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Of course, given the size of the deposits involved, the district would most likely round the
payments to $137,697, or even $137,700. On a payment of this size, rounding to the nearest
dollar does not seem unreasonable. From this point forward in the text, we will feel free to
round large values to the nearest dollar in our final answers. This of course raises the ques-
tion of how large is “large”? This is a judgment call, but from here on in we will use the
following rule of thumb: when a dollar amount in a final answer is larger than $10,000, we
will feel free to round to the nearest dollar. Of course, it is also still correct to keep dollar
and cent accuracy if you prefer.
Notice that the interest rate that the district pays to the investors is completely indepen-
dent from the rate that it receives from the bank on its deposits. While the sinking fund was
required as a term of the loan, financially it is a separate account and there is no necessary
connection between the rates.
Situations such as these are actually the source of the term sinking fund. The purpose
of the deposits is to take care of that looming future value, and so, with each deposit, more
and more of the future obligation is taken care of, and the amount of the obligation that is
not covered declines, or “sinks.” Hence the name.Sinking Funds and Retirement Planning
One particularly useful application of sinking funds is in retirement planning. It is hard to
avoid being bombarded with news reports and commentaries about the pressing need for
all of us to “save more for retirement.” Many companies offer 401(k) or similar retirement
savings plans and encourage workers to use them to save, and the perpetual debate over
the future of the U.S. Social Security system includes plenty of discussion about ways to
require or provide incentives for people to save for their long-term financial needs.
While the benefits of saving money over time are obvious, it is less obvious just how
much saving is enough. On general principle, the answer should probably be “as much as
possible,” but that is hopelessly vague and not especially helpful. Treating a retirement
goal as a sinking fund can help to provide a clearer answer.
Suppose that Joe is now 25 years old, and hopes to be able to retire 45 years from now,
at age 70. To ensure that he can do so comfortably, he has decided to start making deposits
into an investment account which he assumes can earn an average of 9%. The deposits will
be automatically deducted from his paycheck when he receives it on the first and fifteenth
of each month. To get some sense of how large his deposits should be, he has set a goal
of having $1,000,000 in the account. These deposits will form an annuity, and since he is
letting the goal future value determine their size, this is a prime example of a sinking fund.
We can thus work out the size of each payment:Example 4.3.4 Using the scenario described above, determine how much Joe should
have deducted from each paycheck to reach his goal.First, to determine n and i. Joe is paid twice a month (semimonthly), which would be
24 times per year (12 months times 2 paychecks/month). Over the course of 45 years, that
is n 45(24) 1,080 deposits. For i, we assume as usual that interest compounds at the
same frequency as the payments, or semimonthly, so i 0.09/24 0.00375.FV PMT s _n (^) | (^) i
$1,000,000 PMT s ___ 1080 .00375
$1,000,000 PMT(14923.819794054)
PMT $67.01
To reach his goal on the basis of these assumptions, Joe needs to be deducting $67.01 from
each paycheck.
This answer may seem surprising; $67.01 out of every paycheck is not exactly peanuts, but
it seems awfully small compared to future value that sounds like a lottery jackpot. We can
get some insight into how this happens by finding the total interest Joe will earn.
4.3 Sinking Funds 165