I have a calculation which gives me the annual interest rate if I already know the monthly interest rate as follows:
(Monthly interest rate + 1)^12
In this case I have 0.32% as the monthly rate (stated as a number: 0.0032 plus 1 is 1.0032 ) to the power 12 (the number of repayments per year) to give me 3.908% as the annual rate.
However let's say I have the annual rate of 2.549% but I do not know the monthly rate. I would like to find the reverse of this calculation to give me the monthly interest rate.
6 Answers
$\begingroup$Let's denote by $L$ the value of the loan, $m$ the monthly rate and $a$ the annual rate. $M$ is the total number of months of the debt and $A$ is the total number of year of the debt (of course, $A=12M$).
What you owe from beginning of month $\mathcal{M}$ to end of month $\mathcal{M}$ is: $$(1+m)^\mathcal{M}\times \dfrac{L}{M}$$
For example is the interest is 0, you will owe $L/M$ each month.
Similarly, for what you owe for year $\mathcal{A}$ is: $$(1+a)^\mathcal{A}\times\dfrac{L}{A}$$
Now, let's write that the interest rates $a$ and $m$ are such that and the end of the first year, both values should be equal (12 times what your owe monthly and 1 time what you owe annualy):
$$12\times(1+m)^\mathcal{12}\times \dfrac{L}{12}=(1+a)^1\times\dfrac{L}{1}$$
This gives (almost, you forgot to substract 1) your first formula: $$\boxed{a=(1+m)^{12}-1}$$
To express $m$ as a function of $a$, you just have to manipulate this formula: $$(1+a)^{1/12}=(1+m)^{12/12}$$ so $$\boxed{m=(1+a)^{\frac{1}{12}}-1}$$
With $a=0.02549$, the calculation yields: $0.00210$ so the monthly rate is $0.21\%$, which is not equal to $0.02549/12$, because you pay interest on the interest which has not been refunded yet.
$\endgroup$ 2 $\begingroup$If the stated annual rate is $2.549\%$, you would divide by $12$ to get the monthly rate.
However, if the effective annual rate is $2.549\%$, then letting the monthly rate be $i$, we have $(1+i)^{12}=1.02549$. So $i=\sqrt[12]{1.02549}-1\approx .0021$, or $0.21\%$
$\endgroup$ 2 $\begingroup$If you want to turn that into depreciation you can use this minor change to anderstood's answer:
$$\boxed{m=(1-a)^{\frac{1}{M}}-1}$$
$\endgroup$ $\begingroup$I will take a novel approach and just demonstrate two mathematical concepts that are critical to grasp. To keep things down to earth, let's say I use the rate 2, and each time I use it, that represents the number of payment periods in the loan. I hope to convince you that answers like: "Divide the annual interest rate expressed as a percentage by 12 to calculate the monthly interest rate expressed as a percentage." are inaccurate.
Addition can be divided
$$ 2 + 2 + 2 = 6 $$
How would I get from $6$ back to $2$?
$$\frac{6}{3} = 2$$
Multiplication can be rooted
$$2 \times 2 \times 2 = 8$$
How would I get from 8 back to 2?
$$ \sqrt[3]{8} = 2$$
Reread the above stuff and make sure you really understand the importance of roots.
APR If you have monthly periods and you are using an APR based on those, you’re getting jipped. This corresponds to a using division to calculate "Multiplication", which is simply wrong (when there are multiple periods in a year). e.g. $\frac{8}{3} \neq 2$
EIR corresponds to what one might expect, given the rules of operators in algebra (how plus and times work: the properties like associativity, commutativity, distributivity...). e.g. $\sqrt[3]{8} = 2$
Combining this knowledge with the formula
Have a look at anderstood's answer.
$$ m = (1+a)^{\frac{1}{12}} - 1 $$
You accepted his answer on the grounds that he did not use a "weird 12th root". Well, I got news for you, the following is equivalent. It is just reversing the multiplication as I demonstrated under Multiplication can be rooted, to get the monthly rate. Why? The rate will be compounded 12 times a year, which is the same as saying the rate will be multiplied times itself 12 times (like 2 in my example).
$$ m = \sqrt[12]{(1+a)} - 1 $$
which means that
$$ m \times m \times m \times m \times m \times m \times m \times m \times m \times m \times m \times m = a $$
which is what we want (monthly rate $\times$ 12 will be the annual rate).
Why do we add 1 only to remove it later? Because $a$ is an annual percentage rate, by virtue of percentages and rates, $0<a<1$. Anything between 0 and 1 makes multiplication work opposite of what you might first think. Just try it yourself on a calculator:
$$\sqrt{0.02} = 0.14142135623$$
What the is going on? You multiply $.14142135623 \times .14142135623 \neq .02$. Not even rounding can help you there.
$$\sqrt{1.02} = 1.00995049384$$
Much better, $0.00995049384 \times 0.00995049384 = .02$ with rounding.
We want a smaller number: the monthly rate, which when multiplied by itself 12 times, yields the annual rate, $a$. The trick is to just add 1 (which is the multiple operator's neutral element in algebra) and remove it later when doing the root. $\sqrt[12]{1.02}$.
$\endgroup$ $\begingroup$Simple:
Divide the annual interest rate expressed as a percentage by 12 to calculate the monthly interest rate expressed as a percentage. For example, if you have an annual interest rate of 7.8 percent, divide 7.8 by 12 to find the monthly interest rate is .65 percent.
$\endgroup$ $\begingroup$It depends on your country.
Mathematically speaking all the answer you received are correct. In fact, the interest on the monthly interest (compound interest) are legal in some country and not in some other. (For instance this is the case in Italy.)
So if you are really interested in calculate interest of a real loan, you should check the law of your country.
See Compound interest and, if you are curious about the law of Italy, see Anatocismo.
So if in your country compound interest are legal, use the formula $\sqrt[12]{1+a}-1$, if compound interest are not legal use $a/12$.
By the way, in your case you get more or less the same number $$\sqrt[12]{1.02549}-1\approx 0.002099747\qquad 0.02549/12=0.002124167$$
So your monthly interest is about $0.21\%$ in any country :)
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