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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?

ymbirtt
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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.

Amaan M
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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.

Dimitri Vulis
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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.

D Stanley
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