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Let's say I want to borrow $1000 from a bank at an interest rate of 12% per year, for one year. The number of payments in this case is equal to 12 (one year * months per year) and the interest rate per period (month) would be 1% (12% / 12).

According to many calculators online:

the payment per period would be equal to $88.85, while the total interest at the end of the loan would be $66.19.
I also built my own spreadsheet in excel and confirmed these calculations.

Question

I expect the bank to receive exactly $1120 at the end of the loan (1000 * 112%). At least not including inflation.
But the bank receives nominal price of $1066.19 (12 payments of $88.85), only 6.6% of total interest and it's less than 12%, and this is even not including inflation.
If the annual inflation is greater than 6.6%, the bank loses money.

Could you explain me where in my calculations I went wrong? Thanks!

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  • Do last answers about compound interest not answer your question?
    – keshlam
    Nov 22, 2022 at 18:21
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    You pay interest for the time your owed the money, after 1st payment you owe less, after 2nd even less, and so on.
    – littleadv
    Nov 22, 2022 at 18:25
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    @littleadv, oh, I got the point, the bank reinvest the returned money and doesn't lose them due to inflation. Thanks!
    – g00dds
    Nov 22, 2022 at 18:28
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    Roughly speaking, you're making regular payments each month from month 1 through month 12, so the average dollar of the loan lives in your account for about (1+12)/2=6.5 months. Thus you should pay approximately (6.5/12)*12%=6.5% of the initial principal in interest. Note that this is close to but slightly less than the 6.619% of the initial principal that comes out when you do the exact computation, because the early payments have slightly more interest while the later ones have slightly more principal, so actually the average dollar stays in your account for a little longer than 6.5 months.
    – Micah
    Nov 23, 2022 at 3:20
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    If inflation is greater than 6.6% the bank does not lose money. The bank loses pizzas or cars or cellphones but not money. $1066 is always more money than $1000, no matter what.
    – user253751
    Nov 23, 2022 at 9:59

2 Answers 2

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I expect the bank to receive exactly 1120 dollars at the end of the loan (1000 * 112%). At least not including inflation. But the bank receives nominal price of 1066.19 dollars (12 payments of $88.85), only 6.6% of total interest and it's less than 12%, and this is even not including inflation. If the annual inflation is greater than 6.6%, the bank loses money.

If the bank allowed you to make zero payments during the year , and required a lump sum at the end of the 1 year loan duration, then you would make a payment of $1120.

Instead you paid some of it off each month. The first months payment of $88.85 covered the interest for the month (1% * 1000) or $10, and a payment against the loan balance of $78.85. leaving the balance at 921.15. That means the second month the interest will only be ($921.15 * 1%) or $9.21.

That monthly payment does allow them to fund another loan to another customer.

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  • Maybe OP will get a prepayment penalty?
    – gerrit
    Nov 23, 2022 at 17:01
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    @gerrit: A prepayment penalty is not contemplated for payments made on the schedule agreed to in advance. Penalties would only be possible if payments were made in advance of the agreed schedule, and only if there's such a clause in the contract, which would be quite unusual.
    – Ben Voigt
    Nov 23, 2022 at 19:41
  • Prepayment penalties are not that unusual, especially in subprime and aggressively marketed loans (such as from car dealerships), since they protect the lenders ability to generate revenue from the interest. There are certain loans (FHA and USDA) that prohibit prepayment penalties, but by and large, I think they're always something to be on the lookout for. They probably should be unusual or even prohibited entirely, but that's a topic for discussion outside the scope of comments and answers.
    – Jonathon Richardson
    Nov 23, 2022 at 21:49
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    @JonathonRichardson But as BenVoight says, prepayment penalties only come into play if you make a prepayment. The assumption of the question and of this answer is that you don't.
    – Jay
    Nov 24, 2022 at 1:18
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    This is a good answer and I upvoted. I would just add that the original question also brought up inflation. But inflation is irrelevant to the discussion. Sure, the value of the interest the bank receives may be less because of inflation, but that has no effect on the number of nominal dollars the bank receives, nor does it change anything about this answer.
    – Jay
    Nov 24, 2022 at 1:19
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After the first month, you don't owe the bank $1,000 anymore. At the end of the month, you paid $10 interest plus another $78.85, so now you owe only $921.15 after that month. So for the next month, you pay only $9.21 in interest, and your repayment goes up. For the last month, you will pay less than one dollar in interest. So your monthly interest goes down from $10 to less than $1 a month over the year, and that's why you don't pay $120 in interest but only $66.19.

When you compare loans, look for a number named "APR". That is an interest rate calculated according to some very precise laws and includes all costs (fees etc.) of the loan. If you find two loans with 6.20% APR they will be exactly as good/bad for you as each other. A third loan with 6.21% APR is a tiny bit more expensive, and one with 6.19% is a tiny bit cheaper. The APR MUST be calculated correctly, or the bank would be in deep ****.

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