By guzg · October 6, 2026 · 6 min read

Every loan calculator on the internet uses the same formula. It's about eighty years old, it's exact, and it will still disagree with the number your loan servicer quotes you.

That gap isn't a bug in the calculator. It's the six things the formula doesn't know about. I'll cover the math first, then the gap, because the second part is the one that actually explains your statement.

The formula

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

• M is the monthly payment
• P is the principal (what you borrowed)
• r is the monthly interest rate, which is the annual rate divided by 12
• n is the number of monthly payments

The reason it looks complicated is that it's solving backwards. You know the loan amount and the rate; you need the payment that drives the balance to exactly zero after n payments. Every payment covers that month's interest first, and whatever remains reduces the principal.

A worked example

Take a $30,000 loan at 5% annual interest, repaid over 10 years.

P = 30,000
r = 0.05 / 12 = 0.0041667
n = 10 × 12 = 120

M = 30,000 × [ 0.0041667 × (1.0041667)^120 ] / [ (1.0041667)^120 − 1 ]
  = 30,000 × [ 0.0041667 × 1.6470 ] / [ 0.6470 ]
  = 30,000 × 0.010606
  = 318.19

So about $318 a month.

Across 120 payments that's $38,183 in total, which means you pay $8,183 in interest on a $30,000 loan. That number surprises people, and it's worth sitting with before you look at repayment terms.

The other thing this calculation shows: in the first month, $125 of your $318 goes to interest and only $193 reduces the balance. By the final month it's reversed, with almost all of the payment hitting principal. That's why paying extra early saves more than paying extra late.

What the formula doesn't know

Here's where the calculator and your servicer part company.

1. Capitalised interest. If you have unsubsidised loans, interest accrues while you're in school and during the grace period. At repayment, that accumulated interest is usually added to the principal, and you then pay interest on it. A loan calculator starting from your original borrowed amount will understate your payment, sometimes significantly.

2. Origination fees. Many loans deduct a fee before the money reaches you. Borrow $30,000 with a 1% fee and you receive $29,700 while owing $30,000. The calculator doesn't know the fee exists.

3. The grace period isn't free on unsubsidised loans. Six months of accruing interest before repayment starts, and it gets capitalised too.

4. Income-driven repayment. If you're on a plan where the payment is calculated from your income rather than your balance, the amortisation formula is irrelevant. Your payment is set by a percentage of discretionary income, and the term may be extended to 20 or 25 years, with forgiveness of the remainder at the end.

5. Variable rates. The formula assumes a fixed rate. If yours adjusts, the payment you calculate today describes only the current period.

6. Servicer-level details. Fees, rate discounts for automatic payments, and rounding conventions all shift the number slightly.

So why use a calculator at all?

Because the formula is still the right tool for the question it answers: what does this debt cost if nothing changes?

Use it to compare scenarios. What's the difference between a 10-year and a 20-year term? What does an extra $50 a month actually save? Those comparisons are valid even when the absolute number is off, because both sides of the comparison carry the same omission.

What it's not good for is predicting your actual statement. For that, you need your servicer's numbers, which include the capitalised interest and fees that no generic calculator can see.

How I'd use this

1. Run the formula on the amount you'll actually owe at repayment, not what you borrowed. If you have unsubsidised loans, that's your principal plus accumulated interest.
2. Compare terms, not absolute figures. The 10-year versus 20-year comparison is the useful output.
3. Check the early payments. Seeing how little of month one goes to principal is the most honest picture of what you're signing up for.
4. Then check your servicer's actual figure and treat any gap as information about fees and capitalisation rather than an error.

The loan calculator implements the standard formula and shows the full amortisation schedule, so you can see the principal-versus-interest split for every month rather than just the headline payment.