A loan is offered with monthly payments and a 11.75 percent APR. What's the loan's effective annual rate (EAR)? (Do not round intermediate calculations and round your final answer to 2 decimal places.) Effective annual rate. %
A loan is offered with monthly payments and a 11.75 percent APR. What's the loan's effective annual rate (EAR)? (Do not round intermediate calculations and round your final answer to 2 decimal places.) Effective annual rate. %
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![### Calculating the Effective Annual Rate (EAR) from APR with Monthly Payments
When you take a loan, it is important to understand the actual cost of borrowing money. The Annual Percentage Rate (APR) is often quoted, but it doesn't reflect the true annual interest rate if payments are made more frequently than once a year. The Effective Annual Rate (EAR) gives a clearer picture of the real cost of borrowing.
#### Problem Statement
A loan is offered with monthly payments and a 11.75 percent APR. What’s the loan’s effective annual rate (EAR)? (Do not round intermediate calculations and round your final answer to 2 decimal places.)
**Input Field:**
- Effective Annual Rate: `[ ] %`
#### Explanation
The APR is the nominal interest rate, which does not take into account the effect of compounding within the year. The EAR, however, does take this compounding into account and provides a more accurate picture of the cost of the loan over a year.
#### Formula
To convert the APR to EAR, given monthly compounding, you can use the following formula:
\[ \text{EAR} = \left(1 + \frac{\text{APR}}{n}\right)^{n} - 1 \]
Where:
- \(\text{APR}\) is the annual percentage rate (expressed as a decimal),
- \(n\) is the number of compounding periods per year.
For monthly payments, \(n = 12\). Given the APR of 11.75%, the calculation steps are:
1. Convert APR to decimal form: \(\text{APR} = 11.75\% = 0.1175\).
2. Apply the formula:
\[ \text{EAR} = \left(1 + \frac{0.1175}{12}\right)^{12} - 1 \]
3. Perform the calculations:
- Divide the APR by the number of periods: \(\frac{0.1175}{12} = 0.00979\)
- Add 1 to this value: \(1 + 0.00979 = 1.00979\)
- Raise to the power of 12: \((1.00979)^{12}\)
- Subtract 1 from the result
After calculating, you should round your final answer to 2 decimal places:
\[ \text{EAR} \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F136a9916-8a5d-490b-a1c4-57e644d5608a%2F57e1f2b1-b6c8-4b1e-9c74-409d8a4fe3e1%2F8gml2yg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Effective Annual Rate (EAR) from APR with Monthly Payments
When you take a loan, it is important to understand the actual cost of borrowing money. The Annual Percentage Rate (APR) is often quoted, but it doesn't reflect the true annual interest rate if payments are made more frequently than once a year. The Effective Annual Rate (EAR) gives a clearer picture of the real cost of borrowing.
#### Problem Statement
A loan is offered with monthly payments and a 11.75 percent APR. What’s the loan’s effective annual rate (EAR)? (Do not round intermediate calculations and round your final answer to 2 decimal places.)
**Input Field:**
- Effective Annual Rate: `[ ] %`
#### Explanation
The APR is the nominal interest rate, which does not take into account the effect of compounding within the year. The EAR, however, does take this compounding into account and provides a more accurate picture of the cost of the loan over a year.
#### Formula
To convert the APR to EAR, given monthly compounding, you can use the following formula:
\[ \text{EAR} = \left(1 + \frac{\text{APR}}{n}\right)^{n} - 1 \]
Where:
- \(\text{APR}\) is the annual percentage rate (expressed as a decimal),
- \(n\) is the number of compounding periods per year.
For monthly payments, \(n = 12\). Given the APR of 11.75%, the calculation steps are:
1. Convert APR to decimal form: \(\text{APR} = 11.75\% = 0.1175\).
2. Apply the formula:
\[ \text{EAR} = \left(1 + \frac{0.1175}{12}\right)^{12} - 1 \]
3. Perform the calculations:
- Divide the APR by the number of periods: \(\frac{0.1175}{12} = 0.00979\)
- Add 1 to this value: \(1 + 0.00979 = 1.00979\)
- Raise to the power of 12: \((1.00979)^{12}\)
- Subtract 1 from the result
After calculating, you should round your final answer to 2 decimal places:
\[ \text{EAR} \approx
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