10 The Mathematics of Money

1. 10 The Mathematics of Money 10.1 Percentages 10.2 Simple Interest 10.3 Compound Interest 10.4 Geometric Sequences 10.5 Deferred Annuities: Planned Savings for the Future 10.6 Installment Loans: The Cost of Financing the Present

3. Money Matters As a consumer, you make decisions about money every day. Some areminor –“Should I get gas at the station on the right or make a U-turnand go to the station across the highway where gas is 5¢ a gallon cheaper?”–,but others are much more significant – “If I buy that new red Mustang,should I take the $2000 dealer’s rebate or the 0% financing for 60 monthsoption?”.

4. Money Matters Decisions of the first type usually involve just a little arithmetic and some common sense (on a 20 gallon fill-up you are saving $1to make that U-turn–is it worth it?); decisions of the second type involve a more sophisticated understanding of the time value of money (is$2000 up front worth more or less than saving the interest on paymentsover the next five years?). This latter type of question and others similarto it are the focus of this chapter.

5. Fractions A general truism is that people don’t like dealing with fractions.There are exceptions, of course, but most people would ratheravoid fractions whenever possible. The most likely culprit for“fraction phobia” is the difficulty of dealing with fractions withdifferent denominators. One way to get around this difficulty is toexpress fractions using a common, standard denominator, and inmodern life the commonly used standard is the denominator100.

6. Percentages A “fraction” with denominator 100 can be interpreted as apercentage, and the percentage symbol (%) is used to indicate thepresence of the hidden denominator 100. Thus,

7. Percentages Percentages are useful for many reasons. They giveus a common yardstick to compare different ratios andproportions; they provide a useful way of dealing withfees, taxes, and tips; and they help us better understandhow things increase or decrease relative to some givenbaseline. The next few examples explore these ideas.

8. Example 10.1 Comparing Test Scores Suppose that in your English Lit class you scored 19 out of 25 on the quiz, 49.2out of 60 on the midterm, and 124.8 out of 150 on the final exam. Without readingfurther, can you guess which one was your best score? Not easy, right? The numbers 19, 49.2, and 124.8 are called raw scores. Since each raw score isbased on a different total, it is hard to compare them directly, but we can do it easily once we express each score as a percentage of the total number of points possible.

9. Example 10.1 Comparing Test Scores ■Quiz score = 19/25:Here we can do the arithmetic in our heads. If we just multiply both numerator and denominator by 4, we get 19/25 = 76/100 = 76%. ■Midterm score = 49.2/60:Here the arithmetic is a little harder, so one might want to use a calculator:49.2 ÷ 60 = 0.82 = 82%. This score is a definite improvement over the quiz score.

10. Example 10.1 Comparing Test Scores ■Final Exam = 124.8/150: Once again, we use a calculator and get:124.8 ÷ 150 = 0.832 = 83.2%.This score is the best one.

11. Convert Decimals to Percents Example 10.1 illustrates the simple but important relation between decimals andpercentages: decimals can be converted to percentages through multiplication by 100(as in 0.76 = 76%, 1.325 = 132.5%, and 0.005 = 0.5%), and conversely, percentages can be converted to decimals through division by 100 (as in 100% = 1.0,83.2% = 0.832, and 7 1/2% = 0.075).

12. Example 10.2 Is 3/20th a Reasonable Restaurant Tip? Imagine you take an old friend out to dinner at a nice restaurant for her birthday.The final bill comes to $56.80. Your friend suggests that since the service wasgood, you should tip 3/20th of the bill. What kind of tip is that?After a moment’s thought, you realize that your friend, who can be abit annoying at times, is simply suggesting you should tip the standard15%. After all, 3/20 =15/100 = 15%.

13. Example 10.2 Is 3/20th a Reasonable Restaurant Tip? Although 3/20 and 15% are mathematically equivalent, the latteris a much more convenient and familiar way to express theamount of the tip. To compute the actual tip, you simplymultiply the amount of the bill by 0.15. In this case we get0.15  $56.80 = $8.52.

