Question
A bank offers 9.00% on savings accounts. What is the effective annual rate if interest is compounded continuously?
Answers
GPT-4o mini
To calculate the effective annual rate (EAR) when interest is compounded continuously, you can use the formula:
\[
\text{EAR} = e^{r} - 1
\]
where \( r \) is the nominal interest rate (expressed as a decimal) and \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given that the nominal interest rate is 9.00% (or 0.09 in decimal form), we can plug this value into the formula:
\[
\text{EAR} = e^{0.09} - 1
\]
Now calculate \( e^{0.09} \):
\[
e^{0.09} \approx 1.09417
\]
Now, subtract 1:
\[
\text{EAR} \approx 1.09417 - 1 \approx 0.09417
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} \approx 9.42\%
\]
Therefore, the effective annual rate if interest is compounded continuously is approximately **9.42%**.
\[
\text{EAR} = e^{r} - 1
\]
where \( r \) is the nominal interest rate (expressed as a decimal) and \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given that the nominal interest rate is 9.00% (or 0.09 in decimal form), we can plug this value into the formula:
\[
\text{EAR} = e^{0.09} - 1
\]
Now calculate \( e^{0.09} \):
\[
e^{0.09} \approx 1.09417
\]
Now, subtract 1:
\[
\text{EAR} \approx 1.09417 - 1 \approx 0.09417
\]
To express this as a percentage, multiply by 100:
\[
\text{EAR} \approx 9.42\%
\]
Therefore, the effective annual rate if interest is compounded continuously is approximately **9.42%**.