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. 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 otherthings being equal, money invested under compound interest grows a lot fasterthan money invested under simple interest, and this difference gets magnifiedover time. If you are investing for the long haul (a college trust fund, a retirementaccount, etc.), always look for compound interest.

4. Example 10.10 Your Trust Fund Found! Imagine that you have just discovered the following bit of startling news: On theday you were born, your Uncle Nick deposited $5000 in your name in a trust fundthat pays a 6% APR. One of the provisions of the trust fund was that you couldn’ttouch the money until you turned 18. You are now 18 years, 10 months old andyou are wondering, How much money is in the trust fund now? How muchmoney would there be in the trust fund if I waited until my next birthday when Iturn 19?

5. Example 10.10 Your Trust Fund Found! How much money would there be in the trust fund if I left the money infor retirement and waited until I turned 60? Here is an abbreviated timeline of the money in your trust fund, starting withthe 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.

6. Example 10.10 Your Trust Fund Found! ■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.

7. Example 10.10 Your Trust Fund Found! 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 = $14,271.70(rounded to the nearest penny)

8. Example 10.10 Your Trust Fund Found! ■Today: Since the bank only credits interest to your account once a year andyou haven’t turned 19 yet, the balance in the account is still $14,271.70. ■Nineteenth birthday: The future value of the account is (1.06)19$5000 = $15,128(rounded to the nearest penny)

9. Example 10.10 Your Trust Fund Found! Moving further along into the future, ■60th birthday: The future value of the account is (1.06)60$5000 = $164,938.45which is an amazing return for a $5000 investment (if you are willing to wait,of course)!

10. Example 10.10 Your Trust Fund Found! This figure plots the growth of the money in the account for the first 18 years.

11. Example 10.10 Your Trust Fund Found! This figure plots the growth of the money in the account for 60 years.

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

13. Example 10.11 Saving for a Cruise Imagine that you have $875 in savings that you want to invest. Your goal is tohave $2000 saved in 7 1/2 years. (You want to send your mom on a cruise on her 50thbirthday.) Imagine now that the credit union around the corner offers a certificateof deposit (CD) with an APR of 6 3/4% compounded annually. What is the futurevalue of your $875 in 7 1/2 years? If you are short of your $2000 target, how muchmore would you need to invest to meet that target?

14. Example 10.11 Saving for a Cruise To answer the first question, we just apply the annual compounding formulawith P = $875, R = 6.75 (i.e., r = 0.0675), and t= 7 (recall that with annualcompounding, fractions of a year don’t count) and get $875(1.0675)7= $1382.24 (rounded to the nearest penny)

15. Example 10.11 Saving for a Cruise Unfortunately, this is quite a bit short of the $2000 you want to havesaved. To determine how much principal to start with to reach a future valuetarget of F = $2000in 7 years at 6.75% annual interest, we solve for P interms of F in the annual compounding formula. In this case substituting $2000for F gives $2000 = P(1.0675)7

16. Example 10.11 Saving for a Cruise $2000 = P(1.0675)7 and solving for P gives This is quite a bit more than the $875 you have right now, so this option isnot viable. Don’t despair–we’ll explore some other options throughout thischapter.

17. Example 10.12 Saving for a Cruise: Part 2 Let’s now return to our story from Example 10.11: 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 $1382.24 – farshort of your goal of $2000.

18. Example 10.12 Saving for a Cruise: Part 2 Now imagine that you find another bank that is advertising a 6.75% APR that iscompounded monthly (i.e., the interest is computed and added to the principal at theend of each month). It seems reasonable to expect that the monthly compoundingcould make a difference and make this a better investment. Moreover, unlike thecase of annual compounding, you get interest for that extra half a year at the end.

19. Example 10.12 Saving for a Cruise: Part 2 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 compounded12 times a year, the monthly interest rate is 6.75% ÷ 12 = 0.5625%(0.005625when written in decimal form). An abbreviated chronology of how the moneygrows looks something like this: ■Original deposit: $875.

20. Example 10.12 Saving for a Cruise: Part 2 ■Month 1: 0.5625% interest is added to the account. The balance in the account is now(1.005625)$875. ■Month 2: 0.5625% interest is added to the previous balance. The balance inthe account is now(1.005625)2$875. ■Month 3: 0.5625% interest is added to the previous balance. The balance inthe account is now(1.005625)3$875.

