128 Chapter 3 Compound Interest
Example 3.4.2 In Example 3.3.7 we found that a nominal rate of 5.75%
compounded daily is equivalent to a rate of 5.92% compounded annually. Suppose
you deposit $3,475 for 2 years at this nominal rate. Find the future value using the
nominal rate, fi nd the future value using the effective rate, and compare the two
results.Nominal rate: n 2(365)730 FV $3,475 1 + 0.0575___ 365
730
$3,898.47Effective rate: FV $3,475(1 0.0592)^2 $3,898.62Despite the fact that the two rates are supposed to be equivalent, the future values are not
the same. The difference is small, but it is still a difference.What gives? The claim that the rates are equivalent implies that they should give the same
results. In fact, that was the basis of our very definition of equivalent rates! While the dif-
ference between the two results in both examples was quite small, there were differences
nonetheless.
The reason for these differences lies in the fact that both of these effective rates were
rounded to two decimal places. Since these effective rates are rounded, there is a bit of
inaccuracy in them, and so it is actually not surprising that they do not give perfectly
accurate results. Any time rounding is involved, there will exist at least the potential for
these sorts of small discrepancies. In Example 3.4.1 the discrepancy was extremely small
because, even though the rate was rounded, the amount of rounding involved was very
small. In Example 3.4.2, the rounding was more significant, and so the resulting discrep-
ancy was also larger.
One way to get around this problem is to demand more decimal places from our
effective rates. If we went out to, say, five decimal places, the effective rate in Example
3.4.2 would have been 5.91805%. Using this as our effective rate, the future value
comes out to be $3,898.47, and the discrepancy vanishes. Actually, though, the two
future values are still different, but the difference is less than a penny and so is lost
when we round the final answers. With a much larger present value and/or a much
longer term, the difference might once again become large enough to show up in the
final answer.
Another way around the problem is simply to just decide to live with it. Suppose that
you calculated your future value with the 5.92% effective rate, and so expected a future
value of $3,898.62. Your bank, however, used the nominal rate, and so when you arrive
to claim your future value on the maturity date you find that the account is worth 15 cents
less than you expected. While you might wonder why the two numbers differ, you are
hardly likely to be too concerned about such a trivial difference. Is a 15-cent discrepancy
on a nearly $4,000 account balance, over 2 years, really worth worrying about or making
an issue over?^8
Rounding is a necessary evil, and as an unavoidable consequence there will always be
some minor discrepancies in financial calculations. So long as the rounding is not exces-
sive, the discrepancies will not be large enough to be a cause for concern. We have been
using two decimal places for effective rates, since in most cases that will give results that
are close enough; in cases where greater precision is required more decimal places can be
used.
Accountants sum up this fact of financial life as the principle of materiality. Unless we
have some reason why we need to demand a very high degree of precision, we accept small
discrepancies as the unavoidable price of the convenience of being able to do a reasonable
amount of rounding. Rather than lose sleep over these minor discrepancies, we instead
simply decide to live with them.(^8) I would bet that if you did insist on making an issue of this, the branch manager would gladly fork over the extra
15 cents from his own pocket just to make you go away. I would also bet that he would be telling stories about
you and your precious 15 cents at cocktail parties for many years to come!