14. Example 10.3 Shopping for an iPod Imagine you have a little discretionary money saved up and you decide to buyyourself the latest iPod. After a little research you find the following options: ■Option 1: You can buy the iPod at Optimal Buy, a local electronics store. Theprice is $399. There is an additional 6.75% sales tax. Your total cost out thedoor is $399 + (0.0675)$399 = $399 + $26.9325 = $399 + $26.94 = $425.94

15. Example 10.3 Shopping for an iPod The above calculation can be shortened by observing that the originalprice (100%) plus the sales tax (6.75%) can be combined for a total of106.75% of the original price. Thus, the entire calculation can be carriedout by a single multiplication: (1.0675)$399 = $425.94 (rounded up to the nearest penny)

16. Example 10.3 Shopping for an iPod ■Option 2: At Hamiltonian Circuits, another local electronic store, the salesprice is $415, but you happen to have a 5% off coupon good for all electronicproducts. Taking the 5% off from the coupon gives the sale price, which is95% of the original price. Sale price:(0.95)$415 = $394.25

17. Example 10.3 Shopping for an iPod We still have to add the 6.75% sales tax on top of that, and as we sawin Option 1, the quick way to do so is to multiply by 1.0675. Final price including taxes: (1.0675)$394.25 = $420.87 For efficiency we can combine the two separate calculations (take thediscount and add the sales tax) into one: (1.0675)(0.95)$415=$420.87

18. Example 10.3 Shopping for an iPod ■Option 3: You found an online merchant in Portland, Oregon, that will sellyou the iPod for $441. This price includes a 5% shipping/processing chargethat you wouldn’t have to pay if you picked up the iPod at the store inPortland (there is no sales tax in Oregon). The $441 is much higher thanthe price at either local store, but you are in luck: your best friend fromPortland is coming to visit and can pick up the iPod for you and save youthe 5% shipping/processing charge. What would your cost be then?

19. Example 10.3 Shopping for an iPod Unlike option 2, in this situation we do nottake a 5% discount on the$441. Here the 5% was added to the iPod’s base price to come up with the finalcost of $441, that is, 105% of the base price equals $441. Using P for theunknown base price, we have Although option 3 is the cheapest, it is hardly worth the few penniesyou save to inconven-ience your friend. Your best bet is to head to HamiltonianCircuits with your 5% off coupon.

20. PERCENT INCREASE If you start with a quantity Q and increase that quantity by x%, you end upwith the quantity

21. PERCENT INCREASE If you start with a quantity Q and decrease that quantity by x%, you end upwith the quantity

22. PERCENT INCREASE If I is the quantity you get when you increase an unknown quantity Q and by x%, then (Notice that this last formula is equivalent to the formula given in the firstbullet.)

23. Example 10.4 The Dow Jones Industrial Average The Dow Jones Industrial Average (DJIA) is one of the most commonly usedindicators of the overall state of the stock market in the United States. (As of thewriting of this material the DJIA hovered around 13,000.) We aregoing to illustrate the ups and downs of the DJIA with fictitiousnumbers. ■Day 1: On a particular day, the DJIA closed at 12,875.

24. Example 10.4 The Dow Jones Industrial Average ■Day 2: The stock market has a good day and the DJIA closes at 13,029.50. This is an increase of 154.50 from the previous day.To express the increase as a percentage, we ask, 154.50 is whatpercent of 12,875 (the day 1 value that serves as our baseline)?The answer is obtained by simply dividing 154.50 into 12,875(and then rewriting it as a percentage).

25. Example 10.4 The Dow Jones Industrial Average Thus, the percentage increase from day 1 to day 2 is Here is a little shortcut for the same computation, particularly convenientwhen you use a calculator (all it takes is one division): 13,029.50 ÷ 12,875 = 1.02All we have to do now is to mentally subtract 1 from the above number. Thisgives us once again 0.012=1.2%.

