Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 10 The Mathematics of Money 10.1Percentages 10.2Simple Interest 10.3 Compound Interest 10.4Geometric Sequences 10.5Deferred Annuities: Planned Savings for the Future 10.6Installment Loans: The Cost of Financing the Present

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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? 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)? Present Value and Future Value

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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.” Interest Rate

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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 Simple Interest or Compound Interest

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Simple Interest

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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 10.7Savings Bonds

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Thus, Example 10.7Savings Bonds ■ 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).

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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 FORMULA

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Simple Interest

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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 10.8Government Bonds: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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). Example 10.8Government Bonds: Part 2

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Credit Cards

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. ■ 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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. ■ 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.) Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 3. The finance charge of $21 calculated as follows: (i)Periodic rate = 0.02 (ii) Balance subject to finance charge = $1050 (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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. (i)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% = Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. (ii)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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. (iii)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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 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: Example 10.9Credit Card Use: The Good, the Bad and the Ugly 1.The previous balance, in this case $476 (the $876 you previously owed minus the $400 payment you made) 2. The new purchases of $288

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 3.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. Example 10.9Credit Card Use: The Good, the Bad and the Ugly 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.

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1.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

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 2.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. Two Important Lessons