Chapter 26, Problem 9SP

a.

Summary Introduction

To determine: Expected present value, NPV and the coefficient of variation of the present value.

Explanation of Solution

Table that shows the calculation of the expected present values of the cash flow is as follows:

Condition	Probability	Cash Flows ($)	Present value ($)
Good	30%	9	2.70
Medium	40%	4	1.60
Bad	30%	-1	-0.30
Expected present value			4

Table (1)

Table that shows calculation of the Net present value is as follows:

Particulars	Cash Flow	Present value factor	Present value
Initial Cost (A)	4	1	4
Cash Flows: (B)
Year 1	4	0.893	3.572
Year 2	4	0.797	3.188
Year 3	4	0.711	2.844
NPV $\left(\text{A}\right)-\left(\text{B}\right)$			0.396

Table (2)

b.

Summary Introduction

To decide: If the project should be accepted with the scrap value of $6 million and cash flow in first year only.

Explanation of Solution

Table that shows calculation of the NPV with current scenario is as follows:

Cost	Probability	Future cash flows			NPV	$\text{Probability}\times \text{NPV}$
		1 ($)	2 ($)	3 ($)
-$10	30%	9	9	9	11.62	3.48
	40%	4	4	4	-0.39	-0.16
	30%	5			-5.54	-1.66
Expected NPV of future Cash flows						1.67

Table (3)

With the abandonment factor, the expected NPV becomes positive but NPV of medium is negative but higher than the NPV of the medium factor if the project will be abandoned in the year 1 itself.

c.

Summary Introduction

Estimate: The value of the project.

Explanation of Solution

Table that shows calculation of the Expected NPV is as follows:

Cost		Future operating cash flows						NPV ($)	$\text{Probability}\times \text{NPV}$ ($)
0	Probability	1 ($)	2 ($)	3 ($)	4 ($)	5 ($)	6 ($)
-10	30%	9	9	9	9	9	9	27	8.10
	40%	4	4	4	0	0	0	-0.39	-0.16
	30%	-1	-1	-1	0	0	0	-12.40	-3.72
Expected NPV									4.22

Table (4)

Table that shows calculation of expected NPV after the implementation of additional project is as follows:

		Future operating cash flows						NPV ($)	$\text{Probability}\times \text{NPV}$ ($)
0	Probability	1 ($)	2 ($)	3 ($)	4 ($)	5 ($)	6 ($)
	30%	0	0	-10	0	0	0	-8.40	-2.52
	40%	0	0	0	0	0	0	0	0
	30%	0	0	0	0	0	0	0	0
Expected NPV									-2.52

Table (5)

Total of the NPV of the future operating cash flows and NPV after the implementation of additional project is $1.70. 

Hence, as with the additional project the NPV gets positive, therefore, the project with additional projects should be accepted.

d.

Summary Introduction

Estimate: The value of the project if it is delayed for one year.

Explanation of Solution

Table that shows calculation of expected NPV of future cash flows with risk-free rate of 6% and WACC of 12% is as follows:

		Future operating cash flows				NPV ($)	$\text{Probability}\times \text{NPV}$ ($)
0	Probability	1 ($)	2 ($)	3 ($)	4 ($)
		Cost
	30%		9	9	9	21.62	6.48
	40%		0	0	0	0	0
	30%		0	0	0	0	0
Expected NPV							6.48

Table (6)

		Future operating cash flows				NPV ($)	$\text{Probability}\times \text{NPV}$ ($)
0	Probability	1 ($)	2 ($)	3 ($)	4 ($)
		Cost
	30%		0	0	0	-9.43	-2.83
	40%		0	0	0	0	0
	30%		0	0	0	0	0
Expected NPV							-2.83

Table (7)

Total of both the expected NPV is $3.65, which is positive; hence, the project should be accepted without any delaying 1 year.

e.

Summary Introduction

Estimate: The value of the growth option using Black-Scholes model.

Explanation of Solution

Table that shows calculation of the expected NPV is as follows:

Cost		Future operating cash flows						NPV ($)	$\text{Probability}\times \text{NPV}$ ($)
0	Probability	1 ($)	2 ($)	3 ($)	4 ($)	5 ($)	6 ($)
	30%	0	0	0	9	9	9	15.39	4.62
	40%	0	0	0	4	4	4	6.84	2.74
	30%	0	0	0	-1	-1	-1	-1.71	-0.51
Expected NPV									6.84

Table (8)

Formula to calculate total value is as follows:

$\text{V}=\text{P}\left[\text{N}\left({\text{d}}_1\right)\right]-{\text{Xe}}^{-{\text{r}}_{\text{RF}}\text{t}}\left[\text{N}\left({\text{d}}_2\right)\right]$

Substituting the equation with $6.84 for P, 0.56 for N(d₁), 0.26 for N(d₂) and $10.00 for X to calculate value of growth option.

$\begin{array}{c}\text{V}=\text{P}\left[\text{N}\left({\text{d}}_1\right)\right]-{\text{Xe}}^{-{\text{r}}_{\text{RF}}\text{t}}\left[\text{N}\left({\text{d}}_2\right)\right]\\ =\$6.84\left(0.56\right)-\$10{\text{e}}^{-0.06\times 3}\left(0.26\right)\\ =\$1.71\end{array}$

Formula to calculate total value is as follows:

$\text{Total Value}=\text{Value of original project}+\text{Value of growth option}$

Substituting the Equation with -$0.39 for the value of original project and $1.71 for the value of growth option to calculate total value.

$\begin{array}{c}\text{Total Value}=-\$0.39+\$1.71\\ =\$1.31\end{array}$

Hence, the project should be accepted even if the value of original project is negative; as the total value with the growth option is greater than 0.

Working Note:

$\begin{array}{c}{\text{d}}_1=\frac{\left\{\text{In}\left(\frac{\text{P}}{\text{X}}\right)+\left[{\text{r}}_{\text{RF}}+\frac{{\sigma }^2}{2}\right]\text{t}\right\}}{\sigma \left({\text{t}}^{\frac{1}{2}}\right)}\\ =0.160\end{array}$

$\begin{array}{c}{\text{d}}_2={\text{d}}_1-\sigma \left({\text{t}}^{\frac{1}{2}}\right)\\ =-0.65\end{array}$