Chapter 2, Problem 26P

(1):

To determine

Calculate future value factor to the annual value.

Explanation of Solution

Option (a):

Interest rate (i) is 19% and time period (n) is 20.

The future value factor to the annual value (FA) through linear interpolate can be calculated as follows. The interest rate lies between (i1)18% and (i2) 20%.

$\begin{array}{c}\text{FA}=\frac{{\left(1+\text{i1}\right)}^{\text{n}}-1}{\text{i1}}-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left(\frac{{\left(1+\text{i1}\right)}^{\text{n}}-1}{\text{i1}}-\frac{{\left(1+\text{i2}\right)}^{\text{n}}-1}{\text{i2}}\right)\right)\\ =\frac{{\left(1+\text{0}\text{.18}\right)}^{\text{20}}-1}{0.18}-\left(\frac{\left(\text{0}\text{.019}-\text{0}\text{.18}\right)100}{\left(\text{0}\text{.2}-\text{0}\text{.18}\right)100}\left(\frac{{\left(1+\text{0}\text{.18}\right)}^{20}-1}{0.18}-\frac{{\left(1+\text{0}\text{.2}\right)}^{20}-1}{0.2}\right)\right)\\ =\frac{27.393-1}{0.18}-\left(\frac{\left(0.01\right)100}{\left(0.02\right)100}\left(\frac{27.393-1}{0.18}-\frac{38.3376-1}{0.2}\right)\right)\\ =146.6278-\left(\frac{1}{2}\left(146.6278-186.688\right)\right)\\ =146.6278-\left(\frac{1}{2}\left(146.6278-186.688\right)\right)\\ =146.6278-\left(-20.301\right)\\ =166.6579\end{array}$

The future value factor to the annual value is 166.6579.

Option (b):

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

$\begin{array}{c}\text{FA}=\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}}\\ =\frac{{\left(1+\text{0}\text{.19}\right)}^{\text{20}}-1}{\text{0}\text{.19}}\\ =\frac{32.4294-1}{0.19}\\ =165.4179\end{array}$

The future value factor to the annual value is 165.4749.

Option (c):

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

=-FV(i,n,,1)

=-FV(19%,20,,1) This function gives the value of 165.41802.

(2):

To determine

Calculate present value factor to the equivalent annual value.

Explanation of Solution

Option (a):

Interest rate (i) is 26% and time period (n) is 15.

The present value factor to the equivalent annual value (PA) through linear interpolate can be calculated as follows. The interest rate is lies between (i1) 25% and (i2) 27%.

$\begin{array}{c}\text{PA}=\frac{{\left(1+\text{i1}\right)}^{\text{n}}-1}{\text{i1}{\left(1+\text{i1}\right)}^{\text{n}}}-\left(\frac{\left(\text{i}-\text{i1}\right)100}{\left(\text{i2}-\text{i1}\right)100}\left(\frac{{\left(1+\text{i1}\right)}^{\text{n}}-1}{\text{i1}{\left(1+\text{i1}\right)}^{\text{n}}}-\frac{{\left(1+\text{i2}\right)}^{\text{n}}-1}{\text{i2}{\left(1+\text{i2}\right)}^{\text{n}}}\right)\right)\\ =\frac{{\left(1+\text{0}\text{.25}\right)}^{15}-1}{\text{0}\text{.25}{\left(1+\text{0}\text{.25}\right)}^{\text{15}}}-\left(\frac{\left(\text{0}\text{.26}-\text{0}\text{.25}\right)100}{\left(\text{0}\text{.27}-\text{0}\text{.25}\right)100}\left(\frac{{\left(1+\text{0}\text{.25}\right)}^{15}-1}{\text{0}\text{.25}{\left(1+\text{0}\text{.25}\right)}^{\text{15}}}-\frac{{\left(1+\text{0}\text{.27}\right)}^{\text{15}}-1}{\text{0}\text{.27}{\left(1+\text{0}\text{.27}\right)}^{\text{15}}}\right)\right)\\ =\frac{28.4217-1}{\text{0}\text{.25}\left(28.4217\right)}-\left(\frac{\left(0.01\right)100}{\left(0.02\right)100}\left(\frac{28.4217-1}{\text{0}\text{.25}\left(28.4217\right)}-\frac{36.0625-1}{\text{0}\text{.27}\left(36.0625\right)}\right)\right)\\ =\frac{27.4217}{7.1054}-\left(\frac{1}{2}\left(\frac{27.4217}{7.1054}-\frac{35.0625}{9.7369}\right)\right)\\ =3.8593-\left(0.5\left(3.8593-3.601\right)\right)\\ =3.8593-\left(0.1292\right)\\ =3.7301\end{array}$

The present value factor to the equivalent annual value is 3.7301.

Option (b):

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

$\begin{array}{c}\text{PA}=\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{{\left(1+0.26\right)}^{\text{15}}-1}{\text{0}\text{.26}{\left(1+\text{0}\text{.26}\right)}^{\text{15}}}\\ =\frac{32.0301-1}{\text{0}\text{.26}\left(32.0301\right)}\\ =\frac{31.0301}{8.3278}\\ =3.7261\end{array}$

The present value factor to the equivalent annual value is 3.7261.

Option (c):

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

=-PV(i,n,,1)

=-PV267%,15,,1) This function gives the value of 3.72607.