176 Chapter 4 Annuities
This payment includes both principal and interest, but doesn’t reveal how much goes to each
category. To determine the total interest they will pay, we can use an indirect approach similar
to the one we used with future values. This loan requires 360 payments of $1,072.49, and
so the total they will pay over the entire life of the loan is (360)($1,072.49) $386,096.40.
Since they borrowed $158,000, we know how much of that total must go toward principal,
and so the rest must be interest. Thus:To tal interest $386,096.40 $158,000 $228,096.40Over the course of the 30-year loan, Pat and Tracy will be paying an awful lot of interest. In
fact, more of their money will go to pay interest than to pay for the house! It is good to be
aware of this, even if the knowledge is depressing. The good news, though, is that knowing
how to work with annuities allows us to consider alternatives and choose the most favorable
options. Since debt is an unfortunate but unavoidable fact of both business and life, being
able to work through the mathematics that underlies it holds obvious advantages—you
generally do better at games where you know how to play.Example 4.4.9 Suppose that Pat and Tracy from Example 4.4.8 decided instead
to try to save some of the interest by paying off their loan more quickly. If they were
to go with a 15-year loan rather than a 30-year loan, how much higher would their
payments need to be (assuming the same interest rate)? How much interest would
they save?Common sense says that to cut the term in half, they would need to double their payments.
Sometimes, though, common sense is wrong.PV PMT a _n (^) | (^) i
$158,000 PMT a ___ 180 | (^) .006
$158,000 PMT(109.884466016)
PMT $1,437.87
So they would need to increase their payments by $1,437.87 $1,072.49 $365.38.
While that is quite a bit more of a monthly payment, it is nowhere near double the original.
(We will explore the reasons for this in Section 4.5.)
Under this scenario, their total payments would be (180)(1,437.87) $258,816.60, and so their
total interest would be $258,816.60 $158,000 $100,816.60. Compared to the 30-year
schedule, this is an interest savings of $228,096.40 $100,816.60 $127,279.80.
Interest is the incredibly powerful force that allowed us to accumulate astonishingly
large future values from relatively small payments over long periods of time. But when
we are the borrowers, instead of earning that interest, we pay it. By paying extra on
the loan, we reduce the principal more quickly, and thus reduce the interest that we
must pay dramatically. We will be able to see this all worked out in greater detail in
Section 4.5.
Other Applications of Present Value
Loans are by no means the only uses for annuity present values, however. The following
examples illustrate some other uses, which have nothing to do with loan payments.
Example 4.4.10 Charlie has accumulated $557,893 in his 401(k) retirement savings
account. Now that he has retired, he is planning to start using this fund to provide him
with some income. He expects that the account can continue to earn 4^1 ⁄ 2 % interest,
and plans to make monthly withdrawals from the account over the next 20 years.
Under these assumptions, how much can he afford to withdraw each month?
It is tempting to just take the $557,893 and divide it by 240, but that would ignore the inter-
est that the account will be earning over those 20 years. In essence, Charlie is talking about
using his $556,893 as the present value of a 20-year annuity. And so:
cf