How this calculator works
A loan is repaid by level payments at a fixed frequency. The nominal annual rate is divided by the number of payments per year, so a 7.5% loan paid weekly uses 7.5%/52 per week. That convention matches the rate and payment frequency quoted on most consumer loan agreements; it is not the same as compounding a quoted annual rate at a different interval. The calculator can also work backwards: given a payment, it finds the amount that payment supports, or given an amount and payment, it finds how many payments are needed. The final payment in the term mode is usually smaller than the regular one.
Formula
i = annual rate / payments per year; n = term in years × payments per year, rounded. Payment A = P × i / (1 - (1 + i)^(-n)). Amount P = A × (1 - (1 + i)^(-n)) / i. Payments n = ln(A / (A - P × i)) / ln(1 + i), rounded up. Totals: total paid = n × A, total interest = total paid - P. At a zero rate, A = P/n and P = n × A.
Worked example
A $20,000 loan at 7.5% over 5 years paid monthly uses 0.625% a month for 60 payments, giving $400.76 a month and $4,045.54 of interest. Paid bi-weekly instead, there are 130 payments of $184.71 and the interest is $4,012.32. The more frequent schedule reduces the balance sooner, but because the rate per payment adjusts with the frequency the saving is modest. Solving instead for the term of the same loan at $500 a month gives 47 payments, with a final payment of $86.63 instead of $500.
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Frequently asked questions
Why is a bi-weekly payment not exactly half the monthly payment?
There are 26 bi-weekly payments a year against 12 monthly payments. Each bi-weekly payment is therefore smaller than half a monthly payment when the same rate and term are used, because 26 payments together cover more of the year.
Does paying more often reduce interest?
Yes, slightly, because principal is reduced sooner. The saving is smaller than with accelerated bi-weekly payments, which keep half the monthly amount and add two extra payments a year.
What does solving for the term return?
The number of whole payments needed, with the last payment reduced. The total paid and total interest use that reduced final payment, so they are not simply the payment multiplied by the count.
Can I use this for a mortgage?
Yes for simple comparisons, but Canadian mortgages usually compound semi-annually even when paid monthly. Use the mortgage calculator for that convention.