198 Chapter 4 Annuities
Example 4.7.3 For the last 5 years, the management of Watermill Corp. has put
$25,000 from its profi ts each quarter into a fund intended to provide capital for future
expansion plans. Unfortunately, in the last quarter of the third year the company had
some fi nancial diffi culties, and skipped making this deposit. If the account earned
4.8%, how much did the company have at the end of the 5 years?There are quite a few different ways to break this account into buckets, but, just as in
Example 4.7.2, the most effi cient solution is to start with a bucket that assumes that the
regular annuity payments were made for the entire term.Bucket 1: $25,000 per quarter for 5 years.FV PMT s _n (^) | (^) i
FV $25,000 s __ 20 | (^) .012
FV $561,321.59
In the previous example, the second bucket was the extra money, but here we don’t have any
extra money—instead money is missing. Here, though, we can make a second bucket out of
the missing money, and calculate what it would have grown to with interest if the payment
hadn’t been missed.
Bucket 2: The missing $25,000 from the end of the third year. This money was missing for
the last 2 years, so we calculate what it would have grown to in those last 2 years:
FV PV(1 i)n
FV $25,000(1 0.012)^8
FV $27,503.26
Bucket 1 assumes that all the payments were made, Bucket 2 contains the missing payment
together with all the interest it would have earned. The company’s actual account was bucket 1,
take away the contents of bucket#2. So the actual future value is $561,321.59 $27,503.26
$533,818.33.
Annuities with Multiple Missing or Extra Payments
What if an “annuity” has more than one extra and/or missing payment? In that case, we
can follow the approach we used in the previous two examples. Start out assuming that all
of the payments were the same, and then create a separate bucket for each extra or missing
payment. Then at the end, add the extra and subtract the missing. A few of the exercises
will require this.
What if instead of just a few deviations up or down, the “annuity” payments actually
change?
Example 4.7.4 For 15 years I deposited $2,500 each year into an investment
account that earned 7½%. Then, for the next 5 years, I kept making payments, but
only $1,000 each. How much was my account worth at the end of 20 years? How much
total interest did I earn?
Bucket 1: Once again, we start by assuming the $2,500 payments continued for all 20 years.
FV PMT s _n (^) | (^) i
FV $2,500 s __ 20 | (^) .075
FV $108,261.70
Bucket 2: In each of the last 5 years, $1,500 was missing compared to what we assumed for
bucket 1. These missing payments form an annuity:
FV PMT s n (^) | (^) i
FV $1,500s 5 | (^) .075
FV $8,712.59
The overall future value, then, is the bucket 1 annuity less the missing deposits and their
interest. So the future value is $108,261.70 $8,712.59 $99,549.11.