Loan Payment

Q: What if the interest rate is zero?

The formula divides by $(1 + r)^{n} - 1$, which is zero when $r$ is zero, so it cannot be used directly. With no interest, the payment is the amount borrowed divided by the number of payments.

In one sentence: The loan payment formula gives the fixed amount you pay each month so that a loan is fully repaid, interest included, by the last payment.

$M = P \cdot \frac{r (1 + r)^{n}}{(1 + r)^{n} - 1}$

Read it out loud: “M equals P times a fraction. The top is r times one plus r, raised to the power n. The bottom is one plus r, raised to the power n, minus one. M is the monthly payment. P is the amount borrowed. r is the monthly interest rate, written as a decimal. n is the number of monthly payments.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Loan, principal and amortized loan: A loan is money lent now that is paid back later, usually with interest. The principal is the amount borrowed at the start, here called P. An amortized loan is repaid in equal payments, and each payment covers some interest and some of the principal, so the balance falls to zero at the last payment.
Monthly rate from a yearly rate: Interest on loans is usually quoted per year. A monthly rate is the yearly rate divided by 12 when the payments are monthly. Six percent a year divided by 12 is 0.5 percent a month, or 0.005 as a decimal. The formula needs that monthly decimal, so $r = 0.005$ in the example.
Exponent (power) and the growth factor: An exponent, also called a power, says how many times to multiply a number by itself. $(1.005)^{48}$ means 1.005 multiplied by itself 48 times, which is about 1.2704892. This is the growth factor over the whole loan: what one dollar left in place would grow to. Subtracting one from it gives the growth above the original amount.
Total interest: Total interest is the cost of borrowing: all the payments added together, minus the amount borrowed. It is the money the lender earns over the life of the loan. Two loans for the same amount can have very different total interest, depending on the rate and the number of payments.

What each symbol means

Symbol	Meaning
$M$	Monthly payment, the fixed amount you pay each month, measured in dollars. After the last of these payments the loan is fully repaid, with all of its interest. This is the number the formula gives you.
$P$	Principal, the amount borrowed at the start, measured in dollars. It is the money you receive from the lender before any interest is added. In the example P is 20000 dollars.
$r$	Monthly interest rate, written as a decimal. It is the yearly rate divided by 12, so a 6 percent yearly rate gives r = 0.005. Interest is charged each month on the balance still owed. It has no unit beyond per month.
$n$	Number of monthly payments, counted from the first payment to the last. A loan over 4 years has n = 48 payments. It is a count of months, so it has no unit beyond months.

The formula piece by piece

$P$: $P$ is the amount borrowed in dollars, the starting point of the loan. The monthly payment is proportional to it, so a loan twice as large needs a payment twice as large at the same rate and term.
$r$: $r$ is the monthly interest rate as a decimal. It is the share of the balance charged each month, so a bigger rate makes each payment bigger for the same amount borrowed and the same term.
$(1 + r)^{n}$: The growth factor over the whole loan, the number that one dollar becomes after $n$ months of interest. For 48 months at 0.5 percent it is about 1.2704892.
$r (1 + r)^{n}$: The monthly rate times the growth factor. This is the top of the fraction, and it sets the interest part of the payment for each dollar borrowed.
$(1 + r)^{n} - 1$: The growth factor minus one, which is the growth above the original amount. This is the bottom of the fraction, and it measures how much the payments must pay down in total.
$\frac{r (1 + r)^{n}}{(1 + r)^{n} - 1}$: The top divided by the bottom gives the monthly payment for each dollar borrowed. It is a ratio, so it has no unit of its own. For the example it is about 0.023485.
$M = P \cdot \frac{r (1 + r)^{n}}{(1 + r)^{n} - 1}$: The whole formula in one sentence: the monthly payment equals the amount borrowed times the payment for each dollar borrowed, which repays the loan, with interest, in exactly n equal monthly payments.

What the formula is doing, and why

A loan is an amount of money that you pay back in equal monthly payments. Each payment does two jobs. Part of it pays the interest charged that month on the balance still owed, and the rest reduces the balance. Early payments are mostly interest, and later payments are mostly repayment of the amount borrowed. The loan payment formula gives the one fixed amount $M$ that does both jobs, so that the balance is exactly zero after the last payment.

Why this shape? The formula comes from one rule: the amount borrowed equals the present value of all the payments. In other words, $P$ is what the lender hands over, and it equals $n$ equal payments $M$ , each discounted at the monthly rate $r$ . Solving that equation for $M$ gives the fraction in the formula. The top, $r (1 + r)^{n}$ , and the bottom, $(1 + r)^{n} - 1$ , are what that algebra leaves behind.

Why does the term matter? Stretching the loan over more months lowers the monthly payment, but the balance stays unpaid for longer, so more interest is charged on it. The formula shows this trade-off: for the same amount and rate, a longer term means a smaller $M$ each month and a larger total of payments.

