Chapter 2, Problem 27P

(1):

To determine

Calculate value factor to the present value.

Explanation of Solution

Option (a):

Interest rate (i) is 18% and time period (n) is 33.

The future value factor to the present value (FP) through linear interpolate can be calculated as follows. The interest rate lies between (i1)17% and (i2) 19%.

$\begin{array}{c}\text{FP}={\left(1+\text{i1}\right)}^{\text{n}}-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left({\left(1+\text{i1}\right)}^{\text{n}}-{\left(1+\text{i2}\right)}^{\text{n}}\right)\right)\\ ={\left(1+\text{0}\text{.17}\right)}^{\text{33}}-\left(\frac{\left(\text{0}\text{.18}-\text{0}\text{.17}\right)100}{\left(0.19-\text{0}\text{.17}\right)100}\left({\left(1+\text{0}\text{.17}\right)}^{\text{33}}-{\left(1+\text{0}\text{.19}\right)}^{\text{33}}\right)\right)\\ =177.8826-\left(\frac{\left(0.01\right)100}{\left(0.02\right)100}\left(177.8826-311.2073\right)\right)\\ =177.8826-\left(\frac{1}{2}\left(177.8826-311.2073\right)\right)\\ =177.8826-\left(-66.6624\right)\\ =244.545\end{array}$

The future value factor to the present value is 244.545.

Option (b):

The future value factor to the present value (FP) through formula can be calculated as follows:

$\begin{array}{c}\text{FP}={\left(1+\text{i}\right)}^{\text{n}}\\ ={\left(1+\text{0}\text{.18}\right)}^{\text{33}}\\ =235.5625\end{array}$

The future value factor to the present value is 235.5625.

(2):

To determine

Calculate annual growth rate.

Explanation of Solution

Option (a):

Interest rate (i) is 12% and time period (n) is 54.

The annual growth rate value (AG) through linear interpolate can be calculated as follows. The interest rate lies between (i1) 11% and (i2) 13%.

$\begin{array}{c}\text{AG}=\left(\frac{1}{\text{i1}}-\frac{\text{n}}{{\left(1+\text{i1}\right)}^{\text{n}}-1}\right)-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left(\begin{array}{l}\left(\frac{1}{\text{i1}}-\frac{\text{n}}{{\left(1+\text{i1}\right)}^{\text{n}}-1}\right)\\ -\left(\frac{1}{\text{i2}}-\frac{\text{n}}{{\left(1+\text{i2}\right)}^{\text{n}}-1}\right)\end{array}\right)\right)\\ =\left(\frac{1}{\text{0}\text{.11}}-\frac{\text{54}}{{\left(1+\text{0}\text{.11}\right)}^{\text{54}}-1}\right)-\left(\frac{\left(\text{0}\text{.12}-\text{0}\text{.11}\right)100}{\left(\text{0}\text{.13}-\text{0}\text{.11}\right)100}\left(\begin{array}{l}\left(\frac{1}{\text{0}\text{.11}}-\frac{\text{54}}{{\left(1+\text{0}\text{.11}\right)}^{\text{54}}-1}\right)\\ -\left(\frac{1}{0.13}-\frac{\text{54}}{{\left(1+\text{0}\text{.13}\right)}^{\text{54}}-1}\right)\end{array}\right)\right)\\ =\left(9.0909-\frac{\text{54}}{280.1824-1}\right)-\left(\frac{\left(0.01\right)100}{\left(0.02\right)100}\left(\begin{array}{l}\left(9.0909-\frac{\text{54}}{280.1824-1}\right)\\ -\left(7.6923-\frac{\text{54}}{734.913-1}\right)\end{array}\right)\right)\\ =\left(9.0909-0.1934\right)-\left(\frac{1}{2}\left(\begin{array}{l}\left(9.0909-0.1934\right)\\ -\left(7.6923-0.0736\right)\end{array}\right)\right)\\ =8.8975-\left(0.5\left(8.8975-7.6187\right)\right)\\ =8.8975-0.6394\\ =8.3356\end{array}$

The annual growth factor value is 8.3356.

Option (b):

The annual growth factor value (AG) through formula can be calculated as follows:

$\begin{array}{c}\text{PA}=\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =\left(\frac{1}{\text{0}\text{.12}}-\frac{\text{54}}{{\left(1+\text{0}\text{.12}\right)}^{\text{54}}-1}\right)\\ =\left(8.3333-\frac{\text{54}}{454.7505-1}\right)\\ =\left(8.3333-0.119\right)\\ =8.2143\end{array}$

The annual growth factor value is 8.2143.