The Mathematics of Money

172 Chapter 4 Annuities

FORMULA 4.4.3

The Present Value Annuity Factor

a _n|i 

s _n|i

___(1  i) (^) n
Example 4.4.4 Use Formula 4.4.3 to fi nd a n| (^) i when n
36 and I
0.0035.
This formula depends on sn (^) | (^) i , so we will start by fi nding that. We calculate this just the same
as we have been doing all along:
sn _ (^) | (^) i

(1  i)n  1
___i

(1  0.0035)^36  1

_____0.0035
38.29504816

Now, we add on the extra step of dividing. Following Formula 4.4.3, we get

an _ (^) | (^) i

s _n (^) | (^) i
_(1  i) (^) n
__(1 38.29504816 0.0035) 36
33.76891092
For entry in the calculator, your best bet is to calculate the future value factor however you
have been doing it, and then once you have that result on the screen, follow step 2 below:
Operation Result
Steps to evaluate s _n (^) | (^) i 39.29504816
/(1.0035)^36
33.76891092
Notice that the values of i and n here are the same as they were in the example we used in
Example 4.4.3, Jon’s car loan. We can confirm that our work here was correct by using this
to find Jon’s car payment, using the present value factor. We will hopefully end up with the
same monthly payment!
Example 4.4.5 Use the annuity factor calculated in Example 4.4.4 to fi nd the
monthly payment on an $8,000 car loan at 4.2% for 3 years.
PV
PMT a _n (^) | (^) i
$8,000
PMT(33.7681092)
As we have done in many other similar situations, we divide both sides through to arrive at
the payment:
$8,000
___33.76891092

PMT(33.76891092)

__33.76891092

PMT
$236.90

The fact that this is the same payment that we arrived at before should give us some confi-

dence that we are doing things correctly.

An Alternative Formula for a _n (^)  (^) i (Optional)
Formula 4.4.3 is not the traditional formula for a _n (^) |i. It does, however, offer some advantages
over the traditional formula, especially in that it builds on the work that we have already
done to find future value factors. To calculate the present value factor of an annuity, we
calculate the future value annuity factor (which at this point should be old hat), and then
just add the one additional step of dividing it by (1  i)n.
While the author’s personal preference is to take that approach, the more traditional for-
mula can certainly be used just as well. We will briefly derive it here, and then demonstrate
its use.^2 If we begin with the formula:
a _n (^) | (^) i

s _n (^) |i
___(1  i) (^) n
(^2) The reader can skip the derivation and jump to the fi nal formula without losing anything other than the pleasure
of the mathematics.