Chapter 2, Problem 25P

(1):

To determine

Calculate present value factor to the future value.

Explanation of Solution

Option (a):

Interest rate (i) is 8.4% and time period (n) is 15.

The present value factor to the future value (PF) through linear interpolate can be calculated as follows. The interest rate lies between (i1) 8% and (i2) 9%.

$\begin{array}{c}\text{PF}=\frac{1}{{\left(1+\text{i1}\right)}^{\text{n}}}-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left(\frac{1}{{\left(1+\text{i1}\right)}^{\text{n}}}-\frac{1}{{\left(1+\text{i2}\right)}^{\text{n}}}\right)\right)\\ =\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{15}}}-\left(\frac{\left(\text{0}\text{.084}-\text{0}\text{.08}\right)100}{\left(\text{0}\text{.09}-\text{0}\text{.08}\right)100}\left(\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{15}}}-\frac{1}{{\left(1+\text{0}\text{.09}\right)}^{\text{15}}}\right)\right)\\ =\frac{1}{3.1722}-\left(\frac{\left(0.004\right)100}{\left(0.01\right)100}\left(\frac{1}{3.1722}-\frac{1}{3.6425}\right)\right)\\ =0.3152-\left(\frac{0.4}{1}\left(0.3152-0.2745\right)\right)\\ =0.3152-\left(0.4\left(0.0407\right)\right)\\ =0.3152-\left(0.0163\right)\\ =0.2989\end{array}$

The present value factor to the future value is 0.2989.

Option (b):

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

$\begin{array}{c}\text{PF}=\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{1}{{\left(1+\text{0}\text{.084}\right)}^{\text{15}}}\\ =\frac{1}{3.35304}\\ =0.2982\end{array}$

The present value factor to the future value is 0.2982.

Option (c):

The present value factor to the future value (PF) through spreadsheet can be done as follows:

=-PV(i,n,,1)

=-PV(8.4%,15,,1) This function gives the value of 0.29824.

(2):

To determine

Calculate equivalent annual value factor to the future value.

Explanation of Solution

Option (a):

Interest rate (i) is 17% and time period (n) is 10.

The equivalent annual value factor to the future value (AF) through linear interpolate can be calculated as follows. The interest rate is lies between (i1) 16% and (i2) 18%.

$\begin{array}{c}\text{AF}=\frac{\text{i1}}{{\left(1+\text{i1}\right)}^{\text{n}}-1}-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left(\frac{\text{i1}}{{\left(1+\text{i1}\right)}^{\text{-1}}-1}-\frac{\text{i}2}{{\left(1+\text{i2}\right)}^{\text{n}}-1}\right)\right)\\ =\frac{0.16}{{\left(1+\text{0}\text{.16}\right)}^{\text{10}}-1}-\left(\frac{\left(\text{0}\text{.17}-\text{0}\text{.16}\right)100}{\left(\text{0}\text{.18}-\text{0}\text{.16}\right)100}\left(\frac{0.16}{{\left(1+\text{0}\text{.16}\right)}^{\text{10}}-1}-\frac{0.18}{{\left(1+\text{0}\text{.18}\right)}^{\text{10}}-1}\right)\right)\\ =\frac{0.16}{4.4114-1}-\left(\frac{\left(0.01\right)100}{\left(0.02\right)100}\left(\frac{0.16}{4.4114-1}-\frac{0.18}{5.2338-1}\right)\right)\\ =0.0469-\left(\frac{1}{2}\left(0.04690-0.04251\right)\right)\\ =0.0469-\left(0.5\left(0.00439\right)\right)\\ =0.0469-\left(0.0022\right)\\ =0.0447\end{array}$

The equivalent value factor to the future value is 0.0447.

Option (b):

The equivalent annual value factor to the future value (AF) through formula can be calculated as follows:

$\begin{array}{c}\text{AF}=\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\\ =\frac{0.17}{{\left(1+\text{0}\text{.17}\right)}^{\text{10}}-1}\\ =\frac{0.17}{4.8068-1}\\ =0.04466\end{array}$

The present value factor to the future value is 0.04466.

Option (c):

The equivalent annual value factor to the future value (AF) through spreadsheet can be done as follows:

=-PMT(i,n,,1)

=-PMT(17%,10,,1) This function gives the value of 0.04466.