Mortgage Amortization Calculator

A mortgage is an annuity: the same payment every month, sized so that the balance reaches exactly zero on the last one. Setting the present value of all payments equal to the loan gives

$M = P \cdot \frac{i}{1 - (1+i)^{-n}} \;=\; P \cdot \frac{i(1+i)^n}{(1+i)^n - 1}$

• $P$ — amount borrowed, which is the purchase price minus the down payment
• $i$ — the periodic rate: annual rate $\div$ 12 for monthly payments
• $n$ — the number of payments: $12 \times$ years, so 30 years is $n = 360$
• $M$ — principal-and-interest payment, unchanging for the whole term

The payment is fixed; the split inside it is not. Starting from $B_0 = P$, each row of the schedule is three lines of arithmetic:

$I_k = B_{k-1} \cdot i, \qquad P_k = M - I_k, \qquad B_k = B_{k-1} - P_k$

Interest is charged on the balance, which shrinks every month, so $I_k$ falls and $P_k$ rises across the table. A correct schedule closes at $B_n = 0$.

Balance at any point

$B_k = P(1+i)^k - M \cdot \frac{(1+i)^k - 1}{i}$

The first term is what the debt would have grown to untouched; the second is the accumulated value of the payments made. Their difference is what you still owe after $k$ payments.

Total interest

$\text{total interest} = nM - P$

This assumes every scheduled payment is made. Pay the loan off early and the true total is smaller, because the later interest is never charged.

Why the term dominates the total

Halving the term raises $M$ by much less than double while cutting total interest sharply, because a shorter term means the balance — and therefore every $I_k$ — collapses faster. That trade-off is arithmetic, and the example below shows it on real numbers.

What this page does not include

$M$ here is principal and interest only. A real monthly housing payment may also carry property tax, homeowners insurance, mortgage insurance and association dues, and those amounts depend on your property, insurer and jurisdiction. Enter them and the solver will add them; it cannot look them up, and it is a math tool rather than a lender or advisor.

• Using the annual rate as $i$: a 6% loan has $i = 0.06/12 = 0.005$ per month. Leaving $0.06$ in the formula inflates the payment enormously.
• Setting $n$ to the number of years: a 30-year monthly mortgage has $n = 360$.
• Amortising the purchase price: $P$ is the amount financed. A \375{,}000$home with$$75{,}000$down amortises$$300{,}000$.
• Charging interest on the original principal each row: $I_k$ uses $B_{k-1}$, the current balance. Using $P$ every row is simple interest and badly overstates the cost.
• Comparing APR to the note rate: APR folds fees into a single figure and is not the rate that belongs in this formula. Use the note rate for $i$.
• Expecting the last row to land on zero unaided: rounding $M$ to the cent leaves a residue of a few cents; lenders absorb it in the final payment.