Student loan payoff calculator

Enter the balance, the annual rate, the standard term and any amount you add on top of the scheduled payment. It returns the scheduled payment, how long the debt takes to clear at the payment you actually make, the total interest, and how much of that interest the extra removes.

Two separate calculations sit underneath. The scheduled payment comes from the level payment formula: the amount that clears the balance in exactly the standard number of months. The payoff time runs the other way. Fix the payment at whatever you are really sending, and the number of months falls out of it.

That second calculation is the one that matters once you pay more than the schedule asks, because the payment does not change. The finish line moves instead.

The scheduled payment is the amount that, repeated $n$ times, exactly clears the balance:

$$M = B \times \frac{i}{1 - (1 + i)^{-n}}$$

$B$ is the balance, $i$ is the monthly rate and $n$ is the number of months in the standard term. That is the same arithmetic as the loan payment calculator, because a student loan on a standard plan is an ordinary amortised loan.

Dividing by 12 is what makes $i$ monthly, and it works only because a lender quotes a nominal rate rather than an effective one. Twelve months of 0.458333 percent compound to 5.64 percent, so a headline 5.5 percent and the rate the balance actually grows at are two different numbers. Enter the rate the way the lender wrote it and let the formula do the dividing.

Once you decide to pay a different amount, the unknown swaps places. The payment is now fixed and the term is what you solve for:

$$n = \frac{-\ln\left(1 - \frac{i B}{M}\right)}{\ln(1 + i)}$$

$M$ has to be bigger than $i B$, which is the first month's interest. If it is not, the balance grows every month rather than shrinking and there is no payoff date to find.

Interest is charged on the balance still outstanding, so the only thing that reduces future interest is a smaller balance sooner.

• The month's interest is the balance times $i$.
• The scheduled payment covers that interest first, and what is left comes off the balance as principal.
• Anything above the scheduled payment has no interest left to cover, so all of it comes off the balance.

That last line is the whole effect. An extra dollar cancels every future month of interest that dollar would have carried, so its value depends on how much schedule is left in front of it. On a 10 year schedule at 5.5 percent, a dollar paid in month one cancels more than six times as much interest as the same dollar paid in month 96.

It also explains why the payoff date jumps further than the extra looks like it should. Each early dollar shortens the tail of the schedule, and the tail is where the remaining interest lives. The same mechanism running in your favour is the compound interest calculator, and how amortisation works walks through the split payment by payment.

This page models the arithmetic of a debt repaid in fixed instalments: a balance, a rate, a payment, and the point at which the balance reaches zero. What sits on top of that arithmetic is set by law and by contract, and it differs by country, by lender and by loan type.

In the United States, some student loans are federal and some are private, and the two are not governed by the same rules. Plans that set the payment from income, periods where interest is covered by someone other than the borrower, and forgiveness of a remaining balance after a qualifying period all exist. Which of them apply, on what terms and for how long are details of the individual agreement and of the rules in force at the time, and those rules change. The balance, rate and term to enter here are the ones on your own statement.

England, Wales and Australia collect repayment as a percentage of income above a threshold, through the tax or payroll system, and write off any balance still outstanding after a set number of years. That is a different instrument rather than a different plan: the end date comes from statute rather than from a schedule, and what is collected each year moves with income. Student loans explained sets out how each of those systems treats the balance.

Three mechanisms are worth understanding before looking yours up:

• A payment set from income can be smaller than the interest accruing. The balance then rises even though a payment is made every month.
• A subsidy that covers interest for a period stops the balance growing while it lasts, so an extra payment made inside that window does something different from one made outside it.
• Where a remaining balance can be forgiven after a qualifying period, an extra payment reduces a balance that would have been written off, so it can buy nothing.

None of that changes the formula. It changes what goes into it. Where the payment comes from income rather than from the balance, how much room sits on top of it is a budgeting question rather than a loan question, and the debt-to-income calculator is the page that works that side out.