Copyright © 2008, The McGraw-Hill Companies, Inc.
4.3 Sinking Funds
So far we have looked at annuities from the point of view that the payments determine
the future value. If instead we set up an annuity with a future value goal in mind, we
would need to look at things in the opposite direction. We call such annuities sinking
funds.Definition
A sinking fund is an annuity for which the amount of the payments is determined by the
future value desired.The difference between a sinking fund and a garden variety annuity is one of attitude, not
substance. Mathematically, there is no difference between a sinking fund and any other
annuity. The same terms and formulas we developed for annuities in general apply equally
well whether the annuity is a sinking fund or not.Example 4.3.1 Suppose Calvin has set for himself a goal of having $10,000 in a
savings account in 5 years. He plans to make equal deposits to the account at the end
of each month, and expects the account to earn 3.6% interest. How much should each
of his deposits be?Using the future value formula, we get:FV PMT s _n (^) | (^) i
$10,000 PMT s 60 | (^) .003
$10,000 PMT(65.63160098)
Dividing through both sides to solve for PMT gives:
PMT $152.37
Note that in this example, we did not show the work done to calculate s 60 (^) .003. In Section 4.2
we got quite a bit of practice with these, and so we are assuming at this point that everyone
is able to work out the value of any future value annuity factor needed. From this point
forward in the text, when an annuity factor is required we will give its value, but not show
the steps done to obtain it. If you are having difficulty calculating these factors yourself,
you should go back to Section 4.2 for additional practice before continuing on with the
remainder of this chapter.
Sinking funds for annuities due work in much the same way:
Example 4.3.2 Shauna owns a software development company, and as part of a
new product she has licensed the right to include in it some code owned by her friend
Elena. To allow Shauna time to develop and market the product, Elena has agreed to
wait 2 years before getting the $10,000 Shauna has agreed to pay. If, in anticipation
of paying Elena, she decides to make equal deposits at the start of each quarter into
an account paying 4.8% how much should each deposit be?
This represents a sinking fund, since her deposits are being made for the purpose of
accumulating a specifi c future value—the $10,000 she will need to pay Elena. Since she is
making these deposits at the start of each quarter, though, this is an annuity due and so we
need to begin with the annuity due formula.
FV PMT s n (^) | (^) i (1 i)
$10,000 PMT s 8 | (^) .012 (1.012)
$10,000 PMT(8.344186128)(1.012)
$10,000 PMT(8.444316362)
PMT $1,184.23
4.3 Sinking Funds 163
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