26. Misleading Use of Percent Changes Percentage decreases are often used incorrectly, mostly intentionally and inan effort to exaggerate or mislead. The misuse is usually framed by the claim thatif an x% increase changes A to B, then an x% decrease changes B to A. Not true!

27. Example 10.5 The Bogus 200% Decrease With great fanfare, the police chief of Happyville reports that crime decreased by200% in one year. He came up with this number based on reported crimes inHappyville going down from 450 one year to 150 the next year. Since an increasefrom 150 to 450 is a 200% increase (true), a decrease from 450 to 150 must surelybe a 200% decrease, right? Wrong.

28. Example 10.5 The Bogus 200% Decrease The critical thing to keep in mind when computing a decrease (or for that matter an increase) between two quantities is that these quantities are not interchangeable. In this particular example the baseline is 450 and not 150, so the correctcomputation of the decrease in reported crimes is 300/450 = 0.666 . . . ≈ 66.67%.

29. Moral to Example 10.5 Be wary of any extravagant claims aboutthe percentage decrease of something (be it reported crimes, traffic accidents,pollution, or any other nonnegative quantity). Always keep in mind that a percentage decrease can never exceed 100%, once you reduce something by 100%,you have reduced it to zero. An important part of being a smart shopper is understanding how markups(profit margins) and markdowns (sales) affect the price of consumer goods.

30. Example 10.6 Combining Markups and Markdowns A toy store buys a certain toy from the distributor to sell during the Christmas season. The store marks up the price of the toy by 80% (the intended profit margin).Unfortunately for the toy store, the toy is a bust and doesn’t sell well. After Christmas, it goes on sale for 40% off the marked price. After a while, an additional 25%markdown is taken off the sale price and the toy is put on the clearance table.

31. Example 10.6 Combining Markups and Markdowns Withall the markups and markdowns, what is the percentage profit/loss to the toy store? The answer to this question is independent of the original cost of the toy tothe store. Let’s just call this cost C. ■After adding an 80% markup to their cost C, the toy store retails the toy fora price of (1.8)C.

32. Example 10.6 Combining Markups and Markdowns ■After Christmas, the toy is marked down and put on sale with a “40% off”tag. The sale price is 60% of the retail price. This gives (0.6)(1.8)C = (1.08)C,(which represents a net markup of 8% on the original cost to the store).

33. Example 10.6 Combining Markups and Markdowns ■Finally, the toy is put on clearance with an “additional 25% off” tag. Theclearance price is (0.75)(1.08)C = 0.81C. (The clearance price is now 81% ofthe original cost to the store–a net loss of 19%! That’s what happens whentoys don’t sell.)

34. 10 The Mathematics of Money 10.1 Percentages 10.2 Simple Interest 10.3 Compound Interest 10.4 Geometric Sequences 10.5 Deferred Annuities: Planned Savings for the Future 10.6 Installment Loans: The Cost of Financing the Present

35. Present Value and Future Value Money has a present value and a future value. Unless you are lendingmoney to a friend, if you invest $P today (the present value) for apromise of getting $F at some future date (the future value), youexpect F to be more than P. Otherwise, why do it? The same principle also works in reverse. If you are getting a present value of Ptoday from someone else (either in cash or in goods), you expect tohave to pay a future value of F back at some time in the future. If we are given the present value P,how do we find the future value F (and vice versa)?

36. Interest Rate The answerdepends on several variables, the most important of which is the interest rate. Interest is the return the lender or investor expects as a reward for theuse of his or her money, and the standard way to describe an interest rate is as ayearly rate commonly called the annual percentage rate (APR). Thus, we can say,“I am investing my money in an account that pays an APR of 5%,” or “I have to paya 24% APR on the balance on my credit card.”

