Chapter 2, Problem 1P

To determine

Calculate the interest factor.

Explanation of Solution

Option (1):

The interest rate (i) is 10% and time period is 7 years.

The future to present compound interest factor (If) can be calculated as follows:

$\begin{array}{c}\text{If}={\left(1+\text{i}\right)}^{\text{n}}\\ ={\left(1+0.1\right)}^7\\ =1.9487\end{array}$

The future to present compound interest factor is 1.9487.

Option (2):

The interest rate (i) is 12% and time period is 10 years.

The equivalent annual to present interest factor (EA) can be calculated as follows:

$\begin{array}{c}\text{EA}=\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\\ =\frac{\text{0}\text{.12}{\left(1+\text{0}\text{.12}\right)}^{\text{10}}}{{\left(1+\text{0}\text{.12}\right)}^{\text{10}}-1}\\ =\frac{\text{0}\text{.12}\left(3.1058\right)}{3.1058-1}\\ =\frac{0.3727}{2.1058}\\ =0.17699\end{array}$

The equivalent annual to present interest factor is 0.17699.

Option (3):

The interest rate (i) is 15% and time period is 20 years.

The present growth interest factor (Pg) can be calculated as follows:

$\begin{array}{c}\text{Pg}=\frac{1}{\text{i}}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\\ =\frac{1}{0.15}\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{20}}-1}{0.15{\left(1+\text{0}\text{.15}\right)}^{\text{20}}}-\frac{\text{20}}{{\left(1+\text{0}\text{.15}\right)}^{\text{20}}}\right)\\ =6.6667\left(\frac{16.36654-1}{0.15\left(16.36654\right)}-\frac{\text{20}}{16.36654}\right)\\ =6.6667\left(\frac{15.36654}{2.45498}-\frac{\text{20}}{16.36654}\right)\\ =6.6667\left(6.2593-1.222\right)\\ =6.6667\left(5.0373\right)\\ =33.5822\end{array}$

The present growth interest factor is 33.5822.

Option (4):

The interest rate (i) is 2% and time period is 50 years.

The future to annual interest factor (Fa) can be calculated as follows:

$\begin{array}{c}\text{Fa}=\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}}\\ =\frac{{\left(1+\text{0}\text{.02}\right)}^{\text{50}}-1}{\text{0}\text{.02}}\\ =\frac{2.6916-1}{\text{0}\text{.02}}\\ =84.58\end{array}$

The future to annual interest factor is 84.58.

Option (5):

The interest rate (i) is 35% and time period is 15 years.

The annual growth interest factor (Ag) can be calculated as follows:

$\begin{array}{c}\text{Fa}=\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\\ =\frac{1}{\text{0}\text{.35}}-\frac{\text{15}}{{\left(1+\text{0}\text{.35}\right)}^{\text{15}}-1}\\ =2.8571-\frac{\text{15}}{90.1585-1}\\ =2.8571-0.1682\\ =2.6889\end{array}$

The annual growth interest factor is 2.6889.