Question

What interest rate, compounded semiannually, results in an annual effective rate of 6%? Input your answer as a percentage rounded to two decimal places.

Answer 1

Answer: 5.91%

Explanation:

Assuming an initial investment of $1, we start the formula rEFF=A−PP. Next, substituting, we now have 0.06=A−11 which gives us 0.06=A−1 so A=$1.06. We now use this value for A in the formula A=P⋅(1+rn)n⋅t with P=$1, n=2 compounding periods for semiannually compounded interest, t=1 year, and r is the unknown semiannually compounded interest rate for which we are solving. Plugging the values in, we get 1.06=(1+r2)2. Take the square root of both sides to get 1.02956=1+r2. Next, subtract 1 from both sides and multiply both sides by 2. The final step gives us 0.059126, which rounded to two decimal places is 5.91%.

Answer 2

The relation between anual interest and semiannual compounded interest is:

`\begin{gathered} \text{annal interest=y, semiannual interest=x} \\ (1+y)=(1+x)^2 \end{gathered}`

In this case, the annula interest is 6%=0.06, so:

`\begin{gathered} 1+0.06=(1+x)^2 \\ 1+x=\sqrt[]{1.06} \\ x=\sqrt[]{1.06}-1=0.02956 \end{gathered}`

So the semiannual interest rate is 2.95%