Question

a woman went to the beneficial loan company and borrowed $10,000. she must pay $323.53 at the end of each month for the next 60 months. what is the monthly interest rate she is paying? what effective annual interest rate is she paying?

Explanation:

Step1: Recall the formula for effective - annual - rate (EAR) and monthly interest rate relationship.

Let the monthly interest rate be $r$. The annual interest rate $i$ and monthly interest rate are related. First, we can use the loan - payment formula to find the monthly interest rate. The loan - payment formula for a fixed - payment loan is $P = A\times\frac{1-(1 + r)^{-n}}{r}$, where $P$ is the principal amount, $A$ is the monthly payment, $r$ is the monthly interest rate, and $n$ is the number of payments. Here, $P=\$10000$, $A = \$323.53$, and $n = 60$.
We have $10000=323.53\times\frac{1-(1 + r)^{-60}}{r}$. This is a non - linear equation. We can also use a financial calculator or an iterative approach. Using a financial calculator: $N = 60$, $PV=- 10000$, $PMT = 323.53$, $FV = 0$. Solving for $I/Y$ (monthly interest rate), we get $r\approx2\%$.

Step2: Calculate the annual interest rate.

The annual interest rate $i$ (nominal) is related to the monthly interest rate $r$ by $i = 12r$. Since $r=0.02$, then $i = 12\times0.02=0.24$ or $24\%$.

Step3: Calculate the effective annual rate.

The formula for the effective annual rate (EAR) is $EAR=(1 + r)^{12}-1$. Substituting $r = 0.02$ into the formula, we have $EAR=(1 + 0.02)^{12}-1$.
\[

$$\begin{align*} EAR&=(1.02)^{12}-1\\ &=1.26824179 - 1\\ &=0.26824179\approx26.82\% \end{align*}$$

\]

Answer:

The monthly interest rate is approximately $2\%$, and the annual interest rate is $24\%$, and the effective annual rate is approximately $26.82\%$.