# Formula for Amortisation: How Loan Math Really Works

Published: 2026-04-19
Author: Warren Team
URL: https://www.heywarren.com/blog/amortization-formula

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Take out a $300,000 30-year mortgage at 7% and your first month's payment is $1,996, but only $246 of that goes to principal. The other $1,750 is pure interest to the bank. By month 360, the proportions completely flip: nearly the entire payment chips away at principal while interest shrinks to almost nothing. That dramatic shift is not random. It follows a precise mathematical pattern called the amortization formula, and once you understand it, every loan you ever take out becomes transparent.

The problem is that most borrowers sign on the dotted line without ever seeing the math. They know the monthly payment but have no idea how much total interest they will pay, how a single extra $200 monthly principal payment could save tens of thousands, or why refinancing might (or might not) make sense. The formula for amortisation answers all of these questions with arithmetic.

This guide breaks down the amortization formula step by step, walks through worked examples on a real $300K mortgage, shows the Excel functions every analyst uses, and reveals the strategies that can shave years off your loan. Warren is the AI financial advisor at heywarren.com, built to give you conflict-free guidance grounded in the same math used by mortgage brokers, CFPs, and lenders.

## What Loan Amortization Actually Means

Loan amortization is the gradual reduction of a loan's principal balance through regular periodic payments that combine principal and interest. Each payment stays constant in total dollar amount, but the split between what reduces your debt and what pays the lender shifts month after month until the loan reaches zero.

The word comes from the Latin "mort," meaning death, you are slowly killing the debt. Mortgages, auto loans, personal loans, and most student loans amortize. Credit cards do not, which is why they can stay outstanding indefinitely.

### Why Amortization Beats Other Repayment Structures

Amortization spreads the cost evenly, making budgeting predictable. Compare this to interest-only loans (no principal reduction) or balloon loans (a massive final payment). For most consumers, fixed amortization removes payment-shock risk and forces disciplined paydown.

### The Two Forces in Every Payment

Every amortizing payment contains exactly two components: interest on the outstanding balance and principal reduction. Interest is calculated first based on what you still owe, then whatever is left of your payment goes to principal. That ordering is why early payments feel like they barely move the balance.

## The Formula for Amortisation: The Core Equation

The amortization formula calculates a fixed periodic payment that fully retires a loan over a set number of periods at a given interest rate. The standard monthly payment formula is:

**PMT = P × [r(1+r)^n] / [(1+r)^n − 1]**

Where P is the principal (original loan amount), r is the periodic interest rate (annual rate divided by 12 for monthly payments), and n is the total number of payments over the loan's life.

### Decoding Each Variable

P is straightforward, it is the amount you borrow. For a $300,000 mortgage, P = 300,000. The variable r trips people up because lenders quote annual rates. If your APR is 7%, the monthly periodic rate is 0.07 / 12 = 0.005833. And n is months, not years, a 30-year loan has n = 360.

### Why the Formula Has That Shape

The formula derives from the present value of an annuity. You are solving for the payment that makes the present value of all future payments exactly equal the loan amount today. The exponent (1+r)^n captures how compounding inflates the time value of each future payment.

## How the Amortization Formula Works: A $300,000 Mortgage

Let's plug real numbers into the amortization formula. Take a $300,000 mortgage at 7% annual interest for 30 years. We need r = 0.07/12 = 0.005833 and n = 360 monthly payments. Now substitute everything into the equation.

PMT = $300,000 × [0.005833 × (1.005833)^360] / [(1.005833)^360 − 1]

The term (1.005833)^360 ≈ 8.1165. So the numerator is 0.005833 × 8.1165 = 0.04734, and the denominator is 8.1165 − 1 = 7.1165. Dividing gives 0.006653, and multiplying by $300,000 yields a monthly payment of approximately $1,996.

That single number, $1,996, is what you owe every month for 360 consecutive months. Multiply it out and you pay $718,527 total to retire a $300,000 loan. The extra $418,527 is pure interest.

## Splitting Each Payment Into Principal and Interest

Once you know the fixed monthly payment, the principal-and-interest split for any specific month uses two simple formulas. Interest in month t equals the remaining balance times the periodic rate. Principal in month t equals the total PMT minus that interest. The new balance is the old balance minus the principal portion.

![How the principal-interest split flips completely over the life of a 30-year, 7% mortgage on $300,000.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20800%20210%22%20width%3D%22800%22%20height%3D%22210%22%20role%3D%22img%22%3E%3Ctitle%3EComparison%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Ctext%20x%3D%22230%22%20y%3D%2257.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EMonth%201%20Interest%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%2225%22%20width%3D%22450%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22702%22%20y%3D%2257.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%232563eb%22%3E%241.8K%3C%2Ftext%3E%3Ctext%20x%3D%22230%22%20y%3D%22152.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EMonth%20360%20Interest%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%22120%22%20width%3D%226%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%237c3aed%22%2F%3E%3Ctext%20x%3D%22258%22%20y%3D%22152.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%237c3aed%22%3E%2412%3C%2Ftext%3E%3C%2Fsvg%3E)

*How the principal-interest split flips completely over the life of a 30-year, 7% mortgage on $300,000.*

### Worked Example: First Three Months

Starting balance: $300,000. Month 1 interest = $300,000 × 0.005833 = $1,750. Principal = $1,996 − $1,750 = $246. New balance = $300,000 − $246 = $299,754.