21. Example 10.12 Saving for a Cruise: Part 2 ■Month 12: At the end of the first year the balance in the account is(1.005625)12$875 = $935.92 After 7 1/2 years, or 90 months, ■Month 90: The balance in the account is Here, t = total number of months(1.005625)90$875 = $1449.62

22. Example 10.13 Saving for a Cruise: Part 3 The story continues. Imagine you find a bank that pays a 6.75% APR that iscompounded 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 Example 10.12, except now the interest is compounded 365 times a year(never mind leap years–they don’t count in banking), and the numbers arenot as nice.

23. Example 10.13 Saving for a Cruise: Part 3 First, we divide the APR of 6.75% by 365.This gives a daily interest rate of6.75% ÷ 365 ≈ 0.01849315% = 0.0001849315 Next, we compute the number of days in the 7 1/2 year life of the investment 365  7.5 = 2737.5 Since parts of days don’t count, we round down to 2737. Thus, F = (1.0001849315)2737$875 = $1451.47

24. Differences: Compounding Frequency Let’s summarize the results of Examples 10.11, 10.12, and 10.13. Each example represents a scenario in which the present value is P = $875, the APR is6.75% (r= 0.0675),and the length of the investment is t= 7 1/2 years. The difference is the frequency of compounding during the year.

25. Differences: Compounding Frequency ■Annual compounding (Example 10.11): Future value is F = $1382.24. ■Monthly compounding (Example 10.12): Future value is F = $1449.62. ■Daily compounding (Example 10.13): Future value is F = $1451.47.

26. Differences: Compounding Frequency A reasonable conclusion from these numbers is that increasing the frequencyof compounding (hourly, every minute, every second, every nanosecond) is notgoing to increase the ending balance by very much. The explanation for this surprising law of diminishing returns will be givenshortly.

27. GENERAL COMPOUNDING FORMULA The future value of P dollars in t years at anAPR of R% compounded n times a year is

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

29. 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

30. Continuous Compounding One of the remarkable properties of the general compounding formula isthat even as n (the frequency of compounding) grows without limit, the futurevalue 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 infinitelyshort time intervals) and is given by the following continuous compoundingformula.

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

32. Example 10.14 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 thefuture value of your $875 in 7 1/2 years is F = $875(e7.50.0675) = $875(e0.50625) = $1451.68

33. Example 10.14 Saving for a Cruise: Part 4 The most disappointing thing is that when you compare this future valuewith the future value under daily compounding (Example 10.13), the differenceis 21¢.

34. Annual Percentage Yield The annual percentage yield (APY)of an investment (sometimes called theeffective rate) is the percentage of profit that the investment generates in a one-yearperiod. For example, if you start with $1000 and after one year you have $1099.60,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.

35. Example 10.15 Computing an APY Suppose that you invest $835.25. At the end of a year your money grows to$932.80. (The details of how your money grew to $932.80 are irrelevant for thepurposes of our computation.) Here is how you compute the APY:

36. 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.

37. Example 10.16 Comparing Investments Through APY Which of the following three investments is better: (a) 6.7% APR compoundedcontinuously, (b) 6.75% APR compounded monthly, or (c) 6.8% APR compounded quarterly? Notice that the question is independent of the principal Pand the length of the investment t. To compare these investments we will compute their APYs.

38. Example 10.16 Comparing Investments Through APY (a) The future value of $1 in 1 year at 6.7% interest compounded continuously isgiven by e0.067≈ 1.06930. (Here we used the continuous compoundingformula).The APY in this case is 6.93%. (The beauty of using $1 as the principal isthat this last computation is trivial.)

39. Example 10.16 Comparing Investments Through APY (b) The future value of $1 in 1 year at 6.75% interest compounded monthly is (1 + 0.0675/12)12≈ 1.00562512 ≈ 1.06963 (Here we used the general compoundingformula).The APY in this case is 6.963%.

40. Example 10.16 Comparing Investments Through APY • The future value of $1 in 1 year at 6.8% interest compounded quarterly is(1 + 0.068/4)4≈ 1.0174 ≈ 1.06975The APY in this case is 6.975%.

41. Example 10.16 Comparing Investments Through APY 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 differencesbetween the three investments may appear insignificant when we look at the effectover one year, these differences become quite significant when we invest overlonger periods.