Loan Payment (Amortized Loan or Mortgage)

Every amortized loan is a promise to hand over the same amount, on the same day, until the balance is gone. Working out that amount looks hard and is not: the payments form a geometric series, and summing it gives $M = P i / (1 - (1+i)^{-n})$. The classic case is a $200,000 mortgage at 6% nominal over 30 years, which means $i = 0.06/12 = 0.005$ per month and $n = 360$ payments, giving $1,199.10.

The single most common mistake is feeding it an annual rate alongside a monthly count. The rate and the payment count must describe the same period, always. A second trap is national: Canadian mortgages are compounded semi-annually by law, so a 6% Canadian mortgage has a monthly rate of $1.03^{1/6} - 1 \approx 0.004939$, not 0.005, and the payment comes out slightly lower than the American figure above.

A quick sanity check lives inside the formula. Set $n = 1$ and it collapses to $M = P(1+i)$, which is exactly right: one payment repays the principal plus a single period's interest. Note also which variable is missing here. You can solve for the principal, and you can solve for the number of payments by taking logs, but there is no closed-form solution for the interest rate. Recovering $i$ from a payment means finding the root of a degree-$n$ polynomial, which is why every spreadsheet computes it by iteration.