Monthly Loan Repayment Calculator and the Formula Behind It

A fixed-rate loan repaid in equal installments has one payment that never changes, and it comes out of a single line of algebra. Borrow 200,000 at a nominal 5.5 percent over 20 years and the payment is 1,375.77 per month, which totals 330,185.91 and carries 130,185.91 of interest. Below is the formula with every term explained, a calculator you can drive with your own numbers, two fully worked examples, and a plain list of the costs this arithmetic ignores.

What an equal-installment payment covers

Every payment on this kind of loan does two jobs at once. Part of it pays the interest that accrued on the outstanding balance during that period, and the remainder reduces the balance. Because the balance shrinks, next period's interest is smaller, so a larger slice of the same fixed payment goes to principal. The payment itself never moves. It is set so that the balance lands on exactly zero at the last installment, no earlier and no later.

This is the equal-installment structure, also called an amortizing loan or a level-payment loan. It is not the only shape a loan can take. On an equal-principal loan the borrower repays the same slice of principal every period and pays interest on whatever is left, so the first payment is the largest and every later one is smaller. On an interest-only loan the payments cover interest alone and the whole principal falls due at the end. The calculator on this page models the level-payment case, which is the default for most mortgages, car loans, and personal installment loans.

Three inputs decide everything: the amount borrowed, the nominal annual rate, and the number of years. Nothing else enters the formula. That is useful to remember when a quoted payment does not match what you compute. The difference is nearly always something outside those three numbers rather than an error in the arithmetic.

The formula, term by term

The payment is the amount that makes the discounted value of all n installments equal to the sum borrowed. Written out:

i = rate / 1200              periodic (monthly) rate, as a decimal
n = years * 12               total number of payments
m = P * i * (1 + i)^n / ((1 + i)^n - 1)

total paid     = m * n
total interest = total paid - P

P is the principal, the amount actually borrowed. rate is the nominal annual percentage. Dividing it by 1200 does two things in one step: it converts percent to a decimal (divide by 100) and splits the annual rate into twelve equal monthly slices (divide by 12). No compounding conversion happens at that step, which is why a nominal 5.5 percent charged monthly works out to an effective annual rate of 5.6408 percent.

The factor (1 + i)^n appears twice, once on top and once at the bottom. The reason is easier to see if the same formula is rearranged:

m = P * (1 + i)^n / ( ((1 + i)^n - 1) / i )

Now both halves have a meaning. P * (1 + i)^n is what the debt grows to by the end of the term if the borrower pays nothing at all, which for 200,000 over 240 months at 5.5 percent is 599,325.11. The denominator, ((1 + i)^n - 1) / i, is what a stream of one unit paid every period accumulates to by the end, with each payment earning interest for the time it sits there. Over the same 240 months that accumulation factor is 435.627395. Dividing the first by the second gives the size each payment has to be for the stream to cancel the grown debt exactly. The same growth factor shows up twice because two different things are being carried forward at the same rate: the debt on one side, the payments on the other. This is the standard annuity expression used in lending and actuarial work, and it also appears in general references such as the amortization calculator entry on Wikipedia, usually written with a negative exponent instead.

One case needs separate handling. When the rate is zero, both i in the numerator and (1 + i)^n - 1 in the denominator become zero, and the expression turns into 0/0, which cannot be evaluated. The answer is still obvious, since a loan with no interest is repaid in equal slices of principal:

m = P / n                    when rate = 0

The calculator treats this as its own branch, so a rate of 0 returns principal / (years * 12) rather than an error. For 200,000 over 20 years that is 833.33 a month and 200,000 in total, with no interest at all.

Set your own numbers below. The result updates as you type, so you can feel how much a half point of rate or five more years actually costs.

A worked example: 200,000 at 5.5 percent over 20 years

Every intermediate value, in order:

P = 200000, rate = 5.5, years = 20

i = 5.5 / 1200                      = 0.00458333
n = 20 * 12                         = 240
(1 + i)^n                           = 2.99662556
m = 200000 * 0.00458333 * 2.99662556 / (2.99662556 - 1)
                                    = 1375.77

total paid     = 1375.77 * 240      = 330185.91
total interest = 330185.91 - 200000 = 130185.91

The interest comes to 65.1 percent of the amount borrowed, and it is 39.4 percent of every unit that leaves the borrower's account across those 20 years.

The first payment splits like this. Interest for month one is the whole balance times the monthly rate, 200,000 times 0.00458333, or 916.67. The remaining 459.11 comes off the principal, leaving a balance of 199,540.89. So the very first installment retires only a third of its own value in debt.