Month 2 interest = $299,754 × 0.005833 = $1,749. Principal = $1,996 − $1,749 = $247. New balance = $299,754 − $247 = $299,507.

Month 3 interest = $299,507 × 0.005833 = $1,748. Principal = $248. New balance = $299,259. The principal portion grows by about $1 each month at first, painfully slow.

### The Classic Amortization Curve

Plot principal and interest over 360 months and you get a perfect crossover chart. Interest starts at $1,750 and decays toward zero. Principal starts at $246 and grows toward $1,996. They cross around year 19 on a 30-year, 7% mortgage. Before the crossover, you are mostly paying the bank. After it, you are mostly paying yourself.

### Year 15 vs Year 30 Comparison

At month 180 (year 15), your $1,996 payment splits roughly $870 interest and $1,126 principal. By month 360, the very last payment is approximately $12 interest and $1,984 principal. The shift is dramatic, and it's the reason early payoff strategies are so powerful.

## Total Interest Across the Life of a Loan

Total interest over a loan's life equals total payments minus principal. For a $300,000 mortgage at 7% over 30 years, you pay $718,527 total, $418,527 of which is interest. That single number reveals just how expensive borrowing really is, and why even small rate differences matter enormously.

![A 4-percentage-point rate difference on a $300,000 30-year mortgage creates a $263,195 gap in lifetime interest paid.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20800%20210%22%20width%3D%22800%22%20height%3D%22210%22%20role%3D%22img%22%3E%3Ctitle%3EComparison%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Ctext%20x%3D%22230%22%20y%3D%2257.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3E7%25%20Rate%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%2225%22%20width%3D%22450%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22702%22%20y%3D%2257.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%232563eb%22%3E%24419K%3C%2Ftext%3E%3Ctext%20x%3D%22230%22%20y%3D%22152.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3E3%25%20Rate%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%22120%22%20width%3D%22167.01288088940498%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%237c3aed%22%2F%3E%3Ctext%20x%3D%22419.012880889405%22%20y%3D%22152.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%237c3aed%22%3E%24155K%3C%2Ftext%3E%3C%2Fsvg%3E)

*A 4-percentage-point rate difference on a $300,000 30-year mortgage creates a $263,195 gap in lifetime interest paid.*

Drop the rate to 5% and total interest falls to $279,767. Drop it to 3% and you pay only $155,332 in interest, less than half of the 7% scenario. Every percentage point on a 30-year mortgage swings tens of thousands of dollars. This is why locking a low rate during a refinance window can be one of the highest-return financial moves a homeowner ever makes.

## Excel Amortization Formula Functions

Excel and Google Sheets ship with built-in functions that handle the entire amortization formula automatically. You don't need to compute (1+r)^n by hand. The PMT, IPMT, PPMT, and CUMIPMT functions cover every calculation in this article.

### The Core Four Functions

`=PMT(rate, nper, -PV)` returns the fixed periodic payment. For our example: `=PMT(0.07/12, 360, -300000)` returns $1,996.

`=IPMT(rate, period, nper, -PV)` returns interest paid in a specific period. `=IPMT(0.07/12, 1, 360, -300000)` returns $1,750.

`=PPMT(rate, period, nper, -PV)` returns principal in that period. `=PPMT(0.07/12, 1, 360, -300000)` returns $246.

### Cumulative Functions for Lifetime Totals

`=CUMIPMT(rate, nper, PV, start_period, end_period, 0)` returns total interest across a range of months. Use it to find how much interest you will pay in the first five years, or across the entire loan. Pair it with `=CUMPRINC()` for cumulative principal paid.

## Different Types of Amortization

Not every loan amortizes the same way. The four main structures determine how the formula gets applied, and how much risk you take.

![The main loan amortization types and how each applies the repayment formula differently.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20760%20211%22%20width%3D%22760%22%20height%3D%22211%22%20role%3D%22img%22%3E%3Ctitle%3EHierarchy%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Crect%20x%3D%22300%22%20y%3D%2220%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22380%22%20y%3D%2254%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22white%22%3EAmortization%20Types%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20110%20105.5%20L%20110%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%2230%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22110%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EFixed-Rate%3C%2Ftext%3E%3Ctext%20x%3D%22110%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3EConstant%20payment%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20290%20105.5%20L%20290%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22210%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22290%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EARM%3C%2Ftext%3E%3Ctext%20x%3D%22290%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3ERate%20resets%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20470%20105.5%20L%20470%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22390%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22470%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EInterest-Only%3C%2Ftext%3E%3Ctext%20x%3D%22470%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3ENo%20early%20principal%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20650%20105.5%20L%20650%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22570%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22650%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EBalloon%3C%2Ftext%3E%3Ctext%20x%3D%22650%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3ELump-sum%20end%3C%2Ftext%3E%3C%2Fsvg%3E)

*The main loan amortization types and how each applies the repayment formula differently.*

**Standard fixed-rate** loans (most mortgages, all auto loans) lock the rate and apply the amortization formula once at origination. Payment never changes.

