The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

F V 
 PV(1  i)n

F V
$4,265.97  1 + __0.036 360 

1,307

F V
$4,861.58

If we are using the exact method, though, matters are a bit more complicated.

Example 3.2.10 Nigel deposited $4,265.97 in a savings account paying 3.6%

compounded daily using the exact method. He closed the account 3 years, 7 months,

and 17 days later. How much did he have in his account when he closed it?

This problem looks identical to the previous one. Unfortunately, this switch away from bank-

ers’ rule presents a big problem in the conversion of months to days. Since not every month

contains the same number of days, it is actually impossible to know for sure how many

days were in the term, which makes it impossible to answer the question exactly unless we

know which specifi c months were included. (If we knew, for example, the dates on which he

opened and closed his account we could fi nd the exact number of days it was open.) For

that matter, to get an exact count of the days we would also need to know whether or not

the term included a leap year.

There is no defi nitive way to deal with this. One typical approach would be to use 365 days per

year but assume 30 days per month. In most (but not all) cases this will slightly understate the

term, but it will never be too far from the exact value. Using this approach we would have

n 
 (3)(365)  7(30)  17 
 1,322 days

In which case:

FV 
 PV(1 + i)n

FV
$4,265.97  1 + __0.036 360 

1,322

FV
$4,860.07

Another not so commonly used but more accurate approach is to assume that each month

has 30.5 days (sort of “averaging” the 30 and 31 day months), discarding any half days in

the end (to recognize that February has fewer.) Using this approach we would have

n 
 (3)(365)  7(30.5)  17 
 1,325.5 days, which we drop to 1,325 days

In which case:

FV 
 PV(1 + i)n

FV
$4,265.97  1 + __0.036 360 

1,325

FV
$4,861.51

Overall, though, this is all much ado about not much. The difference is quite small, and

in either case the answer can never be any more than an approximation. Whichever

approach is used, the most important thing is to be clear about the fact that the answer is

an approximation, and not necessarily the actual exact future value. Neil’s actual future

value will depend on the actual number of days; while we don’t know this, the bank

knows the actual days and months involved, and so they will. There is nothing wrong

with approximating, so long as we realize that that is what we are doing, and Nigel is not

likely to be much bothered by the small difference between the approximation and actual

values. Approximations can lead to conflicts and misunderstandings, though, when they

are mistaken for exact numbers and so we should make sure to be crystal clear that our

answer is only an approximation.

We will close with one more example, this time illustrating the need to not overthink things!

Example 3.2.11 Suppose I deposit $1,200 in a certifi cate of deposit paying 6%

compounded monthly for 9 months. How much interest will I earn?

Since interest is compounded monthly, i 
 0.06/12. For n, we need to know the term in

months. Don’t overthink this—there is no conversion to do! The term is already given in

months, so n 
 9.

3.2 Compounding Frequencies 109