problem, in which Paul’s balanceof $9500 grows at an annual rate of 21%, compounded daily. We use the compound

APR 5 21%,

Avoiding Credit Card Trouble

Most adults have credit cards for good reason. Usedproperly, credit cards offer many conveniences: They aresafer and easier to carry than cash, they offer monthlystatements that list everything charged to the card, andthey can be used as ID to rent a car. But credit card trou-ble can compound quickly, and many people get intofinancial trouble as a result. A few simple guidelines canhelp you avoid credit card trouble.

• Use only one credit card. People who accumulatebalances on several cards often lose track of theiroverall debt. A lost wallet or purse means more creditcards that must be canceled.

• If possible,pay off your balance in full each month.Then there’s no chance of getting into a financial hole.

• If you plan to pay off your balance in full each month,be sure that your credit card offers an interest-free“grace period” on purchases (usually of about1 month) so that you will not have to pay any interest.

• Compare the interest rate and annual fee (if any) ofyour credit card and others. Fees and rates differgreatly among credit cards, so be sure you are get-ting a good deal. In particular, if you carry a balance,look for a card with a relatively low interest rate.

• When choosing a credit card, watch out for teaserrates. These are low interest rates that are offered fora short period, such as 6 months, after which the cardreverts to very high rates.

• Never use your credit card for a cash advanceexcept in an emergency, because nearly all creditcards charge both fees and high interest rates forcash advances. In addition, most credit cards chargeinterest immediately on cash advances, even if thereis a grace period on purchases. When you need cash,get it directly from your own bank account by cash-ing a check or using an ATM card.

• If you own a home, consider replacing a commoncredit card with a home equity credit line.You’ll gen-erally get a lower interest rate, and the interest maybe tax deductible.

• If you find yourself in a deepening financial hole, con-sult a financial advisor right away. A good place tostart is with the National Foundation for Credit Coun-seling (www.nfcc.org). The longer you wait, the worseoff you’ll be in the long run.

By the WayAmericans hold a totalof more than 500 millionVISA, MasterCard, andAmerican Express cards,plus another 800 millionstore credit cards anddebit cards. Americanscharge more than $1 tril-lion each year to theircredit cards, and paymore than $50 billion ininterest on thesecharges. The averageadult carries nearly$10,000 in credit carddebt.

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interest formula with a starting balance of year, and(for daily compounding). At the end of the year, his loan balance will be

During his year of unemployment, interest alone will make Paul’s credit card balancegrow from $9500 to over $11,700, an increase of more than $2200. Clearly, thisincrease will only make it more difficult for Paul to get back on his financial feet.


MortgagesOne of the most popular types of installment loans is designed specifically to help youbuy a home. It’s called a home mortgage. Mortgage interest rates generally are lowerthan interest rates on other types of loans because your home itself serves as a pay-ment guarantee. If you fail to make your payments, the lender (usually a bank or mort-gage company) can take possession of your home and sell it to recover the amountloaned to you.

There are several considerations in getting a home mortgage. First, the lenderwill probably require a down payment, typically 10% to 20% of the purchase price.Then the lender will loan you the rest of the money needed to purchase the home.Most lenders also charge fees, or closing costs, at the time you take out a loan.Closing costs can be substantial and may vary significantly between lenders, so youshould be sure that you understand them. In general, there are two types of closingcosts:

• Direct fees, such as fees for getting the home appraised and checking your credithistory, for which the lender charges a fixed dollar amount. These fees typicallyrange from a few hundred dollars to a couple thousand dollars.

• Fees charged as points, where each point is 1% of the loan amount. Many lendersdivide points into two categories: an “origination fee” that is charged on all loansand “discount points” that vary for loans with different rates. For example, alender might charge an origination fee of 1 point (1%) on all loans, then offer youa choice of adding 1 discount point for a loan at 8% or 2 discount points for a loanat 7.75%. Despite their different names, there is no essential difference betweenan origination fee and discount points.

As always, you should watch out for any fine print that may affect the cost of yourloan. For example, you should check to make sure that there are no prepayment penal-ties if you decide to pay off your loan early. Most people pay off mortgages early,either because they sell the home or because they decide to refinance the loan to get abetter interest rate or to change their monthly payments.

➽

5 $11,719.23

5 $9500 3 a1 10.21365

bA36531B

A 5 P 3 a1 1APR

nbAnY B

n 5 365Y 5 1APR 5 0.21,P 5 $9500,

By the WayThe idea of a mortgagecontract originated inearly British real estatelaw. The curious wordmortgage comes fromLatin and old French. Itliterally means “deadpledge.”

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Fixed Rate MortgagesThe simplest type of home loan is a fixed rate mortgage, in which you are guaranteedthat the interest rate will not change over the life of the loan. Most fixed rate loans havea term of either 15 or 30 years, with lower interest rates on the shorter-term loans. Wecan calculate payments on fixed rate loans with the loan payment formula.

❉EXAMPLE 6 Fixed Rate Payment OptionsYou need a loan of $100,000 to buy your new home. The bank offers a choice of a30-year loan at an APR of 8% or a 15-year loan at 7.5%. Compare your monthlypayments and total loan cost under the two options. Assume that the closing costsare the same in both cases and therefore do not affect the choice.

SOLUTION The starting loan principal is and we set formonthly payments. For the 30-year loan, we have and Themonthly payments are

Over the 30-year life of the loan, your total payments are

For the 15-year loan, we have and The monthly payments are

Over the 15-year life of the loan, your total payments are

15 yr 312 mo

yr 3$927.01

mo< $166,860

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$100,000 3 a0.07512

bc1 2 a1 1

0.07512

bA212315B d5 $927.01

Y 5 15.APR 5 0.075

30 yr 312 mo

yr 3$733.76

mo< $264,150

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$100,000 3 a0.0812

bc1 2 a1 1

0.0812

bA212330B d5 $733.76

Y 5 30.APR 5 0.08n 5 12P 5 $100,000

MORTGAGE BASICS

If you are seeking a home mortgage, be sure to keep the following considerationsin mind as you compare lenders:

• What interest rate and down payment are required for the loan?• What closing costs will be charged? Be sure you identify all closing costs, includ-

ing origination fees and discount points, since different lenders may quote theirfees differently.

• Watch out for fine print, such as prepayment penalties, that may make the loanmore expensive than it seems on the surface.

By the WayThe average mortgagein the United States ispaid off after 7 years,usually because thehome is sold.

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Note that the payments of $927.01 on the 15-year loan are almost $200 higher thanthe payments of $733.76 on the 30-year loan. However, the 15-year loan saves youalmost $100,000 in total payments. Thus, the 15-year loan saves you a lot in the longrun, but it’s a good plan only if you are confident that you can afford the additional$200 per month that it will cost you for the next 15 years. (See Example 9 for an alter-native payment strategy.) Now try Exercises 47–50. ➽

By the WayWhen you pay points,most lenders give you achoice between payingthem up front and fold-ing them into the loan.For example, if you pay2 points on a $100,000loan, your choice is topay $2000 up front or tomake the loan amount$102,000 rather than$100,000. For the exam-ples in this book, weassume that you paythe points up front.

Time out to thinkDo a quick Web search to find today’s average interest rate for 15-year and 30-yearfixed mortgage loans. How would the payments in Example 6 differ with the currentrates?

❉EXAMPLE 7 Closing CostsGreat Bank offers a $100,000, 30-year, 8% fixed rate loan with closing costs of $500plus 2 points. Big Bank offers a lower rate of 7.9%, but with closing costs of $1000plus 2 points. Evaluate the two options.

SOLUTION In Example 6, we calculated the payments on the 8% loan to be $733.76.At the lower 7.9% rate, the payments are

Thus, you’ll save about $7 per month with Big Bank’s lower interest rate. Now wemust consider the difference in closing costs. Both banks charge the same 2 points, sothis portion of the closing costs won’t affect your decision. (Note, however, that2 points means 2% of the $100,000 loan, which is a $2000 fee!) But you must considerBig Bank’s extra $500 in direct fees.

The choice comes down to this: Big Bank costs you an extra $500 now, but savesyou $7 per month in payments. Dividing $500 by $7 per month, we find the time itwill take to recoup the extra $500:

Thus, it will take you about 6 years to save the extra $500 that Big Bank charges upfront. Unless you are sure that you will be staying in your house (and keeping thesame loan) for much more than 6 years, you probably should go with the lower clos-ing costs at Great Bank, even though your monthly payments will be slightly higher.


❉EXAMPLE 8 Points DecisionContinuing Example 7, suppose you’ve decided to go with Great Bank’s lower closingcosts. You learn that Great Bank actually offers two options for 30-year loans: an 8%interest rate with 2 points or a 7.5% rate with 4 points. Evaluate your options.

➽

$500$7>mo

5 71.4 mo < 6 yr

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$100,000 3 a0.07912

bc1 2 a1 1

0.07912

bA212330B d5 $726.81

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By the WayMortgage rates varysubstantially with time. Inthe 1980s, average U.S.rates for new mortgages(30-year fixed) werealmost always above10%, peaking at morethan 18% in 1981. From2003 to early 2005, aver-age rates were oftennear or below —lower than during anyother period in the past40 years.

5 12 %

SOLUTION We already know that the monthly payments for the 8% loan are$733.76. For the 7.5% loan, we have and Setting

we find that the monthly payments are

The 7.5% loan lowers the monthly payments by How-ever, this loan has 2 additional points in closing costs, which means 2% of your$100,000 loan, or $2000. Thus, you must decide whether it is worth an extra $2000up front for a monthly savings of just under $35. Let’s calculate how long it will taketo make up the added up-front costs:

This is not quite 5 years. If you think it’s likely that you will sell or refinance within5 years, you should not pay the extra points. However, if you expect to keep the loanfor a long time, the added points might be worth it. For example, if you keep theloan for the full 30 years (360 months), you’ll save inmonthly payments over the life of the loan—far more than the extra $2000 you payfor the lower rate today. Now try Exercises 53–54.

Prepayment StrategiesBecause of the long loan term, the early payments on a mortgage tend to be almostentirely interest. For example, Figure 4.8 shows the portion of each payment going toprincipal and interest for a 30-year, $100,000 loan at 8%. In addition, the total inter-est paid on mortgages is often much more than the principal. In Example 6, we foundthat the total payments for this $100,000 loan would be about $264,000—more than

times the starting principal!2 12

➽

360 3 $34.55 5 $12,438

$2000$34.55>mo

5 57.9 mo

$733.76 2 $699.21 5 $34.55.

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$100,000 3 a0.07512

bc1 2 a1 1

0.07512

bA212330B d5 $699.21

APR 5 7.5% 5 0.075,Y 5 30.n 5 12,P 5 $100,000,

By the WayAlthough it may seemstrange at first, makingprepayments on ahome loan is not alwaysa good idea eventhough it reduces thetotal payments. Forexample, if you have$200 per month tospare, you might chooseto invest it rather thanpay down the loan. Ifthe investment return isgreater than the effec-tive loan interest rate,you will come outahead. For home mort-gages, where tax bene-fits can make youreffective interest ratemuch lower than theactual rate (because ofthe mortgage interestdeduction; see Unit 4E),it may be relatively easyto come out ahead byinvesting.

0

734$800

600

400

200

. . . while only about $100—the height of the principal (pink) portion—goes toward reducing the loan principal.

PrincipalInterest

30252015105Years

Pay

men

t

1009080

706050403020100

Per

cent

ageFor any month during the 30-year loan period, the height of the

interest (blue) portion tells you the part of the $734 payment going to interest; here, we see that after five years, about $734 � $100 � $634 goes to interest . . .

FIGURE 4.8 Portions of monthly payments going to principal and interest over the life of a 30-year,$100,000 loan at 8%.

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Clearly, you can save a lot if you can reduce your interest payments. One way to dothis is to pay extra toward the principal, particularly early in the term. For example,suppose you pay an extra $100 toward principal in the first monthly payment of your$100,000 loan. That is, instead of paying the required $734 (see Example 6), you pay$834. Because you’ve reduced your loan balance by $100, you will save the com-pounded value of this $100 over the rest of the 30-year loan term—which is nearly$1100. In other words, paying an extra $100 in the first month saves you about $1100in interest over the 30 years.

❉EXAMPLE 9 An Alternative StrategyAn alternative strategy to the mortgage options in Example 6 is to take the 30-yearloan at 8%, but to try to pay it off in 15 years by making larger payments than arerequired. How much would you have to pay each month? Discuss the pros and consof this strategy.

SOLUTION To reflect paying off an 8% loan in 15 years, we set andwe still have and The monthly payments are

In Example 6, we found that the 30-year loan requires payments of $733.76. Thus, topay off the loan in 15 years, you must make payments that are more than the mini-mum required by per month.

Note that this payment is about $30 per month more than the payment of $927.01required with the 15-year loan (see Example 6), because the 15-year loan had a lowerinterest rate. Clearly, if you know you’re going to pay off the loan in 15 years, youshould take the lower-interest 15-year loan. However, taking the 30-year loan has oneadvantage: Because your required payments are only $733.76, you can always dropback to this level if you find it difficult to afford the extra needed to pay off the loan in15 years. Now try Exercises 55–56. ➽

$955.65 2 $733.76 5 $221.89

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d5

$100,000 3 a0.0812

bc1 2 a1 1

0.0812

bA212315B d5 $955.65

n 5 12.P 5 $100,000Y 5 15;APR 5 0.08

Adjustable Rate MortgagesA fixed rate mortgage is advantageous for you because your monthly payments neverchange. However, it poses a risk to the lender. Imagine that you take out a fixed, 30-yearloan of $100,000 from Great Bank at a 6% interest rate. Initially, the loan may seemlike a good deal for Great Bank. But suppose that, 2 years later, prevailing interestrates have risen to 8%. If Great Bank still had the $100,000 that it lent to you, itcould lend it out to someone else at this higher 8% rate. Instead, it’s stuck with the

Time out to thinkConsider two options for paying off a loan in 15 years: taking out a 15-year loan ortaking out a 30-year loan and making an extra principal payment each month.Assuming that you would like to pay off the loan in 15 years, how would you decidewhich strategy is better for you?

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6% rate that you are paying. In effect, Great Bank loses potential future income ifprevailing rates rise substantially and you have a fixed rate loan.

