Mortgage Rate Calculator

This one runs the mortgage backwards

Nearly every mortgage calculator asks for a rate and gives you a payment. This one asks for the payment and gives you the rate.

You tell it how much is being borrowed, over how long, and what the monthly payment is. It works out the interest rate that makes those three numbers agree.

That sounds like a small reversal. It is not, and the reason is interesting enough that it gets its own section below. Solving a mortgage forwards takes one line of algebra. Solving it backwards cannot be done with algebra at all.

It is also the more useful direction more often than you would think. Somebody quotes you a monthly figure without mentioning a rate. A seller offers to finance the house themselves. A dealer advertises a payment in big print and the rate in small print, or not at all. In every one of those you have the payment and you want the rate.

Filling in the three fields

1. Loan Amount. The sum being borrowed.
2. Loan Term. How long there is to repay it, and a dropdown to say whether you are counting in years or months.
3. Monthly Payment. The regular payment against the loan. Principal and interest only, so leave out any escrow for taxes or insurance.

Press Calculate and you get the implied rate, plus a summary showing the total of all payments and the total interest across the term.

That third field is the one to be careful with. If you feed it a bundled figure that includes property tax and insurance, the calculator will faithfully tell you the rate that would justify such a large payment, and it will be far higher than your real rate. There is more on that below.

Why there is no formula for the rate

The standard mortgage equation ties four things together: the loan L, the monthly rate i, the number of payments n, and the payment M.

M = L × i ÷ (1 - (1 + i)^-n)

Give it any three of those and you can find the fourth. Except that is not quite true, and the exception is the interesting bit.

Rearranging for L is trivial. Rearranging for n takes a logarithm but it works out fine. Rearranging for i is impossible, and not in a "nobody has managed it yet" way. It is impossible in the same way that solving certain polynomial equations by radicals is impossible. The rate appears both as a plain multiplier and inside an exponent, and there is no way to untangle it so that i sits alone on one side.

Mathematicians call an equation like this transcendental. It has an answer, exactly one sensible answer in fact, and you simply cannot write that answer down as a formula.

Which is a nice reminder that "there is no formula" and "there is no answer" are completely different statements. There is an answer. You just have to go and find it.

So the calculator guesses, cleverly

The method here is binary search, and it is the same trick you use without thinking when somebody says pick a number between 1 and 100.

You do not start at 1. You guess 50, get told "lower", and immediately throw away half the possibilities. Then you guess 25, get told "higher", and throw away half of what is left. Every guess halves the search space, so a hundred options fall to nothing in about seven questions.

The calculator does exactly this with the interest rate. It knows the answer sits somewhere between 0 percent and 100 percent, so:

1. Guess halfway between the two bounds.
2. Run the ordinary forward formula with that guessed rate and see what payment it produces.
3. If that payment is smaller than yours, the real rate must be higher, so raise the lower bound. If it is larger, lower the upper bound.
4. Guess halfway again, and keep going.

It stops when the payment produced by its guess is within a ten-millionth of the payment you typed, and it will not run more than a hundred rounds. In practice it never needs anywhere near that. Ten halvings narrow a 100 point range to about a tenth of a percentage point. Thirty five halvings narrow it to roughly three billionths.

What makes this work is that the relationship between rate and payment only ever moves one way. Push the rate up and the payment goes up, always, with no wobbles. That guarantee is why halving the range is safe, and why the search cannot get stuck.

Watching it solve one

Loan of 300,000, term 30 years, monthly payment 1,798.65. Here are the calculator's first guesses, exactly as it makes them:

Notice guess four. It is already within fifty dollars of the target after only four attempts, which is the whole point of halving. From there it needs another thirty or so rounds to nail the last few decimal places, and it settles on 6.000 percent.

A few more, so you can see the shape of the thing:

Look at the first two rows. On the same loan over the same term, 400 more per month is the difference between roughly 4.9 percent and roughly 7 percent. Payments compress a lot of information into one number, which is precisely why sellers like quoting them.

Why some payments get rejected

Enter a payment that is too small and the calculator tells you so rather than returning an answer, and there is real mathematics behind that rather than fussiness.

The floor is the loan divided by the number of months. On 300,000 over 360 months that is 833.33. At exactly that payment you would be repaying pure principal with no interest whatsoever, which means a rate of zero.

Anything below it is not a low rate, it is an impossible one. A payment of 700 on that loan cannot clear 300,000 in 360 months even if the lender charges you nothing at all. There is no rate that makes those numbers work, so the honest response is to say so.

If you meet that message in the wild, it usually means one of the inputs is off. A term entered in years while the payment was quoted for a much longer schedule, a loan amount with an extra digit, or a payment that was actually a fortnightly figure rather than a monthly one.

What the rate it gives you does and does not include

The figure that comes back is the interest rate implied by your three numbers. It is derived purely from the payment, so it reflects exactly what the payment reflects, and nothing more.

