Copyright © 2008, The McGraw-Hill Companies, Inc.
Since this is a 30-year loan with monthly payments, n (30)(12) 360 and i 0.084/12
0 .007. We know that we will need the present value annuity factor for this, so we fi rst will
fi nd that. This factor may be found from a table, from calculator programs, or by using one
of our formulas. We will show the calculation for each of the two formulas; of course, when
doing these problems yourself, you do not need to calculate the factor more than once.Using Formula 4.4.3:We fi rst fi nd the future value annuity factor just as we have been doing:sn _ (^) | (^) i
(1 i)n 1
___i
(1 0.007)^360 1
_____0.007 1,617.137554
Then, plugging this into Formula 4.4.3, we get:an _ (^) | (^) i
s _n (^) | (^) i
___(1 i) (^) n
1,617.137554
_____________
(1 0.007)^360131.2615606
Using Formula 4.4.4:Plugging values into the formula we get:an _ (^) | (^) i
1 (1 i)n
___i
1 (1 0.007)^360
__0.007 131.2615606
Whichever way we fi nd the annuity factor, we can now use it to complete the problem.PV PMT a _n (^) | (^) i
PV $650 a 360 | (^) .007
PV ($650)(131.2615606)
PV $85,320.01
Assuming that on an amount this large we can ignore the pennies, we conclude the most she
can afford to borrow is $85,320. In addition to what she borrows, she has her $7,500 to put
toward the purchase price, and so the most she can afford to pay is $85,320 $7,500
$92,820.
In the remainder of this section, we will not show the work to calculate the annuity factors,
but you should calculate them yourself and make sure that your factors agree with those
used in these examples. The exercises at the end of this section also offer many opportuni-
ties to practice and get comfortable with calculating the present value annuity factor.
Finding Total Interest For a Loan
Finding the total interest paid on a loan is similar to, but not exactly the same as, finding
the total interest in an annuity’s future value.
Example 4.4.8 Pat and Tracy are buying a house, and will need to take out a
$158,000 mortgage loan. They plan to take out a standard 30-year mortgage loan,
on which their interest rate will be 7.2%. If they make all their payments as scheduled,
how much total interest will they pay over the course of the 30 years?
First, we need to determine their monthly payments. (While the problem did not explicitly
state that payments would be monthly, that is usually the case and can be assumed unless
otherwise specifi ed.)
We know that we will need the present value annuity factor. Since the payments are monthly
for 30 years, n 30(12) 360 and i 0.072/12 0.06. Using a table, calculator pro-
gram, or one of the annuity factor formulas, we can calculate that a 360 (^) | (^) .006 147.3213568.
Putting this to use, we get:
PV PMT a _n (^) | (^) i
$158,000 PMT a ___ 360 | (^) .006
$158,000 PMT(147.321356802)
PMT $1,072.49
4.4 Present Values of Annuities 175
cf