Lenders can lessen the risk of rising interest rates by charging higher rates forlonger-term loans. That is why rates generally are higher for 30-year loans than for15-year loans. But an even lower-risk strategy for the lender is an adjustable ratemortgage (ARM), in which the interest rate you pay changes whenever prevailingrates change. Because of the reduced long-term risk to lenders, ARMs generally havemuch lower initial interest rates than fixed rate loans. For example, a bank offering a6% rate on a fixed 30-year loan might offer an ARM that begins at 4%. Most ARMsguarantee their starting interest rate for the first 6 months or 1 year, but interest ratesin subsequent years move up or down to reflect prevailing rates. Most ARMs alsoinclude a rate cap that cannot be exceeded. For example, if your ARM begins at aninterest rate of 4%, you may be promised that your interest rate can never go higherthan a rate cap of 10%. Making a decision between a fixed rate loan and an ARM canbe one of the most important financial decisions of your life.

❉EXAMPLE 10 Rate Approximations for ARMsYou have a choice between a 30-year fixed rate loan at 8% and an ARM with a first-year rate of 5%. Neglecting compounding and changes in principal, estimate yourmonthly savings with the ARM during the first year on a $100,000 loan. Suppose thatthe ARM rate rises to 11% by the fourth year. How will your payments be affected?

SOLUTION Because mortgage payments are mostly interest in the early years of a loan, we can make approximations by pretending that the principal remainsunchanged. For the 8% fixed rate loan, the interest on the $100,000 loan for the firstyear will be approximately With the 5% loan, your first-year interest will be approximately Thus, the ARM willsave you about $3000 in interest during the first year, which means a monthly savingsof about

By the fourth year, when rates reach 11%, the situation is reversed. The rate on theARM is now 3 percentage points above the rate on the fixed rate loan. Instead of sav-ing $250 per month, you’d be paying $250 per month more on the ARM than on the8% fixed rate loan. Moreover, if interest rates remain high on the ARM, you will con-tinue to make these high payments for many years to follow. Thus, while ARMsreduce risk for the lender, they add risk for the borrower. Now try Exercises 57–58. ➽

$3000 4 12 5 $250.

5% 3 $100,000 5 $5000.8% 3 $100,000 5 $8000.

By the WayWatch out for “teaser”rates on adjustable ratemortgages. Under nor-mal circumstances, yourrate on an ARM risesonly if prevailing interestrates rise. However, somelenders offer low teaserrates—rates below theprevailing rates—for thefirst few months of anARM. Teaser rates arecertain to rise as soon asthe teaser period is over.Thus, while teaser ratesmay be attractive, thelonger-term policies ofthe ARM are far moreimportant.

Time out to thinkIn the past few years, another type of mortgage loan has become popular: theinterest only loan, in which you pay only interest and pay nothing toward principal.Most financial experts advise against these loans, because your principal nevergets paid off. Can you think of any circumstances under which such a loan mightmake sense for a home buyer? Explain.

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1. In the loan payment formula, assuming all other variablesare constant, the monthly payment

a. increases as P increases.

b. increases as APR decreases.

c. increases as Y increases.

2. With the same APR and principal, a 15-year loan will have

a. a higher monthly payment than a 30-year loan.

b. a lower monthly payment than a 30-year loan.

c. a payment that could be greater or less than that of a 30-year loan.

3. With the same term and principal, a loan with a higherAPR will have

a. a lower monthly payment than a loan with a lower APR.

b. a higher monthly payment than a loan with a lowerAPR.

c. a payment that could be greater or less than that of aloan with a lower APR.

4. In the early years of a 30-year mortgage loan,

a. most of the payment goes to the principal.

b. most of the payment goes to interest.

c. equal amounts go to principal and interest.

5. If you make monthly payments of $1000 on a 10-year loan,your total payments over the life of the loan amount to

a. $10,000. b. $100,000. c. $120,000.

6. Credit card loans are different from installment loans inthat

a. credit card loans always have higher interest rates.

b. credit card loans do not have a fixed APR.

c. credit card loans do not have a set loan term.

7. A loan of $200,000 that carries a 2-point origination feerequires an advance payment of

a. $2000. b. $40,000. c. $4000.

8. A $120,000 loan with $500 in closing costs plus 1 pointrequires an advance payment of

a. $1500. b. $1700. c. $500.

9. You are currently paying off a student loan with an interestrate of 9% and a monthly payment of $450. You areoffered the chance to refinance the remaining balance witha new 10-year loan with an interest rate of 8% that willgive you a significantly lower monthly payment. Refinanc-ing in this way

a. is always a good idea.

b. is a good idea if it lowers your monthly payment by atleast $100.

c. is a good idea only if closing costs are low and your cur-rent loan has many years remaining in its loan term.

10. Consider two mortgage loans with the same principal andthe same APR. Loan 1 is fixed for 15 years, and Loan 2 isfixed for 30 years. Which statement is true?

a. Loan 1 will have higher monthly payments, but you’llpay less total interest over the life of the loan.

b. Loan 1 will have lower monthly payments, and you’llpay less total interest over the life of the loan.

c. Both loans will have the same monthly payments, butyou’ll pay less total interest with Loan 1.

REVIEW QUESTIONS11. Suppose you pay only the interest on a loan. Will the loan

ever be paid off? Why not?

12. What is an installment loan? Explain the meaning and useof the loan payment formula.

13. Explain, in general terms, how the portions of loan pay-ments going to principal and interest change over the lifeof the loan.

14. Suppose that you need a loan of $10,000 and are offered achoice of a 3-year loan at 7% interest or a 5-year loan at8% interest. Discuss the pros and cons of each choice.

15. How do credit card loans differ from ordinary installmentloans? Why are credit card loans particularly dangerous?

16. What is a mortgage? What is a down payment on a mort-gage? Explain how closing costs, including points, canaffect loan decisions.

EXERCISES 4D

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17. The interest rate on my student loan is only 7%, yet morethan half of my payments are currently going toward inter-est rather than principal.

18. My student loans were all 20-year loans at interest ratesof 8% or above, so when my bank offered me a 20-yearloan at 7%, I took it and used it to pay off the studentloans.

19. I make only the minimum required payments on my creditcard balance each month, because that way I’ll have moreof my own money to keep.

20. I carry a large credit card balance, and I had a credit cardthat charged an annual interest rate of 12%. So when Ifound another credit card that promised a 3% interest ratefor the first 3 months, it was obvious that I should switchto this new card.

21. I had a choice between a fixed rate mortgage at 6% and anadjustable rate mortgage that started at 3% for the firstyear with a maximum increase of 1.5 percentage points ayear. I took the adjustable rate, because I’m planning tomove within three years.

22. Fixed rate loans with 15-year terms have lower interestrates than loans with 30-year terms, so it always makessense to take the 15-year loan.

BASIC SKILLS & CONCEPTSLoan Terminology. For the loans described in Exercises 23–24,do the following:

a. Clearly identify the starting loan principal, the annual inter-est rate, the number of payments per year, the loan term,and the payment amount.

b. How many payments will you make in total? What totalamount will you pay over the full term of the loan?

c. Of the total amount you pay, how much will go towardprincipal and how much toward interest?

23. You borrowed $80,000 at an APR of 7%, which you arepaying off with monthly payments of $620 for 20 years.

24. You borrowed $15,000 at an APR of 9%, which you arepaying off with monthly payments of $190 for 10 years.

Loan Payments. For the loans described in Exercises 25–34,do the following:

a. Calculate the monthly payment.b. Determine the total payment over the term of the loan.c. Determine how much of the total payment over the loan

term goes to principal and how much to interest.25. A student loan of $50,000 at a fixed APR of 10% for

20 years

26. A student loan of $12,000 at a fixed APR of 8% for10 years

27. A home mortgage of $200,000 with a fixed APR of 7.5%for 30 years


29. A home mortgage of $200,000 with a fixed APR of 9% for15 years


31. You borrow $10,000 over a period of 3 years at a fixed APRof 12%.

32. You borrow $10,000 over a period of 5 years at a fixed APRof 10%.

33. You borrow $150,000 over a period of 15 years at a fixedAPR of 8%.

34. You borrow $100,000 over a period of 30 years at a fixedAPR of 7%.

Principal and Interest Payments. For the loans described inExercises 35–36, calculate the monthly payment and the por-tions of the payments that go to principal and to interest duringthe first 3 months. (Hint: Use a table as in Example 2.)


36. A student loan of $24,000 at a fixed APR of 8% for 15 years

37. Choosing an Auto Loan. You need to borrow $12,000 tobuy a car and you determine that you can afford monthlypayments of $250. The bank offers three choices: a 3-yearloan at 7% APR, a 4-year loan at 7.5% APR, or a 5-yearloan at 8% APR. Which loan best meets your needs?Explain your reasoning.

38. Choosing a Personal Loan. You need to borrow $4000to pay off your credit cards and you can afford monthlypayments of $150. The bank offers three choices: a 2-year

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New Month Payment Expenses Interest Balance

0 — — — $1200

1 $200 $75

2 $200 $75

3 $200 $75

5 $10931 $75 1 $185 $18

$1200 2 $2001.5% 3 $1200

Month Payment Expenses Interest Balance

0 — — — $300

1 $300 $175 $179.50

2 $150 $150

3 $400 $350

4 $500 $450

5 0 $100

6 $100 $100

7 $200 $150

8 $100 $80

5 $4.501.5% 3 $300

loan at 8% APR, a 3-year loan at 9% APR, or a 4-year loanat 10% APR. Which loan best meets your needs? Explainyour reasoning.

Credit Card Debt. For Exercises 39–42, assume you have abalance of $5000 on your credit card that you want to pay off.Calculate your monthly payment and total payment under theconditions listed. Assume you make no additional charges to the card.

39. The credit card APR is 18% and you want to pay off thebalance in 1 year.

40. The credit card APR is 20% and you want to pay off thebalance in 2 years.

41. The credit card APR is 21% and you want to pay off thebalance in 3 years.

42. The credit card APR is 22% and you want to pay off thebalance in 1 year.

43. Credit Card Debt. Assume you have a balance of $1200on a credit card with an APR of 18%, or 1.5% per month.You start making monthly payments of $200, but at thesame time you charge an additional $75 per month to thecredit card. Assume that interest for a given month is basedon the balance for the previous month. The followingtable shows how you can calculate your monthly balance.

est for a given month is charged on the balance for theprevious month. Complete the table. After 8 months, whatis the balance on the credit card? Comment on the effectof the interest and the initial balance, in light of the factthat for 7 of the 8 months expenses never exceeded payments.

46. Teaser Rate. You have a total credit card debt of $4000.You receive an offer to transfer this debt to a new card withan introductory APR of 6% for the first 6 months. Afterthat, the rate becomes 24%.

a. What is the monthly interest payment on $4000 duringthe first 6 months? (Assume you pay nothing towardprincipal and don’t charge any further debts.)

b. What is the monthly interest payment on $4000 afterthe first 6 months? Comment on the change from theteaser rate.

Fixed Rate Options. Compare your monthly payments andtotal loan cost under the two options listed in each of Exer-cises 47–50. Assume that the loans are fixed rate and that closingcosts are the same in both cases. Briefly discuss the pros andcons of each option.

47. You need a $200,000 loan.Option 1: a 30-year loan at an APR of 8%Option 2: a 15-year loan at 7.5%

48. You need a $75,000 loan.Option 1: a 30-year loan at an APR of 8%Option 2: a 15-year loan at 7%

49. You need a $60,000 loan.Option 1: a 30-year loan at an APR of 7.15%Option 2: a 15-year loan at 6.75%

Complete and extend the table to show your balance at theend of each month until the debt is paid off. How longdoes it take to pay off the credit card debt?

44. Credit Card Debt. Repeat the table of Exercise 43, butthis time assume that you make monthly payments of$300. Extend the table as long as necessary until your debtis paid off. How long does it take to pay off your debt?

45. Credit Card Woes. The following table shows theexpenses and payments for 8 months on a credit cardaccount with an initial balance of $300. Assume that theinterest rate is 1.5% per month (18% APR) and that inter-

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50. You need a $180,000 loan.Option 1: a 30-year loan at an APR of 7.25%Option 2: a 15-year loan at 6.8%

Closing Costs. You need a loan of $120,000 to buy a home.Each of Exercises 51–54 offers two choices. Calculate yourmonthly payments and total closing costs in each case. Brieflydiscuss how you would decide between the two choices.

51. Choice 1: 30-year fixed rate at 8% with closing costs of$1200 and no points

Choice 2: 30-year fixed rate at 7.5% with closing costs of$1200 and 2 points

52. Choice 1: 30-year fixed rate at 8.5% with no closing costsand no points


53. Choice 1: 30-year fixed rate at 7.25% with closing costs of$1200 and 1 point


54. Choice 1: 30-year fixed rate at 7.5% with closing costs of$1000 and no points


55. Accelerated Loan Payment. Suppose you have a studentloan of $30,000 with an APR of 9% for 20 years.

a. What are your required monthly payments?

b. Suppose you would like to pay the loan off in 10 yearsinstead of 20. What monthly payments will you need tomake?

c. Compare the total amounts you’ll pay over the loanterm if you pay the loan off in 20 years versus 10 years.

56. Accelerated Loan Payment. Suppose you have a studentloan of $60,000 with an APR of 8% for 25 years.

a. What are your required monthly payments?

b. Suppose you would like to pay the loan off in 15 yearsinstead of 25. What monthly payments will you need tomake?

c. Compare the total amounts you’ll pay over the loanterm if you pay the loan off in 25 years versus 15 years.

57. ARM Rate Approximations. You have a choice betweena 30-year fixed rate loan at 7% and an ARM with a first-year rate of 5%. Neglecting compounding and changes inprincipal, estimate your monthly savings with the ARMduring the first year on a $150,000 loan. Suppose that the

ARM rate rises to 8.5% at the start of the third year.Approximately how much extra will you then be payingover what you would have paid if you had taken the fixedrate loan?

58. ARM Rate Approximations. You have a choice betweena 30-year fixed rate loan at 8.5% and an ARM with a first-year rate of 5.5%. Neglecting compounding and changesin principal, estimate your monthly savings with the ARMduring the first year on a $125,000 loan. Suppose that theARM rate rises to 10% at the start of the second year.Approximately how much extra will you then be payingover what you would have paid if you had taken the fixedrate loan?

FURTHER APPLICATIONS59. How Much House Can You Afford? You can afford

monthly payments of $500. If current mortgage rates are9% for a 30-year fixed rate loan, what loan principal canyou afford? If you are required to make a 20% down pay-ment and you have the cash on hand to do it, what pricehome can you afford? (Hint: You will need to solve theloan payment formula for P.)

60. How Much House Can You Afford? You can affordmonthly payments of $1200. If current mortgage rates are7.5% for a 30-year fixed rate loan, what loan principal canyou afford? If you are required to make a 20% downpayment and you have the cash on hand to do it, whatprice home can you afford? (Hint: You will need to solvethe loan payment formula for P.)

