Financial Mathematics

Financial Mathematics
Chapter 10: Financial Mathematics 10.1 Percentages

Money Matters As a consumer, you make decisions about money every day.
Some are minor –“Should I get gas at the station on the right or make a U-turn and 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 months option?”. 2

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 $1 to 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 payments over the next five years?). This latter type of question and others similar to it are the focus of this chapter. 3

Fractions A general truism is that people don’t like dealing with fractions. There are exceptions, of course, but most people would rather avoid fractions whenever possible. The most likely culprit for “fraction phobia” is the difficulty of dealing with fractions with different denominators. One way to get around this difficulty is to express fractions using a common, standard denominator, and in modern life the commonly used standard is the denominator 100. 4

Percentages A “fraction” with denominator 100 can be interpreted as a percentage, and the percentage symbol (%) is used to indicate the presence of the hidden denominator 100. Thus, 5

Percentages Percentages are useful for many reasons.
They give us a common yardstick to compare different ratios and proportions; they provide a useful way of dealing with fees, taxes, and tips; and they help us better understand how things increase or decrease relative to some given baseline. The next few examples explore these ideas. 6

Example: Comparing Test Scores
Suppose that in your English Lit class you scored 19 out of 25 on the quiz, 49.2 out of 60 on the midterm, and out of 150 on the final exam. Without reading further, can you guess which one was your best score? Not easy, right? The numbers 19, 49.2, and are called raw scores. Since each raw score is based 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. 7

■ 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. 8

■ Final Exam = 124.8/150: Once again, we use a calculator and get: ÷ 150 = = 83.2%. This score is the best one. 9

Convert Decimals to Percents
The previous example illustrates the simple but important relation between decimals and percentages: decimals can be converted to percentages through multiplication by 100 (as in 0.76 = 76%, = 132.5%, and = 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). 10

Example: 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 $ Your friend suggests that since the service was good, 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 a bit annoying at times, is simply suggesting you should tip the standard 15%. After all, 3/20 =15/100 = 15%. 11

Example: Is 3/20th a Reasonable Restaurant Tip?
Although 3/20 and 15% are mathematically equivalent, the latter is a much more convenient and familiar way to express the amount of the tip. To compute the actual tip, you simply multiply the amount of the bill by 0.15. In this case we get 0.15  $56.80 = $8.52. 12

Example: Shopping for an iPod
Imagine you have a little discretionary money saved up and you decide to buy yourself 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. The price is $399. There is an additional 6.75% sales tax. Your total cost out the door is $399 + (0.0675)$399 = $399 + $ = $399 + $26.94 = $425.94 13

The above calculation can be shortened by observing that the original price (100%) plus the sales tax (6.75%) can be combined for a total of % of the original price. Thus, the entire calculation can be carried out by a single multiplication: (1.0675)$399 = $425.94 (rounded up to the nearest penny) 14

■ Option 2: At Hamiltonian Circuits, another local electronic store, the sales price is $415, but you happen to have a 5% off coupon good for all electronic products. Taking the 5% off from the coupon gives the sale price, which is 95% of the original price. Sale price: (0.95)$415 = $394.25 15

We still have to add the 6.75% sales tax on top of that, and as we saw in Option 1, the quick way to do so is to multiply by Final price including taxes: (1.0675)$ = $420.87 For efficiency we can combine the two separate calculations (take the discount and add the sales tax) into one: (1.0675)(0.95)$415=$420.87 16

■ Option 3: You found an online merchant in Portland, Oregon, that will sell you the iPod for $441. This price includes a 5% shipping/processing charge that you wouldn’t have to pay if you picked up the iPod at the store in Portland (there is no sales tax in Oregon). The $441 is much higher than the price at either local store, but you are in luck: your best friend from Portland is coming to visit and can pick up the iPod for you and save you the 5% shipping/processing charge. What would your cost be then? 17

Unlike option 2, in this situation we do not take a 5% discount on the $441. Here the 5% was added to the iPod’s base price to come up with the final cost of $441, that is, 105% of the base price equals $441. Using P for the unknown base price, we have Although option 3 is the cheapest, it is hardly worth the few pennies you save to inconvenience your friend. Your best bet is to head to Hamiltonian Circuits with your 5% off coupon. 18

