# Percentage conversion

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.

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…)

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?

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

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?