**Adjustable-rate mortgages (ARMs)** start with a fixed period, then recalculate the formula whenever the rate adjusts using the new rate, remaining balance, and remaining term.

**Interest-only loans** pay no principal for the first 5–10 years, then convert to a fully amortizing loan over the remaining term, meaning payments jump significantly when the interest-only period ends.

**Balloon loans** charge low monthly payments calculated on a long amortization schedule but require a massive lump-sum payment at the end. **Negative amortization** loans, banned in many jurisdictions after 2008, allow payments smaller than the interest charge so the balance actually grows.

## Strategies That Bend the Amortization Curve

Once you understand the formula, you can game it. Two strategies dominate: extra principal payments and bi-weekly schedules. Both work by accelerating the principal-vs-interest crossover described earlier.

### Extra Principal Payments

Add an extra $200 each month to that $300K mortgage at 7% and the loan pays off in roughly 23 years instead of 30, saving about $130,000 in interest. The math works because every extra dollar of principal reduces the base on which next month's interest is calculated, compounding savings forward.

### Bi-Weekly Payment Strategy

Pay half your monthly payment every two weeks instead of the full amount monthly. Because there are 52 weeks per year, you make 26 half-payments, equivalent to 13 full payments instead of 12. That single extra payment annually shaves roughly 4–6 years off a 30-year mortgage and tens of thousands in interest, with no formal refinance required.

### Refinance Breakeven

When rates drop, refinancing replaces your old amortization schedule with a new one at a lower rate. Calculate breakeven as closing costs divided by monthly savings. If a refinance costs $4,000 and saves $200/month, breakeven is 20 months. Stay in the home longer than that and you win.

## Amortization Across Loan Types and Tax Treatment

The amortization formula is identical for mortgages, auto loans, and student loans, but conventions differ. Mortgages typically run 15 or 30 years. Auto loans usually 36–84 months. Federal student loans default to 10 years on the standard plan, with extended and income-driven options available.

Tax treatment varies sharply. Mortgage interest is deductible up to $750,000 of acquisition debt for primary and secondary homes (2026 limits). Auto loan interest is generally not deductible unless the vehicle is used for business. Student loan interest is deductible up to $2,500 per year, subject to income phaseouts.

Don't confuse loan amortization with depreciation. Both systematically allocate cost over time, but amortization applies to loans and intangible assets (patents, goodwill), while depreciation applies to tangible fixed assets like buildings and equipment.

## Common Mistakes With the Amortization Formula

Borrowers consistently trip over four pitfalls. First, confusing APR with the periodic rate, always divide the annual rate by the number of payments per year before plugging into the formula. Second, focusing only on the monthly payment and ignoring lifetime interest cost.

Third, never modeling extra-principal scenarios, most borrowers have no idea how much a small additional payment could save. Fourth, forgetting that real mortgage payments include escrow for property taxes and homeowners insurance, often adding $300–$800 to the calculated PMT figure.

## Authoritative Sources

For deeper background and primary-source data on this topic, the following authoritative sources are useful starting points:

- [Federal Deposit Insurance Corporation](https://www.fdic.gov/)
- [Federal Reserve](https://www.federalreserve.gov/)
- [Office of the Comptroller of the Currency](https://www.occ.treas.gov/)
- [National Credit Union Administration](https://www.ncua.gov/)

## Conclusion

The amortization formula is one of the most useful equations in personal finance. Five takeaways to remember: PMT = P × [r(1+r)^n] / [(1+r)^n − 1] gives you the fixed payment for any installment loan; interest in any month equals remaining balance times the periodic rate; early payments are overwhelmingly interest, and the crossover takes years to arrive; small rate differences create enormous lifetime cost swings; and extra principal or bi-weekly payments can shave years and tens of thousands in interest with zero refinance paperwork.

Master Excel's PMT, IPMT, PPMT, and CUMIPMT functions and you can build a full amortization schedule for any loan in minutes. Compare scenarios side by side, different rates, terms, extra-payment strategies, before signing anything.

As rates continue to shift through 2026 and beyond, the borrowers who understand the formula for amortisation will consistently outperform those who don't. They will refinance when the math favors it, accelerate payoff when cash flow allows, and choose loan structures aligned with their actual goals rather than whatever the lender suggests.

Ready to put this knowledge to work? Try Warren, your AI financial advisor, get personalized, conflict-free guidance at heywarren.com

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**Authoritative sources**:
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