PERCENT INCREASE If you start with a quantity Q and increase that quantity by x%, you end up with the quantity 19

PERCENT INCREASE If you start with a quantity Q and decrease that quantity by x%, you end up with the quantity 20

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 first bullet.) 21

Example: The Dow Jones Industrial Average
The Dow Jones Industrial Average (DJIA) is one of the most commonly used indicators of the overall state of the stock market in the United States. ( As of the writing of this material the DJIA hovered around 13,000.) We are going to illustrate the ups and downs of the DJIA with fictitious numbers. ■ Day 1: On a particular day, the DJIA closed at 12,875. 22

■ Day 2: The stock market has a good day and the DJIA closes at 13, This is an increase of from the previous day. To express the increase as a percentage, we ask, is what percent of 12,875 (the day 1 value that serves as our baseline)? The answer is obtained by simply dividing into 12,875 (and then rewriting it as a percentage). 23

Thus, the percentage increase from day 1 to day 2 is Here is a little shortcut for the same computation, particularly convenient when you use a calculator (all it takes is one division): 13, ÷ 12,875 = 1.02 All we have to do now is to mentally subtract 1 from the above number. This gives us once again 0.012=1.2%. 24

Misleading Use of Percent Changes
Percentage decreases are often used incorrectly, mostly intentionally and in an effort to exaggerate or mislead. The misuse is usually framed by the claim that if an x% increase changes A to B, then an x% decrease changes B to A. Not true! 25

Example: The Bogus 200% Decrease
With great fanfare, the police chief of Happyville reports that crime decreased by 200% in one year. He came up with this number based on reported crimes in Happyville going down from 450 one year to 150 the next year. Since an increase from 150 to 450 is a 200% increase (true), a decrease from 450 to 150 must surely be a 200% decrease, right? Wrong. 26

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 correct computation of the decrease in reported crimes is 300/450 = ≈ 66.67%. 27

The moral of this story? Be wary of any extravagant claims about the 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. 28

An important part of being a smart shopper is understanding how markups (profit margins) and markdowns (sales) affect the price of consumer goods. 29

Example: 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. 30

With all 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 to the store. Let’s just call this cost C. ■ After adding an 80% markup to their cost C, the toy store retails the toy for a price of (1.8)C. 31

■ 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). 32

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

Chapter 10: Financial Mathematics 10.2 Simple Interest

Present Value and Future Value
Money has a present value and a future value. Unless you are lending money to a friend, if you invest $P today (the present value) for a promise of getting $F at some future date (the future value), you expect F to be more than P. Otherwise, why do it?

Present Value and Future Value
The same principle also works in reverse. If you are getting a present value of P today from someone else (either in cash or in goods), you expect to have 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)?

Interest Rate The answer depends 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 the use of his or her money, and the standard way to describe an interest rate is as a yearly 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 pay a 24% APR on the balance on my credit card.”

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 be considered. Is the interest simple or compounded? If compounded, how often is it compounded? Are there additional fees? If so, are they in addition to the interest or are they included in the APR? We will consider these questions in Sections 10.2 and 10.3.

Simple Interest In simple interest, only the original money invested or borrowed (called the principal) 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.

Example: Savings Bonds
Imagine that on the day you were born your parents purchased a $1000 savings bond that pays 5% annual simple interest. What is the value of the bond on your 18th 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%. This means 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 by the bond is the same every year.

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

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

Simple Interest You should think of the simple interest formula as a formula relating four variables: 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 find the 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.

Example: Government Bonds: Part 2
Government bonds are often sold based on their future value. Suppose that you want 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 the future value of the bond, and the price you pay for this bond is its present value.

Example: Government Bonds: Part 2
To find the present value of the bond, we let F = $1000, R = 4.28%, and t = 5 and 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).

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

Credit Cards In the latter case you got a free, short-term loan from the credit 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, a credit card is a financial trap.

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

Like with most credit cards, you have a little time from the time you got the statement to the payment due date (this grace period is usually around 20 days). You can pay a minimum payment of $20, the full balance of $876, or any other amount in between. Let’s consider these three different scenarios separately.

