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For the fi rst 2 years, this account was an annuity, and so we can fi nd its future value by using
the annuity formula:FV PMT s _n (^) | (^) i
FV $175 s __ 24 | (^) .003
FV $175(24.84650638)
FV $4,348.14
So at the end of the fi rst 2 years, the account had grown to $4,348.14.
Now for the last 3 years, no payments are being made, but that $4,348.14 does continue
to grow at compound interest:
FV PV(1 i)n
FV $4,348.14(1.003)^36
FV $4,348.14(1.113867644)
FV $4,843.25
So at the end of 5 years, the account was worth $4,843.25.
Don’t be confused by the fact that $4,348.14 was the FV in the first step of this solution,
but the PV in the second step. This amount came at the end of the annuity portion, so there
it was the FV. In the plain compound interest portion it came at the beginning, and so there
it was the PV. Using time lines can help keep this straight.
Breaking the life of an account up chronologically allows us to take on all sorts of situ-
ations like this one. Before moving on to try it out with other types of problems, let’s use
this approach on another problem similar to this example.
We have seen the surprising power of compound interest throughout Chapters 3 and 4,
and we will see it again in Chapter 5. We have noted that compounding is especially power-
ful over long terms, where interest on interest (on interest on interest... ) has the chance to
really pile up. Comparatively modest sums can build up into shockingly large future values,
and most of those large future values come from interest, not the payments themselves. This
might lead us to wonder about questions like the one proposed in the following example:
Example 4.6.2 Wanda is 25 years old. She has decided to put $2,500 each year into
a retirement account starting this year, but she plans to make these deposits for only
10 years. Her twin brother Wayne is also 25 years old. More of a procrastinator, he
plans to wait 10 years before putting anything into his retirement account. But once he
starts, he plans to keep up deposits of $2,500 per year until he is 65.
Assuming that Wayne and Wanda both earn 8.4% interest, how much will each have at
age 65?
Wanda’s stream of payments is similar to Example 5.1.1. If we look at the fi rst 10 years, her
account is an annuity, since she is making equal payments every year. However, for the last
30 years no payments are being made at all, but compound interest does accumulate.
Again, we don’t have a single formula that will allow us to fi nd her future value at the end of the
40 years. But let’s look at Wanda’s account chronologically. The fi rst 10 years of Wanda’s
account can be looked at as an annuity, and so we can fi nd the future value for those:
FV PMT s _n (^) | (^) i
FV $2,500 s 10 __ (^) | (^) .084
FV $2,500(14.76465595)
FV $36,911.64
Of course, this represents the value of her account at the end of the 10 years, or when she is
- The account still will be open for another 30 years, with no payments made. While that
isn’t an annuity, that money will still be earning compound interest. And so the last 30 years
can be dealt with by using the regular compound interest formula:
FV PV(1 i)n
FV $36,911.64(1.084)^30
FV $414,993.99
4.6 Future Values with Irregular Payments: The Chronological Approach (Optional) 193