The eight factors
Each factor turns one kind of amount into another at interest rate i per period over n periods, with cash flows at the end of each period. Read (P/A, i, n) as “find P given A”: the amount you have, times the factor, gives the amount you want.
| Factor | Finds | Given | Formula |
|---|---|---|---|
| (F/P, i, n) | F at year n | one amount P at year 0 | x |
| (P/F, i, n) | P at year 0 | one amount F at year n | 1 / x |
| (F/A, i, n) | F at year n | A at years 1 to n | (x − 1) / i |
| (A/F, i, n) | A at years 1 to n | F at year n | i / (x − 1) |
| (P/A, i, n) | P at year 0 | A at years 1 to n | (x − 1) / (i x) |
| (A/P, i, n) | A at years 1 to n | P at year 0 | i x / (x − 1) |
| (P/G, i, n) | P at year 0 | gradient 0, G, 2G … (n − 1)G | (x − 1 − i n) / (i² x) |
| (A/G, i, n) | A at years 1 to n | the same gradient | 1/i − n / (x − 1) |
Tables are to four places; answers from the table or the exact formulas are both marked right.
An example from set EX4DFR
A fund pays a scholarship of $13,500 at the end of each year from year 5 to year 9, at 12% a year. That is 9 − 5 + 1 = 5 payments. (P/A, 12%, 5) = 3.6048 values them at year 4, one year before the first, and (P/F, 12%, 4) = 0.6355 brings that to year 0: P = 13,500 × 3.6048 × 0.6355 = 30,926.48, or $30,926. Discounting from year 5 instead gives $27,612, a common slip. Open set EX4DFR
Questions
How do I read factor notation like (P/A, 8%, 6)?
Read it as “find P given A” at 8% per period over 6 periods. Multiply the amount you have (A) by the factor to get the amount you want (P). If the units in the slash would not cancel, you have the factor upside down.
Why is the P/F for a deferred annuity one year less than the first payment?
(P/A) always gives the value one period before the first payment. For payments from year 5 to year 9, (P/A, i, 5) lands at year 4, so (P/F, i, 4) brings it back to year 0. Using year 5 is the most common slip.
When do I use the effective rate instead of the nominal rate?
Factor tables need the rate for one compounding period. For 12% compounded monthly with monthly payments, use 1% and count n in months. The effective annual rate, (1.01)^12 − 1 = 12.68%, is for comparing offers or for yearly cash flows.
My answer is a few units off the key. Is it wrong?
Probably not. Answers from the table, the exact formulas or a standard route such as (P/G)(A/P) for (A/G) are all marked right, and a unit or so off is marked Close.
Sources, checked 2 November 2026: the factor tables in C. S. Park, Contemporary Engineering Economics, Appendix A, Interest Factors for Discrete Compounding, and Blank and Tarquin, Engineering Economy, 4% table; B. Maddah, AUB INDE 301 notes on factors and gradients and nominal and effective rates; University of Alberta, Introduction to Engineering Economics: interest.