Find the nominal annual rate of interest compounded semi-annually that is equivalent to $7.7 \%$ compounded quarterly.
The nominal annually compounded rate of interest is $\square \%$. (Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as needed.)
Answer
Final Answer: The nominal annual rate of interest compounded semi-annually that is equivalent to 7.7% compounded quarterly is \(\boxed{3.8871\%}\).
Step 1 ：Given that the nominal interest rate is 7.7% compounded quarterly, we need to find the equivalent nominal annual rate of interest compounded semi-annually.
Step 2 ：We start by calculating the Effective Annual Rate (EAR) for the quarterly compounded rate. The formula to calculate the EAR is: \(EAR = (1 + \frac{i}{n})^{nt} - 1\), where \(i\) is the nominal interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the number of years.
Step 3 ：Substituting the given values into the formula, we get \(EAR = (1 + \frac{0.077}{4})^{4*1} - 1\), which simplifies to \(EAR = 0.0792520456290664\).
Step 4 ：We then use this EAR to find the semi-annually compounded rate that would give the same EAR. We rearrange the EAR formula to solve for \(i\), the nominal interest rate: \(i = n((1 + EAR)^{\frac{1}{nt}} - 1)\).
Step 5 ：Substituting the values into the formula, we get \(i = 2((1 + 0.0792520456290664)^{\frac{1}{2*1}} - 1)\), which simplifies to \(i = 0.038870562500000094\).
Step 6 ：Converting this to a percentage gives us the nominal annual rate of interest compounded semi-annually: \(i = 3.8871\%\).
Step 7 ：Final Answer: The nominal annual rate of interest compounded semi-annually that is equivalent to 7.7% compounded quarterly is \(\boxed{3.8871\%}\).