■ 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 card company’s money for a short period of time. When your next monthly bill comes, the only charges will be for your new purchases.

■ 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 your payment of $20.)

2. The charges for your new purchases. Let’s say, for the sake of argument, that you were a little more careful with your card and your new purchases for this period were $288.

3. The finance charge of $21 calculated as follows: (i) Periodic rate = (ii) Balance subject to finance charge = $ (iii) Finance charge = (0.02)$1050 = $21 You might wonder, together with millions of other credit card users, where these numbers come from. Let’s take them one at a time.

The periodic rate is obtained by dividing the annual percentage rate (APR) by the number of billing periods. Almost all credit cards use monthly billing periods, so the periodic rate on a credit card is the APR divided by 12. Your credit card has an APR of 24%, thus yielding a periodic rate of 2% = 0.02.

The balance subject to finance charge (an official credit card term) is obtained by taking the average of the daily balances over the previous billing period. Since this balance includes your new purchases, all of a sudden you are paying interest on all your purchases and lost your grace period! In your case, the balance subject to finance charge came to $1050.

The finance charge is obtained by multiplying the periodic rate times the balance subject to finance charge. In this case, (0.02)$1050 = $21. ■ Option 3: You make a payment that is more than the minimum payment but less than the full payment.

Let’s say for the sake of argument that you make a payment of $400. When your next monthly bill comes, you have a new balance of $ As in option 2, this new balance consists of: The previous balance, in this case $476 (the $876 you previously owed minus the $400 payment you made) 2. The new purchases of $288

The finance charges, obtained once again by multiplying the periodic rate (2% = 0.02) times the balance subject to finance charges, which in this case came out to $682. Thus, your finance charges turn out to be (0.02)$682 = $13.64, less than under option 2 but still a pretty hefty finance charge.

Two Important Lessons Make sure you understand the terms of your credit card agreement. Know the APR (which can range widely from less than 10% to 24% or even more), know the length of your grace period, and try to understand as much of the fine print as you can.

Two Important Lessons Make a real effort to pay your credit card balance in full each month. This practice will help you avoid finance charges and keep you from getting yourself into a financial hole. If you can’t make your credit card payments in full each month, you are living beyond your means and you may consider putting your credit card away until your balance is paid.

Chapter 10: Financial Mathematics 10.3 Compound Interest

Compound Interest Under simple interest the gains on an investment are constant–only the principal generates interest. Under compound interest, not only does the original principal generate interest, so does the previously accumulated interest. All other things being equal, money invested under compound interest grows a lot faster than money invested under simple interest, and this difference gets magnified over time. If you are investing for the long haul (a college trust fund, a retirement account, etc.), always look for compound interest. 62

Example: Your Trust Fund Found!
Imagine that you have just discovered the following bit of startling news: On the day you were born, your Uncle Nick deposited $5000 in your name in a trust fund that pays a 6% APR. One of the provisions of the trust fund was that you couldn’t touch the money until you turned 18. You are now 18 years, 10 months old and you are wondering; how much money is in the trust fund now? How much money would there be in the trust fund if I waited until my next birthday when I turn 19? 63

How much money would there be in the trust fund if I left the money in for retirement and waited until I turned 60? Here is an abbreviated timeline of the money in your trust fund, starting with the day you were born: ■ Day you were born: Uncle Nick deposits $5000 in trust fund. ■ First birthday: 6% interest is added to the account. Balance in account is (1.06)$5000. 64

■ Second birthday: 6% interest is added to the previous balance (in red). Balance in account is (1.06)(1.06)$5000 = (1.06)2$5000. ■ Third birthday: 6% interest is added to the previous balance (again in red). Balance in account is (1.06)(1.06)2$5000 = (1.06)3$5000. At this point you might have noticed that the exponent of (1.06) in the right-hand expression goes up by 1 on each birthday and in fact matches the birthday. 65

Thus, ■ Eighteenth birthday: The balance in the account is (1.06)18$5000. It is now finally time to pull out a calculator and do the computation: (1.06)18$5000 = $14, (rounded to the nearest penny) 66

