First National Bank charges 11.6 percent compounded monthly on its business loans. First United Bank charges 11.8 percent compounded semiannually. Calculate the EAR for each bank.

The **Effective Annual Rate (EAR)** is the actual interest rate a borrower pays after accounting for compounding within a year. The formula to calculate the EAR is:

\[
EAR = \left(1 + \frac{i}{n}\right)^n - 1
\]

Where:
- \( i \) is the nominal interest rate (annual percentage rate or APR),
- \( n \) is the number of compounding periods per year.

### For First National Bank:
- Nominal interest rate (\(i\)) = 11.6% or 0.116
- Compounding periods per year (\(n\)) = 12 (monthly)

\[
EAR_{\text{FNB}} = \left(1 + \frac{0.116}{12}\right)^{12} - 1
\]

### For First United Bank:
- Nominal interest rate (\(i\)) = 11.8% or 0.118
- Compounding periods per year (\(n\)) = 2 (semiannually)

\[
EAR_{\text{FUB}} = \left(1 + \frac{0.118}{2}\right)^2 - 1
\]

Now, let's calculate both EARs.

I'll compute the values.The Effective Annual Rates (EAR) are:

- **First National Bank**: 12.24% (approximately)
- **First United Bank**: 12.15% (approximately)

The First National Bank offers a slightly higher effective interest rate than the First United Bank.

Would you like more details or have any questions? Here are five related questions you might find useful:

1. How is EAR affected by increasing the number of compounding periods?
2. How would the EAR change if First National Bank compounded daily?
3. Can you calculate the monthly payment for a loan with these EARs?
4. How does nominal interest differ from effective interest?
5. What impact does compounding frequency have on the cost of a loan?

**Tip:** When comparing loan rates, always consider the EAR, as it accounts for the effects of compounding.

Learn how to calculate the Effective Annual Rate (EAR) for two different banks: First National Bank with 11.6% interest compounded monthly, and First United Bank with 11.8% interest compounded semiannually. This problem involves understanding interest rates, compounding, and using the EAR formula. Suitable for Grades 11-12 or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation