Question

Consider an investment opportunity with an option to grow that requires a $10m investment today. In one year we will find out whether the project is successful or not. The probability that the project will generate $1M per year in perpetuity (starting from one year after the investment is made; that is, starting from year 1) is 50%. Otherwise, the project will generate nothing. You can double the size of the project in year 1 if the original project is successful. Proceeding with the extension will require an additional $10m investment which will allow the firm to generate an incremental annual cash flow $1M in perpetuity starting from year 2 (that is, there is no uncertainty about future cash flows from the extended project). Assume the risk-adjusted discount rate is 6% p.a. and do not consider any other factors.

(a) What is the NPV of the project without growth option?

(b) What is the NPV of the project with growth option? (Hint: compute the PV of the growth option first by considering cash flows from the expanded project only)

Answer 1

Formula sheet

A	B	C	D	E	F	G	H	I	J	K	L	M	N
2
3		To calculate NPV of the project, free cash flow needs to be calculated as follows:
4		Free Cash Flow = Operating Cash Flow - Capital Expenditures - Change in working capital
5		All numbers in million $
6		Probability	Cash Flow
7		0.5	1
8		0.5	0
9
10		Expected Cash Flow from the project	=0.5*1+0.5*0
11			0.5	=C7*D7+C8*D8
12
13		Calculation of NPV without growth option:
14		Free Cash flow of project without growth option can be represented as below:
15		Year	0	1	2	3	4	…
16		Investment	($10)
17		Expected Cash flow		$0.5	$0.5	$0.5	$0.5	$0.5
18		Free Cash Flow	($10)	$0.5	$0.5	$0.5	$0.5	$0.5
19
20		NPV of the project is present value of future cash flows discounted at required rate of return less the initial investment.
21
22		Year	0	1	2	3	4	…
23		Free cash flow	($10)	$0.5	$0.5	$0.5	$0.5	$0.5
24		Required rate of retun	6%
25		Project Life	Perpetuity
26		Present value of future cash flows	=0.5*(1/6%)	(Using perpetuity formula C/r)
27			$8.33	=E23/D24
28
29		Net Present Value	=Present Value of future cash flows - Initial investment
30		Net Present Value	($1.67)	=D27+D23
31
32		Hence NPV of the project without the growth option is	($1.67)
33
34
35		Calculation of NPV with growth option:
36		First present value of growth option needs to be calculated as below:
37		Since growth option will be persued only if project is successful in year 1.
38		Therefore growth option will also be considered successful and will have fixed cash flow of 1 million $ in perpetuity.
39
40		Cash flow for growth option is as below:
41		Year	0	1	2	3	4	…
42		Investment		($10)
43		Cash flow			$1.0	$1.0	$1.0	$1.0
44		Free Cash Flow		($10)	$1.0	$1.0	$1.0	$1.0
45		Required rate of retun	6%
46		Project Life	Perpetuity
47		Present value of Growth Option	=(-10+1*(1/6%))*(P/F,6%,1)
48			$6.29	=(E44+F44*(1/D45))*(1/(1+D45))
49
50		Thus project will have additional value of 6.29 million $ at year zero if the project is successful.
51
52		Now project with growth option cash flows can be represented as follows:
53		Year	Probability	0	1	2	3	4	…
54		Cash Flow if project is successful	0.5	($3.7)	$1.0	$1.0	$1.0	$1.0	$1.0
55		Cash Flow If project is not successful	0.5	($10.0)	$0.0	$0.0	$0.0	$0.0	$0.0
56		Expected Cash Flow		($6.9)	$0.5	$0.5	$0.5	$0.5	$0.5	=$D$54*J54+$D$55*J55
57
58		Calculation of NPV for project with growth option:
59		Year	0	1	2	3	4	…
60		Free cash flow	($6.9)	$0.5	$0.5	$0.5	$0.5	$0.5
61		Required rate of retun	6%
62		Project Life	Perpetuity
63		Present value of future cash flows	=0.5*(1/6%)	(Using perpetuity formula C/r)
64			$8.33	=E60/D61
65
66		Net Present Value	=Present Value of future cash flows - Initial investment
67		Net Present Value	$1.48	=D64+D60
68
69		Hence NPV of the project with the growth option is	$1.48
70