Math Dude Web Bonus
By
Jason Marshall, PhD
Jason Marshall, PhD
June 20, 2011
1-minute read
How to Derive the Recurring Compound Interest Formula
The value of a recurring investment after 1, 2, and 3 years can be found using:
- Value at the End of Year 1 = $2000 x 1.05^1
- Value at the End of Year 2 = $2000 x 1.05^1 + $2000 x 1.05^2
- Value at the End of Year 3 = $2000 x 1.05^1 + $2000 x 1.05^2 + $2000 x 1.05^3
Do you see a pattern here? We can use the distributive property to write it:
- Value at the End of Year 1 = $2000 x (1.05)
- Value at the End of Year 2 = $2000 x (1.05 + 1.05^2)
- Value at the End of Year 3 = $2000 x (1.05 + 1.05^2 + 1.05^3)
Now do you see the pattern? After what we’ll call N years, the value would be:
- Value at the End of Year N = $2000 x (1.05 + 1.05^2 + 1.05^3 + … + 1.05^N)
And that last part in brackets turns out to be a mathematical series that can be rewritten in a very convenient way which allows us to write the general formula for the value of a recurring investment:
(future value) = (previous value) x [ (1+rate) / rate ] x [ (1+rate)^years – 1 ]
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