# APR and 12th Root (Recursive Formulas)

I was going over recursive formulas and interest rates with my students the other day and I stumbled across something that I wasn't suspecting. I remember watching the following video here awhile back and now notice at 1 minute in that this problem is mentioned by taking the twelfth roof of the number vs taking the APR and dividing it by 12. Which is it? I mean looking at it here makes me think that you just divide by twelve but it still feels wrong.

Let's say your $$APR=0.15$$ and the balance you owe is $$5000$$. You have a minimum payment each payment of $$100$$ dollars. The amount you owe next month would be $$C_{i+1}=(C_i-100)(1+0.15/12)$$ where $$C_0=5000$$. Your monthly interest rate would be your APR divided by 12 as there are twelve months in a year.

Now, I assumed this was true for the longest time. However, consider the formula where $$i$$ is the number of years and $$P$$ is the principal one owe's on a loan. Ignoring the amount you pay each month implies $$P_{i+1}=1.15P_i \implies P_i=1.15^i\cdot 5000$$ where $$P_o=5000$$ (easily proven by induction). This implies $$P_n=1.15^{n/12}\cdot 5000$$ would hold where $$n$$ is the number of months. In other words, $$P_n=(1.15^{1/12})^n\cdot 5000=(\sqrt[12]{1.15})^n\cdot 5000\implies P_{n+1}=\sqrt[12]{1.15}P_n$$ where $$P_0=5000$$. So, then clearly $$\sqrt[12]{1.15}=\text{monthly interest rate}$$. This is surely a problem as the monthly interest rate cannot simply be calculated by dividing the number by $$12$$. So, why is the monthly interest rate on a credit card simply calculated by taking the number and dividing it by twelve? The two formulas give different results which is where I am confused because only one is correct. What am I doing wrong?

Note $$APR$$ is really useless to see what's going on. By definition, $$APR:=\text{monthly periodic interest rate}\cdot 12$$. It has nothing to do with compounded interest; it's just defined to be as such. The monthly periodic interest rate is what is important as it is related to the $$APY$$ below.
As noted for the twelfth root, $$APY=(1+\text{monthly periodic interest rate})^{12}-1$$.
Note $$\sqrt[12]{1+APY}=\sqrt[12]{(1+\text{monthly periodic interest rate})^{12}}=1+\text{monthly periodic interest rate}$$
Also, $$C_{i+1}=(C_i−100)(1+\text{monthly periodic interest rate})=(C_i−100)(1+APR/12)=(C_i−100)(\sqrt[12]{1+\text{APY}})$$
Hence we can convert between $$APY$$ and $$APR$$ by the last two expressions of the last equation.