Amortization Calculator - Derivation of The Formula

Derivation of The Formula

The formula for the periodic payment amount is derived as follows. For an amortization schedule, we can define a function that represents the principal amount remaining at time . We can then derive a formula for this function given an unknown payment amount and .

We can generalize this to

Applying the substitution (see geometric progressions)

We end up with

For payment periods, we expect the principal amount will be completely paid off at the last payment period, or

Solving for A, we get

\; A = P \frac{r^n (r-1)}{r^n-1} = P \frac{(i+1)^n ((i+\cancel{1})-\cancel{1})}{(i+1)^n-1} = P \frac{i (1 + i)^n}{(1 + i)^n-1}

or

After substitution and simplification we get

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