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.
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.