In my economics class, our textbook provides these massive tables for looking up these compound interest factors to put in worth equations. They look something like this:

table for compound interest factors of 1/4% interest, with column names "n,F/P,P/F,A/F,A/P,F/A,P/A,A/G,P/G,n"

There has to be some formula for deriving these factors, right? Where are they getting these values from?

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    Math? I mean, really, these exponential formulae have been known for centuries (even if not applied to money). The only reason they're still in a book in 2022 is anachronism (or a visual example of the time value of money) because for the past 40 years they'd be calculated by a spreadsheet.
    – RonJohn
    Nov 2, 2022 at 16:36
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    @RonJohn - don't forget the HP-12c calculator! OK, not nearly as much fun as the HP-15, but still...
    – Jon Custer
    Nov 2, 2022 at 21:12
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    One reason for including tables in textbooks is so you're familiar with them for exams. Exams often have restrictions on the calculators that can be used, and some of the formulae used here would take a fair few steps on a simple scientific calculator such as is often allowed. P×(1+i)^n for interest i and time period n is basic enough but some of the others less so
    – Chris H
    Nov 3, 2022 at 13:15

1 Answer 1


It's actually really simple, despite how they make it appear!

Compound interest means nothing more than the fact that the interest of each month is calculated based on the outstanding balance at that month, rather than at the beginning. So if you start with a principal of P, at an interest rate of x (phrased as a plain number, not a percent -- 3% is x=0.03), the amount grows to P(1+x) after one month. The "1" is from the original amount and the "+x" accounts for the additional amount due to the interest from previous months. It becomes P(1+x)² after 2 months, P(1+x)³, and in general, after n months it is P(1+x)ⁿ. Of course, we typically quote the interest rate as a yearly figure, but compound more often than that. In that case, we just have to compute the interest per compounding period. If we compound C times per year, it's simply P(1+x/C)ⁿ.

This is a geometric series. Using the letters wikipedia uses, the initial value, a, is equal to P, and the ratio, r, is equal to (1+x/C). With that information in hand, we can use the equation for the sum of a geometric series to compute all of the values needed.

There is also "continuously compounded" interest. In this case, we ask the question "what if C just gets larger and larger, compounding faster and faster?" It turns out that that doesn't make the outstanding balances baloon without bound. It actually approaches a finite value (which I think is kind of cool).

In practice we use tables like this because there's a lot of room for error when using the equations. Fat-fingering is pretty easy. We also often leverage Excel's functions for calculating these things.

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    A key point here is that in the limit as $C$ goes to infinity you get exponential growth $e^{rt}$. Nov 2, 2022 at 14:48
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    " If we compound C times per year, it's simply P(1+x/C)^n" That should be P(1+x/C)^C if it's the amount after one year. Nov 2, 2022 at 21:13

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