Copyright © 2008, The McGraw-Hill Companies, Inc.
It’s nice to have a formula, but at this point alarms should be going off in your head. Sure,
we do have a general formula now, but in order to use it we need to calculate the annuity
factor, and the way we did that was not much of an improvement over just using the bucket
approach in the first place! At least this formula would nicely allow us to change the pay-
ment amount if we wanted to without having to recalculate everything from scratch. But a
far greater improvement would be some more efficient way of getting the annuity factors.
Fortunately, we can do this.Finding Annuity Factors Efficiently—Tables
One way to find annuity factors is simply to have a table listing the factors for certain inter-
est rates and numbers of payments. An example of such a table is given below:SAMPLE TABLE OF ANNUITY FACTORS
Number of
Payments
(n)RATE PER PERIOD (i)
0.50% 1.00% 2.00% 4.00% 6.00% 8.00%1 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
2 2.00500000 2.01000000 2.02000000 2.04000000 2.06000000 2.08000000
3 3.01502500 3.03010000 3.06040000 3.12160000 3.18360000 3.24640000
4 4.03010012 4.06040100 4.12160800 4.24646400 4.37461600 4.50611200
5 5.05025063 5.10100501 5.20404016 5.41632256 5.63709296 5.86660096
8 8.14140879 8.28567056 8.58296905 9.21422626 9.89746791 10.63662763
10 10.22802641 10.46221254 10.94972100 12.00610712 13.18079494 14.48656247
15 15.53654752 16.09689554 17.29341692 20.02358764 23.27596988 27.15211393
20 20.97911544 22.01900399 24.29736980 29.77807858 36.78559120 45.76196430
25 26.55911502 28.24319950 32.03029972 41.64590829 54.86451200 73.10593995
30 32.28001658 34.78489153 40.56807921 56.08493775 79.05818622 113.28321111
40 44.15884730 48.88637336 60.40198318 95.02551570 154.76196562 259.05651871
50 56.64516299 64.46318218 84.57940145 152.66708366 290.33590458 573.77015642This table allows us to look up the value of the appropriate annuity factor for any of the
interest rates and values of n that the table includes. For example, if we need the annuity
factor for a 15-year annual annuity at 8%, we would look in the n 15 row and 8% column
and see that the value is 27.15211393. Using our annuity factor notation, we would writes 15 __ (^) (^) .08 27.15211393.
The following example will illustrate the use of a table factor to find the future value
of an annuity.
Example 4.2.1 How much will I have as a future value if I deposit $3,000 at the end
of each year into an account paying 6% compounded annually for 30 years?
The payments are equal and at regular intervals, and their timing is at the end of each
period, so we have an ordinary annuity. Using Formula 4.2.1 with the annuity factor from
the table above:
FV PMT s _n (^) | (^) i
FV ($3,000)s __ 30 | (^) .06
We can fi nd the value of the needed annuity factor from the table. We look in the n 30 row
and the 6% column and fi nd that the value of the annuity factor is 79.05818622. We can
then plug this into our formula and fi nish the calculation:
FV ($3,000)(79.05818622)
FV $237,174.56
4.2 Future Values of Annuities 149