To calculate the nominal annual interest rate based on monthly compounding given the annual effective interest rate, we can use the relationship between the two.

The formula relating the nominal interest rate $i_{nom}$, the number of compounding periods per year $n$, and the annual effective interest rate $i_{eff}$ is:

$i_{eff} = \left(1 + \frac{i_{nom}}{n}\right)^n - 1$

In this case:

• $i_{eff} = 25.3\% = 0.253$
• $n = 12$ (since interest is compounded monthly)

We need to solve for $i_{nom}$, the nominal annual interest rate.

Let's calculate this now.Let me manually walk through the calculation steps for the nominal annual interest rate.

We have the formula:

$0.253 = \left(1 + \frac{i_{nom}}{12}\right)^{12} - 1$

1. Add 1 to both sides:

$1.253 = \left(1 + \frac{i_{nom}}{12}\right)^{12}$

2. Take the 12th root of both sides:

$(1.253)^{\frac{1}{12}} = 1 + \frac{i_{nom}}{12}$

3. Solve for $i_{nom}$:

$\frac{i_{nom}}{12} = (1.253)^{\frac{1}{12}} - 1$

$i_{nom} = 12 \times \left((1.253)^{\frac{1}{12}} - 1\right)$

Now let's calculate this value.

$i_{nom} = 12 \times (1.019135 - 1) = 12 \times 0.019135 = 0.22962$

The nominal annual interest rate is approximately 22.96%.

Would you like further details or clarifications? Here are some related questions to explore:

1. What is the difference between nominal and effective interest rates?
2. How does compounding frequency affect the total interest paid?
3. What would be the nominal rate if interest compounded daily instead of monthly?
4. Can you explain the process of converting an effective interest rate to a nominal one for quarterly compounding?
5. How do credit card interest rates compare to loan rates in terms of compounding?

Tip: Always check the compounding frequency when comparing different interest rates—it can significantly impact the cost of borrowing!