MENULoan Calculator
Enter the loan amount, the term and the interest rate. The calculator shows the monthly payment, the total cost of the loan, the APR and a month-by-month amortization schedule. The results update as you type.
What is an amortized loan?
An amortized loan is paid back in equal monthly payments. Every payment does two jobs: one part pays the interest for the month, and the rest pays down the principal, the amount you still owe. At the start the balance is large, so most of the payment is interest. As the balance shrinks, so does the interest, and more of each payment goes to the principal. You can see this in the schedule above: the Interest column falls month by month while the Principal column grows.
Where the formula comes from
Let P be the amount borrowed, n the number of monthly payments and i the monthly rate: the annual rate divided by 12 and written as a decimal. For 6% a year, [tex]i = 0.06 \div 12 = 0.005[/tex].
Every month the balance first grows by the interest and then drops by the payment A:
[tex]B_k = B_{k-1}(1+i) - A[/tex]
After n payments nothing is owed. Writing the balances out one by one gives
[tex]P(1+i)^n - A\left[(1+i)^{n-1} + \dots + (1+i) + 1\right] = 0[/tex]
The bracket is a geometric series with ratio [tex]1+i[/tex], so its sum is [tex]\dfrac{(1+i)^n - 1}{i}[/tex]. Solving for A:
[tex]A = P \times \dfrac{i}{1-(1+i)^{-n}}[/tex]
At 0% there is no interest and the payment is simply [tex]A = P \div n[/tex].
Worked example: a 30-year mortgage
You buy a $400,000 home with a 20% down payment of $80,000, so you borrow P = $320,000 at 6.5% for 30 years. Then n = 360 and [tex]i = 0.065 \div 12 \approx 0.0054167[/tex].
[tex]A = 320{,}000 \times \dfrac{0.0054167}{1-1.0054167^{-360}} \approx 2{,}022.62[/tex]
In month 1 the interest is [tex]320{,}000 \times 0.0054167 \approx 1{,}733.33[/tex], so only [tex]2{,}022.62 - 1{,}733.33 = 289.29[/tex] dollars go to the principal. Over 30 years you pay $728,140.64 in total, of which $408,140.64 is interest, more than the amount you borrowed. The last payment is $2,020.06: it settles the cents left over from rounding. Enter these numbers in the calculator to see the whole schedule.
Equal or declining payments?
With declining payments you repay the same principal every month, [tex]320{,}000 \div 360 \approx 888.89[/tex] dollars, plus the interest on what you still owe. The first payment is 888.89 + 1,733.33 = $2,622.22 and the last only $893.30. The balance falls faster, so the total interest is $312,866.27, which is $95,274.37 less than with equal payments. The price is a first payment $599.60 higher.
What is APR?
The interest rate alone does not tell you what a loan costs, because lenders also charge fees. The annual percentage rate (APR) includes them. Suppose the lender charges 2 points at closing: 2% of the loan, or $6,400. You receive $313,600 but still repay $2,022.62 a month, as if you had received $320,000. The APR comes from the monthly rate j at which all the payments Ck are worth exactly the money you received:
[tex]P - F = \displaystyle\sum_{k=1}^{n} \frac{C_k}{(1+j)^k}, \qquad \text{APR} = 12 \times j[/tex]
Here F is the upfront fee, and j is found step by step, not with a formula. For this mortgage the APR is 6.70%, so 2 points turn a 6.5% loan into a 6.70% APR. This is the US method from the Truth in Lending Act. In the European Union the monthly rate is compounded instead, [tex](1+j)^{12} - 1[/tex], and the same loan would show 6.90%.
Practice problems
1. You borrow $12,000 for 3 years at 9% a year with equal monthly payments. What is the monthly payment?
Show the answer
[tex]i = 0.09 \div 12 = 0.0075[/tex] and n = 36, so [tex]A = 12{,}000 \times \dfrac{0.0075}{1-1.0075^{-36}} \approx 381.60[/tex]. The payment is $381.60.
2. A $20,000 loan at 6% is repaid over 4 years with declining payments. How big is the first payment?
Show the answer
The principal part is [tex]20{,}000 \div 48 \approx 416.67[/tex] and the interest in month 1 is [tex]20{,}000 \times 0.005 = 100[/tex]. The first payment is $516.67.
3. In the first month of a $250,000 mortgage at 7.2%, how much of the payment is interest?
Show the answer
[tex]i = 0.072 \div 12 = 0.006[/tex], so the interest is [tex]250{,}000 \times 0.006 = 1{,}500[/tex] dollars. Only the part of the payment above $1,500 reduces the balance.
Frequently asked questions
How is the monthly payment calculated?
With the formula A = P × i / (1 − (1 + i)−n), where P is the loan amount, i the monthly rate (the annual rate divided by 12) and n the number of payments. For $10,000 at 6% over 5 years the payment is $193.33.
What is APR?
The annual percentage rate is the yearly cost of the loan with the fees included. With no fees it equals the interest rate. A fee withheld at the start raises it, because you receive less money but repay the same amount.
Should an extra payment shorten the term or lower the payment?
Shortening the term saves more interest. On a $10,000 loan at 6% for 5 years, an extra $2,000 in month 12 saves $467.04 of interest if it shortens the term and $254.60 if it lowers the payment.
Are declining payments cheaper than equal payments?
They cost less interest, because the balance falls faster. The same $10,000 loan costs $1,525.00 of interest with declining payments and $1,599.68 with equal payments, but the first payment is $216.67 instead of $193.33.
See also: Compound interest calculator Geometric progression formulas
