# Percentage conversion

**URL:** <https://boards.straightdope.com/t/percentage-conversion/133739>\
**Category:** Factual Questions\
**Created:** [October 24, 2002, 8:35pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739 "2002-10-24T20:35:13Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![Carnal\_knowledge](https://avatars.discourse-cdn.com/v4/letter/c/c57346/32.png) [@Carnal\_knowledge](https://boards.straightdope.com/u/Carnal_knowledge)\
**Post date:** [October 24, 2002, 8:35pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739/1 "2002-10-24T20:35:13Z")

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I’ve blanked and the search engines are spewing financial reports for my queries.

What’s the equation to convert an annual percentage rate, say 12%, to a quarterly (or semi, or monthly) percentage?

Thanks.

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**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\
**Post date:** [October 24, 2002, 8:57pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739/2 "2002-10-24T20:57:28Z")

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Well, if the interest is paid monthly and compounded, it’s:

(1 + [sup]x[/sup]/[sub]100[/sub])[sup]12[/sup] = 1.12

So x would be (1.12 - 1)[sup]1/12[/sup] x 100

Er, I think.

You might need a scientific calculator, to calculate that 12th root… (1/12 = 0.08333…)

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [October 24, 2002, 9:02pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739/3 "2002-10-24T21:02:56Z")

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If you have a compounded annually rate **r** and want to convert it to or from some equivalent interest rate **I** compounded **P** times a year:

r = [1 + (I/P)][sup]P[/sup] - 1

So 12% compounded annually, for example would be equivalent to about 11.49% compounded quarterly.

Is that what you’re looking for?

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<div class="post-metadata">

**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\
**Post date:** [October 24, 2002, 9:04pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739/4 "2002-10-24T21:04:36Z")

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Sorry, getting confused with all the VB code. That should be:

monthly percentage rate = {(1.12)[sup]1/12[/sup] - 1} x 100 (assuming APR is 12%).

More generally, where M is the monthly rate and A is the APR:

M = {(1 + [sup]A[/sup]/[sub]100[/sub])[sup]1/12[/sup] - 1} x 100

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<div class="post-metadata">

**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\
**Post date:** [October 24, 2002, 9:19pm UTC](https://boards.straightdope.com/t/percentage-conversion/133739/5 "2002-10-24T21:19:50Z")

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And I _think_ (although it’s getting late and my head is hurting) that this would also work for interest compounded at other intervals, as **Cabbage** posted:

Where M is the rate per time interval of [sup]1[/sup]/[sub]T[/sub] of a year, and A is the APR:

(1 + [sup]A[/sup]/[sub]100[/sub]) = (1 + [sup]M[/sup]/[sub]100[/sub])[sup]T[/sup]

==\> M = {(1 + [sup]A[/sup]/[sub]100[/sub])[sup]1/T[/sup] - 1} x 100

(NB this assumes A and M are quoted as whole figures, hence the division by 100 to convert the “percentage” interest rate to a fraction")

I notice that **Cabbage** used the term “11.49% compounded quarterly” to mean an overall annual rate of 11.49%, but compoounded each quarter. I’d call that a rate of somewhere under 3% (roughly - I haven’t done the math) per quarter, but I guess it’s all a matter of semantics.

Similarly, a rate of around 0.9% (very roughly) per month will yield an APR of 12%.

Any accountants in the house?