Future Value of an Ordinary (Simple) Annuity

Consider the ordinary annuity diagrammed previously, where you deposit $100 at the end of each year for 3 years and earn 5% per year.

To see how the annuity process works, consider the following table. 

Since this is an ordinary annuity the payments are made at the end of the month. The first payment of $100 is made at the end of the year and therefore no interest is earned for that year. At the beginning of year 2 there is $100 in the account. Interest for year 2 is $100 x 5% = $5. At the end of year 2 another $100 is deposited so the balance at the end of year 2 is $100 + $5 + $100 = $205. At the beginning of year 3 there is $205 in the account. Interest for year 3 is $205 x 5% = $10.25. At the end of year 3 another $100 is deposited so the total balance at the end is $205 + $10.25 + $100 = $315.25.

Here's the same example presented as a timeline

As you can see from the time line diagram, with the step-by-step approach we apply the following equation, with n = 3 and i = 5%: 

FVA = PMT(1 + i)ⁿ⁻¹ + PMT(1 + i)ⁿ⁻² + PMT(1 + i)ⁿ⁻³

FVA = $100(1.05)³⁻¹ + $100(1.05)³⁻² + $100(1.05)³⁻³

FVA = $100(1.05)² + $100(1.05)¹ + $100(1.05)⁰

FVA = $100(1.1025) + $100(1.05) + $100(1)

FVA = $110.25 + $105.00 + $100

FVA = $315.25

From the above process we derive the future value formula of:

FVA = PMT[(1+i)ⁿ -1/i]

Here's a clearer depiction

Where FVA (or FV) = Future Value of Annuity

PMT = Payment

i = Interest rate (adjust if compounded more than once a year)

n = number of periods (adjust if compounded more than once a year)

Going back to our earlier example, using the formula would look like this:

FVA = $100[(1+0.05)³ -1/0.05]

FVA = $100[(1.05)³ -1/0.05]

FVA = $100[1.157625 - 1/0.05]

FVA = $100[0.157625/0.05]

FVA = $100[3.1525]

FVA = $315.25

Adjusting for Multiple Compounding Periods

If the 

https://efficientminds.com/wp-content/uploads/2012/08/web_chapter_28_tvm.pdf

http://www.csun.edu/~jpd45767/303/4%20-%20The%20Time%20Value%20of%20Money.pdf

https://www.cuemath.com/ordinary-annuity-formula/

https://financetrain.com/future-value-and-present-value-of-ordinary-annuity

https://opentextbc.ca/businesstechnicalmath/chapter/9-4-annuities/