■ Today: Since the bank only credits interest to your account once a year and you haven’t turned 19 yet, the balance in the account is still $14, ■ Nineteenth birthday: The future value of the account is (1.06)19$5000 = $15,128 (rounded to the nearest penny) 67

Moving further along into the future, ■ 60th birthday: The future value of the account is (1.06)60$5000 = $164, which is an amazing return for a $5000 investment (if you are willing to wait, of course)! 68

This figure plots the growth of the money in the account for the first 18 years. 69

This figure plots the growth of the money in the account for 60 years. 70

ANNUAL COMPOUNDING FORMULA
The future value F of P dollars compounded annually for t years at an APR of R% is given by F = P(1 + r)t 71

Example: Saving for a Cruise
Imagine that you have $875 in savings that you want to invest. Your goal is to have $2000 saved in 7 1/2 years. (You want to send your mom on a cruise on her 50th birthday.) Imagine now that the credit union around the corner offers a certificate of deposit (CD) with an APR of 6 3/4% compounded annually. 72

What is the future value of your $875 in 7 1/2 years? If you are short of your $2000 target, how much more would you need to invest to meet that target? To answer the first question, we just apply the annual compounding formula with P = $875, R = 6.75 (i.e., r = ), and t = 7 (recall that with annual compounding, fractions of a year don’t count) and get $875(1.0675)7 = $ (rounded to the nearest penny) 73

Unfortunately, this is quite a bit short of the $2000 you want to have saved. To determine how much principal to start with to reach a future value target of F = $2000 in 7 years at 6.75% annual interest, we solve for P in terms of F in the annual compounding formula. In this case substituting $2000 for F gives $2000 = P(1.0675)7 74

and solving for P gives This is quite a bit more than the $875 you have right now, so this option is not viable. Don’t despair–we’ll explore some other options throughout this chapter. 75

Example: Saving for a Cruise, Part 2
Let’s now return to our story from the previous example: You have $875 saved up and a 7 1/2 -year window in which to invest your money. As discussed in Example 10.11, the 6.75% APR compounded annually gives a future value of only $ – far short of your goal of $2000. 76

Now imagine that you find another bank that is advertising a 6.75% APR that is compounded monthly (i.e., the interest is computed and added to the principal at the end of each month). It seems reasonable to expect that the monthly compounding could make a difference and make this a better investment. Moreover, unlike the case of annual compounding, you get interest for that extra half a year at the end. 77

To do the computation we will have to use a variation of the annual compounding formula. The key observation is that since the interest is compounded 12 times a year, the monthly interest rate is 6.75% ÷ 12 = % ( when written in decimal form). An abbreviated chronology of how the money grows looks something like this: ■ Original deposit: $875. 78

■ Month 1: % interest is added to the account. The balance in the account is now ( )$875. ■ Month 2: % interest is added to the previous balance. The balance in the account is now ( )2$875. ■ Month 3: % interest is added to the previous balance. The balance in the account is now ( )3$875. 79

■ Month 12: At the end of the first year the balance in the account is ( )12$875 = $935.92 After 7 1/2 years, or 90 months, ■ Month 90: The balance in the account is ( )90$875 = $ 80

The story continues. Imagine you find a bank that pays a 6.75% APR that is compounded daily. You are excited! This will surely bring you a lot closer to your $2000 goal. Let’s try to compute the future value of $875 in 7 1/2 years. The analysis is the same as in part 2, except now the interest is compounded 365 times a year (never mind leap years–they don’t count in banking), and the numbers are not as nice. 81

First, we divide the APR of 6.75% by 365. This gives a daily interest rate of 6.75% ÷ 365 ≈ % = Next, we compute the number of days in the 7 1/2 year life of the investment 365  7.5 = Since parts of days don’t count, we round down to 2737. Thus, F = ( )2737$875 = $ 82

Differences: Compounding Frequency
Let’s summarize the results of the “Saving for a Cruise” examples. Each example represents a scenario in which the present value is P = $875, the APR is 6.75% (r = ), and the length of the investment is t = 7 1/2 years. The difference is the frequency of compounding during the year. 83

