Questions: Find the nominal annual rate of interest compounded semi-annually that is equivalent to 9.3% compounded quarterly. The nominal annually compounded rate of interest is %. (Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as needed.)

Solution Steps

Given the nominal annual interest rate compounded quarterly is \( r_q = 9.3\% = 0.093 \), we can calculate the Effective Annual Rate (EAR) using the formula:

\[ EAR = \left(1 + \frac{r_q}{n_q}\right)^{n_q} - 1 \]

where \( n_q = 4 \) (the number of compounding periods per year for quarterly compounding). Substituting the values:

\[ EAR = \left(1 + \frac{0.093}{4}\right)^{4} - 1 = \left(1 + 0.02325\right)^{4} - 1 \approx 0.0962939395203164 \]

Next, we need to find the nominal annual interest rate compounded semi-annually that is equivalent to the calculated EAR. The formula for the nominal rate \( r_s \) compounded semi-annually is given by:

\[ r_s = n_s \left( (1 + EAR)^{\frac{1}{n_s}} - 1 \right) \]

where \( n_s = 2 \) (the number of compounding periods per year for semi-annual compounding). Substituting the values:

\[ r_s = 2 \left( (1 + 0.0962939395203164)^{\frac{1}{2}} - 1 \right) \approx 0.09408112500000021 \]

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

\[ \text{Nominal Annual Rate} = r_s \times 100 \approx 9.408112500000021 \]

Rounding this value to four decimal places gives:

\[ \text{Nominal Annual Rate} \approx 9.4081\% \]

Final Answer