10 The Mathematics of Money

2. 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. 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)?

4. 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.”

5. 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.

6. 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.

7. 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.

8. 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,

9. 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).

10. 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.

11. 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.

12. 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).

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.)

19. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly 2. The charges for your new purchases. Let’s say, for the sake of argument, thatyou were a little more careful with your card and your new purchases for thisperiod were $288.

20. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly 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 = $21You might wonder, together with millions of other credit card users,where these numbers come from. Let’s take them one at a time.

21. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly (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 billingperiods, so the periodic rate on a credit card is the APR divided by 12. Yourcredit card has an APR of 24%, thus yielding a periodic rate of 2% = 0.02.

22. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly (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 billingperiod. Since this balance includes your new purchases, all of a suddenyou are paying interest on all your purchases and lost your grace period!In your case, the balance subject to finance charge came to $1050.

23. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly • The finance charge is obtained by multiplying the periodic rate times thebalance subject to finance charge. In this case,(0.02)$1050 = $21. ■Option 3: You make a payment that is more than the minimum payment butless than the full payment.

24. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly Let’s say for the sake of argument that you make a payment of $400. Whenyour next monthly bill comes, you have a new balance of $777.64. As in option 2,this new balance consists of: 1. The previous balance, in this case $476 (the $876 you previously owed minusthe $400 payment you made) 2. The new purchases of $288

25. Example 10.9 Credit Card Use: The Good, the Bad and the Ugly 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 casecame out to $682. Thus, your finance charges turn out to be (0.02)$682 = $13.64,less thanunder option 2 but still a pretty hefty finance charge.

26. Two Important Lessons 1. Make sureyou understand the terms of your credit card agreement.Know the APR (which canrange widely from less than 10% to 24% or even more),know the length of your grace period, and try to understandas much of the fine print as you can.

27. Two Important Lessons 2. Make a real effort topay your credit card balance in full each month.This practicewill help you avoid finance charges and keep you from getting yourself into a financial hole. If you can’t make yourcredit card payments in full each month, you are living beyond your means and you may consider putting your creditcard away until your balance is paid.