The Math Behind a Mortgage: From Interest Rates to Amortization

A 30-year mortgage payment stays exactly the same from month 1 to month 360. But what that payment buys changes every single month. On a typical loan, your first payment is more than 85% interest, and you don't pay more principal than interest in a month until you are nearly 20 years in.

That isn't a trick the bank plays on you. It falls straight out of one formula. Once you understand that formula, you'll see why an extra $200 a month can wipe out years of payments.

Equal Payments, Unequal Progress

Most borrowers assume a fixed payment chips away at the balance at a steady pace. It doesn't. Each month's interest is charged on the balance that is still outstanding. Early on the balance is huge, so interest eats most of the payment. As the balance shrinks, interest shrinks with it, and more of the same payment goes to principal.

Take a $400,000 loan at 6.5% for 30 years:

• Monthly payment: $2,528.27
• Month 1 interest: $2,166.67
• Month 1 principal: $361.61
• Total interest over 30 years: about $510,178

You read that right. Over the life of the loan you pay more in interest than you borrowed. And the month where principal finally beats interest is month 233, more than 19 years in.

The Amortization Formula

The fixed payment comes from the annuity formula:

M = P × r(1 + r)^n / ((1 + r)^n − 1)

Where:

• M = monthly payment
• P = principal (the amount borrowed)
• r = monthly interest rate (annual rate ÷ 12)
• n = number of payments (years × 12)

For our example, r = 0.065 / 12 ≈ 0.0054167 and n = 360. Plug them in and you get $2,528.27.

Where does this formula come from? It's the payment that makes the present value of all 360 payments equal to the amount you borrowed. Each future payment is discounted back to today by (1 + r)^k, and the sum of those discounted payments is a geometric series. Solve that series for M and the formula above drops out. You'll find the same structure in the financial formulas reference.

Building an Amortization Schedule

An amortization schedule is just this loop, repeated n times:

interest  = balance × r
principal = M − interest
balance   = balance − principal

Here are a few rows for the $400,000 loan:

Notice how the principal column grows. Each month it rises by a factor of exactly (1 + r). Principal payments form a geometric sequence, the same exponential growth you see with compound interest, just running in the borrower's favor.

Why the Rate Matters More Than It Looks

A 1-percentage-point change in the rate feels small. On a 30-year loan, it isn't:

Going from 6% to 7% adds about $263 a month, and roughly $94,700 over the life of the loan. The exponent n = 360 magnifies every basis point. It's the same exponential effect covered in our post on the mathematics of compound interest.

Term Length: 15 Years vs. 30 Years

Shortening the term cuts total interest dramatically:

• 30-year at 6.5%: $2,528.27/month, ~$510,178 interest
• 15-year at 6.5%: $3,484.43/month, ~$227,197 interest

The 15-year payment is about 38% higher, but total interest falls by more than $280,000. Real 15-year loans usually come with a lower rate too, which widens the gap even further.

The Power of Extra Payments

Here is the most practical takeaway. Every extra dollar you pay goes 100% to principal, and it removes all the future interest that dollar would have generated.

Add $200 a month to the $400,000 loan at 6.5%:

• Payoff drops from 360 months to 293 months (about 5.5 years sooner)
• Total interest drops from $510,178 to **$398,286**

That's roughly $112,000 saved for about $58,600 in extra payments. Extra payments made early do the most work, because they come off the balance while it is largest.

An Insider View: Why Banks Love the 30-Year Loan

The 30-year fixed-rate mortgage is mostly an American invention. It spread after the Federal Housing Administration was created in 1934 and Fannie Mae in 1938 made long, fully amortizing loans standard. Before that, many home loans were 5- to 10-year interest-only loans with a large balloon payment at the end, and that structure contributed to mass foreclosures in the Great Depression.

Economists Richard Green and Susan Wachter trace this history in their 2005 Journal of Economic Perspectives paper, "The American Mortgage in Historical and International Context." A key point from that work is that the long fixed-rate loan moves interest-rate risk from households to lenders and investors. That's one reason it usually costs more than shorter or adjustable loans. You're paying for the certainty that the payment will never change.

Two Concepts Worth Knowing

Present Value

Money received in the future is worth less than money today. The present value of a payment M made k months from now is M / (1 + r)^k. A mortgage is a promise to pay a stream of payments whose total present value equals the loan amount.

APR vs. Interest Rate

The Annual Percentage Rate (APR) folds fees and points into an effective rate by finding the r that makes the present value of your payments equal to the cash you actually received. Solving for it usually takes numerical methods, because the annuity formula can't be rearranged to isolate r algebraically.

Quick Answer: How Is a Mortgage Payment Calculated?

A fixed mortgage payment is M = P·r(1+r)^n / ((1+r)^n − 1), where P is the loan amount, r is the monthly rate and n is the number of months. Each payment first covers that month's interest (balance × r), and the remainder reduces principal.

Try Them Yourself

• Compound Interest Calculator: see exponential growth from the saver's side
• Simple Interest Calculator: compare with non-compounding interest
• Financial Formulas: annuities, present value and future value
• Logarithm Calculator: solve for the number of payments with logs
• Scientific Calculator: plug in your own P, r and n
• Finance Calculators: plan what happens after the mortgage is paid off

Run your own loan through the formula. Then try adding $100 or $200 a month and watch how many years disappear.