Question
An effective annual rate of 7.12 percent is equal to 7 percent compounded
Answers
GPT-4o mini
To determine the compounding frequency needed so that an effective annual rate (EAR) of 7.12% is equal to a nominal rate of 7% compounded at that frequency, we use the formula for converting a nominal interest rate to an effective annual rate:
\[
EAR = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) is the nominal annual interest rate (in decimal form),
- \( n \) is the number of compounding periods per year,
- \( EAR \) is the effective annual rate (in decimal form).
In this case, \( r = 0.07 \) (for 7%) and \( EAR = 0.0712 \) (for 7.12%). We want to find \( n \).
We rewrite the formula:
\[
0.0712 = \left(1 + \frac{0.07}{n}\right)^n - 1
\]
To solve for \( n \), we can rearrange:
\[
1 + 0.0712 = \left(1 + \frac{0.07}{n}\right)^n
\]
\[
1.0712 = \left(1 + \frac{0.07}{n}\right)^n
\]
To find \( n \), trial and error or numerical methods (e.g., using a financial calculator or spreadsheet software) can be applied.
However, we can provide a few calculations for commonly used compounding frequencies:
1. **Annual (n = 1)**:
\[
1 + \frac{0.07}{1} = 1.07
\]
\[
(1.07)^1 = 1.07 \quad (EAR = 7.00\%)
\]
2. **Semi-Annual (n = 2)**:
\[
1 + \frac{0.07}{2} = 1.035
\]
\[
(1.035)^2 = 1.071225 \quad (EAR \approx 7.12\%)
\]
3. **Quarterly (n = 4)**:
\[
1 + \frac{0.07}{4} = 1.0175
\]
\[
(1.0175)^4 \approx 1.071859 \quad (EAR \approx 7.19\%)
\]
4. **Monthly (n = 12)**:
\[
1 + \frac{0.07}{12} = 1.0058333
\]
\[
(1.0058333)^{12} \approx 1.072290 \quad (EAR \approx 7.23\%)
\]
From the calculations, we see that a nominal rate of 7% compounded semi-annually gives an effective annual rate of approximately 7.12%. Thus, the answer would be:
**7 percent compounded semi-annually.**
\[
EAR = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) is the nominal annual interest rate (in decimal form),
- \( n \) is the number of compounding periods per year,
- \( EAR \) is the effective annual rate (in decimal form).
In this case, \( r = 0.07 \) (for 7%) and \( EAR = 0.0712 \) (for 7.12%). We want to find \( n \).
We rewrite the formula:
\[
0.0712 = \left(1 + \frac{0.07}{n}\right)^n - 1
\]
To solve for \( n \), we can rearrange:
\[
1 + 0.0712 = \left(1 + \frac{0.07}{n}\right)^n
\]
\[
1.0712 = \left(1 + \frac{0.07}{n}\right)^n
\]
To find \( n \), trial and error or numerical methods (e.g., using a financial calculator or spreadsheet software) can be applied.
However, we can provide a few calculations for commonly used compounding frequencies:
1. **Annual (n = 1)**:
\[
1 + \frac{0.07}{1} = 1.07
\]
\[
(1.07)^1 = 1.07 \quad (EAR = 7.00\%)
\]
2. **Semi-Annual (n = 2)**:
\[
1 + \frac{0.07}{2} = 1.035
\]
\[
(1.035)^2 = 1.071225 \quad (EAR \approx 7.12\%)
\]
3. **Quarterly (n = 4)**:
\[
1 + \frac{0.07}{4} = 1.0175
\]
\[
(1.0175)^4 \approx 1.071859 \quad (EAR \approx 7.19\%)
\]
4. **Monthly (n = 12)**:
\[
1 + \frac{0.07}{12} = 1.0058333
\]
\[
(1.0058333)^{12} \approx 1.072290 \quad (EAR \approx 7.23\%)
\]
From the calculations, we see that a nominal rate of 7% compounded semi-annually gives an effective annual rate of approximately 7.12%. Thus, the answer would be:
**7 percent compounded semi-annually.**