Chapter 18, Problem 37P

(a):

To determine

Calculate the expected value.

Explanation of Solution

The decision tree is given below:

(b):

To determine

Calculate the expected present worth.

Explanation of Solution

Rate of return is denoted by ‘i’. Probability is denoted by ‘P’. Expected present worth (EPW) for the expansion option can be calculated as follows:

$\begin{array}{c}\text{EPW}=\left(\begin{array}{l}{\text{P}}_{\text{Top node}}\times \left(-\text{Investment}+\frac{{\text{Cash flow}}_{\text{Top node}}}{{\left(1+\text{i}\right)}^1}\right)+{\text{P}}_{\text{Middle node}}\times \left(-\text{Investment}+\frac{{\text{Cash flow}}_{\text{Bottom node}}}{{\left(1+\text{i}\right)}^1}\right)\\ +{\text{P}}_{\text{Bottom node}}\times \left(-\text{Investment}+\frac{{\text{Cash flow}}_{\text{Bottom node}}}{{\left(1+\text{i}\right)}^1}\right)\end{array}\right)\\ =\left(\begin{array}{l}\text{0}\text{.3}\times \left(-\text{100,000}+\frac{\text{120,000}}{{\left(1+\text{0}\text{.15}\right)}^1}\right)+\text{0}\text{.3}\times \left(-100,000+\frac{\text{140,000}}{{\left(1+\text{0}\text{.15}\right)}^1}\right)\\ +\text{0}\text{.4}\times \left(-\text{100,000}+\frac{\text{175,000}}{{\left(1+0.15\right)}^1}\right)\end{array}\right)\\ =\left(\text{0}\text{.3}\times \left(4,352\right)+\text{0}\text{.3}\times \left(21,174\right)+\text{0}\text{.4}\times \left(52,180\right)\right)\\ =28,700\end{array}$

The expected present worth for the expansion is $28,700.

The expected present worth (EPW) for the no expansion option can be calculated as follows:

$\begin{array}{c}\text{EPW}=\frac{{\text{Cash flow}}_{\text{No epansion}}}{{\left(1+\text{i}\right)}^{\text{1}}}\\ =\frac{\text{100,000}}{{\left(1+\text{0}\text{.15}\right)}^1}\\ =\left(\text{0}\text{.3}\times \left(4,352\right)+\text{0}\text{.3}\times \left(21,174\right)+\text{0}\text{.4}\times \left(52,180\right)\right)\\ =86,957\end{array}$

The expected present worth for the no expansion option is $86,957. Since the expected present worth of the no expansion option is greater, select the no expansion option.

(c):

To determine

Calculate the expected present worth.

Explanation of Solution

Time period is denoted by ‘n’, which is equal to 3. The expected present worth (EPW) for the produce option can be calculated as follows:

$\begin{array}{c}\text{EPW}=-\text{Cost}+\left(\begin{array}{l}\left(\begin{array}{l}{\text{P}}_{\text{Top node}}\\ \times {\text{Cash flow}}_{\text{Top node}}\end{array}\right)+\left(\begin{array}{l}{\text{P}}_{\text{Middle node}}\\ \times {\text{Cash flow}}_{\text{Bottom node}}\end{array}\right)\\ +\left(\begin{array}{l}{\text{P}}_{\text{Bottom node}}\times \\ {\text{Cash flow}}_{\text{Bottom node}}\end{array}\right)\end{array}\right)\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)\\ =-250,000+\left(\begin{array}{l}\text{0}\text{.5}\times \text{75,000}+\text{0}\text{.4}\times \text{90,000}\\ +\text{0}\text{.1}\times 150,000\end{array}\right)\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{3}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{3}}}\right)\\ =-250,000+\left(88,500\right)\left(2.28323\right)\\ =-47,934\end{array}$

The expected present worth for the produce option is -$47,934.

Time period 1 (n1) is 2. Cash flow is denoted by ‘CF”. Expected present worth (EPW) for the buy option can be calculated as follows:

$\begin{array}{c}\text{EPW}=-\text{Cost}+\left(\begin{array}{l}\left({\text{P}}_{\text{Sales up}}\times \left({\text{CF}}_{\text{Expansion}}\left(\frac{{\left(1+\text{i}\right)}^{\text{n1}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n1}}}\right)+\frac{{\text{EPW}}_{\text{Expansion}}}{{\left(1+\text{i}\right)}^{\text{n1}}}\right)\right)\\ +\left({\text{P}}_{\text{Sales down}}\times \left(\frac{{\text{CF}}_{\text{Year l}}+{\text{CF}}_{\text{Sell}}}{{\left(1+\text{i}\right)}^{\text{1}}}\right)\right)\end{array}\right)\\ =-\text{450,000}+\left(\begin{array}{l}\left(\text{0}\text{.55}\times \left(\text{100,000}\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{2}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{2}}}\right)+\frac{\text{86,957}}{{\left(1+\text{0}\text{.15}\right)}^{\text{2}}}\right)\right)\\ +\left(\text{0}\text{.45}\times \left(\frac{25,000+200,000}{{\left(1+\text{0}\text{.15}\right)}^{\text{1}}}\right)\right)\end{array}\right)\\ =-\text{450,000}+\left(\begin{array}{l}\left(\text{0}\text{.55}\times \left(\text{100,000}\left(1.62571\right)+\frac{\text{86,957}}{1.3225}\right)\right)\\ +\left(\text{0}\text{.45}\times \left(\frac{225,0000}{1.15}\right)\right)\end{array}\right)\\ =-\text{450,000}+\left(\left(\text{0}\text{.55}\times \left(162,571+65,752\right)\right)+\left(\text{0}\text{.45}\times \left(195,652.17\right)\right)\right)\\ =-\text{450,000}+\left(125,578+88,043\right)\\ =-236,379\end{array}$

The expected present worth for the buy option is -$236,379. Since the expected present worth for the produce option is greater, select the produce option.

(d):

To determine

Change in the expected present worth.

Explanation of Solution

As the time period is extended for three more years, the return on investment for both the option increases. However, the increase in return for buy option is greater than the produce option.