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. Fixed Annuity A fixed annuity is a sequence of equal payments made or received over regular(monthly, quarterly, annually) time intervals. Annuities (often disguised underdifferent names) are so common in today’s financial world that there is a goodchance you may be currently involved in one or more annuities and not evenrealize it. You may be making regular deposits to save for a vacation, a wedding, or college, or you may be making regular payments on a car loan or ahome mortgage.

4. Deferred Annuity - Installment Loan You could also be at the receiving end of an annuity,gettingregular payments from an inheritance, a college trust fund set up on your behalf, or a lottery jackpot. When payments are made so as to produce a lump-sum payout at a later date(e.g., making regular payments into a college trust fund), we call the annuity adeferred annuity; when a lump sum is paid to generate a series of regular payments later (e.g., a car loan), we call the annuity an installment loan.

5. Deferred Annuity - Installment Loan Simplystated, in a deferred annuity the pain (in the form of payments) comes first andthe reward (a lump-sum payout) comes in the future, whereas in an installmentloan the reward (car, boat, house) comes in the present and the pain (paymentsagain) is stretched out into the future.In this section we will discuss deferred annuities. In the next section we willtake a look at installment loans.

6. Example 10.21 Setting Up a College Trust Fund Given the cost of college, parents often set up college trust funds for their children by setting aside a little money each month over the years. A college trustfund is a form of forced savings toward a specific goal, and it is generallyagreed to be a very good use of a parent’s money–it spreads out the pain ofcollege costs over time, generates significant interest income, and has valuabletax benefits.

7. Example 10.21 Setting Up a College Trust Fund Let’s imagine a mother decides to set up a college trust fund for her new-born child. Her plan is to have $100 withdrawn from her paycheck each monthfor the next 18 years and deposited in a savings account that pays 6% annualinterest compounded monthly. What is the future value of this trust fund in 18 years?

8. Example 10.21 Setting Up a College Trust Fund What makes this example different from Uncle Nick’s trust fund example(Example 10.10) is that money is being added to the account in regular installments of $100 per month. Each $100 monthly installment has a different “lifespan”:The first $100 compounds for 216 months (12 times a year for 18 years), thesecond $100 compounds for only 215 months, the third $100 compounds for only214 months, and so on.

9. Example 10.21 Setting Up a College Trust Fund Thus, the future value of each $100 installment is different.To compute the future value of the trust fund we will have to compute the futurevalue of each of the 216 installments separately and add. Sounds like a tall order,but the geometric sum formula will help us out.

10. Example 10.21 Setting Up a College Trust Fund Critical to our calculations are that each installment is for a fixed amount($100) and that the periodic interest rate p is always the same (6% ÷ 12 =0.5% = 0.005). Thus, when we use the general compounding formula, each futurevalue looks the same except for the compounding exponent: Future value of the first installment ($100 compounded for 216 months): (1.005)216$100

11. Example 10.21 Setting Up a College Trust Fund Future value of the second installment ($100 compounded for 215 months): (1.005)215$100 Future value of the third installment ($100 compounded for 214 months): (1.005)214$100 … Future value of the last installment ($100 compounded for one month): (1.005)$100 = $100.50

12. Example 10.21 Setting Up a College Trust Fund The future value F of this trust fund at the end of 18 years is the sum of all theabove future values. If we write the sum in reverse chronological order (startingwith the last installment and ending with the first), we get (1.005)$100 + (1.005)2$100 + … + (1.005)215$100 + (1.005)216$100

13. Example 10.21 Setting Up a College Trust Fund A more convenient way to deal with the above sum is to first observe that thelast installment of (1.005)$100 = $100.50is a common factor of every term in thesum; therefore, $100.50[1 + (1.005) + (1.005)2 + … + (1.005)214 + (1.005)215]

14. Example 10.21 Setting Up a College Trust Fund You might now notice that the sum inside the brackets is a geometric sumwith common ratio c = 1.005and a total of N = 216 terms. Applying thegeometric sum formula to this sum gives

15. FIXED DEFERRED ANNUITY FORMULA The future value F of a fixed deferred annuity consisting of T payments of$P having a periodic interest of p (written in decimal form) is where L denotes the future value of the last payment.

16. Example 10.22 Setting Up a College Trust Fund: Part 2 In Example 10.21 we saw that an 18-year annuity of $100 monthly payments at anAPR of 6% compounded monthly is $38,929. For the same APR and the samenumber of years, how much should the monthly payments be if our goal is an annuity with a future value of $50,000?

17. Example 10.22 Setting Up a College Trust Fund: Part 2 If we use the fixed deferred annuity formula with F = $50,000,we get Solving for L gives

18. Example 10.22 Setting Up a College Trust Fund: Part 2 Recall now that L is the future value of the last payment, and since the payments are made at the beginning of each month,L = (1.005)P. Thus,

19. Relationship Between F and P The main point of Example 10.22 is to illustrate that the fixed deferredannuity formula establishes a relationship between the future value F of theannuity and the fixed payment P required to achieve that future value. If weknow one, we can solve for the other (assuming, of course, a specified number ofpayments T and a specified periodic interest rate p).

20. Example 10.23 Saving for a Cruise: Part 5 We saw (Example 10.12) that if you invest the $875 at a 6.75% APR compounded monthly, the future value of your investment is $1449.62–for simplicity,let’s call it $1450. This is $550 short of the $2000 you will need. Imagine you wantto come up with the additional $550 by making regular monthly deposits into thesavings account, essentially creating a small annuity. How much would you haveto deposit each month to generate the $550 that you will need?

21. Example 10.23 Saving for a Cruise: Part 5 Using the fixed deferred annuity formula with F = $550, T = 90(12 installments a year for 7 1/2 years), and a periodic rate of p = 0.005625(obtained by taking the 6.75% APR and dividing by 12), we have

22. Example 10.23 Saving for a Cruise: Part 5 Solving: In conclusion, to come up with the $2000 that you will needto send Mom on a cruise in 71/2 years do the following: (1) Invest your $875 savingsin a safe investment such as a CD offered by a bank or a credit union and (2) saveabout $5 a month and put the money into a fixed deferred annuity.