To calculate the Annual Percentage Rate (APR) given an effective annual rate (EAR) of 5.89% with quarterly compounding, we can use the relationship between the EAR, the APR, and the number of compounding periods per year.
The formula connecting these values is:
\[ EAR = \left(1 + \frac{APR}{n}\right)^n - 1 \]
Where:
- \( EAR \) is the effective annual rate,
- \( APR \) is the annual percentage rate,
- \( n \) is the number of compounding periods per year.
In this case, the number of compounding periods \( n = 4 \) (quarterly compounding).
We rearrange the formula to solve for APR:
\[ 1 + EAR = \left(1 + \frac{APR}{n}\right)^n \]
Substituting the values:
\[ 1 + 0.0589 = \left(1 + \frac{APR}{4}\right)^4 \]
Calculating \(1 + EAR\):
\[ 1.0589 = \left(1 + \frac{APR}{4}\right)^4 \]
Next, we take the fourth root of both sides:
\[ 1 + \frac{APR}{4} = (1.0589)^{\frac{1}{4}} \]
Calculating \(1.0589^{\frac{1}{4}}\):
\[ 1 + \frac{APR}{4} \approx 1.01423 \]
Now, subtract 1 from both sides:
\[ \frac{APR}{4} \approx 0.01423 \]
Now, multiplying by 4 to find APR:
\[ APR \approx 0.01423 \times 4 \approx 0.05692 \]
Converting to percentage:
\[ APR \approx 5.692% \]
Thus, the APR is approximately 5.69%.