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I've been reading about loans and interest rates, and I've run across a mathematical sticking point. The crux is that, if my loan's outstanding balance is £100,000, I make no payments, and it has an annual interest of 8%, I expect that the outstanding balance after one year will be £108,000.

Sites like Investopedia give examples like the following:

The interest on a mortgage is compounded or applied on a monthly basis. If the annual interest rate on that mortgage is 8%, the periodic interest rate used to calculate the interest assessed in any single month is 0.08 divided by 12, working out to 0.0067 or 0.67%.

This example does not support my intuition though. If the balance increases by 0.67% per month and I make no payments, then after a year I will have an outstanding balance of £100,000 * (1 + 0.08/12)^12, which is around £108,300. The actual annual interest is closer to 8.3%, which is somewhat higher than 8%.

If we wanted the monthly interest rate that would cause the yearly interest to actually be 8%, then we should compute the 12th root of 1.08; the monthly interest ought to be around 0.643%.

I've looked around for answers to this discrepancy - a maths.stackexchange post clarifies how the interest rate works, but the root of my question is why the interest rate works like this. If it is indeed the case that an 8% yearly interest actually means that a loan's balance increases by 8.3% per year, what use is the number 8% here?

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    The terminology is dependent on the country you're in, but it's basically this: investopedia.com/terms/a/… – Paul Sep 4 '20 at 14:11
  • "If it is indeed the case that an 8% yearly interest actually means that a loan's balance increases by 8.3% per year, what use is the number 8% here?" It sounds like a better rate than it really is, so the lender is likely to get more business. – Paul Sep 4 '20 at 14:16
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    @Paul It sounds like your answer to this is "lenders are allowed to quote either because of lax laws, so they quote the number that makes them look better". I'm happy with that as an answer, but could you actually post it as an answer so I can upvote it and accept it? – ymbirtt Sep 4 '20 at 14:17
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    US Federal Reserve R1314 : "The 'Truth in Lending Act' passed in 1968 did not incorporate the mathematically-true annual percentage rate, because the true calculation used compounding (sometime fraction compounding), which was not readily available." - money.stackexchange.com/a/124507/11768 – Chris Degnen Sep 4 '20 at 14:36
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    @ymbirtt: Not everyone had access to, or could easily us, those 14-place log tables, but most people can divide by 12. The divide-by-12 rule is also much simpler and applicable to typical situations, since the great majority of loans are paid in monthly installments. – jamesqf Dec 24 '20 at 4:42
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In the United States, this is the difference between APR and APY. APR is typically the annual percentage rate without taking compounding into effect, so in your example, 8%. If the loan quotes you an APR of 8%, then the true interest that will accrue over the course of a year will potentially be different, depending on how frequently the loan is compounded. That's the 8.3% you calculated.

APY is the annual percentage yield, which does take compounding into effect. This is, again, the 8.3% you calculated given an APR of 8% and monthly compounding. If the loan quotes an APY of 8%, then that 8% takes compounding into effect, and would have an APR of 7.72% (12 times the 0.643% you calculated).

A loan can specify either APR or APY, so that's where you need to read the fine print to know whether the interest rate they're quoting you includes compounding or not.

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  • Thanks for the answer - knowing that there are different names for the two calculations is certainly useful, but I can't accept it since this doesn't really get to the heart my question. I'm hoping to understand why APR might be a useful figure to quote. As far as I can tell, the only useful thing either I or a bank can do with APR is convert it into APY. Do you happen to know of a common use-case for APR which doesn't simply involve computing APY? – ymbirtt Jan 10 at 20:43
  • You should always use APY to calculate interest revenues/expenses, and when comparing any two loans or investments, you should always make that comparison on an APY basis. In certain contexts, interest rates are quoted as APRs (e.g., most credit card terms). In such contexts, it's the borrower's task to calculate the APY to determine the actual cost of borrowing. If you're given an APY, there's rarely any need to calculate APR. – Amaan M Jan 13 at 22:51
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The market convention just a tradition, and depends on the country. In the U.S. and most other countries, if a bond or a loan paying 6% a year with semi-annual frequency means each coupon is exactly 6/2=3%. (Or perhaps daycounted to almost exactly 3%.) But in Brazil, for example, 6% a year semi-annually means that each coupon is (1+6%)^(1/2)-1=2.956301% (see, for example, https://sisweb.tesouro.gov.br/apex/f?p=2501:9::::9:P9_ID_PUBLICACAO:27710 , page 8). Clearly, a fraction 1/frequency is easier than the frequency'th root, which may be why the former convention has been more widely adopted - but not universally.

