If a bank says that it has a 5% nominal interest rate, but money is compounded monthly in that bank, what is the effective annual interest rate? What is the value of an account 10 years after it started? The initial balance in the account was $5000, and the account has 5% annual interest rate that is compounded monthly.
If a bank says that it has a 5% nominal interest rate, but money is compounded monthly in that bank, what is the effective annual interest rate? What is the value of an account 10 years after it started? The initial balance in the account was $5000, and the account has 5% annual interest rate that is compounded monthly.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![**Transcription for Educational Website**
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**Understanding Compound Interest:**
1. If a bank says that it has a 5% nominal interest rate, but money is compounded monthly in that bank, what is the effective annual interest rate?
2. What is the value of an account 10 years after it started? The initial balance in the account was $5000, and the account has a 5% annual interest rate that is compounded monthly.
**Explanation of Concepts:**
- **Nominal Interest Rate vs. Effective Annual Rate (EAR):**
- *Nominal Interest Rate* is the stated interest rate without considering compounding.
- *Effective Annual Rate* takes compounding into account and provides a more accurate measure of annual interest earned.
- **Compounding Monthly:**
- Compounding involves earning interest on both the initial principal and the accumulated interest from previous periods.
- Monthly compounding results in interest being added to the principal 12 times a year.
**Calculations:**
1. **Effective Annual Interest Rate:**
\[
\text{EAR} = \left(1 + \frac{\text{Nominal Rate}}{n}\right)^n - 1
\]
Where \( n \) is the number of compounding periods per year.
2. **Future Value of the Account:**
\[
\text{Future Value} = P \times \left(1 + \frac{r}{n}\right)^{n \times t}
\]
Where:
- \( P \) is the principal amount ($5000).
- \( r \) is the nominal annual interest rate (0.05).
- \( n \) is the number of times interest is compounded per year (12).
- \( t \) is the number of years the money is invested (10).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c12b0d1-7ded-4bb5-9f1d-3bec03db509e%2F1f6c4a1f-2ffb-413d-bab0-8207ffaaf895%2Femqcf3f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**Understanding Compound Interest:**
1. If a bank says that it has a 5% nominal interest rate, but money is compounded monthly in that bank, what is the effective annual interest rate?
2. What is the value of an account 10 years after it started? The initial balance in the account was $5000, and the account has a 5% annual interest rate that is compounded monthly.
**Explanation of Concepts:**
- **Nominal Interest Rate vs. Effective Annual Rate (EAR):**
- *Nominal Interest Rate* is the stated interest rate without considering compounding.
- *Effective Annual Rate* takes compounding into account and provides a more accurate measure of annual interest earned.
- **Compounding Monthly:**
- Compounding involves earning interest on both the initial principal and the accumulated interest from previous periods.
- Monthly compounding results in interest being added to the principal 12 times a year.
**Calculations:**
1. **Effective Annual Interest Rate:**
\[
\text{EAR} = \left(1 + \frac{\text{Nominal Rate}}{n}\right)^n - 1
\]
Where \( n \) is the number of compounding periods per year.
2. **Future Value of the Account:**
\[
\text{Future Value} = P \times \left(1 + \frac{r}{n}\right)^{n \times t}
\]
Where:
- \( P \) is the principal amount ($5000).
- \( r \) is the nominal annual interest rate (0.05).
- \( n \) is the number of times interest is compounded per year (12).
- \( t \) is the number of years the money is invested (10).
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