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Sample question: We invest $100,000 in an account earning interest at a rate of 7.5% for 54 months. How much money will be in the account if interest is compounded quarterly?

To me "compounded quarterly" means we apply the interest rate to the current balance every three months, so $100000 \cdot (1 + .075)$ after three months, $100000 \cdot (1 + .075)^2$ after six months, etc.

But apparently that's not correct at all and I don't understand why.

Lukas Heger
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user525966
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    The issue is that the rate is the annual rate (roughly), not the rate per compounding time. – Aaron Feb 05 '18 at 00:36
  • You're wrong simply because that's not the general commercial (and legal) usage of the term, at least in the United States. It might mean what you think it means, but as it happens, it doesn't. – saulspatz Feb 05 '18 at 00:36
  • What does it mean then? – user525966 Feb 05 '18 at 00:37
  • Also see https://math.stackexchange.com/questions/2627519/why-do-we-divide-by-the-number-of-compounding-periods-when-calculating-compound/2627541 – David K Feb 05 '18 at 00:51

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If the interest rate is $7.5$ percent per year, then we don't want it to be $7.5$ percent per quarter. The idea is to take the interest rate, and split it up, applying one quarter of it each quarter (or $1/12$ of it each month, etc.).

If you get $\frac14$ of $7.5$ percent after one quarter, then you're on pace to earn $7.5$ percent per year, but you go ahead and get a partial payment after one quarter.

This is because the given interest rate is an "APR" - an "Annual Percentage Rate". It's not a QPR, if that makes sense.

G Tony Jacobs
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  • Is that how it works by definition? If $r$ is an annualized rate then we can take the quarterly rate by dividing it by $4$, the semi-annual rate by dividing it by $2$, etc. Is this a definition thing or is there a mathematical basis for justifying this basic division? – user525966 Feb 05 '18 at 00:59
  • It's by definition. I mean, it's kind of weird, because (1+r/4)^4 is more than 1+r, so you end up with more than the nominal growth in a year. On the other hand, if you only leave your investment in the account for one quarter, you might complain if you don't have a quarter of the interest after that time. – G Tony Jacobs Feb 05 '18 at 02:24
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7.5% is the APR (annual percentage rate). The monthly percentage rate is $\dfrac rm$ where $m$ is the number of times per year that the interest is applied. Note that ther actual percent of principal paid per year is therefore more than 7.5%.

Here is what $\color{red}{\text{Wikipedia}}$ has to say about it.

Steven Alexis Gregory
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The interest rate of $ 7.5 $ percent is an annual rate.

Thus for a quarterly rate you need to divide it by $4.$

The new rate is $1.875$ percent and you need to compound it for $18$ quarters.

The final result is $$100000(1.01875)^{18} =139706.6862 $$

Mohammad Riazi-Kermani
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