61. Student Loan Consolidation. Suppose you have thefollowing three student loans: $10,000 with an APR of 8%for 15 years, $15,000 with an APR of 8.5% for 20 years,and $12,500 with an APR of 9% for 10 years.

a. Calculate the monthly payment for each loan individu-ally.

b. Calculate the total you’ll pay in payments during the lifeof all three loans.

c. A bank offers to consolidate your three loans into a singleloan with an APR of 8.5% and a loan term of 20 years.What will your monthly payments be in that case?What will your total payments be over the 20 years?Discuss the pros and cons of accepting this loanconsolidation.

62. Bad Deals: Car-Title Lenders. Some “car-title lenders”offer quick cash loans in exchange for being allowed tohold the title to your car as collateral (you lose your car ifyou fail to pay off the loan). In many states, these lendersoperate under pawnbroker laws that allow them to chargefees as a percentage of the unpaid balance. Suppose you

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need $2000 in cash, and a car-title company offers you aloan at an interest rate of 2% per month plus a monthly feeof 20% of the unpaid balance.

a. How much will you owe in interest and fees on your$2000 loan at the end of the first month?

b. Suppose that you pay only the interest and fees eachmonth. How much will you pay over the course of a fullyear?

c. Suppose instead that you obtain a loan from a bank witha term of 3 years and an APR of 10%. What are yourmonthly payments in that case? Compare these to thepayments to the car-title lender.

63. Other Than Monthly Payments. Suppose you want toborrow $100,000 and you find a bank offering a 20-yearloan with an APR of 6%.

a. Find your regular payments if you pay 12, 26,52 times a year.

b. Compute the total payout for each of the loans in part a.

c. Compare the total payouts computed in part b. Discussthe pros and cons of the plans.

64. 13 Payments (challenge). Suppose you want to borrow$100,000 and you find a bank offering a 20-year loan withan APR of 6%.

a. What are your monthly payments?

b. Instead of making 12 payments per year, you saveenough money to make a 13th payment each year (in theamount of your regular monthly payment of part a).How long will it take to retire the loan?

65. Project: Choosing a Mortgage. Imagine that you workfor an accounting firm and a client has told you that he isbuying a house and needs a loan of $120,000. His monthlyincome is $4000 and he is single with no children. He has$14,000 in savings that can be used for a down payment.

n 5 1,

Find the current rates available from local banks for bothfixed rate mortgages and adjustable rate mortgages(ARMs). Analyze the offerings and summarize orally or inwriting the best options for your client, along with thepros and cons of each option.


66. Credit Card Comparisons. Visit a Web site that givescomparisons between credit cards. Briefly explain the fac-tors that are considered in the comparisons. How doesyour own credit card compare to other credit cards? Basedon this comparison, do you think you would be better offwith a different credit card?

67. Home Financing. Visit a Web site that offers onlinehome financing. Describe the terms of a particular homemortgage. Discuss the advantages and disadvantages offinancing a home online rather than at a local bank.

68. Online Car Purchase. Find a car online that you mightwant to buy. Find a loan that you would qualify for, andcalculate your monthly payments and total payments overthe life of the loan. Next, suppose that you started a sav-ings plan instead of buying the car, depositing the sameamounts that would have gone to car payments. Estimatehow much you would have in your savings plan by the timeyou graduate from college. Explain your assumptions.

69. Student Financial Aid. There are many Web sites thatoffer student loans. Visit a Web site that offers studentloans and describe the terms of a particular loan. Discussthe advantages and disadvantages of financing a studentloan online rather than through a bank or through youruniversity or college.

70. Scholarship Scams. The Federal Trade Commissionkeeps track of many financial scams related to collegescholarships. Read about two different types of scams, andreport on how they work and how they hurt people whoare taken in by them.

71. Financial Scams. Many Web sites keep track of currentfinancial scams. Visit some of these sites and report on onescam that has already hurt a lot of people. Describe how thescam works and how it hurts those who are taken in by it.

IN THE NEWS72. Mortgage Rates. Find advertisements in the newspaper

for two different home mortgages companies. Using theideas of this unit, evaluate the terms of loans from eachcompany and decide which company you would use for ahome mortgage.

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4E Income Taxes 289

73. Credit Card Statement. Look carefully at the terms offinancing explained on your most recent credit card state-ment. Explain all the important terms, including the inter-est rates that apply, annual fees, and grace periods.

74. Bank Rates. Find the interest rates that your bank (oranother local bank) charges for different types of loans, suchas auto loans, personal loans, and home mortgages. Why doyou think the rates are different in the different cases?

In this world, nothing iscertain but deathand taxes.

—BENJAMIN FRANKLIN

UNIT 4E Income Taxes

There are many different types of taxes, including sales tax, gasoline tax, and propertytax. But for most Americans, the largest tax burden comes from taxes on wages andother income. In this unit, we explore a few of the many aspects of federal incometaxes.

Income Tax BasicsIt’s quite possible that no one fully understands federal income taxes. The complete taxcode consists of thousands of pages of detailed regulations. Many of the regulationsare difficult to interpret, and disputes about their meaning are often taken to court.Congress frequently tinkers with tax laws and occasionally undertakes major reforms.For example, tax laws were greatly simplified by Congress in 1986. Unfortunately,politicians were unable to resist making modifications to the simplified tax code, so itgradually became more complex once again.

Nevertheless, the many arcane tax laws generally apply only to relatively small seg-ments of the population. Most people not only can file their own taxes—which usuallyrequires little more than filling in a few boxes and looking up numbers in a table—butcan understand how their taxes work. This is important, because understanding yourtaxes not only will allow you to make intelligent decisions about your personal financesbut also will help you understand the political issues that you vote on.

Figure 4.9 summarizes the steps in a basic tax calculation. We’ll follow the flow ofthe steps, defining terms as we go along.

• The process begins with your gross income, which is all your income for theyear, including wages, tips, profits from a business, interest or dividends frominvestments, and any other income you receive.

gross income

EQUALStaxable income

total taxtax computation

based on ratesor tables

MINUSadjustments to

income

MINUSdeductions and

exemptions

adjusted grossincome

MINUStax credits

MINUSpayments orwithholding

EQUALSadjusted gross

income

EQUALStotal tax

EQUALSamount owed

(or refund)

FIGURE 4.9 Flow chart showing the basic steps in calculating income tax.

The hardest thing inthe world to under-stand is the incometax.

—ALBERT EINSTEIN

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By the WayUnited States federalincome taxes are col-lected by the InternalRevenue Service (IRS),which is part of theUnited States Depart-ment of the Treasury.Most people file federaltaxes by completing atax form,such asForm 1040,1040A,or1040EZ.

• Some gross income is not taxed (at least not in the year it is received), such as con-tributions to IRAs and other tax-deferred savings plans. These untaxed portions ofgross income are called adjustments. Subtracting adjustments from your grossincome gives your adjusted gross income.

• Most people are entitled to certain exemptions and deductions—amounts thatyou subtract from your adjusted gross income before calculating your taxes. (Theamounts you can subtract depend on factors that we’ll discuss shortly.) Once yousubtract the exemptions and deductions, you are left with your taxable income.

• A tax table or tax rate computation allows you to determine how much tax youowe on your taxable income. However, you may not actually have to pay thismuch tax if you are entitled to any tax credits. For example, you may be entitled toa tax credit of $1000 per child. From your tax rate computation, you subtract theamount of any credits to find your total tax.

• Finally, most people have already paid part or all of their tax bill during the year,either through withholdings (by your employer) or through paying quarterlyestimated taxes (if you are self-employed). You subtract the taxes that you’ve alreadypaid to determine how much you still owe. In many cases, you may have paidmore than you owe, in which case you should receive a tax refund.

❉EXAMPLE 1 Income on Tax FormsKaren earned wages of $34,200, received $750 in interest from a savings account, andcontributed $1200 to a tax-deferred retirement plan. She was entitled to a personalexemption of $3300 and to deductions totaling $5400. Find her gross income,adjusted gross income, and taxable income.

SOLUTION Karen’s gross income is the sum of all her income, which means the sumof her wages and her interest:

Her $1200 contribution to a tax-deferred retirement plan counts as an adjustment toher gross income, so her adjusted gross income (AGI) is

To find her taxable income, we subtract her exemptions and deductions:

Her taxable income is $25,050. Now try Exercises 29–32.

Filing StatusTax calculations depend on your filing status, such as single or married. Most peoplefall into one of four filing status categories:

• Single applies if you are unmarried, divorced, or legally separated.

• Married filing jointly applies if you are married and you and your spouse file a sin-gle tax return. (In some cases, this category also applies to widows or widowers.)

➽

5 $33,750 2 $3300 2 $5400 5 $25,050 taxable income 5 AGI 2 exemptions 2 deductions

AGI 5 gross income 2 adjustments 5 $34,950 2 $1200 5 $33,750

gross income 5 $34,200 1 $750 5 $34,950

HISTORICAL NOTE

An income tax was firstlevied in the UnitedStates in 1862 (duringthe Civil War), but wasabandoned a few yearslater. The 16th Amend-ment to the Constitution,ratified in 1913, gave thefederal government fullauthority to levy anincome tax.

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4E Income Taxes 291

By the WayNot all taxpayers get thefull advantage ofexemptions and deduc-tions. For example, theamounts of exemptionsbegin to “phase out” forsingle people earningmore than about$150,000, and manymiddle- to high-incometaxpayers are subject tothe alternative minimumtax (AMT), which disal-lows most or all deduc-tions.

• Married filing separately applies if you are married and you and your spouse file twoseparate tax returns.

• Head of household applies if you are unmarried and are paying more than half thecost of supporting a dependent child or parent.

We will use these four categories in the rest of our discussion.

Exemptions and DeductionsBoth exemptions and deductions are subtracted from your adjusted gross income.However, they are calculated differently, which is why they have different names.

Exemptions are a fixed amount per person ($3300 in 2006). You can claim theamount of an exemption for yourself and each of your dependents (for example, chil-dren whom you support).

Deductions vary from one person to another. The most common deductionsinclude interest on home mortgages, contributions to charity, and taxes you’ve paid toother agencies (such as state income taxes or local property taxes). However, youdon’t necessarily have to add up all your deductions. When you file your taxes, youhave two options for deductions:

• You can choose a standard deduction, the amount of which depends on your fil-ing status.

• You can choose itemized deductions, in which case you add up all the individualdeductions to which you are entitled.

Note that you get either the standard deduction or itemized deductions, not both.Because deductions lower your tax bill, you should choose whichever option is larger.

❉EXAMPLE 2 Should You Itemize?Suppose you have the following deductible expenditures: $2500 for interest on ahome mortgage, $900 for contributions to charity, and $250 for state income taxes.Your filing status entitles you to a standard deduction of $5150. Should you itemizeyour deductions or claim the standard deduction?

SOLUTION The total of your deductible expenditures is

If you itemize your deductions, you can subtract $3650 when finding your taxableincome. But if you take the standard deduction, you can subtract $5150. You are bet-ter off with the standard deduction. Now try Exercises 33–38.

Tax RatesThe United States has a progressive income tax, meaning that people with highertaxable incomes pay at a higher tax rate. The system works by assigning differentmarginal tax rates to different income ranges (or margins). For example, suppose youare single and your taxable income is $25,000. Under 2006 tax rates, you would pay10% tax on the first $7550 and 15% tax on the remaining $17,450. In this case, we saythat your marginal rate is 15%, or that you are in the 15% tax bracket. For each majorfiling status, Table 4.10 shows the marginal tax rate, standard deduction, and exemp-tions for 2006.

➽

$2500 1 $900 1 $250 5 $3650

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The income levels for the tax brackets, the standard deductions, and the exemptionamounts all rise each year to keep pace with inflation. In addition, Congress and thePresident tend to change the rates for the tax brackets every few years. Thus, if youare calculating taxes for a year other than 2006, you must get an updated tax ratetable. You can find current tax rates on the IRS Web site.

❉EXAMPLE 3 Marginal Tax ComputationsUsing 2006 rates, calculate the tax owed by each of the following people. Assume thatthey all claim the standard deduction and neglect any tax credits.

a. Deirdre is single with no dependents. Her adjusted gross income is $80,000.b. Robert is a head of household taking care of two dependent children. His

adjusted gross income also is $80,000.c. Jessica and Frank are married with no dependents. They file jointly. They

each have $80,000 in adjusted gross income, making a combined income of$160,000.

SOLUTION

a. First, we must find Deirdre’s taxable income. She is entitled to a personalexemption of $3300 and a standard deduction of $5150. We subtract theseamounts from her adjusted gross income to find her taxable income:

Now we calculate her taxes using the single rates in Table 4.10. She is in the25% tax bracket because her taxable income is above $30,650 but below the28% threshold of $74,200. Thus, she owes 10% on the first $7550 of her

taxable income 5 $80,000 2 $3300 2 $5150 5 $71,550

TABLE 4.10 2006 Marginal Tax Rates, Standard Deductions, and Exemptions*

Tax Rate Single Married Filing Jointly Married Filing Separately Head of Household

10% up to $7550 up to $15,100 up to $7550 up to 10,750

15% up to $30,650 up to $61,300 up to $30,650 up to $41,050




35% above $336,550 above $336,550 above $168,275 above $336,550

standard deduction $5150 $10,300 $5150 $7550

exemption (per person) $3300 $3300 $3300 $3300

*Each higher marginal rate begins where the prior one leaves off. For example, for a single person, the 15%marginal rate affects income starting at $7550 at which the 10% rate leaves off and continuing up to$30,650.

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4E Income Taxes 293

By the WayIn Example 3, Jessicaand Frank (part c) eachearned the sameamount as Deirdre (parta), but together theypaid more than twice asmuch tax (by almost$600). This feature of thetax code, whereby peo-ple pay more when theyare married than theywould if they were sin-gle, is called themarriage penalty. Not allcouples are affectedthe same way by themarriage penalty. Somecouples even get a mar-riage bonus instead,especially if one spouseearns much more thanthe other (or only one isemployed).

taxable income, 15% on her taxable income above $7550 but below$30,650, and 25% on her taxable income above $30,650.

