The Mathematics of Money

(Darren Dugan) #1

148 Chapter 4 Annuities


of a final way to handle these situations, though, as we will see before long, the bucket
approach can be adapted to efficiently handle those other situations.

Approach 3: The Annuity Factor Approach


There is a third alternative. Suppose that instead of payments of $1,200, the payment were
instead $3,600, three times as much. It makes sense that the future value would then be
three times as much. So to find the future value of the $3,600 annuity, we wouldn’t need
to start from scratch, we could just multiply the $1,200 annuity’s future value by 3 to get
FV  3($6,928.48)  $20,785.44. If you have doubts about this, you can verify it for your-
self by working through the future value of a $3,600 annuity using either of the approaches
we used above.
In general, a larger or smaller payment changes the future value proportionately.
Exploiting this, we can define the future value annuity factor.

Definition 4.2.1
For a given interest rate, payment frequency, and number of payments, the future value annu-
ity factor is the future value that would accumulate if each payment were $1. We denote this fac-

tor with the symbol s n (^) |i , where n is the number of payments and i is the interest rate per payment
period. [For convenience, this symbol can be pronounced “snigh” (rhymes with “sigh”).]
When we find annuity factors, we carry the results out to more than the usual two decimal
places, to avoid losing too much in the rounding. It is conventional not to write a dollar
sign in front of the annuity factor, since, among other reasons, using the dollar sign will
look strange with numbers carried out to more than two decimal places. Now, let’s revisit
our problem by using an annuity factor. Since there are 5 annual payments, and since the
interest rate is 7.2% per year, the annuity factor we are looking for is s
5  (^) .072. Using the
bucket approach, this time with payments of $1, we get
Payment from Year Payment Amount Years of Interest Future Value
1 1.00 4 1.32062387
2 1.00 3 1.23192525
3 1.00 2 1.14918400
4 1.00 1 1.07200000
5 1.00 0 1.00000000
Grand total 5.77373311
Since our payments were actually $1,200, or 1,200 times as much, it is logical that the
future value we want is:
FV  $1,200(5.77373311)
FV  $6,928.48
The Future Value Annuity Formula
It should be apparent that if the amount of the payments were something other than $1,200, if the
time period were something other than 5 years, and if the interest rate were something other than
7.2%, the same principles would apply. This allows us to generalize a future value formula:


FORMULA 4.2.1


The Future Value of an Ordinary Annuity

FV  PMTs _n|i

where
FV represents the FUTURE VALUE of the annuity
PMT represents the amount of each PAYMENT
and

s _n (^) | (^) i is the FUTURE VALUE ANNUITY FACTOR (as defi ned in Defi nition 4.2.1)

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