That ratio moves slowly. By month 60 the split is 774.48 interest against 601.29 principal. Principal does not overtake interest until month 90, seven and a half years into a 20-year loan. By month 180 the payment is 334.89 interest and 1,040.89 principal, and the last installment is 6.28 interest and 1,369.50 principal. Across the first twelve months the borrower pays 16,509.30 and reduces the balance by only 5,650.32, with the other 10,858.98 going to interest.

The front loading is not a fee or a penalty. It falls straight out of the rule that interest accrues on the outstanding balance, and that balance is at its largest on day one.

The same loan over 30 years, and one point more rate

Stretching the term lowers the payment and raises the total. Here is the same 200,000 at the same 5.5 percent, repaid over 30 years:

n = 30 * 12                         = 360
(1 + i)^n                           = 5.18738784
m = 200000 * 0.00458333 * 5.18738784 / (5.18738784 - 1)
                                    = 1135.58

total paid     = 1135.58 * 360      = 408808.08
total interest = 408808.08 - 200000 = 208808.08

The monthly payment falls by 240.20, which is real relief in a monthly budget. The price is 78,622.17 of extra interest, and the interest now exceeds the sum borrowed: 104.4 percent of principal, and slightly more than half of everything paid. The amortization curve stretches with the term. On the 30-year schedule principal does not overtake interest until month 210, and the first twelve months reduce the balance by only 2,694.18, against 5,650.32 on the 20-year schedule.

Four versions of the same 200,000 loan:

The table reads two ways. Term is the bigger lever on total cost: moving from 15 to 30 years cuts the payment by 498.59 but more than doubles the interest, from 94,150.04 to 208,808.08. Rate is the sharper lever on the monthly figure, since the extra point in the last row adds 115.38 a month and 27,689.20 of interest over an unchanged 20 years. A borrower choosing between offers usually has to weigh a payment they can afford now against a total they will not see until the schedule ends.

What this calculation leaves out

This is a principal-and-interest calculation and nothing else. It does not include property tax, home or mortgage insurance, origination and arrangement fees, appraisal or legal costs, servicing charges, or any premium a lender bundles into the monthly bill. On a mortgage the amount that actually leaves a bank account each month is often visibly larger than what this formula produces, because taxes and insurance are collected alongside the loan payment.

It also assumes the rate never changes. A variable or adjustable loan resets on a schedule, and every reset produces a new payment computed from the balance and the term remaining at that moment. The formula still applies at each reset, but one run of it cannot describe the whole life of such a loan.

Two more exclusions matter for planning. Early repayment is not modeled: paying extra cuts the balance and therefore every unit of interest that balance would have generated afterward, and many contracts charge a prepayment penalty for doing exactly that. Fee financing is not modeled either. When fees are rolled into the amount borrowed, the real cost is better captured by an annual percentage rate that includes them than by the nominal rate alone.

Lenders also round. Payments are usually rounded to the smallest currency unit, and the final installment is nudged up or down to clear the balance to zero, so a contractual schedule can differ from this arithmetic by a small amount. Treat this page as a calculation aid, not as financial advice. No result here is an offer, a quote, or a guarantee of terms, and the numbers should be confirmed with the lender before anything is signed.

Common questions

Which currency does the calculator use?

None, and that is deliberate. The formula contains no currency, so 200,000 of any unit produces a payment in that same unit. The only requirement is consistency between the principal you enter and the result you read. It also means the calculator works for figures quoted in thousands or millions, as long as every input uses the same scale.

What happens if I enter a rate of zero?

The calculator switches to principal / (years * 12) and returns the payment for an interest-free loan. This branch exists because the general formula cannot produce that answer on its own, since it collapses to 0/0 when i is zero. Interest-free installment plans are common in retail and vehicle financing, so the case is not only a mathematical curiosity.

Why does my lender quote a different number?

Almost always because the quote contains something the formula does not. Taxes, insurance, and fees are the usual additions, and on a mortgage they can be a large share of the monthly bill. Day-count conventions explain smaller gaps, because some lenders accrue interest on actual days rather than twelve equal months. When the difference is large, ask for a written breakdown of what the quoted figure includes.

Does paying extra each month help?

Yes, and it helps most early in the term. An extra amount goes straight against the balance, which removes all the interest that slice of balance would have generated over the rest of the schedule. On the 20-year example above, an extra payment made in month one erases principal that would otherwise have accrued interest for another 239 months. This calculator does not model overpayments, so read its result as the schedule you are committed to rather than the fastest route to a zero balance.