5 $14,445 5 $755 1 $3465 1 $10,225

(''''''''''')'''''''''''*

25% marginal rate on taxable income above $30,650

(''''''''''')'''''''''''*

15% marginal rate on taxable incomebetween $7550 and $30,650

(''''')'''''*

10% marginal rate on first$7550 of taxable income

A10% 3 $7550B 1 A15% 3 3$30,650 2 $7550 4 B 1 A25% 3 3$71,550 2 $30,650 4 B

Deirdre’s tax is $14,445.

b. Robert is entitled to three exemptions of $3300 each—one for himself andone for each of his two children. As a head of household, he is also entitledto a standard deduction of $7550. We subtract these amounts from hisadjusted gross income to find his taxable income:

We calculate Robert’s taxes using the head of household rates. His taxableincome of $62,550 puts him in the 25% tax bracket, so his tax is

Robert’s tax is $10,995.

c. Jessica and Frank are each entitled to one exemption of $3300. Because theyare married filing jointly, their standard deduction is $10,300. We subtractthese amounts from their adjusted gross income to find their taxableincome:

We calculate their taxes using the married filing jointly rates. Their taxableincome of $143,100 puts them in the 28% tax bracket, so their tax is

Jessica and Frank’s combined tax is $29,472, equivalent to $14,736 each.Now try Exercises 39–46. ➽

5 $29,4725 $1510 1 $6930 1 $15,600 1 $5432

(''''''''''''')'''''''''''''*

28% marginal rate on taxableincome above $123,700

(''''''''''''')''''''''''''*

25% marginal rate on taxable income between $61,300 and $123,700

1 A25% 3 3$123,700 2 $61,300 4 B 1 A28% 3 3$143,100 2 $123,700 4 B

('''''''''''')''''''''''''*

15% marginal rate on taxable incomebetween $15,100 and $61,300

('''''')''''''*

10% marginal rate on first$15,100 of taxable income

A10% 3 $15,100 B 1 A15% 3 3$61,300 2 $15,100 4 B

taxable income 5 $160,000 2 A2 3 $3300 B 2 $10,300 5 $143,100

5 $10,9955 $1075 1 $4545 1 $5375

(''''''''''')'''''''''''*

25% marginal rate on taxable income above $41,050

(''''''''''')'''''''''''*

15% marginal rate on taxable incomebetween $10,750 and $41,050

('''''')''''''*

10% marginal rate on first$10,750 of taxable income

1 A25% 3 3$62,550 2 $41,050 4 BA10% 3 $10,750B 1 A15% 3 3$41,050 2 $10,750 4 B

taxable income 5 $80,000 2 A3 3 $3300 B 2 $7550 5 $62,550

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Time out to thinkNote that all four individuals in Example 3 have the same $80,000 in adjusted grossincome, yet they each pay a different amount in taxes. Explain why this is the case.Do you believe the outcomes are fair? Why or why not? (Bonus: Could their grossincomes have differed even though their adjusted gross incomes were the same?Explain.)

Tax Credits and DeductionsTax credits and tax deductions may sound similar, but they are very different. Supposeyou are in the 15% tax bracket. A tax credit of $500 reduces your total tax bill by thefull $500. In contrast, a tax deduction of $500 reduces your taxable income by $500,which means it saves you only in taxes. As a rule, tax credits aremore valuable than tax deductions.

Congress authorizes tax credits for only specific situations, such as a (maximum)$1000 tax credit for each child. In contrast, your spending determines how much youclaim in deductions, at least if you are itemizing. The most valuable deduction formost people is the mortgage interest tax deduction, which allows you to deduct allthe interest (but not the principal) you pay on a home mortgage. Many people also getsubstantial deductions from donating money to charities.

❉EXAMPLE 4 Tax Credits vs. Tax DeductionsSuppose you are in the 28% tax bracket. How much does a $1000 tax credit save you?How much does a $1000 charitable contribution (which is tax deductible) save you?Answer these questions both for the case in which you itemize deductions and for thecase in which you take the standard deduction.

SOLUTION The entire $1000 tax credit is deducted from your tax bill and thereforesaves you a full $1000, whether you itemize deductions or take the standard deduc-tion. In contrast, a $1000 deduction reduces your taxable income, not your total taxbill, by $1000. Thus, for the 28% tax bracket, at best your $1000 deduction will saveyou However, you will save this $280 only if you are itemizingdeductions. If your total itemized deductions are less than the standard deduction (seeExample 2), you will still be better off with the standard deduction. In that case, the$1000 contribution will save you nothing at all. Now try Exercises 47–52.

❉EXAMPLE 5 Rent or Own?Suppose you are in the 28% tax bracket and you itemize your deductions. You are try-ing to decide whether to rent an apartment or buy a house. The apartment rents for$1400 per month. You’ve investigated your loan options, and you’ve determined thatif you buy the house, your monthly mortgage payments will be $1600, of which anaverage of $1400 goes toward interest during the first year. Compare the monthlyrent to the mortgage payment. Is it cheaper to rent the apartment or buy the house?

SOLUTION The monthly cost of the apartment is $1400 in rent. For the house,however, we must take into account the value of the mortgage deduction. Themonthly interest of $1400 is tax deductible. Because you are in the 28% tax bracket,

➽

28% 3 $1000 5 $280.

15% 3 $500 5 $75

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4E Income Taxes 295

this deduction saves you Thus, the true monthly cost of themortgage is the payment minus the tax savings, or

Despite the fact that the mortgage payment is $200 higher than the rent, its true costto you is almost $200 per month less because of the tax savings from the mortgageinterest deduction. Of course, as a homeowner, you will have other costs, such as formaintenance and repairs, that you may not have to pay if you rent. (This exampleassumes you would be itemizing deductions regardless of whether you rent or buy.)


$1600 2 $392 5 $1208

28% 3 $1400 5 $392.

❉EXAMPLE 6 Varying Value of DeductionsDrew is in the 15% marginal tax bracket. Marian is in the 35% marginal tax bracket.They each itemize their deductions. They each donate $5000 to charity. Comparetheir true costs for the charitable donation.

SOLUTION The $5000 contribution to charity is tax deductible. Because Drew is inthe 15% tax bracket, this contribution saves him in taxes.Thus, its true cost to him is the contributed amount of $5000 minus his tax savingsof $750, or $4250. For Marian, who is in the 35% tax bracket, the contributionsaves in taxes. Thus, its true cost to her is

The true cost of the donation is considerably lower for Marian because she isin a higher tax bracket. Now try Exercises 55–56. ➽$3250.

$5000 2 $1750 535% 3 $5000 5 $1750

15% 3 $5000 5 $750

Social Security and Medicare TaxesIn addition to being subject to taxes computed with the marginal rates, some incomeis subject to Social Security and Medicare taxes, which are collected under theobscure name of FICA (Federal Insurance Contribution Act) taxes. Taxes collectedunder FICA are used to pay Social Security and Medicare benefits, primarily to peo-ple who are retired.

FICA applies only to income from wages (including tips) and self-employment. Itdoes not apply to income from such things as interest, dividends, or profits from salesof stock. In 2006, the FICA tax rates for individuals who were not self-employed were

Time out to thinkAs shown in Example 6, tax deductions are more valuable to people in higher taxbrackets. Some people argue that this is unfair because it means that tax deduc-tions save more money for richer people than for poorer people. Others argue thatit is fair, because richer people pay a higher tax rate in the first place. What do youthink? Defend your opinion.

Time out to thinkAside from the lower monthly cost, what other factors would affect your decisionabout whether to rent or buy in Example 5?

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By the WayThe portion of FICAgoing to Social Securityis called OASDI (OldAge, Survivors, and Dis-ability Insurance). Theportion going toMedicare is called HI(Hospital Insurance).

By the WayWhen the portion ofFICA taxes paid byemployers (and by theself-employed) is takeninto account, mostAmericans pay more inFICA tax than in ordinaryincome tax. Ordinaryincome tax rates havebeen cut substantiallysince 2001. FICA rateshave not changed.

• 7.65% on the first $94,200 of income from wages

• 1.45% on any income from wages in excess of $94,200

In addition, the individual’s employer is required to pay matching amounts of FICAtaxes.

Individuals who are self-employed must pay both the employee and the employershares of FICA. Thus, the rates for self-employed individuals are double the rates paidby individuals who are not self-employed.

FICA is calculated on all wages, tips, and self-employment income. You may notsubtract any adjustments, exemptions, or deductions when calculating FICA taxes.

❉EXAMPLE 7 FICA TaxesIn 2006, Jude earned $22,000 in wages and tips from her job waiting tables. Calculateher FICA taxes and her total tax bill including marginal taxes. What is her overall taxrate on her gross income, including both FICA and income taxes? Assume she is sin-gle and takes the standard deduction.

SOLUTION Jude’s entire income of $22,000 is subject to the 7.65% FICA tax:

Now we must find her income tax. We get her taxable income by subtracting her$3300 personal exemption and $5150 standard deduction:

From Table 4.10, her income tax is 10% on the first $7550 of her taxable income and15% on the remaining amount of Thus, her income tax is

Her total tax, including both FICAand income tax, is

Her overall tax rate, including both FICA and income tax, is

Jude’s overall tax rate is 15.2%. Note that she pays slightly more in FICA tax than inincome tax. Now try Exercises 57–62.

Dividends and Capital GainsNot all income is created equal, at least not in the eyes of the tax collector! In particu-lar, dividends (on stocks) and capital gains—profits from the sale of stock or otherproperty—get special tax treatment. Capital gains are divided into two subcategories.Short-term capital gains are profits on items sold within 12 months of their pur-chase, and long-term capital gains are profits on items held for more than 12 monthsbefore being sold.

➽

total taxgross income

5$3338

$22,0005 0.152

total tax 5 $1683 1 $1655 5 $3338

A10% 3 $7550 B 1 A15% 3 $6000 B 5 $1655.$13,550 2 $7550 5 $6000.

taxable income 5 $22,000 2 $3300 2 $5150 5 $13,550

FICA tax 5 7.65% 3 $22,000 5 $1683

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4E Income Taxes 297

By the WayThe rationale behind alower tax on capitalgains is that it encour-ages investment in newbusinesses and productsthat involve risk on thepart of the investor.

Long-term capital gains and most dividends are taxed at lower rates than otherincome such as wages and interest earnings. As of 2006, the rates were

• a maximum of 5% for income in the 10% and 15% tax brackets

• a maximum of 15% for income in all higher tax brackets

In a few cases, capital gains get even better tax treatment. For example, capital gainson the sale of your home are often tax exempt.

❉EXAMPLE 8 Dividend and Capital Gains IncomeIn 2006, Serena was single and lived off an inheritance. Her gross income consistedsolely of $90,000 in dividends and long-term capital gains. She had no adjustments toher gross income, but had $12,000 in itemized deductions and a personal exemptionof $3300. How much tax does she owe? What is her overall tax rate?

SOLUTION She owes no FICA tax because her income is not from wages. She hadno adjustments to her gross income, so we find her taxable income by subtracting heritemized deductions and personal exemption:

Because her income is all dividends and long-term capital gains, she pays tax at thespecial rates for these types of income. The special 5% rate for dividends and long-term capital gains applies to the income on which she would have been taxed at 10%or 15% if it had been ordinary income. From Table 4.10, therefore, we see that this5% rate applies to her first $30,650 of income. The rest of her income is taxed at thespecial 15% rate. Thus, her total tax is

Her overall tax rate is

Serena’s overall tax rate is 9.0%. Now try Exercises 63–64. ➽

total taxgross income

5$8140

$90,0005 0.090

('''''''''')'''''''''''*

15% capital gains rate(''''')'''''*

5% capital gains rate

A5% 3 $30,650 B 1 A15% 3 3$74,700 2 $30,650 4 B 5 $1532.50 1 $6607.50 5 $8140

taxable income 5 $90,000 2 $12,000 2 $3300 5 $74,700

Time out to thinkNote that Serena in Example 8 had a gross income more than quadruple that ofJude in Example 7. Compare their tax payments and overall tax rates. Who paysmore tax? Who pays at a higher tax rate? Explain.

Tax-Deferred IncomeThe tax code tries to encourage long-term savings by allowing you to defer incometaxes on contributions to certain types of savings plans, called tax-deferred savingsplans. Money that you deposit into such savings plans is not taxed now. Instead, it willbe taxed in the future when you withdraw the money.

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By the WayWith tax-deferred sav-ings, you will eventuallypay tax on the moneywhen you withdraw it.With tax-exempt invest-ments, you never haveto pay tax on the earn-ings. Some governmentbonds are tax-exempt.A Roth IRA is a specialtype of individual retire-ment account in whichyou pay taxes on moneyyou deposit now, but allearnings on the accountare tax-exempt whenyou withdraw them.

$0

$50,000

$100,000

$150,000

$200,000

Valu

e o

f in

vest

men

ts

Years

Taxable vs. tax-deferred savings plan

$250,000

$300,000

$350,000

5 10 15 20 25 30

Chart assumes• $2000 invested per year,• 10% APR, and• 31% marginal tax rate.

Tax deferred

Taxable

FIGURE 4.10 This graph compares the values of a tax-deferred savings plan and an ordinary savingsplan, assuming that tax on the interest is paid from the plan in the latter case. After 30years, the tax-deferred savings plan is worth over $100,000 more than the ordinary plan.

Tax-deferred savings plans go by a variety of names, such as individual retirementaccounts (IRAs), qualified retirement plans (QRPs), 401(k) plans, and more. All are sub-ject to strict rules. For example, you generally are not allowed to withdraw moneyfrom any of these plans until you reach age Anyone can set up a tax-deferred sav-ings plan, and you should, regardless of your current age. Why? Because they offer twokey advantages in saving for your long-term future.

First, contributions to tax-deferred savings plans count as adjustments to your pres-ent gross income and are not part of your taxable income. As a result, the contribu-tions cost you less than contributions to savings plans without special tax treatment.For example, suppose you are in the 28% marginal tax bracket. If you deposit $100 inan ordinary savings account, your tax bill is unchanged and you have $100 less tospend on other things. But if you deposit $100 in a tax-deferred savings account, youdo not have to pay tax on that $100. With your 28% marginal rate, you therefore save$28 in taxes. Thus, the amount you have to spend on other things decreases by only

The second advantage of tax-deferred savings plans is that their earnings are alsotax deferred. With an ordinary savings plan, you must pay taxes on the earnings eachyear, which effectively reduces your earnings. With a tax-deferred savings plan, all ofthe earnings accumulate from one year to the next. Over many years, this tax savingmakes the value of tax-deferred savings accounts rise much more quickly than that ofordinary savings accounts (Figure 4.10).

$100 2 $28 5 $72.

59 12 .

❉EXAMPLE 9 Tax-Deferred Savings PlanSuppose you are single, have a taxable income of $65,000, and make monthly pay-ments of $500 to a tax-deferred savings plan. How do the tax-deferred contributionsaffect your monthly take-home pay?

