Chapter 7, Problem 22P

To determine

Calculate the rate of return.

Explanation of Solution

Additional investment (C) is -$950,000. Additional revenue in the 11^th year (S) is $450,000 and additional revenue increases (SI) by $50,000 thereafter till 15^th year. Time period (n) is 5 years and time period 1(n1) is 10 years.

Rate of return (i) for infinitive time period can be calculated as follows:

$\begin{array}{c}-\text{C}=\left(\text{S}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)+\text{SI}\frac{1}{\text{i}}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\right)\left(\frac{1}{{\left(1+\text{i}\right)}^{\text{n1}}}\right)\\ 950,000=\left(\text{450,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{5}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{5}}}\right)+\text{50,000}\times \frac{1}{\text{i}}\left(\frac{{\left(1+\text{i}\right)}^{\text{5}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{5}}}-\frac{\text{5}}{{\left(1+\text{i}\right)}^{\text{5}}}\right)\right)\left(\frac{1}{{\left(1+\text{i}\right)}^{\text{10}}}\right)\end{array}$

Substitute the rate of return as 8% by trial and error method in the above Equation.

$\begin{array}{l}950,000=\left(\text{450,000}\left(\frac{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{5}}}\right)+\text{50,000}\times \frac{1}{\text{0}\text{.08}}\left(\frac{{\left(1+\text{0}\text{.08}\right)}^{\text{5}}-1}{\text{0}\text{.08}{\left(1+\text{0}\text{.08}\right)}^{\text{5}}}-\frac{\text{5}}{{\left(1+\text{.0}\text{.08}\right)}^{\text{5}}}\right)\right)\left(\frac{1}{{\left(1+\text{0}\text{.08}\right)}^{\text{10}}}\right)\\ 950,000=\left(\text{450,000}\left(\frac{1.469328-1}{\text{0}\text{.08}\left(1.469328\right)}\right)+\text{50,000}\times 12.5\left(\frac{1.469328-1}{\text{0}\text{.08}\left(1.469328\right)}-\frac{\text{5}}{1.469328}\right)\right)\left(\frac{1}{2.158925}\right)\\ 950,000=\left(\text{450,000}\left(\frac{0.469328}{0.117546}\right)+\text{50,000}\times 12.5\left(\frac{0.469328}{0.117546}-3.402916\right)\right)\left(0.463193\right)\\ 950,000=\left(\text{450,000}\left(3.992718\right)+\text{50,000}\times 12.5\left(3.992718-3.402916\right)\right)\left(0.463193\right)\\ 950,000=\left(1,976,273.1+\text{50,000}\times 12.5\left(0.589802\right)\right)\left(0.463193\right)\\ 950,000=\left(1,976,273.1+368,626.25\right)\left(0.463193\right)\\ 950,000<1,086,140.96\end{array}$

When the rate of return is substituted as 8%, the calculated value is greater than the additional investment. Thus, the rate of return increases to 8.45%.

$\begin{array}{l}950,000=\left(\begin{array}{l}\text{450,000}\left(\frac{{\left(1+\text{0}\text{.0845}\right)}^{\text{5}}-1}{\text{0}\text{.0845}{\left(1+\text{0}\text{.0845}\right)}^{\text{5}}}\right)+\text{50,000}\\ \times \frac{1}{\text{0}\text{.0845}}\left(\frac{{\left(1+\text{0}\text{.0845}\right)}^{\text{5}}-1}{\text{0}\text{.0845}{\left(1+\text{0}\text{.0845}\right)}^{\text{5}}}-\frac{\text{5}}{{\left(1+\text{.0}\text{.0845}\right)}^{\text{5}}}\right)\end{array}\right)\left(\frac{1}{{\left(1+\text{0}\text{.0845}\right)}^{\text{10}}}\right)\\ 950,000=\left(\begin{array}{l}\text{450,000}\left(\frac{1.500195-1}{\text{0}\text{.0845}\left(1.500195\right)}\right)+\text{50,000}\\ \times 11.83432\left(\frac{1.500195-1}{\text{0}\text{.0845}\left(1.500195\right)}-\frac{\text{5}}{1.500195}\right)\end{array}\right)\left(\frac{1}{2.250586}\right)\\ 950,000=\left(\begin{array}{l}\text{450,000}\left(\frac{0.500195}{0.126766}\right)+\text{50,000}\\ \times 11.83432\left(\frac{0.500195}{0.126766}-\frac{\text{5}}{1.500195}\right)\end{array}\right)\left(0.444329\right)\\ 950,000=\left(\begin{array}{l}\text{450,000}\left(3.945814\right)+\text{50,000}\\ \times 11.83432\left(3.945814-3.3329\right)\end{array}\right)\left(0.444329\right)\\ 950,000=\left(1,775,616.3+\text{50,000}\times 11.83432\left(0.612914\right)\right)\left(0.444329\right)\\ 950,000=\left(1,775,616.3+363,590.39\right)\left(0.444329\right)\\ 950,000\approx 950,511.57\end{array}$

The calculated value is nearly equal to the additional investment with rate of return 8.45%. Thus, it is confirmed that the rate of return is 8.45%.