That distinction matters when you compare it against a lender's paperwork, because two different rates appear there.

• The interest rate is the cost of borrowing the money. It is the number that drives your monthly payment, and it is the one this calculator recovers.
• The APR is broader. It folds in points, origination charges, broker fees and other costs of obtaining the loan, which is why it is nearly always higher than the interest rate.

So if a lender charged you 4,000 in points and fees on the loan above, your interest rate would still be 6 percent and your APR would be somewhat above it. The payment has no way of knowing about those fees, so neither does anything derived from the payment.

Which makes this the right tool for one question and the wrong one for another. To work out what rate is behind a quoted payment, this is exactly it. To compare two offers on total cost, you want the APR from page three of each Loan Estimate, because that is the number designed for the comparison.

Four things worth pointing it at

Checking a payment somebody quoted you. If a broker gives you a monthly figure and a rate, run the payment through and see whether the rate that comes back matches what they said. A gap usually means fees are financed into the loan, or the payment includes escrow.

Seller or family financing. These deals are often agreed as "X a month for Y years" with no rate ever discussed. Running it gives both sides a number they can sanity check against what a bank would charge.

Advertised payments. Any advert that leads with a monthly figure has made a set of assumptions about the rate and the term. This tells you what rate they assumed.

An old loan you are still paying. If you have the original balance, the term and the payment but the paperwork is long gone, this reconstructs the rate.

One habit worth keeping. Whatever comes back, compare it against what a normal lender would offer today for the same kind of borrowing. A rate meaningfully above that is telling you something about the deal, and it is generally worth understanding what.

Years or months

The dropdown beside the term takes either, and it is there because the two halves of this problem tend to arrive in different units.

Mortgages come in years. Car loans, personal loans and seller financing usually get quoted in months, and 60 months is a much more natural thing to type than 5 years when that is what the paperwork says. Either way the calculator converts to months internally before it starts searching, so the answer is identical.

The summary echoes it back both ways when you enter years, showing 30 years and 360 months side by side, which is a quick way to confirm you typed what you meant.

Questions people ask

What does this calculate?

The interest rate. You give it the loan amount, the term and the monthly payment, and it finds the rate that makes those three consistent.

Why does it search instead of using a formula?

Because no formula exists. The mortgage equation can be rearranged to solve for the loan amount, the payment or the term, but not for the rate, since the rate appears both as a multiplier and inside an exponent. The answer has to be found by narrowing in on it.

How accurate is the answer?

It keeps halving the range until the payment produced by its guess is within a ten-millionth of the payment you entered, then reports the rate to three decimal places. For any practical purpose it is exact.

Is the result my APR?

It is the interest rate implied by the payment. APR additionally includes points, origination charges and broker fees, so if your loan carries those, your APR will be higher than this figure. Both numbers appear on a Loan Estimate, the interest rate on page one and the APR on page three.

Should the payment include taxes and insurance?

No. Enter principal and interest only. A payment bundled with escrow will produce a rate well above your real one, because the calculator has no way to know part of that money never reaches the loan.

Why was my payment rejected as too low?

Because it cannot repay the loan in the time given even at zero interest. The floor is the loan divided by the number of months, so 300,000 over 360 months needs more than 833.33 a month before any rate exists at all.

Can I use it for a car loan or a personal loan?

Yes. The mathematics is the same for any fixed rate loan repaid in equal instalments. Switch the term dropdown to months, which is how those loans are usually quoted.

References

The forward payment relation used inside the search is the standard actuarial amortisation equation, under which the unpaid balance is increased each period by the finance charge earned and reduced by the payment made, with the annual rate expressed as a periodic rate multiplied by the number of periods in a year. Regulation Z sets this out and publishes its equations so they can be used to program calculators. The distinction between an interest rate and an APR, and where each appears on a Loan Estimate, follows the Consumer Financial Protection Bureau's guidance.

1. Consumer Financial Protection Bureau (CFPB), Regulation Z, Appendix J to Part 1026: Annual Percentage Rate Computations for Closed-End Credit Transactions. https://www.consumerfinance.gov/rules-policy/regulations/1026/j/
2. Consumer Financial Protection Bureau (CFPB), Regulation Z, § 1026.22 Determination of Annual Percentage Rate. https://www.consumerfinance.gov/rules-policy/regulations/1026/22/
3. Consumer Financial Protection Bureau (CFPB), What Is the Difference Between a Mortgage Interest Rate and an APR? https://www.consumerfinance.gov/ask-cfpb/what-is-the-difference-between-a-mortgage-interest-rate-and-an-apr-en-135/
4. Consumer Financial Protection Bureau (CFPB), On a Mortgage, What Is the Difference Between My Principal and Interest Payment and My Total Monthly Payment? https://www.consumerfinance.gov/ask-cfpb/on-a-mortgage-whats-the-difference-between-my-principal-and-interest-payment-and-my-total-monthly-payment-en-1941/