SOLUTION Table 4.10 shows that your marginal tax rate is 25%. Each $500 contri-bution to a tax-deferred savings plan therefore reduces your tax bill by

25% 3 $500 5 $125

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In other words, $500 goes into your tax-deferred savings account each month, butyour monthly paychecks go down by only The special taxtreatment makes it significantly easier for you to afford the monthly contributionsneeded to build your retirement fund. (The monthly saving found here is youraverage monthly saving for the year, after any tax refund or tax bill. It will be your pre-cise monthly saving only if your withholding is computed so that you have zero taxdue at year end.) Now try Exercises 65–68. ➽

$500 2 $125 5 $375.


1. The total amount of income you receive is called your

a. gross income.

b. net income.

c. taxable income.

2. If your taxable income puts you in the 25% marginal taxbracket,

a. your tax is 25% of your taxable income.

b. your tax is 25% of your gross income.

c. your tax is 25% of only a portion of your income; therest is taxed at a lower rate.

3. Suppose you are in the 25% marginal tax bracket. Then atax credit of $1000 will reduce your tax bill by

a. $1000. b. $150. c. $500.

4. Suppose you are in the 15% marginal tax bracket and earn$25,000. Then a tax deduction of $1000 will reduce your taxbill by

a. $1000. b. $150. c. $500.

5. Suppose that in the past year your only deductible expenseswere $4000 in mortgage interest and $2000 in charitablecontributions. If you are entitled to a standard deduction of$5150, then the total deduction you can claim is

a. $5150. b. $6000. c. $11,150.

6. Assume you are in the 25% tax bracket and you are enti-tled to a standard deduction of $5150. If you have no otherdeductible expenses, by how much will a $1000 charitablecontribution reduce your tax bill?

a. $0 b. $250 c. $1000

7. What is the FICA tax?

a. a tax on investment income

b. another name for the marginal tax rate system

c. a tax collected primarily to fund Social Security andMedicare

8. Based on the FICA rates for 2006, which of the followingpeople pays the highest percentage of his or her income inFICA taxes?

a. Joe, whose income consists of $12,000 from his job atBurger Joint

b. Kim, whose income is $150,000 in wages from her job asan aeronautical engineer

c. David, whose income is $1,000,000 in capital gains frominvestments

9. Jerome, Jenny, and Jacqueline all have the same taxableincome, but Jerome’s income is entirely from wages at hisjob, Jenny’s income is a combination of wages and short-term capital gains, and Jacqueline’s income is all from divi-dends and long-term capital gains. If you count bothincome taxes and FICA, how do their tax bills compare?

a. They all pay the same amount in taxes.

b. Jerome pays the most, Jenny the second most, andJacqueline the least.

c. Jacqueline pays the most, Jenny the second most, andJerome the least.

10. When you place money into a tax-deferred retirementplan,

a. you never have to pay tax on this money.

b. you pay tax on this money now, but not when you with-draw it later.

c. you do not pay tax on this money now, but you pay taxon money you withdraw from the plan later.

EXERCISES 4E

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REVIEW QUESTIONS11. Explain the basic process of calculating income taxes, as

shown in Figure 4.9. What is the difference between grossincome, adjusted gross income, and taxable income?

12. What is meant by filing status? How does it affect tax cal-culations?

13. What are exemptions and deductions? How should youchoose between taking the standard deduction and itemiz-ing deductions?

14. What is meant by a progressive income tax? Explain theuse of marginal tax rates in calculating taxes. What ismeant by a tax bracket?

15. What is the difference between a tax deduction and a taxcredit? Why is a tax credit more valuable?

16. Explain how a deduction, such as the mortgage interest taxdeduction, can save you money. Why do deductions bene-fit people in different tax brackets differently?

17. What are FICA taxes? What type of income is subject toFICA taxes?

18. How are dividends and capital gains treated differentlythan other income by the tax code?

19. Explain how you can benefit from a tax-deferred savingsplan.

20. Why do tax-deferred savings plans tend to grow faster thanordinary savings plans?


21. We’re both single with no children and we both have thesame total (gross) income, so we must both pay the sameamount in taxes.

22. The $1000 child tax credit sounds like a good idea, but itdoesn’t help me because I take the standard deductionrather than itemized deductions.

23. When I calculated carefully, I found that it was cheaperfor me to buy a house than to continue renting, eventhough my rent payments were lower than my new mort-gage payments.

24. My husband and I paid $12,000 in mortgage interest thisyear, but we didn’t get any tax benefit from it.

25. Bob and Sue were planning to get married in December ofthis year, but they postponed their wedding until Januarywhen they found it would save them money in taxes.

26. The top marginal tax rate may be 35%, but I never paymore than 15% because I live off the dividends from myinheritance.

27. I didn’t owe any ordinary income tax because my business(self-employed) made only a $7000 profit, but my total taxbill still came to 15.3% of my income.

28. I started contributing $400 each month to my tax-deferredsavings plan, but my take-home pay declined by only $300.

BASIC SKILLS & CONCEPTSIncome on Tax Forms. For each situation described in Exer-cises 29–32, find the person’s gross income, adjusted grossincome, and taxable income.

29. Antonio earned wages of $47,200, received $2400 in inter-est from a savings account, and contributed $3500 to a tax-deferred retirement plan. He was entitled to a personalexemption of $3300 and had deductions totaling $5150.

30. Marie earned wages of $28,400, received $95 in interestfrom a savings account, and was entitled to a personalexemption of $3300 and a standard deduction of $5150.

31. Isabella earned wages of $88,750, received $4900 in inter-est from a savings account, and contributed $6200 to a tax-deferred retirement plan. She was entitled to a personalexemption of $3300 and had deductions totaling $9050.

32. Lebron earned wages of $3,452,000, received $54,200 ininterest from savings, and contributed $30,000 to a tax-deferred retirement plan. He was not allowed to claim apersonal exemption (because of his high income) but wasallowed deductions totaling $674,500.

Should You Itemize? In Exercises 33–34, decide whether youshould itemize your deductions or claim the standard deduction.Explain your reasoning.

33. Your deductible expenditures are $8600 for interest on ahome mortgage, $2700 for contributions to charity, and$645 for state income taxes. Your filing status entitles youto a standard deduction of $10,300.

34. Your deductible expenditures are $3700 for contributionsto charity and $760 for state income taxes. Your filing sta-tus entitles you to a standard deduction of $5150.

Income Calculations. In Exercises 35–38, compute the indi-vidual’s (or couple’s) gross income, adjusted gross income, andtaxable income. Use the 2006 values for exemptions and stan-dard deductions in Table 4.10. Be sure to explain how you

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4E Income Taxes 301

decide to claim standard or itemized deductions. (Note: Do yourcalculations based only on the given data, which may not includeall credits and deductions.)

35. Suzanne is single and earned wages of $33,200. Shereceived $350 in interest from a savings account. Shecontributed $500 to a tax-deferred retirement plan. Shehad $450 in itemized deductions from charitablecontributions.

36. Malcolm is single and earned wages of $23,700. He had$4500 in itemized deductions from interest on a housemortgage.

37. Wanda is married, but she and her husband filed sepa-rately. Her salary was $35,400, and she earned $500 ininterest. She had $1500 in itemized deductions andclaimed three exemptions for herself and two children.

38. Emily and Juan are married and filed jointly. Their com-bined wages were $75,300. They earned $2000 from arental property they own, and they received $1650 ininterest. They claimed four exemptions for themselves andtwo children. They contributed $3240 to their tax-deferred retirement plans, and their itemized deductionstotaled $9610.

Marginal Tax Calculations. In Exercises 39–46, use the 2006marginal tax rates in Table 4.10 to compute the tax owed.

39. Gene is single and had a taxable income of $35,400.

40. Sarah and Marco are married filing jointly with a taxableincome of $87,500.

41. Bobbi is married filing separately with a taxable income of$77,300.

42. Abraham is single with a taxable income of $23,800.

43. Paul is a head of household with a taxable income of$89,300. He is entitled to a $1000 tax credit.

44. Pat is a head of household with a taxable income of$57,000. She is entitled to a $1000 tax credit.

45. Winona and Jim are married filing jointly with a taxableincome of $105,500. They also are entitled to a $2000 taxcredit.

46. Chris is married filing separately with a taxable income of$127,300.

Tax Credits and Tax Deductions. In Exercises 47–52, statehow much each individual or couple will save in taxes with thetax credit or tax deduction specified.

47. Midori and Tremaine are in the 28% tax bracket and claimthe standard deduction. How much will their tax bill bereduced if they qualify for a $500 tax credit?

48. Vanessa is in the 35% tax bracket and itemizes her deduc-tions. How much will her tax bill be reduced if she quali-fies for a $500 tax credit?

49. Rosa is in the 15% tax bracket and claims the standarddeduction. How much will her tax bill be reduced if shemakes a $1000 contribution to charity?

50. Shiro is in the 15% tax bracket and itemizes his deduc-tions. How much will his tax bill be reduced if he makes a$1000 contribution to charity?

51. Sebastian is in the 28% tax bracket and itemizes his deduc-tions. How much will his tax bill be reduced if he makes a$1000 contribution to charity?

52. Santana is in the 35% tax bracket and itemizes her deduc-tions. How much will her tax bill be reduced if she makes a$1000 contribution to charity?

Rent or Own? Exercises 53–54 state a tax bracket, an apart-ment rent, and a house payment, along with the average amountgoing toward interest in the first year. Including savings throughthe mortgage interest deduction, determine whether renting orbuying is cheaper (in terms of monthly payments) during thefirst year. Assume you are itemizing deductions in all cases.

53. You are in the 33% tax bracket. The apartment rents for$1600 per month. Your monthly mortgage paymentswould be $2000, of which an average of $1800 per monthgoes toward interest during the first year.

54. You are in the 15% tax bracket. The apartment rents for$600 per month. Your monthly mortgage payments wouldbe $675, of which an average of $600 per month goestoward interest during the first year.

55. Varying Value of Deductions. Maria is in the 33% taxbracket. Steve is in the 15% tax bracket. They each item-ize their deductions and pay $10,000 in mortgage interestduring the year. Compare their true costs for mortgageinterest.

56. Varying Value of Deductions. Yolanna is in the 35% taxbracket. Alia is in the 10% tax bracket. They each itemizetheir deductions, and they each donate $4000 to charity.Compare their true costs for charitable donations.

FICA Taxes. Exercises 57–62 each describe a person’s income.In each case, calculate the person’s FICA taxes and total tax bill,including marginal income taxes. Then find the person’s overalltax rate on his or her gross income, including both FICA andincome taxes. Assume all individuals are single and take the stan-dard deduction. Use the 2006 tax rates in Table 4.10. (Round taxcalculations to the nearest dollar.)

57. Luis earned $28,000 from wages as a computer program-mer and made $2500 in tax-deferred contributions to aretirement fund.

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58. Carla earned $34,500 in salary and $750 in interest andmade $3000 in tax-deferred contributions to a retirementfund.

59. Jack earned $44,800 in salary and $1250 in interest andmade $2000 in tax-deferred contributions to a retirementfund.

60. Alejandro earned $102,400 in salary and $4450 in interestand made $9500 in tax-deferred contributions to a retire-ment fund.

61. Brittany earned $48,200 in wages and tips. She had no otherincome and made no contributions to retirement plans.

62. Larae earned $21,200 in wages and tips. She had no otherincome and made no contributions to retirement plans.

Dividends and Capital Gains. In Exercises 63–64, calculatethe total tax owed by each of the two people, including bothFICA and income taxes. Compare their overall tax rates, includ-ing both FICA and income taxes. Assume all individuals are sin-gle and take the standard deduction. Use the 2006 tax rates inTable 4.10 for ordinary income and the special rates for divi-dends and capital gains listed in the text.

63. Pierre earned $120,000 in wages. Katarina earned$120,000, all in dividends and long-term capital gains.

64. Deion earned $60,000 in wages. Josephina earned $60,000,all in dividends and long-term capital gains.

Tax-Deferred Savings Plans. In Exercises 65–68, calculate theeffect on monthly take-home pay of the tax-deferred contribu-tions described. Use the 2006 tax rates in Table 4.10.

65. You are single and have a taxable income of $18,000. Youmake monthly contributions of $400 to a tax-deferred sav-ings plan.

66. You are single and have a taxable income of $45,000. Youmake monthly contributions of $600 to a tax-deferred sav-ings plan.

67. You are married filing jointly and have a taxable income of$90,000. You make monthly contributions of $800 to a tax-deferred savings plan.

68. You are married filing jointly and have a taxable income of$200,000. You make monthly contributions of $800 to atax-deferred savings plan.

FURTHER APPLICATIONSMarriage Penalty. Exercises 69–72 give the adjusted grossincomes of a couple that is engaged to be married. Calculate thetax owed by the couple in two ways: (1) if they delay their mar-riage until next year so that they can each file a tax return at the

single tax rate this year and (2) if they marry before the end ofthe year and file a joint return. Assume that each person takesone exemption and the standard deduction. Use the 2006 taxrates in Table 4.10. Does the couple face a “marriage penalty” ifthey marry before the end of the year? Explain. (Note: Marriedrates apply for the entire year, no matter when during a year youare married.)

69. Gabriella and Roberto have adjusted gross incomes of$44,500 and $33,400, respectively.

70. Joan and Paul have adjusted gross incomes of $32,500 and$29,400, respectively.

71. Mia and Steve each have an adjusted gross income of$185,000.

72. Lisa has an adjusted gross income of $85,000, and Patrickis a student with no income.

73. Estimating Your Taxes. List all the gross income youexpect for the coming year, along with any expenses youare entitled to deduct from gross income. Then calculateyour adjusted gross income and taxable income.a. Based on your estimates, how much tax will you owe this

year? Use the 2006 tax rates in Table 4.10, or findupdated rates on the Web.

b. How much (if any) tax is being withheld from your pay-checks each month? Should you expect a tax refund nextyear? Explain.

c. Suppose you begin making a $100 monthly contributionto a tax-deferred retirement plan. How will it affect yourtake-home pay? Explain.

d. Suppose you make a $1000 contribution to charity. Byhow much, if at all, will this contribution reduce yourtax bill? Explain.


74. Tax Simplification Plans. Use the Web to investigate arecent proposal to simplify federal tax laws and filing pro-cedures. What are the advantages and disadvantages of thesimplification plan, and who supports it?

75. Fairness Issues. Choose a tax question that has issues offairness associated with it (for example, capital gains rates,the marriage penalty, or the alternative minimum tax[AMT]). Use the Web to investigate the current status ofthis question. Have new laws been passed that affect it?What are the advantages and disadvantages of recent orproposed changes, and who supports the changes? Summa-rize your own opinion about whether current tax law isunfair and, if so, what should be done about it.

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4F Understanding the Federal Budget 303

76. The Digital Daily. The electronic news publication ofthe Internal Revenue Service is the Digital Daily. Visit theWeb site for the publication. Choose a current “frontpage” issue and report on it in terms of how it affects youas a taxpayer.

