What is the EAR of a quarterly compounded rate of \( 3.75 \% \) ? \( 15.87 \% \) (Round to two decimal places.) What is the EAR of a semiannually compounded rate of \( 9 \% \) ? \( \% \) (Round to two decimal places.)

To find the Effective Annual Rate (EAR) for a given interest rate compounded quarterly, we can use the formula: \[ EAR = \left(1 + \frac{r}{n}\right)^n - 1 \] where: - \( r \) is the annual interest rate (in decimal form) - \( n \) is the number of times the interest is compounded per year Given that the quarterly compounded rate is \( 3.75\% \), we can calculate the EAR as follows: 1. Convert the quarterly compounded rate to decimal form: \( 3.75\% = 0.0375 \) 2. Calculate the EAR using the formula: \[ EAR = \left(1 + \frac{0.0375}{4}\right)^4 - 1 \] Now, let's calculate the EAR for the quarterly compounded rate of \( 3.75\% \). Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.0375}{4}\right)^{4}-1\) - step1: Divide the terms: \(\left(1+\frac{3}{320}\right)^{4}-1\) - step2: Add the numbers: \(\left(\frac{323}{320}\right)^{4}-1\) - step3: Rewrite the expression: \(\frac{323^{4}}{320^{4}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{323^{4}}{320^{4}}-\frac{320^{4}}{320^{4}}\) - step5: Transform the expression: \(\frac{323^{4}-320^{4}}{320^{4}}\) The EAR for a quarterly compounded rate of \( 3.75\% \) is approximately \( 3.80\% \) (rounded to two decimal places). Now, let's calculate the EAR for a semiannually compounded rate of \( 9\% \). Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.09}{2}\right)^{2}-1\) - step1: Divide the terms: \(\left(1+\frac{9}{200}\right)^{2}-1\) - step2: Add the numbers: \(\left(\frac{209}{200}\right)^{2}-1\) - step3: Rewrite the expression: \(\frac{209^{2}}{200^{2}}-1\) - step4: Evaluate the power: \(\frac{43681}{40000}-1\) - step5: Reduce fractions to a common denominator: \(\frac{43681}{40000}-\frac{40000}{40000}\) - step6: Transform the expression: \(\frac{43681-40000}{40000}\) - step7: Subtract the numbers: \(\frac{3681}{40000}\) The EAR for a semiannually compounded rate of \( 9\% \) is approximately \( 9.20\% \) (rounded to two decimal places).