Questions: Find the effective interest rate. Round to two decimal places when needed. Rate: 5% Compounded: Quarterly The effective rate is % .

Find the effective interest rate. Round to two decimal places when needed. Rate: 5% Compounded: Quarterly The effective rate is % .
Transcript text: Find the effective interest rate. Round to two decimal places when needed. Rate: 5\% Compounded: Quarterly The effective rate is $\square$ $\%$ 。
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Solution

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Solution Steps

To find the effective interest rate given a nominal interest rate and the number of compounding periods per year, we can use the formula for the effective annual rate (EAR):

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

where \( r \) is the nominal annual interest rate (expressed as a decimal), and \( n \) is the number of compounding periods per year. In this case, \( r = 0.05 \) and \( n = 4 \) (since the interest is compounded quarterly).

Step 1: Identify Given Values

We are given the nominal annual interest rate and the number of compounding periods per year:

  • Nominal rate (\( r \)): 5\% or 0.05
  • Compounding periods (\( n \)): Quarterly or 4 times per year
Step 2: Apply the Effective Annual Rate (EAR) Formula

The formula to calculate the effective annual rate (EAR) is: \[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \]

Substituting the given values: \[ \text{EAR} = \left(1 + \frac{0.05}{4}\right)^4 - 1 \]

Step 3: Calculate the Effective Annual Rate

Perform the calculation: \[ \text{EAR} = \left(1 + 0.0125\right)^4 - 1 \] \[ \text{EAR} = (1.0125)^4 - 1 \] \[ \text{EAR} \approx 1.0509453369140622 - 1 \] \[ \text{EAR} \approx 0.0509453369140622 \]

Step 4: Convert to Percentage and Round

Convert the effective annual rate to a percentage and round to two decimal places: \[ \text{Effective Rate Percentage} = 0.0509453369140622 \times 100 \] \[ \text{Effective Rate Percentage} \approx 5.09\% \]

Final Answer

The effective interest rate is \( \boxed{5.09\%} \).

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