77. Current Tax Rates. Use the IRS Web site to find thecurrent tax rates. Recast Table 4.10 with these tax rates.

IN THE NEWS78. Alternative Minimum Tax (AMT). The calculations

described in this unit all assume that a person pays taxesaccording to the “normal” tax code. However, some peoplewill be subject to the alternative minimum tax (AMT) incoming years. The AMT calculates taxes in a very differentway, making taxes higher for people who pay it. The AMT

is becoming a hot political issue, because more and morepeople are expected to owe it over the next few years. Finda recent article about the impact of the AMT. Write ashort report on what you learn.

79. Tax Changes. Find a recent news article about proposedchanges to federal tax laws. Briefly describe the proposedchanges and their impact. What parties support andoppose the changes?

80. Your Tax Return. Briefly describe your own experienceswith filing a federal income tax return. Do you file yourown returns? If so, do you use a computer software pack-age or a professional tax advisor? Will you change your fil-ing method in the future? Why or why not?

UNIT 4F Understanding the Federal Budget

So far in this chapter, we have discussed issues of financial management that affect usdirectly as individuals. But we are also affected by the way our government managesits finances. In this unit, we will discuss a few of the basic concepts needed to under-stand the federal budget.

Federal Budget BasicsIn theory, the federal budget works much like your personal budget (Unit 4A) or thebudget of a small business. All have receipts, or income, and outlays, or expenses.Net income is the difference between receipts and outlays. When receipts exceedoutlays, net income is positive and the budget has a surplus (profits). When outlaysexceed receipts, net income is negative and the budget has a deficit (losses).

There can be no free-dom or beauty abouta home life thatdepends on borrow-ing and debt.

—HENRIK IBSEN, 1879

DEFINITIONS

Receipts, or income, represent money that has been collected.

Outlays, or expenses, represent money that has been spent.

If net income is positive, the budget has a surplus.

If net income is negative, the budget has a deficit.

Net income 5 receipts 2 outlays

Note that a deficit means spending more money than was collected. The only wayyou (or a business or government) can survive a deficit is by spending savings or bor-rowing money. When you borrow, you accumulate a debt. Every year that you bor-row to cover a deficit, your debt grows. In addition, the lender will surely charge

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DEBT VERSUS DEFICIT

A deficit represents money that is borrowed (or taken from savings) during asingle year.

The debt is the total amount of money owed to lenders, which may result fromaccumulating deficits over many years.

�300,000

1955

1965

1975

1985

1995

2005

1955

1965

1975

1985

1995

2005

�400,000

�200,000

Def

icit

Surp

lus

Gross Federal Debt (millions of dollars)Surplus or Deficit (millions of dollars)

�100,000

0

100,000

200,000

300,000

400,000

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

(a) (b)FIGURE 4.11 (a) Annual deficits or surpluses since 1955. (b) Accumulated gross federal debt since 1955. In both cases,

the value for 2007 is an estimate (unshaded bar). Data are based on fiscal years, which end on September 30.Source: Budget of the United States Government, 2007.

A national debt, if it isnot excessive, will beto us a nationalblessing.

—ALEXANDER HAMILTON, 1781

interest on your debt. Thus, in addition to accumulating a debt, you will also facegrowing interest payments as your debt rises. In contrast, a surplus means collectingmore money than was spent. If you have a surplus, you can use it either to add to yoursavings or to reduce your debt.

For over half a century, the U.S. government has run a deficit almost every year,with the notable exception of the years 1998–2001. Figure 4.11a shows the deficitsand surpluses since 1955. Figure 4.11b shows the national debt that has accumulatedduring this period.

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❉EXAMPLE 1 Personal BudgetSuppose your gross income last year was $40,000. Your expenditures were as follows:$20,000 for rent and food, $2000 for interest on your credit cards and student loans,$6000 for car expenses, and $9000 for entertainment and miscellaneous expenses. Youalso paid $8000 in taxes. Did you have a deficit or a surplus?

SOLUTION The total of your outlays, including tax, was

Because your outlays were greater than your $40,000 income by $5000, your personalbudget had a $5000 deficit. Therefore, you must have either withdrawn $5000 fromsavings or borrowed $5000 to cover your expenditures. Now try Exercises 25–26.

❉EXAMPLE 2 The Federal DebtThe federal debt at the end of 2006 was nearly $9 trillion. If this debt were dividedevenly among the roughly 300 million citizens of the United States, how much wouldyou owe?

SOLUTION This question is easiest to answer by putting the numbers in scientificnotation. We divide the debt of $9 trillion by the 300 million population:

Your personal share of the total debt is roughly or $30,000.Now try Exercises 27–28. ➽

$3 3 104,

$9 3 1012

3 3 108 persons5 $3 3 104>person

A3 3 108 BA$9 3 1012 B

➽

$20,000 1 $2000 1 $6000 1 $9000 1 $8000 5 $45,000

Time out to thinkHow does your share of the national debt compare to personal debts that youowe? Explain.

A Small-Business AnalogyBefore we focus on the federal budget, let’s investigate the simpler books of an imagi-nary company with not-so-imaginary problems. Table 4.11 summarizes four years ofbudgets for the Wonderful Widget Company, which started with a clean slate at thebeginning of 2004.

The first column shows that, during 2004, the company had receipts of $854,000and total outlays of $1,000,000. Thus, the company’s net income was

The negative sign tells us that the company had a deficit of $146,000. The companyhad to borrow money to cover this deficit and ended the year with a debt of $146,000.The debt is shown as a negative number because it represents money owed to some-one else.

$854,000 2 $1,000,000 5 2$146,000

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In 2005, receipts increased to $908,000, while outlays increased to $1,082,000.These outlays included a $12,000 interest payment on the debt from the first year.Thus, the deficit for 2005 was

The company had to borrow $174,000 to cover this deficit. Further, it had no moneywith which to pay off the debt from 2004. Thus, the total debt at the end of 2005 was

Here is the key point: Because the company again failed to balance its budget in itssecond year, its total debt continued to grow. As a result, its interest payment in 2006increased to $26,000.

In 2007, the company’s owners decided to change strategy. They froze operatingexpenses and employee benefits (relative to 2006) and actually cut security expenses.However, the interest payment rose substantially because of the rising debt. Despitethe attempts to curtail outlays and despite another increase in receipts, the companystill ran a deficit in 2007 and the total debt continued to grow.

$146,000 1 $174,000 5 $320,000

$908,000 2 $1,082,000 5 2$174,000

❉EXAMPLE 3 Growing Interest PaymentsConsider Table 4.11 for the Wonderful Widget Company. Assume that the $47,000interest payment in 2007 was for the prior debt of $566,000. What was the annualinterest rate? If the interest rate remains the same, what will the payment be on the

Time out to thinkSuppose you were a loan officer for a bank in 2008, when the Wonderful WidgetCompany came asking for further loans to cover its increasing debt. Would youlend it the money? If so, would you attach any special conditions to the loan?Explain.

TABLE 4.11 Budget Summary for the Wonderful Widget Company (in thousands of dollars)

2004 2005 2006 2007

Total Receipts $854 $908 $950 $990

Outlays

Operating 525 550 600 600

Employee Benefits 200 220 250 250

Security 275 300 320 300

Interest on Debt

Total Outlays 1000 1082 1196 1197

Surplus/Deficit

Debt (accumulated) 2773256623202146

2207224621742146

4726120

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debt at the end of 2007? What will the payment be if the interest rate rises by 2 per-centage points?

SOLUTION Paying $47,000 interest on a debt of $566,000 means an interest rate of

The interest rate was 8.3%. At the end of 2007, the debt stands at $773,000. At thesame interest rate, the next interest payment will be

If the interest rate rises by 2 percentage points, to 10.3%, the next interest paymentwill be

A 2-percentage-point change in the interest rate increases the interest payment bymore than $15,000. Now try Exercises 29–30.

The Federal BudgetThe Widget Company example shows that a succession of deficits leads to a risingdebt. The increasing interest payments on that debt, in turn, make it even easier torun deficits in the future. The Widget Company story is a mild version of what hap-pened to the U.S. budget. Table 4.12 shows a summary of the federal budget in recentyears. Moreover, as the debt has risen, interest payments have increased. Low interestrates have helped ease this burden in recent years, but interest payments still make upclose to 10% of federal outlays. For example, interest on the debt cost the govern-ment about $220 billion in 2006—more than double what the government spent onall education, training, and social services combined, and nearly 15 times as much as itspent for NASA.

The future of the federal budget is notoriously difficult to predict. Over the pastcouple decades, each year’s budget projections for the following year have been offby an average of about 11%. Projections more than one year out have been even fur-ther off.

➽

0.103 3 $773,000 5 $79,619

0.083 3 $773,000 5 $64,159

$47,000$566,000

5 0.083

TABLE 4.12 U.S. Federal Budget Summary, 1999–2006 (all amounts in billions of dollars)

1999 2000 2001 2002 2003 2004 2005 2006

Total Receipts $1827 $2025 $1991 $1853 $1783 $1880 $2153 $2407

Total Outlays 1703 1789 1863 2011 2160 2293 2472 2654

Net Income 124 236 128

Source: United States Office of Management and Budget.

22482319241323772158

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❉EXAMPLE 4 Budget ProjectionsAs of 2006, the government projected total receipts of $2590 billion ($2.590 trillion)and a deficit of $223 billion in 2008. How would net income change if the projectionof receipts turned out to be too high by 11%? How would it change if the projectionof receipts were too low by 11%? Assume that outlays are unchanged.

SOLUTION An error of 11% of the projected receipts of $2590 billion is

(rounded to nearest $1 billion)

Thus, if receipts were 11% lower than expected, net income would be $285 billionless than projected, thereby increasing the deficit from its projected $223 billion to

That is, the deficit would be more thandouble the projection. On the other hand, if receipts were 11% higher than expected,net income would be $285 billion higher than projected, turning the projected$223 billion deficit into a $62 billion surplus. Now try Exercises 31–32. ➽

$223 billion 1 $285 billion 5 $508 billion.

0.11 3 $2590 billion 5 $285 billion

By the WayThe government alsocollects revenues from afew “business-like” activi-ties, such as chargingentrance fees atnational parks. However,for historical reasons,these revenues are sub-tracted from outlaysinstead of being addedto receipts when thegovernment publishes itsbudget. Although thismethod of accountingmay seem odd, it doesnot affect overall calcu-lations of the surplus ordeficit.

Time out to thinkDo you think it is wise to base long-term spending or taxing plans on long-termbudget projections? Why or why not?

Federal Government ReceiptsTo understand the federal budget more deeply, we need to understand how the gov-ernment gets its receipts and how it spends its outlays. Figure 4.12 shows the basicmakeup of government receipts as of 2006. The categories are

• Individual income taxes, as we discussed in Unit 4E

• Corporate income taxes, which are income taxes paid by businesses

• Social insurance taxes, which primarily represent FICA taxes (seeUnit 4E) for Social Security and Medicare but also include paymentsinto retirement plans by federal employees and taxes for unemploymentinsurance

• Excise taxes, which include taxes on alcohol, tobacco, gasoline, andother products

• Other, which includes such things as gift taxes and fines collected bythe government

Note that most of the receipts currently come from income taxes. How-ever, social insurance taxes are expected to represent a rising share of totalreceipts in the future.

Federal Government OutlaysFigure 4.13 shows the basic makeup of government outlays as of 2006. For purposesof projecting budgets, the government generally groups spending into two majorareas.

Corporateincome

taxes

Other

Excisetaxes

Social Security,Medicare, and other

social insurancereceipts

37%

Individualincome

taxes44%

12%

4%3%

FIGURE 4.12 Approximate makeup of fed-eral government receipts, 2006.Source: United States Office of Managementand Budget.

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• Mandatory outlays are expenses that will be paid auto-matically unless Congress acts to change them. Most ofthe mandatory outlays are for “entitlements” such asSocial Security, Medicare, and other payments to indi-viduals. (They are called entitlements because the lawspecifically states the conditions under which individualsare entitled to them.) Interest on the debt is also amandatory outlay, because it must be paid to prevent thegovernment from being in default on its loans.

• Discretionary outlays are decided on a year-to-yearbasis. The amounts for discretionary programs must beapproved by Congress in authorization bills, which thenmust be signed by the President to become law. Discre-tionary outlays are subdivided into programs for defense(military and homeland security) and non-defense. Non-defense discretionary outlays include everything except themandatory outlays and defense. For example, non-defense discretionary outlays include education, trans-portation, housing, international aid, the space program,and scientific research.

❉EXAMPLE 5 Discretionary SqueezeThe portion of the budget going to Social Security is expected to grow as more peo-ple retire in coming decades. Suppose that Social Security rises to 30% of total out-lays while all other programs except non-defense discretionary spending hold steadyat the proportions shown in Figure 4.13. As a percentage of total outlays, how muchwould non-defense discretionary spending have to decrease to cover the increase inSocial Security? Comment on how this scenario would affect Congress’s power tocontrol the surplus or deficit.

SOLUTION Figure 4.13 shows that 21% of outlays currently go to Social Security,so a rise to 30% would be a rise of 9 percentage points. Thus, the proportion ofspending for all other programs would have to drop by 9 percentage points for thetotal to remain 100%. If this drop came entirely from non-defense discretionaryspending, non-defense discretionary spending would fall from 18% to 9% of totaloutlays.

If non-defense discretionary spending were only 9% of total outlays, Congresswould lose much of its power to control surpluses or deficits. Here’s why: First,remember that Congress authorizes only discretionary spending (as opposed tomandatory spending) on a year-to-year basis. In essence, this is the only portion ofthe budget that Congress can easily control. Second, 9% is smaller than the averageerror in budget projections (see Example 4). Thus, the proportion of the budget thatCongress can easily control would be smaller than the uncertainty that Congressmust deal with in making a budget. Clearly, this would make it nearly impossible forCongress to predict a surplus or deficit accurately.


By the WayIf you’ve ever paid SocialSecurity taxes, then youhave your own privateSocial Security account.The Social SecurityAdministration automati-cally sends annual state-ments to wage earnersage 25 or older.Yourstatement should arriveabout 3 months beforeyour birthday. If youdon’t receive an auto-matic statement,youcan request a statementfrom the Social SecurityAdministration Web site.You should check yourstatement carefully, tomake sure that yourSocial Security taxeshave been properlycredited to youraccount.

