The Mathematics of Money

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

In an ideal financial world we might all like all of our answers to be infinitely

precise, but it is actually just common sense to temper that expectation. Small errors,

while undesirable, are not worth the cost and effort of chasing down. If you find a

$500 discrepancy when balancing your checkbook, it is well worth investing a couple

of hours to chase down the problem. But wasting an evening to chase down a nickel is

a waste of time. It would be nice to have everything match to the penny, but it is a far

better decision to write off the nickel difference than to waste valuable time and effort

to chase it down.

This is not license for sloppiness, nor is it a statement that we don’t really care all that

much about getting the right answer. Taking a materiality point of view simply means that

we agree to accept some small inaccuracies as a regrettable fact of life. It does not mean

that we throw accuracy to the wind and stop caring about having the right answers. We still

will make every reasonable attempt to see everything work out exactly. We just decline to

exert unreasonable effort for to-the-penny exactness.

For most purposes, effective rates rounded to two, or at most three, decimal places result

in discrepancies that are small enough to be ignored for all practical purposes.

Example 3.4.3 Tr is deposited $5,000 at 6.38% compounded daily for 16 years. Find

his account’s future value in two ways: (a) using the nominal rate and (b) by fi nding the

effective rate and using it to fi nd the future value.

(a) n 
 (16 years)(365 days/year) 
 5,840 days. Then:

FV
$5,000  1  ___0.0638 365 

5,840

FV
$13,875.83

(b) We fi rst need to fi nd the effective rate. Even though we know that Tris actually deposited

$5,000 and the account was actually open for 16 years, we do not need (or want) these

details when fi nding the effective rate.

FV 
 PV(1  i)n

FV
$100  1 + 0.0638___ 365 

365

FV
$106.59

From which we conclude that the effective rate is 6.59%.

Now to use the effective rate. Remember that even though we know that Tris’s interest actu-

ally compounds daily, the effective rate is always annually compounded.

FV 
 PV(1  i)n

FV 
 $5,000(1  0.0659)^16

(^) FV
$13,881.40
The actual balance in Tris’s account would be $13,875.83, since the nominal rate would be
the one really used. The disagreement between these two answers is once again due to the
rounding of the effective rate.
Example 3.4.3 illustrates the fact that “close enough” depends on context. An error
of more than $5 would be quite serious if the Tris’s account’s size were $20. How-
ever, with an account balance of many thousands of dollars over a term of 16 years,
this isn’t much. This is only common sense, but it does demand that we think about
the situation and use sound judgment when asking whether or not a difference is big
enough to matter. Considering the size of the account and time period, the discrepancy
between the two answers is unlikely to be much of an issue. If, however, a situation
did demand greater precision, we could get it by taking the effective rate out to more
decimal places.
3.4 Comparing Effective and Nominal Rates 129