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  • Thanks for the answer - it sounds as though you're saying that this difference between APR and APY exists as a tradition, which I suppose I can accept, but isn't totally satisfying. I'm not sure I follow your argument, though. Are you saying, then, that institutions across borders sometimes agree on particular interest rates and compounding schedules, but some favour a simpler number per compound, so pick a method that gives them a simple number and accept that they'll be out by a few dollars/real compared to the other institution? – ymbirtt Jan 10 at 21:05
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Interest on loans, bonds, and other financial instruments is typically quoted as an annualized, uncompounded figure. So a loan with a quoted rate of 8% that is charged monthly will have a monthly rate of 8%/12 or 0.6667%. Some other "loans" (like credit cards) compute a daily interest rate that is used to calculate interest based on the average daily balance. The loan does not compound daily, but the amount of interest charged it calculated by taking the annual rate divided by 365 (or 360, or 366 depending on the terms of the loan and how many days are in the year). Bond interest rates are quoted that way too, even though interest is paid every 3 or 6 months.

The effective rate is found by taking the periodic rate and annualizing it by compounding it using the compounding period (e.g. monthly). So a loan with a 0.66667% monthly rate, after compounding for 12 months, will have an effective rate of (1.00666667 ^ 12) - 1, or about 8.3%, meaning your 100,000 loan will have a balance of 108,300 after one year if no payments are made (plus any late fees, of course).

The reason banks quote rates this way is to make different types of loans comparable. Not all loans compound monthly (bonds can compound every 3 or 6 months) and it allows for more round numbers than would be practical with a monthly rate.

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  • As a non-American, this is another one of those things that just don't make sense. So banks can advertise a rate that is lower than the actual rate you will be charged? Insane – Jon Dec 24 '20 at 9:45
  • @Jon. This is not an America thing. This is a most-of-the-world thing. Also, as this answer already states, this is also a simplicity thing, not just an advertising thing. – AxiomaticNexus Dec 24 '20 at 14:31
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    @Jon: No, the advertised rate is the rate you will be charged if you make the payments. If you allow the interest to be added to the principal, you of course pay more. E.g. if I take out a $100K loan at 3% and make monthly payments, then the first month I will be paying $100K * 3% / 12, or $250. But I will have also paid some principal, so next month my interest will be a bit less, since it's charged only on the outstanding balance. – jamesqf Dec 24 '20 at 18:02
  • I understand your point that different products compound at different rates for various reasons - a credit card that compounds interest every six months obviously isn't much good for the bank! I don't see how this way of quoting rates makes loans more "comparable", though. Could you clarify that? If I see one bank offer 8% interest and another offer 8.1% interest, 8% looks more appealing. If, though, the 8% loan compounds every month while the 8.1% compounds every 6, then the APY of the 8% loan is 8.30% and the APY of the 8.1% is 8.26% - using APR here has made the products less comparable – ymbirtt Jan 10 at 21:14
  • I was talking more about annualizing the interest by multiplying by the periodicity - a 1.15% loan that bills monthly and a 7% bond that pays every 6 months are not directly comparable without a little arithmetic.It's not hard math but normalizing everything to 12-months gets rid of it with no ambiguity. And most people don't understand the subtleties of yield enough to understand compounded bond rates. So simple interest just makes things, well, simpler. – D Stanley Jan 11 at 14:23

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