SocialSecurity

21% 20%

18%

20%8%

13%

Defense andHomeland Security

Non-DefenseDiscretionary

Medicaid, GovernmentPensions, and OtherMandatory Spending

Intereston Debt

Medicare

FIGURE 4.13 Approximate makeup of federalgovernment outlays, 2006. All categories except“Defense and Homeland Security” and “Non-Defense Discretionary” are considered mandatory.Source: United States Office of Management andBudget.

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Strange Numbers: Publicly Held and Gross DebtTake another look at Figure 4.11 and you may notice something rather strange: Evenin the years when the government ran a surplus (1998–2001), the debt still continuedto increase. Why did the debt keep rising, even when the government collected moremoney than it spent? More generally, why does the debt tend to rise from one year tothe next by more than the amount of the deficit for the year? To answer these ques-tions, we must investigate government accounting in a little more detail.

Financing the DebtRemember that whenever you run a deficit, you must cover it either by withdrawingfrom savings or by borrowing money. The federal government does both. It with-draws money from its “savings,” and it borrows money from people and institutionswilling to lend to it.

Let’s consider borrowing first. The government borrows money by selling Trea-sury bills, notes, and bonds (see Unit 4C) to the public. If you buy one of these Trea-sury issues, you are effectively lending the government money that it promises to payback with interest. Because Treasury issues are considered to be very safe investments,the government has never had trouble finding people or institutions willing to buythem. By the end of 2006, the government had borrowed a total of about $5 trillionthrough the sale of Treasury issues. This debt, which the government must eventuallypay back to those who hold the Treasury issues, is called the publicly held debt(sometimes called the net debt or the marketable debt). Nearly half of this debt is cur-rently held by foreign individuals and banks, with China as the largest holder of U.S.securities.

The government’s “savings” consist of special accounts designed to meet futureobligations. These accounts are called trust funds. The biggest trust fund by far is forSocial Security, which is primarily a retirement program. People “invest” by payingSocial Security taxes (most of the FICA taxes; see Unit 4E) and then collect SocialSecurity benefits after they retire.

Currently, the government is collecting much more in Social Security taxes than itis paying out in Social Security benefits (see Figures 4.12 and 4.13). This reflects thefact that, today, many more people are working and paying Social Security taxes thanare collecting benefits. However, as today’s workers retire, the government will haveto pay more and more in Social Security benefits. Therefore, to make sure there isenough money to pay future Social Security benefits, the government should investthe excess Social Security taxes that it collects today.

Legally, the government must invest the excess Social Security money in the SocialSecurity trust fund. It does the same for several other trust funds, including those forthe pensions of government workers. In a sense, these trust funds are like the govern-ment’s savings accounts. But there’s a catch: Before the government borrows from thepublic to finance a deficit, it first tries to cover the deficit by borrowing from its owntrust funds.

In fact, the government has to date borrowed every penny it ever deposited intothese trust funds. Thus, there is no money in any of the trust funds, including Social Secu-rity. Instead, the trust fund is filled with the equivalent of a stack of IOUs (more tech-nically, with Treasury bills), in which the government has promised to return themoney it borrowed, with interest.

The trust fund moreaccurately repre-sents a stack of IOUsto be presented tofuture generations forpayment, rather thana build-up ofresources to fundfuture benefits.

—JOHN HAMBOR, FORMER

RESEARCH DIRECTOR FOR THE

SOCIAL SECURITY ADMINISTRATION

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As of the end of 2006, the government’s debt to its own trust funds was approach-ing $4 trillion. Adding this amount to the publicly held debt of $5 trillion, we get agross debt of almost $9 trillion. This is the total debt shown in Figure 4.11b, and itrepresents the total amount that the government is eventually obligated to repay fromother government receipts (that is, receipts besides those collected for Social Securityand other trust funds).

On-Budget and Off-Budget: Effects of Social SecurityAs an example of how trust funds affect the two kinds of debt, consider 2001—whenthe federal government ran a $128 billion surplus (see Table 4.12). The governmentused this surplus money to buy back some of the Treasury notes and bonds it had soldto the public, which reduced the publicly held debt.

However, remember that the government also collected excess Social Security taxes,which legally had to be deposited in the Social Security trust fund. In addition, the gov-ernment owed the trust fund interest for all the money it had borrowed from the trustfund in the past. When we add both the excess Social Security taxes and the owed inter-est, it turns out that the government was supposed to deposit $161 billion in the SocialSecurity trust fund in 2001. But the government had already spent the $161 billion,leaving no cash available to deposit in the trust fund. The government therefore“deposited” $161 billion worth of IOUs in the trust fund, adding to the stack of IOUsalready there from the past. Because IOUs represent loans, the government effectivelyborrowed $161 billion from the Social Security trust fund. When we subtract this bor-rowed amount from the $128 billion surplus, the government’s income for 2001becomes

With Social Security counted, the $128 billion surplus turns into a $33 billion deficit!In government-speak, Social Security is said to be off-budget. Because the govern-

ment really did collect $128 billion more than it spent, this number is called the unifiednet income. The on-budget net income is what remains after we subtract the portionof the unified net income that came from Social Security. It represents the amount bywhich the government overspent its revenue when Social Security is included.

Although Social Security is the only major expenditure that is legally consideredoff-budget, other trust funds also represent future repayment obligations. Because thegovernment borrowed from all these other trust funds as well, the gross debt rose byconsiderably more than the $33 billion on-budget deficit. In fact, when all was saidand done, the gross debt rose by $141 billion during 2001. Despite the surplus, thedebt to be repaid in the future grew substantially.

$128 billion('')''*

unified net income

2 $161 billion(''')'''*

off-budget net income

5 2$33 billion(''')'''*

on-budget net income

By the WayIf you want completedetails of debt account-ing, you can downloadthe entire federalbudget (typically a cou-ple thousand pages)from the Web site for theUnited States Office ofManagement and Bud-get. The site also offersmany simplified sum-maries and other usefuldata.

TWO KINDS OF NATIONAL DEBT

The publicly held debt (or net debt) represents money the government mustrepay to individuals and institutions that bought Treasury issues.

The gross debt includes both the publicly held debt and money that the govern-ment owes to its own trust funds, such as the Social Security trust fund.

OF

FIC

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F

MANAGE M N

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GD

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DNATE

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By the WaySocial Security benefitsdiffer from private retire-ment benefits in at leasttwo major ways. First,Social Security benefitsare guaranteed. Privateretirement accountsmay rise or fall in value,thereby changing howmuch you can afford towithdraw during retire-ment, but Social Securitypromises a particularbenefit payment in anycircumstances. Second,Social Security benefitsare paid as long as youlive, but cannot bepassed on to your heirs.In contrast, privateretirement accountscan be passed onthrough your will.


❉EXAMPLE 6 On- and Off-BudgetThe federal government ran a $318 billion deficit (unified deficit) in 2005. However,this number does not separate the effects of excess Social Security taxes. For 2005, thegovernment collected $175 billion more in Social Security revenue than it paid out inSocial Security benefits. What do we call this excess $175 billion of Social Securityrevenue, and what happened to it? What was the government’s on-budget deficit for2005? Explain.

SOLUTION The $175 billion excess Social Security revenue represents what we callthe off-budget net income (a surplus) for 2005, because it is counted separately fromthe rest of the budget (that’s what makes it “off” budget). By law, this $175 billion hadto be added to the Social Security trust fund. Unfortunately, it had already been spent(on programs other than Social Security), so the government instead added $175 bil-lion worth of IOUs (Treasury bills) to the trust fund. Since this $175 billion worth ofIOUs will have to be repaid eventually, it should be included in the calculation ofwhat we call the on-budget deficit—the amount by which the government actuallyoverspent in 2005. That is, the on-budget net income for 2005 was

In summary, the unified deficit of $318 billion means the government’s total revenuefell $318 billion short of its outlays. But because the government added $175 billionto its long-term obligation to repay its own Social Security trust fund, the govern-ment’s future repayment obligations really rose by the on-budget deficit of $493 bil-lion. (In fact, because of the government’s other trust funds and other accountingdetails, the debt rose by even more than this amount.) Now try Exercises 39–40. ➽

2$318 billion('')''*

unified net income

2 $175 billion('')''*

off-budget net income

5 2$493 billion('')''*

on-budget net income

UNIFIED BUDGET, ON BUDGET, AND OFF BUDGET

The U.S. government’s unified budget represents all federal revenues and spend-ing. For accounting purposes, the government divides this unified budget into twoparts:

• The portion of the unified budget that is involved in Social Security (that is, rev-enue from Social Security tax and spending on Social Security benefits) is con-sidered off-budget.

• The rest of the unified budget (that is, everything that is not involved in SocialSecurity) is considered on-budget.

Thus, the following relationships hold for any surplus or deficit in the budget:

Or, equivalently,

unified net income 2 on-budget net income 5 off-budget net income

unified net income 5 on-budget net income 1 off-budget net income

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The Future of Social SecurityImagine that you decide to set up a retirement savings plan that will allow you toretire comfortably at age 65. Using the savings plan formula (see Unit 4C), you deter-mine that you can achieve your retirement goal by making monthly deposits of $250into your retirement plan. So you start the plan today by making your first $250deposit.

However, in the first month you see a new music system that looks really cool.Being short on cash, you decide to buy it by withdrawing the $250 that you had putinto your retirement plan. Because you don’t want to fall behind on your retirementsavings, you write yourself an IOU stating that you “owe” your retirement plan $250.

Next month, you again deposit $250 in your retirement plan—but soon withdrawit for a nice weekend getaway. As before, you write an IOU to remind yourself thatyou owe $250 to your retirement plan. Moreover, recognizing that you would haveearned interest on the previous month’s deposit if you hadn’t withdrawn it, you writeyourself an IOU for the lost interest.

Month after month and year after year, you continue in the same way. Because youalways spend the money you had planned to put in your retirement plan, you keepwriting yourself IOUs for the payments plus interest. When you finally reach age 65,your retirement plan contains IOUs that say you owe yourself enough money toretire on—but your retirement account contains no actual money. Obviously, it will be dif-ficult to live off the IOUs you wrote to yourself.

This method of “saving” for retirement may sound silly, but it essentially describesthe Social Security trust fund. Officially, the Social Security trust fund is growinglarger and larger because of excess Social Security taxes and interest on past IOUs.According to recent projections, its balance will grow to over $3 trillion by 2015. But,in reality, the trust fund contains no cash today and will contain no cash in 2015. Itwill just be filled with $3 trillion worth of IOUs from the government to itself.

Now comes the bad news. Sometime around or after 2015, the increasing numberof retirees will mean that Social Security payments will exceed the receipts fromSocial Security taxes. In order to pay benefits, the government will have to beginwithdrawing money from the trust fund. This means the government will somehowhave to start redeeming the IOUs that it has written to itself.

To see the problem vividly, consider the year 2040 when the “intermediate” projec-tions (meaning those that are neither especially optimistic nor especially pessimistic)say the Social Security trust fund will go bankrupt. By then, projected Social Securitypayments will be about $900 billion more than collections from Social Security taxes.The government will therefore have to redeem $900 billion in IOUs from the SocialSecurity trust fund, which means it will somehow have to find $900 billion in cold,hard cash. Generally speaking, the government could do this through some combina-tion of the following three options:

1. It could cut spending on discretionary programs, such as the military or educa-tion, in order to free up money to redeem the IOUs. Unfortunately, $900 bil-lion roughly equals the amount currently spent on all discretionary spending.Thus, the government would have to eliminate virtually all discretionaryspending—including eliminating the military—in order to cover the SocialSecurity payments.

Technical NoteProjections are madein current dollars, so itis not necessary toadjust projectednumbers for futureinflation.

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By the WaySocial Security is some-times called the “thirdrail” of politics. The termcomes from the NewYork City subways,where the trains run ontwo rails and the third railcarries electricity at veryhigh voltage. Touchingthe third rail generallycauses instant death.

Time out to thinkSome proposals for solving the Social Security problem call for converting part or allof the program to private savings accounts. This would have the advantage ofmaking sure that the government couldn’t keep borrowing from Social Security. Inaddition, private investments have historically grown at a faster rate than the trustfund interest the government pays itself. However, the fact that private accountscan also lose value makes them at least somewhat risky. What’s your opinion of pri-vatizing Social Security? Explain.

2. It could borrow the money from the public by issuing more Treasury notes andbonds. But the needed $900 billion would be larger than any single-year deficitin history.

3. It could raise taxes to collect extra cash.

You may notice that any of these three possibilities could have a dramatic impacton you. Programs such as education for your kids may be cut, or the economy will behurt by huge deficits, or you’ll be taxed much more heavily than you are today.Clearly, something must be done to solve this problem before it arrives. Unfortu-nately, the politics of Social Security makes a solution hard to come by. Worse yet,Medicare is expected to face a similar crisis, and this crisis may hit within a decade.

❉EXAMPLE 7 Tax IncreaseIn 2006, individual income taxes made up about 44% of total government receipts ofmore than $2.3 trillion. Suppose that the government needed to raise an additional$900 billion through individual income taxes. How much would taxes have to increase?Neglect any economic problems that the tax increase might cause.

SOLUTION To bring in 44% of the $2.3 trillion in receipts for 2006, individualincome taxes had to account for about $1.0 trillion in government revenue. Thus,raising an additional $900 billion ($0.9 trillion) would require an additional 90% inrevenue from individual income taxes. To generate an extra $900 billion, overallincome taxes would have to rise by 90%. Now try Exercises 41–42. ➽

EXERCISES 4F


1. In 2006, Bigprofit.com had $1 million more in outlaysthan in receipts, bringing the total amount it owed lendersto $7 million. We say that at the end of 2006 Bigprofit.comhad

a. a deficit of $7 million and a debt of $1 million.

b. a deficit of $1 million and a debt of $7 million.

c. a surplus of $1 million and a deficit of $7 million.

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2. If the U.S. government decided to pay off the federal debtby asking for an equal contribution from all U.S. citizens,you’d be asked to pay approximately

a. $300. b. $3000. c. $30,000.

3. Suppose the government predicts that for next year taxreceipts will be $2.5 trillion and net income will be

billion (a deficit). Based on historical errors in pre-dicting the budget for the following year, you can expectnext year’s actual net income to be

a. a deficit of between about $90 billion and $110 billion.

b. a deficit of between about $50 billion and $150 billion.

c. somewhere between a deficit of $350 billion and a sur-plus of $150 billion.

4. In terms of the U.S. budget, what do we mean bydiscretionary outlays?

a. money that the government spends on things that aren’treally important

b. money that the government spends on programs thatCongress must authorize every year

c. programs funded by FICA taxes

5. Which of the following is not considered a mandatoryexpense in the U.S. federal budget?

a. national defense b. interest on the debt

c. Medicare

6. Currently, the majority of government spending goes to

a. mandatory expenses. b. national defense.

c. science and education.

