Chapter 3, Problem 45P

(a):

To determine

Calculate the value x.

Explanation of Solution

The cash flow is given in Table 1 as follows:

Year	1	2	3	4	5	6	7	8
CF	2,000	2,200	X	2x	2,800	3,000	3,200	3,400

.

The equivalent annual value (A) of the cash flow is $30,000. Interest rate (i) is 10%. The time period (n) is 8 years.

The missed value of x can be calculated as follows:

$\begin{array}{c}\text{A}=\left(\begin{array}{l}\frac{\text{CF1}}{{\left(1+\text{i}\right)}^{\text{1}}}+\frac{\text{CF2}}{{\left(1+\text{i}\right)}^{\text{2}}}+\frac{\text{CF3}}{{\left(1+\text{i}\right)}^{\text{3}}}+\frac{\text{CF4}}{{\left(1+\text{i}\right)}^{\text{4}}}\\ +\frac{\text{CF5}}{{\left(1+\text{i}\right)}^5}+\frac{\text{CF6}}{{\left(1+\text{i}\right)}^{\text{6}}}\frac{\text{CF7}}{{\left(1+\text{i}\right)}^{\text{7}}}+\frac{\text{CF8}}{{\left(1+\text{i}\right)}^{\text{8}}}\end{array}\right)\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ 30,000=\left(\begin{array}{l}\frac{\text{2,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{1}}}+\frac{\text{2,200}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}+\frac{\text{x}}{{\left(1+\text{0}\text{.1}\right)}^{\text{3}}}+\frac{\text{2x}}{{\left(1+\text{0}\text{.1}\right)}^{\text{4}}}\\ +\frac{\text{2,800}}{{\left(1+\text{0}\text{.1}\right)}^5}+\frac{\text{3,000}}{{\left(1+\text{0}\text{.1}\right)}^{\text{6}}}+\frac{\text{3,200}}{{\left(1+\text{0}\text{.1}\right)}^{\text{7}}}+\frac{\text{3,400}}{{\left(1+\text{0}\text{.1}\right)}^{\text{8}}}\end{array}\right)\left(\frac{0.1{\left(1+\text{0}\text{.1}\right)}^{\text{8}}}{{\left(1+\text{0}\text{.1}\right)}^{\text{8}}-1}\right)\\ 30,000=\left(\begin{array}{c}\frac{\text{2,000}}{1.1}+\frac{\text{2,200}}{1.21}+\frac{\text{x}}{1.331}+\frac{\text{2x}}{1.4641}\\ +\frac{\text{2,800}}{1.6105}+\frac{\text{3,000}}{1.7716}+\frac{\text{3,200}}{1.9487}+\frac{\text{3,400}}{2.1436}\end{array}\right)\left(\frac{\text{0}\text{.1}\left(2.14359\right)}{2.14359-1}\right)\\ 30,000=\left(\begin{array}{c}1,818.1818+1,818.1818+0.75131\text{x}+1.36603\text{x}\\ +1,738.5905+1,693.3845+1,642.1204+1,586.1168\end{array}\right)\left(\frac{\text{0}\text{.21436}}{1.14359}\right)\\ 30,000=\left(10,296.5758+2.1173\text{x}\right)\left(0.187478\right)\\ \frac{30,000}{0.187444}=\left(10,296.5758+2.1173\text{x}\right)\end{array}$

$\begin{array}{c}160,047.8009=\left(10,296.5758+2.1173\text{x}\right)\\ 2.1173\text{x}=160,047.8009-10,296.5758\\ \text{x}=\frac{149,751.2251}{2.1173}\\ =70,727.45\end{array}$

The value of x is $70,727.45.

(b):

To determine

Calculate the value of x through spreadsheet.

Explanation of Solution