150 Chapter 4 Annuities
The obvious shortcoming of this approach is that a reasonably sized table can list only a limited
number of interest rates and number of payments. If the interest rate was 6^1 ⁄ 2 %, or the time
period 35 years, we’d be out of luck. The obvious solution is to get a bigger table, but a table
that gave every possible rate together with every possible number of payments would be cum-
bersome. Still, this has historically been a commonly used approach, and many libraries still
have in their collections books containing lengthy tables of annuity and other financially use-
ful factors. Fortunately technology has largely eliminated the need for such monstrous tables.
That being said, such tables do exist. At most large bookstores you can find books for
sale that include such annuity factor tables, and they are available on some Internet sites
as well.Finding Annuity Factors Efficiently—Calculators and Computers
Some calculators and software packages have features to automatically calculate annu-
ity factors. This is not a standard feature on most calculators on the market today, but it
is not an unusual feature either, particularly on models sold specifically as “business” or
“financial” calculators. For the course that you are taking, you may or may not be using a
calculator with this feature. If you are, your instructor or the calculator’s owner’s manual
can provide the details of how to obtain these factors from your calculator.
It is also possible to quickly obtain annuity factors by using many common types of
business software, such as Microsoft Excel.
It is not necessary to buy a special calculator or have access to a computer to findannuity factors, however. We will derive a formula for s _n (^) |i that can be worked out on almost
any calculator.
A Formula for s _n (^) (^) i
Deriving the s _n (^) |i formula requires a bit of algebra and a bit of cleverness. Readers without
a solid algebra background can skip over this derivation can go directly to Formula 4.2.2
(though doing so will deprive you of all the fun... .)
Let’s look back at the bucket approach’s calculation of s _n (^) |i. The values that we added up
to get s _n (^) |i were:
s _n|i 1(1.072)^4 1(1.072)^3 1(1.072)^2 1(1.072) 1
Now (for entirely nonobvious reasons) let’s multiply both sides of this equation by 1.072
to get:
(1.072)s _n|i 1.072[1(1.072)^4 1(1.072)^3 1(1.072)^2 1(1.072) 1]
Distributing on the right side, and neatening things up a bit, we get:
(1.072)s _n|i (1.072)^5 (1.072)^4 (1.072)^3 (1.072)^2 (1.072)
Now (once again mysteriously) suppose we add 1 to both sides to get:
(1.072)s n|i 1 (1.072)^5 (1.072)^4 (1.072)^3 (1.072)^2 (1.072) 1
Now it looks like we’ve have really made a mess—that is, until we notice that most of the
right-hand side is something we’ve seen before:
(1.072)s n (^) |i 1 (1.072)^5 (1.072)^4 (1.072)^3 (1.072)^2 (1.072) 1
Hey, that’s s n (^) (^) i!
So:
(1.072)s n (^) |i 1 (1.072)^5 s _n (^) |i
Now, if we subtract 1 from both sides, and also subtract s n (^) |i from both sides, this becomes:
(1.072)s n (^) |i s _n (^) |i (1.072)^5 1