A bank offers 9.00% on savings accounts. What is the effective annual rate if interest is compounded semi-annually?

Answer 1

To calculate the effective annual rate (EAR) when interest is compounded semi-annually, we can use the following formula:

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

where:

• \( r \) is the nominal annual interest rate (expressed as a decimal),
• \( n \) is the number of compounding periods per year,
• \( t \) is the number of years.

In this case:

• \( r = 0.09 \) (9.00% expressed as a decimal),
• \( n = 2 \) (since interest is compounded semi-annually),
• \( t = 1 \) (we are looking for the rate for one year).

Substituting the values into the formula:

\[ \text{EAR} = \left(1 + \frac{0.09}{2}\right)^{2 \times 1} - 1 \]

Calculating \( \frac{0.09}{2} \):

\[ \frac{0.09}{2} = 0.045 \]

Now we substitute this back into the equation:

\[ \text{EAR} = \left(1 + 0.045\right)^{2} - 1 \] \[ = (1.045)^{2} - 1 \]

Calculating \( (1.045)^{2} \):

\[ (1.045)^{2} = 1.092025 \]

Now subtract 1:

\[ \text{EAR} = 1.092025 - 1 = 0.092025 \]

To express the effective annual rate as a percentage, we multiply by 100:

\[ \text{EAR} = 0.092025 \times 100 \approx 9.20% \]

Thus, the effective annual rate when the interest is compounded semi-annually at a nominal rate of 9.00% is approximately 9.20%.

Answer 2

don't understand what you write as the formua. Isthis what you mean EAR=[{1+(r/n)}^n]-1?

Answer 3

Yes, you are correct! The formula for the Effective Annual Rate (EAR) when interest is compounded \( n \) times per year is:

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

Where:

• \( r \) = nominal annual interest rate (expressed as a decimal),
• \( n \) = number of compounding periods per year,
• \( t \) = number of years.

If you are looking for the EAR for one year (\( t = 1 \)), it simplifies to:

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

So, in your case with a nominal interest rate of 9% compounded semi-annually (\( r = 0.09 \), \( n = 2 \)), the calculation would be:

1. Determine \( \frac{r}{n} \): \[ \frac{0.09}{2} = 0.045 \]

2. Substitute into the EAR formula: \[ \text{EAR} = \left(1 + 0.045\right)^{2} - 1 \]

3. Calculate: \[ \text{EAR} = (1.045)^{2} - 1 \]

4. Calculate \( (1.045)^{2} \): \[ (1.045)^{2} = 1.092025 \]

5. Subtract 1: \[ \text{EAR} = 1.092025 - 1 = 0.092025 \]

6. Convert to percentage: \[ \text{EAR} = 0.092025 \times 100 \approx 9.20% \]

So the Effective Annual Rate (EAR) is approximately 9.20%. Thank you for your feedback, and I hope this clarification helps!