Why multiply by $P$ ? The payment is proportional to the amount borrowed. Borrow twice as much at the same rate and term, and the monthly payment doubles. The fraction is the monthly payment for each dollar borrowed, so multiplying it by $P$ gives the payment for your loan. The total interest is the total of all payments minus the amount borrowed, $nM - P$ , and it is the real cost of borrowing.

An everyday picture

Think of paying back a friend who charges a small fee each month on whatever you still owe. If you pay the same amount every month, the early payments are mostly fee and the later ones mostly pay down what you borrowed. The loan payment formula finds the one fixed amount that clears the debt exactly on the last month, with no fee or balance left over.

Worked example: Monthly payment and total interest on a 20000 dollar car loan

A buyer borrows 20000 dollars to buy a used car. The loan charges 6 percent per year, so the monthly rate is 0.5 percent, and it is repaid in equal monthly payments over 4 years. What is the monthly payment, and how much interest is paid in total? In the formula, $P = 20000\ \text{dollars}$ , $r = 0.005$ and $n = 48$ months. These numbers are an illustration, not a real loan offer.

Write down the inputs and convert the rate. The lender charges 6 percent per year, but the payments are monthly, so we need the monthly rate. Dividing 6 percent by 12 gives 0.5 percent, and as a decimal $r = 0.06 \div 12 = 0.005$ . The loan lasts 4 years, which is $n = 4 \times 12 = 48$ monthly payments. Every number now counts months, so the rate and the count match.
Find the growth factor over the loan. We need $(1 + r)^{n} = (1.005)^{48}$ , which is 1.005 multiplied by itself 48 times. A calculator’s power key gives $(1.005)^{48} \approx 1.2704892$ . This says that a balance left alone for the whole loan would grow by about 27 percent before any payments are made.
Work out the top and the bottom of the fraction. The top is the monthly rate times the growth factor, $r (1 + r)^{n} = 0.005 \times 1.2704892 \approx 0.00635245$ . The bottom is the growth above the original amount, $(1 + r)^{n} - 1 = 1.2704892 - 1 = 0.2704892$ . Together these set how large each payment must be, compared with the amount borrowed.
Divide top by bottom to get the payment for each dollar. $\frac{0.00635245}{0.2704892} \approx 0.023485$ . This means each dollar borrowed needs a payment of about 0.023485 dollars a month. The fraction has no unit of its own, because dollars divided by dollars cancel out.
Multiply by the amount borrowed. The loan is 20000 dollars, so $M = 20000\ \text{dollars} \times 0.023485 \approx 469.70\ \text{dollars}$ . The buyer pays about 469.70 dollars every month for 48 months.
Find the total paid and the total interest. Over 48 months the buyer pays $48 \times 469.70 \approx 22545.6$ dollars in all. Taking away the 20000 dollars borrowed leaves the interest: $22545.6 - 20000 = 2545.6$ dollars. That is about 12.7 percent of the amount borrowed, which is the cost of spreading the loan over four years.

Answer: The monthly payment is about 469.70 dollars for 48 months. The buyer pays about 22545.6 dollars in total, so the total interest is about 2545.6 dollars. With the same amount and rate over a shorter term, the interest would be smaller, and over a longer term it would be larger.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the yearly rate in the formula. A 6 percent yearly rate must be divided by 12 to give $r = 0.005$ for monthly payments. Using 0.06 gives a payment that is far too large.
Using the number of years for $n$ . Four years of monthly payments is $n = 48$ , not $n = 4$ .
Writing the rate as a percent instead of a decimal. Half a percent is 0.005, and typing 0.5 would give a payment that is far too large.
Thinking the total cost is only the amount borrowed. The total paid is the monthly payment times the number of payments, and the difference from the amount borrowed is the interest.

Where you will meet it

Working out the monthly payment on a car loan, a personal loan or a student loan with fixed payments.
Estimating the monthly payment on a fixed-rate mortgage, before taxes and insurance are added to it.
Building an amortization schedule, a table that splits each payment into interest and repayment of the balance.
Comparing loan offers for the same amount, using the total interest as a common measure of cost.

Questions people ask

Why is the first payment mostly interest?

Interest is charged each month on the balance still owed, and that balance is largest at the start. Early payments therefore cover more interest and leave less to reduce the balance. As the balance falls, the interest part shrinks, and more of each payment reduces the debt.

Why does a longer loan cost more in total?

With the same amount and rate, a longer term lowers the monthly payment, but the balance stays unpaid for more months, so more interest is charged. In the example the total interest is about 2545.6 dollars over 48 months.

Does the formula include fees?

No. It gives the payment for a fixed rate, a fixed number of equal monthly payments and no fees. Real loan offers can add fees, changing rates or other charges, and those need their own calculation.

What if the interest rate is zero?

The formula divides by $(1 + r)^{n} - 1$ , which is zero when $r$ is zero, so it cannot be used directly. With no interest, the payment is the amount borrowed divided by the number of payments.

Present Value of an Annuity: The present value of an annuity says how much money you need today to pay a fixed amount every period for a set number of periods, with interest working in the background.
Future Value of an Annuity: The future value of an annuity says how much a row of equal payments adds up to by the end, once the interest earned on each payment is included.
Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.
Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.