To calculate the effective annual rate (EAR) when interest is compounded daily, we can use the formula for EAR:
\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{n \cdot t} - 1 \]
where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year (for daily compounding, \( n = 365 \)),
- \( t \) is the number of years (for one year, \( t = 1 \)).
Given:
- \( r = 0.09 \) (9.00% expressed as a decimal)
- \( n = 365 \)
- \( t = 1 \)
Substituting these values into the formula, we get:
\[ \text{EAR} = \left(1 + \frac{0.09}{365}\right)^{365 \cdot 1} - 1 \]
Calculating it step by step:
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Calculate \( \frac{0.09}{365} \): \[ \frac{0.09}{365} \approx 0.00024657534 \]
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Calculate \( 1 + \frac{0.09}{365} \): \[ 1 + 0.00024657534 \approx 1.00024657534 \]
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Raise it to the power of 365: \[ \left(1.00024657534\right)^{365} \approx 1.094171 \]
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Subtract 1: \[ 1.094171 - 1 \approx 0.094171 \]
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Convert to percentage: \[ 0.094171 \times 100 \approx 9.42% \]
So, the effective annual rate (EAR) when interest is compounded daily is approximately 9.42%.