7. Suppose the government collects $100 billion more inSocial Security taxes than it pays out in Social Securitybenefits. Under current policy, what happens to this“extra” $100 billion?

a. It is physically deposited into a bank that holds it to beused for future Social Security benefits.

b. It is used to fund other government programs.

c. It is returned in the form of rebates to those who paidthe excess taxes.

8. If the government were able to pay off the publicly held debt,who would receive the money?

a. The money would be distributed among all U.S citizens.

b. The money would go to holders of Treasury bills, notes,and bonds.

c. The money would go to future retirees through theSocial Security Trust Fund.

2$100

9. Which of the following best describes the total amount ofmoney that the government has obligated itself to pay backin the future?

a. the publicly held debt b. the gross debt

c. the off-budget debt

10. By the year 2030, the government is expected to owe sev-eral hundred billion dollars more in Social Security bene-fits each year than it will collect in Social Security taxes.Although all options for covering this shortfall might bepolitically difficult, which of the following is not an optioneven in principle?

a. The shortfall could be covered by tax increases.

b. The shortfall could be covered by additional borrowingfrom the public.

c. The shortfall could be covered by reducing the amountof education grants offered.

REVIEW QUESTIONS11. Define receipts, outlays, net income, surplus, and deficit as

they apply to annual budgets.

12. What is the difference between a deficit and a debt? Howlarge is the federal debt?

13. Explain why years of running deficits makes it increasinglydifficult to get a budget into balance.

14. How large is the deficit at present? Should we assumefuture deficit projections are correct? Explain.

15. Briefly summarize the makeup of federal receipts and fed-eral outlays. Distinguish between mandatory outlays anddiscretionary outlays.

16. How does the federal government finance its debt? Distin-guish between the publicly held debt and the gross debt.

17. Briefly describe the Social Security trust fund. What’s in it?What problems may this cause in the future?

18. Distinguish between an off-budget deficit (or surplus) andan on-budget deficit (or surplus). What is the unifieddeficit (or surplus)?

DOES IT MAKE SENSE?

Decide whether each of the following statements makes sense(or is clearly true) or does not make sense (or is clearly false).Explain your reasoning.

19. My share of the federal government’s debt is greater thanthe cost of a weekend in Miami.

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20. My share of the federal government’s annual interest pay-ments on the federal debt is greater than what I need tobuy a new car in cash.

21. Because Social Security is off-budget, we could cut SocialSecurity taxes with no impact on the rest of the federalgovernment.

22. The government collected more money than it spent, butits total debt still increased.

23. I read today that in 10 years the government will return tosurpluses (from deficits), so we should start planning howwe’ll use the surplus.

24. The Social Security trust fund will have a positive balancefor at least 40 years to come, so there’s no need to be con-cerned about how the government will pay Social Securitybenefits.

BASIC SKILLS & CONCEPTS25. Personal Budget Basics. Suppose your after-tax annual

income is $38,000. Your annual expenses are $12,000 forrent, $6000 for food and household expenses, $1200 forinterest on credit cards, and $8500 for entertainment,travel, and other.

a. Do you have a surplus or a deficit? Explain.

b. Next year, you expect to get a 3% raise. You think youcan keep your expenses unchanged, with one exception:You plan to spend $8500 on a car. Explain the effect ofthis purchase on your budget.

c. As in part b, assume you get a 3% raise for next year. Ifyou can limit your expenses to a 1% increase (over theprior year), could you afford $7500 in tuition and feeswithout going into debt?

26. Personal Budget Basics. Suppose your after-tax incomeis $28,000. Your annual expenses are $8000 for rent, $4500for food and household expenses, $1600 for interest oncredit cards, and $10,400 for entertainment, travel, andother.

a. Do you have a surplus or a deficit? Explain.

b. Next year, you expect to get a 2% raise, but plan to keepyour expenses unchanged. Will you be able to pay off$5200 in credit card debt? Explain.

c. As in part b, assume you get a 2% raise for next year. Ifyou can limit your expenses to a 1% increase, could youafford $3500 for a wedding and honeymoon withoutgoing into debt?

27. Per Worker Debt. Suppose the government decided topay off the $9 trillion debt with a one-time charge distrib-uted equally among all workers. Assuming the total U.S.

work force is 170 million people, how much would eachworker be charged?

28. Per Family Debt. Suppose the government decided topay off the $9 trillion debt with a one-time charge distrib-uted equally among all families. Assuming there are120 million families in the United States, how muchwould each family be charged?

29. The Wonderful Widget Company Future. Extendingthe budget summary of the Widget Company (Table 4.11),assume that, for 2008, total receipts are $1,050,000, oper-ating expenses are $600,000, employee benefits are$200,000, and security costs are $250,000.

a. Based on the accumulated debt at the end of 2007, cal-culate the 2008 interest payment. Assume an interestrate of 8.2%.

b. Calculate the total outlays for 2008, the year-end surplusor deficit, and the year-end accumulated debt.

c. Based on the accumulated debt at the end of 2008, cal-culate the 2009 interest payment, again assuming an8.2% interest rate.

d. Assume that in 2009 the Widget Company has receiptsof $1,100,000, holds operating costs and employee ben-efits to their 2008 levels, and spends no money on secu-rity. Calculate the total outlays for 2009, the year-endsurplus or deficit, and the year-end accumulated debt.

e. Imagine that you are the CFO (Chief Financial Officer)of the Wonderful Widget Company at the end of 2009.Write a three-paragraph statement to shareholdersabout the company’s future prospects.

30. The Wonderful Widget Company Future. Extendingthe budget summary of the Widget Company (Table 4.11),assume that, for 2008, total receipts are $975,000, operat-ing expenses are $850,000, employee benefits are$290,000, and security costs are $210,000.

a. Based on the accumulated debt at the end of 2007, cal-culate the 2008 interest payment. Assume an interestrate of 8.2%.

b. Calculate the total outlays for 2008, the year-end surplusor deficit, and the year-end accumulated debt.

c. Based on the accumulated debt at the end of 2008, cal-culate the 2009 interest payment, again assuming an8.2% interest rate.

d. Assume that in 2009 the Widget Company has receiptsof $1,050,000, holds operating costs and employee ben-efits to their 2008 levels, and spends no money on secu-rity. Calculate the total outlays for 2009, the year-endsurplus or deficit, and the year-end accumulated debt.

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e. Imagine that you are the CFO (Chief Financial Officer)of the Wonderful Widget Company at the end of 2009.Write a three-paragraph statement to shareholdersabout the company’s future prospects.

31. Budget Projections. Refer to the 2006 data in Table 4.12.How would the deficit have been affected by a 1% decreasein total receipts? How would it have been affected by a0.5% increase in total outlays?

32. Budget Projections. Refer to the 2006 data in Table 4.12.How would the deficit have been affected by a 0.5%decrease in total receipts? How would it have been affectedby a 1% increase in total outlays?

Budget Analysis. Consider the 2006 total receipts and outlaysshown in Table 4.12. Based on Figures 4.12 and 4.13, answer thequestions in Exercises 33–38.

33. How much income came from individual income taxes?

34. How much income came from social insurance taxes?

35. How much income came from excise taxes?

36. How much was spent on Social Security?

37. How much was spent on Medicare?

38. How much was spent on defense?

39. On- and Off-Budget. Suppose the government has aunified net income of $40 billion, but was supposed todeposit $180 billion in the Social Security trust fund. Whatwas the on-budget surplus or deficit? Explain.

40. On- and Off-Budget. Suppose the government has aunified net income of billion, but was supposed todeposit $205 billion in the Social Security trust fund. Whatwas the on-budget deficit? Explain.

41. Social Security Finances. Suppose the year is 2020, andthe government needs to pay out $350 billion more inSocial Security benefits than it collects in Social Securitytaxes. Briefly discuss the options for finding this money.

42. Social Security Finances. Suppose the year is 2025, andthe government needs to pay out $525 billion more inSocial Security benefits than it collects in Social Securitytaxes. Briefly discuss the options for finding this money.

FURTHER APPLICATIONS43. Counting the Federal Debt. Suppose you began count-

ing the $9 trillion federal debt, $1 at a time. If you couldcount $1 each second, how long would it take to completethe count?

2$220

44. Paving with the Federal Debt. Suppose you began cov-ering the ground with $1 bills. If you had the $9 trillionfederal debt in $1 bills, how much total area could youcover? Compare this area to the total land area of theUnited States, which is about 10 million square kilometers.(Hint: Measure the length and width of a $1 bill in cen-timeters. Then compute its area in square centimeters andconvert the area to square kilometers.)

45. Rising Debt. Suppose the federal debt increases at anannual rate of 1% per year. Use the compound interestformula to determine the size of the debt in 10 years and in50 years. Assume that the current size of the debt (theprincipal for the compound interest formula) is $9 trillion.

46. Rising Debt. Suppose the federal debt increases at anannual rate of 2% per year. Use the compound interestformula to determine the size of the debt in 10 years and in50 years. Assume that the current size of the debt (theprincipal for the compound interest formula) is $9 trillion.

47. Budget 2008. Consider the 2008 projection described inExample 4. What are the projected outlays? Suppose thatoutlays turn out to be higher than projected by 5%, whilereceipts are lower by 5%. In that case, what is the 2008surplus or deficit?

48. Budget 2008. Consider the 2008 projection described inExample 4. What are the projected outlays? Suppose thatoutlays turn out to be higher than projected by 10%, whilereceipts are lower by 11%. In that case, what is the 2008surplus or deficit?

49. Retiring the Public Debt. Consider the publicly helddebt of $5.0 trillion in 2006. Use the loan payment for-mula to determine the annual payments needed to pay thisdebt off in 10 years. Assume an annual interest rate of 4%.

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50. Retiring the Public Debt. Consider the publicly helddebt of $5.0 trillion in 2006. Use the loan payment for-mula to determine the annual payments needed to pay thisdebt off in 15 years. Assume an annual interest rate of 2%.

51. National Debt Lottery. Imagine that, through somepolitical or economic miracle, the gross debt stopped ris-ing. To retire the gross debt, the government decided tohave a national lottery. Suppose that every U.S. citizenbought a $1 lottery ticket every week, thereby generatingabout $300 million in weekly lottery revenue. Because lot-teries typically use half their revenue for prizes and lotteryoperations, half the $300 million, or $150 million, wouldgo toward debt reduction each week. How long would ittake to retire the debt through this lottery? Use the 2006gross debt of $9 trillion.

52. National Debt Lottery. Suppose the government hopesto pay off the gross debt of $9 trillion with a national lot-tery. For the debt to be paid off in 50 years, how muchwould each citizen have to spend on lottery tickets eachyear? Assume that half of the lottery revenue goes towarddebt reduction and that there are 300 million citizens.


53. Federal Budget Deficit/Surplus. Use the Web to findthe most recent projections of the federal deficit/surplusfor the next 10 years.

54. Debt Problem. How serious of a problem is the grossdebt? Use the Web to find arguments on both sides of thisquestion. Summarize the arguments and state your ownopinion.

55. Social Security Problems. Using information availableon the Web, research the current status of the Social Secu-rity trust fund and potential future problems in paying outbenefits. For example, when is the fund projected to startpaying out more than it takes in each year? Write a one- totwo-page report that summarizes your findings.

56. Social Security Solutions. Research various proposals forsolving the problems with Social Security. Choose oneproposal that you think is worthwhile and write a one- totwo-page report summarizing it and describing why youthink it is a good idea.

57. Privatizing Social Security. One proposal for saving theSocial Security program is privatization—removing it fromthe government and running it like a for-profit business.Find an argument for and an argument against privatiza-tion of Social Security. Summarize each argument and dis-cuss which case you think is stronger.

IN THE NEWS58. Federal Budget. Choose one of the many current news

stories concerning federal finances. Summarize the storyand the issues involved.

59. Social Security. Find a news article that concerns eitherthe present or the future state of the Social Security sys-tem. Briefly summarize the article and interpret it in lightof what you learned in this unit.

60. Relying on Projections. Find a news story in whichCongress or the President is relying on projections severalyears into the future to make a budget today. Report onhow the uncertainty in the projections is being dealt with,and discuss whether the decisions are being made wisely.

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Chapter 4 Summary 319

CHAPTER 4 SUMMARY

UNIT KEY TERMS KEY IDEAS AND SKILLS

4A

4B

4C

4D

budgetcash flow

principalsimple interestcompound interestannual percentage

rate (APR)annual percentage

yield (APY)variable definitions:

A, P, i, N, n, Y

savings plantotal returnannual returnmutual fundinvestment

considerationsliquidityriskreturn

bond characteristicsface valuecoupon ratematurity ratecurrent yield

installment loanmortgages

down paymentclosing costpointsfixed rate mortgageadjustable rate

mortgage

Understand the importance of controlling your finances.Know how to make a budget.Be aware of factors that help determine whether your spending patterns make sense for your situation.

General form of the compound interest formula:

Compound interest formula for interest paid once a year:

Compound interest formula for interest paid n times a year:

Compound interest formula for continuous compounding:

Know when and how to apply these formulas.

Savings plan formula:

Return on investments:

Understand investment types: stock, bond, cash.Read financial tables for stocks, bonds, and mutual funds.Remember important principles of investing, such as

Higher returns usually involve higher risk.High commissions and fees can dramatically lower returns.Build an appropriately diversified portfolio.

Loan payment formula:

Understand the uses and dangers of credit cards.Understand strategies for early payment of loans.Understand considerations in choosing a mortgage.

PMT 5

P 3 aAPRn

bc1 2 a1 1

APRn

bA2nY B d


2 1

total return 5AA 2 P B

P

A 5 PMT 3

c a1 1APR

nbAnY B

2 1 daAPR

nb

A 5 P 3 eAAPR3YB

A 5 P a1 1APR

nbAnY B


A 5 P 3 A1 1 i B N

(Continues on the next page)

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4E

4F

gross incomeadjusted gross incomeexemptions,

deductions, creditstaxable incomefiling statusprogressive income taxmarginal tax ratesSocial Security, FICA,

self-employment taxcapital gains

receipts, outlaysnet income

surplusdeficit

debtmandatory outlaysdiscretionary outlayspublicly held debtgross debton budget, off budgetunified budget

Define different types of income as they apply to taxes.Use tax rate tables to calculate taxes.Distinguish between tax credits and tax deductions.Calculate FICA taxes.Be aware of special tax rates for dividends and capital gains.Understand the benefits of tax-deferred savings plans.

Distinguish between a deficit and a debt.Understand basic principles of the federal budget.Distinguish between publicly held debt and gross debt.Be familiar with major issues concerning the future of Social Security.

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Chapter 4