The Mathematics of Money

Example 4.2.10 Suppose $100 per month is deposited in an account for which the

interest is 6% compounded quarterly. Find the future value after 5 years.

Each month is one-third of a quarter, and so the interest compounding for a month would be

(1  0.06/4)

(^1) / 3

. Proceeding as in the previous example, we get

sn _ (^) | (^) i

(1  i)n  1
___i

(^)  (^)  1  _0.06
4
(^) 
1/3
(^) 
60
 1

(^)  1  _____0.06 4 
(^1) ⁄ 3
 1

69.7167087

And so the future value would be ($100)(69.7167087) = $6,971.67

If we have instead just used 6% compounded monthly, the future value would have been

$6,977.00. Once again, there is a difference, but it is not large.

If, however, the between-compounding payments earn no between-compounding inter-

est, then the situation is the same as if all of the payments for a given compounding period

were made all at once at the end of the compounding period.

Example 4.2.11 Suppose that $100 per month is deposited in an account for which

the interest is 6% compounded quarterly. Find the future value after 5 years, assuming

that each quarter’s interest is paid only on the funds that were in the account at the

start of the quarter.

It makes no difference here whether you make $100 payments each month or instead just

keep the payments in a coffee can until the end of the quarter and then make a single $300

deposit then. So the future value can be found by assuming payments of $300 per quarter,

in which case the future value will be $6,937.10.

Recall from Chapter 3 that, while the compounding frequency does matter, the difference

between monthly and quarterly compounding is not huge. In all of the previous examples, had

we just assumed that interest compounded at the same frequency as the payments, the differ-

ence would not have been all that great. If the situation demands an exact value, this difference

would of course matter, and so we would not be able to ignore it. However, if the future value is

a projection or illustration, where it is understood that the figure given is not meant to be taken

as exact, using quarterly compounding would probably be close enough. It is often understood

that the future value in question is not to be taken too literally, and so it is usually reasonable

to stick with the same frequency assumption. With the exception of a few clearly marked exer-

cises in this section, we will assume matching frequencies for the remainder of this book.

EXERCISES 4.2

In all of the exercises in this section, assume that there are no additional deposits or withdrawals from the accounts other

than those described.

A. The Chronological and Bucket Approaches

1. Suppose that you deposit $1,359.55 at the end of each year into an investment account that earns 5.7% interest
   compounded annually for 4 years. Determine the future value of your payments using (a) the chronological approach
   and (b) the bucket approach by completing the tables below:

a. Chronological

Year Starting Balance Interest Earned Deposit Ending Balance

1 $0.00 $0.00 $1,359.55

2 $1,359.55

3 $1,359.55

4 $1,359.55

Exercises 4.2 157

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