The Mathematics of Money

18. Krystof took out a personal loan for $3,569.74, with monthly payments for 5 years. The interest rate was 11.39%.

a. Construct an amortization table for the fi rst 6 months of this loan.

Month Payment Interest Principal

Remaining

Balance

1 2 3 4 5 6

b. Use the “present value of the remaining payments” method to fi nd the amount he would owe after 6 months of

payments.

c. Why doesn’t your answer to (b) agree with the balance shown in the last row of the amortization table?

192 Chapter 4 Annuities

4.6 Future Values with Irregular Payments: The

Chronological Approach (Optional)

Annuities provide a powerful tool to work with streams of payments, but they have some

serious limitations. The most serious of these is the requirement that all the payments must

be the same. Our annuity formulas do not allow us to skip payments or change them, even

though it is not hard to imagine realistic situations in which that might happen.

In this section, we will consider some of these situations. It turns out that the tools we

have already can be used to find future vales of “annuities” whose payments change or

stop. In particular, we will see that these problems can often be tackled by looking at the

series of payments in pieces, in the order in which they are made. For convenience, we will

refer to this method of attack as the chronological approach.

“Annuities” Whose Payments Stop

Suppose that you start funding an investment or savings account by making equal pay-

ments at regular intervals (required for an annuity), but then at some point along the way

decide to keep the account open but stop making any more payments (not allowed for an

annuity.) This situation doesn’t fit the requirements we demand of an annuity, but it’s not

hard to imagine it happening. Can we adapt the mathematics of annuities to figure out how

much the account will grow to?

The account we are talking about here is not an annuity for its entire life, but until the

payments stop it is. So we can at least use our annuity formulas to look at this first part of

the story. Then notice that, once the payments stop, the account becomes a sum of money

growing at compound interest for a period of time. That’s something we can handle by

using the compound interest formula. By looking at the account “as time goes by” we can

break the problem up into two pieces, each of which can be handled with tools we already

have. As noted, we will refer to this way of approaching problems as the chronological

approach. The following example will illustrate:

Example 4.6.1 For 2 years, I deposited $175 monthly into a savings account. Then

I stopped, but kept the account open for 3 more years. The interest rate was 3.6%.

How much did I have in the account at the end of the full 5 years?