Question

Adjust the annual formula for a future value of a single amount at 12 percent for 10 years to a semiannual compounding formula. What are the interest factors (FVIF) before and after? Why are they different?

Short answer

The interest factor before compounding is 3.11, whereas, the factor is 3.21 after the compounding. These interest factors are different because of the frequency of compounding.

Step-by-step solution

Future value interest factor (FVIF) before

$$\begin{gathered} \text{Future Value}\mathbf{=}\text{Present Value}\times {\left(1+\text{Interest Rate}\right)}^{\text{t}\times \text{n}} \\ \mathbf{=}\text{Present Value}\times {\left(1+12\%\right)}^{10\times 1} \\ \mathbf{=}\text{Present Value}\times 3.11 \end{gathered}$$

Future value interest factor (FVIF) after

$$\begin{gathered} \text{Future Value}\mathbf{=}\text{Present Value}\times {\left(1+\frac{\text{Interest Rate}}{\text{n}}\right)}^{\text{t}\times \text{n}} \\ \mathbf{=}\text{Present Value}\times {\left(1+\frac{12\%}{2}\right)}^{10\times 2} \\ \mathbf{=}\text{Present Value}\times 3.21 \end{gathered}$$

Reason for difference

The interest factor after semi-annual compounding, 3.21, is higher than the interest factor before the compounding, 3.11. This is because of the frequency of compounding in both of these cases. The frequency of compounding is lower in the case of annual compounding (i.e., 10) than that of the semi-annual compounding (i.e., 20). This is why the interest rate has been adjusted accordingly which has led to different interest factors in both cases.