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wondering if someone can help a non-mathematician out.

I am looking for the formula for calculating the total interest payable over the life of a loan.

Given that we know:

P: Principal (amount) of loan

R: The monthly repayments on the loan

T: The term of the loan (i.e. the number of repayments)

r: The annual interest rate

Assume that interest is accrued monthly and that repayments are made in arrears (at the end of each loan period).

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  • $\begingroup$ Most loans I am familiar with accrue daily, not monthly...not that it is a huge difference with smaller loans. They will quote you an APR then charge interest on a daily basis (esp for credit cards) dividing that annual rate by 365 or 365.something or other. $\endgroup$ – user340642 May 19 '16 at 1:01
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If you know the monthly payment and the number of payments, the total of payments is $RT$, so the total interest you pay is $RT-P$. The fact that the interest is accrued monthly and payments are made in arrears only go into calculating the payment from the interest rate, which has already been done for us.

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  • $\begingroup$ @CrescentFresh: the interest is whatever money you pay beyond the starting principal. I added up all the payments and deducted the principal, so that is the total interest. We were not asked to find when the interest is paid, so it doesn't matter how each payment is allocated. Say you borrow 2500 and pay 100/mo for 36 mo, total 3600. Then the interest is 1100. We don't need to know the interest rate-it just goes into figuring the payment. $\endgroup$ – Ross Millikan Mar 17 '16 at 23:37
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"If you know the monthly payment and the number of payments, the total of payments is RT, so the total interest you pay is RT−P. The fact that the interest is accrued monthly and payments are made in arrears only go into calculating the payment from the interest rate, which has already been done for us."

Part of Ross's answer must be missing, otherwise it makes no sense.

RT-P. Let's say R, monthly payments on loan, are \$1,000. T, total number of payments is 36. RT then equals 36,000, minus the Principal, which is \$36,000 = \$0. So, you're paying $0 in interest, and you've factored in the interest rate nowhere. Great!

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  • $\begingroup$ What makes you think that the principal is \$36,000? That's only true if you implicitly assume that you are paying no interest. $\endgroup$ – Nick Peterson Oct 28 '13 at 21:13
  • $\begingroup$ You don't need the interest rate to calculate this. You may not have a calculated interest rate even and this formula would apply. FYI, your "answer" should have been posted as a comment to Ross' (correct) answer, not as a separate answer. $\endgroup$ – Jared Jul 4 '15 at 23:22
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You said Given that we know: P: Principal (amount) of loan R: The monthly repayments on the loan T: The term of the loan (i.e. the number of repayments) r: The annual interest rate

On a \$50,000 loan at an interest rate of 8% for 5 years, then your monthly payment is \$1013.82 each month for 60 months.

You then multiply \$1013.82 x 60 payments to determine you will pay \$60,829.20 over the life of the loan.

By subtracting your original loan amount of \$50,000 (P) from \$60,829.20 (RT) you will see that over the life of the loan you will pay \$10,829.20 in total interest.

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