Question

The effective rate which is equivalent to stated rate of \(6\%\) compounded semiannually is:

• A. \(6.09\%\)
• B. \(1.03\%\)
• C. \(4.09\%\)
• D. \(5.09\%\)

Answer 1

The Correct Option is A

To find the effective annual rate equivalent to a stated rate of \(6\%\) compounded semiannually, we use the formula for the effective rate (ER):

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

where \(r\) is the nominal rate (in decimal form), and \(n\) is the number of compounding periods per year.

In this problem:

• \(r=0.06\) (since \(6\%\) is \(0.06\) in decimal form)
• \(n=2\) (since it's compounded semiannually)

Plug these values into the formula:

\[ER=\left(1+\frac{0.06}{2}\right)^2-1\]

\[ER=\left(1+0.03\right)^2-1\]

\[ER=1.03^2-1\]

\[ER\approx1.0609-1\]

\[ER\approx0.0609\]

To express this as a percentage, we multiply by \(100\):

\(ER\approx6.09\%\)

Therefore, the effective rate equivalent to a \(6\%\) stated rate compounded semiannually is \(6.09\%\).