108 Chapter 3 Compound Interest
FORMULA 3.2.1
The Continuous Compound Interest Formula
(Alternative Version)FV PVe(rt)where
FV represents the FUTURE VALUE (the ending amount)
PV represents the PRESENT VALUE (the starting amount)
e is a mathematical constant (approximately 2.71828)
r represents the ANNUAL INTEREST RATE
and
t represents the NUMBER OF YEARSCompound Interest with “Messy” Terms
So far, all of the problems we have looked at in this section have had terms measured in whole
years. This is of course not required in the real world, and so we will need to consider other
terms as well—terms that do not happen to be a whole number of years. Those sorts of terms
don’t require any new formulas, but require a bit more effort in determining the value of n.
The key fact to remember is that n must give the term in the time units of the compound-
ing. The following examples will illustrate:Example 3.2.8 Luisa deposited $2,850 in a credit union certifi cate of deposit, paying
4.8% compounded monthly for 2½ years. What will the maturity value of her certifi cate be?We must convert years to months. In previous problems, we have multiplied years by 12 to
get months; that doesn’t change here. So n (2^1 ⁄ 2 years)(12 months/year) (2.5 years)
(12 months/year) 30 months.
FV PV(1 i)n
FV $2,850^ ^1 ^ ______0.048 12 ^(^30)
FV $3,212.60
For some students, a non–whole number in the term is sometimes distracting enough to
cause confusion about whether to multiply or divide when they are finding n. Remember
that however we calculate it, n is supposed to represent the term in months. If you had gotten
confused and divided here instead of multiplying, you would have come up with 2.5/12
0.208333333333. It should be obvious that 2^1 ⁄ 2 years is not the same as 0.2083333333
months, and so thinking about whether the value you are using for n makes sense will spot
any mistakes of this type.
There is never any ambiguity in converting between months and years. There are always
12 months in every year, so the number we use in our conversion will always be 12. Convert-
ing between months and days, though, presents a bit of a problem, since a month may have
28, 29, 30, or 31 days, and unless we know the specific months in question we can’t know
the correct number of days. One way of dealing with this is to extend bankers’ rule. We can
pretend that the year has 360 days, and it is divided into 12 months of 30 days each.Example 3.2.9 Nigel deposited $4,265.97 in a savings account paying 3.6%
compounded daily using bankers rule. He closed the account 3 years, 7 months, and
17 days later. How much did he have in his account when he closed it?Here we have a term expressed in a variety of units. The simplest approach is to convert each
to days and then fi nd the total. Under bankers’ rule we pretend that the year has 360 days,
which we divide equally into months of 30 days each. So:(3 years)(360 days/year) 1,080 days
(7 months)(30 days/month) 210 days
17 days 17 days
And so in total n 1,080 210 17 1,307 dayscb
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