■ Annual compounding (Example 10.11): Future value is F = $ ■ Monthly compounding (Example 10.12): Future value is F = $ ■ Daily compounding (Example 10.13): Future value is F = $ 84

A reasonable conclusion from these numbers is that increasing the frequency of compounding (hourly, every minute, every second, every nanosecond) is not going to increase the ending balance by very much. The explanation for this surprising law of diminishing returns will be given shortly. 85

GENERAL COMPOUNDING FORMULA
The future value of P dollars in t years at an APR of R% compounded n times a year is 86

A Better Looking Form In the general compounding formula, r/n represents the periodic interest rate expressed as a decimal, and the exponent n • t represents the total number of compounding periods over the life of the investment. If we use p to denote the periodic interest rate and T to denote the total number of times the principal is compounded over the life of the investment, the general compounding formula takes the following particularly nice form. 87

GENERAL COMPOUNDING FORMULA (VERSION 2)
The future value F of P dollars compounded a total of T times at a periodic interest rate p is 88

Continuous Compounding
One of the remarkable properties of the general compounding formula is that even as n (the frequency of compounding) grows without limit, the future value F approaches a limiting value L. This limiting value represents the future value of an investment under continuous compounding (i.e., the compounding occurs over infinitely short time intervals) and is given by the following continuous compounding formula. 89

CONTINUOUS COMPOUNDING FORMULA
The future value F of P dollars compounded continuously for t years at an APR of R% is 90

Example: Saving for a Cruise: Part 4
You finally found a bank that offers an APR of 6.75% compounded continuously. Using the continuous compounding formula and a calculator, you find that the future value of your $875 in 7 1/2 years is F = $875(e7.50.0675) = $875(e ) = $ 91

Example: Saving for a Cruise: Part 4
The most disappointing thing is that when you compare this future value with the future value under daily compounding, the difference is 21¢. 92

Annual Percentage Yield
The annual percentage yield (APY) of an investment (sometimes called the effective rate) is the percentage of profit that the investment generates in a one-year period. For example, if you start with $1000 and after one year you have $ , you have made a profit of $99.60. The $99.60 expressed as a percentage of the $1000 principal is 9.96%–this is your APY. 93

Example: Computing an APY
Suppose that you invest $ At the end of a year your money grows to $ (The details of how your money grew to $ are irrelevant for the purposes of our computation.) Here is how you compute the APY: 94

Annual Percentage Yield
In general, if you start with S dollars at the beginning of the year and your investment grows to E dollars by the end of the year, the APY is the ratio (E – S)/S. You may recognize this ratio from Section 10.1–it is the annual percentage increase of your investment. 95

Example: Comparing Investments Through APY
Which of the following three investments is better: 6.7% APR compounded continuously, 6.75% APR compounded monthly, or 6.8% APR compounded quarterly? Notice that the question is independent of the principal P and the length of the investment t. To compare these investments we will compute their APYs. 96

The future value of $1 in 1 year at 6.7% interest compounded continuously is given by e0.067 ≈ (Here we used the continuous compounding formula). The APY in this case is 6.93%. (The beauty of using $1 as the principal is that this last computation is trivial.) 97

The future value of $1 in 1 year at 6.75% interest compounded monthly is ( /12)12 ≈ ≈ (Here we used the general compounding formula). The APY in this case is 6.963%. 98

The future value of $1 in 1 year at 6.8% interest compounded quarterly is ( /4)4 ≈ ≈ The APY in this case is 6.975%. 99

Although they are all quite close, we can now see that (c) is the best choice, (b) is the second-best choice, and (a) is the worst choice. Although the differences between the three investments may appear insignificant when we look at the effect over one year, these differences become quite significant when we invest over longer periods. 100

Chapter 10: Financial Mathematics 10.4 Consumer Debt

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

Credit Cards In the latter case you got a free, short-term loan from the credit 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, a credit card is a financial trap.

Imagine that you recently got a new credit card. Like most people, you did not pay much attention to the terms of use or to the APR, which with this card is a whopping 24%. To make matters worse, you went out and spent a little more than you should have the first month, and when your first statement comes in you are surprised to find out that your new balance is $876.