Why Is There A Minus 1 In the Annuity Formula

I was learning about the annuity formula in a math class I was taking. 

Math and formulas are fun and really cool. Formulas, especially, are cool because they allow you to do complicated, time-consuming things much more quickly than you could without them. 

I am a nerd and I love that moment when I fully understand a formula and how it came to be, the proof of the formula. 

But I was struggling to understand the Annuity Formula. Much of it made sense. It is basically just compounding interest applied to many equal payments over time, and then summing the payments and various amounts of interest accrued over time. 

But I didn’t understand the -1 (minus one) part of the equation. 

The equation looks like this:

Annuity Formula

Here are the variable declarations.
A = the final value of all investment and accrued interest
m = the amount of each annuity payment
r = the annual interest rate
n = the number of times per year that payments are made
t = time, in years, over which the investment spans

So all of it makes sense. 

The formula is based on the general “Sum of a Geometric Series” formula. In the “Sum of a Geometric Series”, each consecutive term is connected by a common ratio. Each term gets the next term in the series when multiplied by the same amount (which is different from the “Sum of an Arithmetic Series” in which each term gets the next by adding or subtracting the same amount). 

Compound Interest Formula
part is just the compound interest formula. 

I just couldn’t figure out why there was the -1 in the numerator. 

I searched Google and Youtube for an explanation video which would prove the equation, and luckily, I found one that cleared everything up for me.

What I realized after I watched the video was that in an ordinary annuity, payments are made at the end of each month. That means that for month number 0, there is no payment and no accrued interest. 

The equation is based on numbers on a number line, and the number line has to start at zero, but since nothing happens until the end of that month a minus one subtracts that month. 

That is where I was getting hung up. 

I hope that this clarification helps you to understand and apply this very excellent formula.

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