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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?

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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.

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