37. Simple Interest or Compound Interest The APR is the most important variable in computing the return on an investment or the cost of a loan, but several other questions come into play and must beconsidered. Is the interest simple or compounded? If compounded, how often is itcompounded? Are there additional fees? If so, are they in addition to the interestor are they included in the APR? We will consider these questions in Sections 10.2and 10.3.

38. Simple Interest In simple interest, only the original money invested or borrowed (called theprincipal) generates interest over time. This is in contrast to compound interest,where the principal generates interest, then the principal plus the interest generate more interest, and so on.

39. Example 10.7 Savings Bonds Imagine that on the day you were born your parents purchased a $1000 savingsbond that pays 5% annual simple interest. What is the value of the bond on your18th birthday? What is the value of the bond on any given birthday?Here the principal is P = $1000 and the annual percentage rate is 5%. Thismeans that the interest the bond earns in one year is 5% of $1000, or(0.05)$1000 = $50.Because the bond pays simple interest, the interest earned bythe bond is the same every year.

40. Example 10.7 Savings Bonds ■Value of the bond on your 1st birthday= $1000 +$50 = $1050. ■Value of the bond on your 2nd birthday= $1000 + (2  $50)= $1100 … ■Value of the bond on your 18th birthday= $1000 + (18  $50)= $1900. ■Value of the bond when you becomet years old = $1000 + (t $50). Thus,

41. SIMPLE INTEREST FORMULA The future value F of P dollars invested under simple interest for t years at anAPR of R% is given by F = P(1 + r •t) (where r denotes the R% APR written as a decimal).

42. Simple Interest You should think of the simple interest formula as a formula relating fourvariables: P (the present value), F (the future value), t(the length of the investment in years), and r (the APR). Given any three of these variables you can findthe fourth one using the formula. The next example illustrates how to use the simple interest formula to find a present value P given F, t, and r.

43. Example 10.8 Government Bonds: Part 2 Government bonds are often sold based on their future value. Suppose that youwant to buy a five-year $1000 U.S.Treasury bond paying 4.28% annual simple interest (so that in five years you can cash in the bond for $1000). Here $1000 is thefuture value of the bond, and the price you pay for this bond is its present value.

44. Example 10.8 Government Bonds: Part 2 To find the present value of the bond, we let F = $1000, R = 4.28%,and t = 5and use the simple interest formula. This gives $1000 = P[1 + 5(0.0428)]= P(1.214) Solving the above equation for P gives (rounded to the nearest penny).

45. Credit Cards Generallyspeaking, credit cards charge exceptionally high interest rates, but you only have topay interest if you don’t pay your monthly balance in full. Thus, a credit card is atwo-edged sword: if you make minimum payments or carry a balance from onemonth to the next, you will be paying a lot of interest; if you pay your balance infull, you pay no interest.

46. Credit Cards In the latter case you got a free, short-term loan from thecredit card company. When used wisely, a credit card gives you a rare opportunity–you get to use someone else’s money for free. When used unwisely and carelessly, acredit card is a financial trap.

47. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly Imagine that you recently got a new credit card. Like most people, youdid not pay much attention to the terms of use or to the APR, whichwith this card is a whopping 24%. To make matters worse, you wentout and spent a little more than you should have the first month, andwhen your first statement comes in you are surprised to find out thatyour new balance is $876.

48. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly Like with most credit cards, you have a little time from the time yougot the statement to the payment due date (this grace period is usually around 20 days). You can pay a minimum paymentof $20, the full balance of$876, or any other amount in between. Let’s consider these three different scenarios separately.

49. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly ■Option 1: Pay the full balance of $876 before the payment due date.This one is easy. You owe no interest and you got free use of the credit cardcompany’s money for a short period of time. When your next monthly billcomes, the only charges will be for your new purchases.

50. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly ■Option 2: Pay the minimum payment of $20.When your next monthly bill comes, you have a new balance of $1165 consisting of: 1. The previous balance of $856. (The $876 you previously owed minus yourpayment of $20.)