An Example Of How To Value An Option To Scale Back Production By 50%

A firm considers investing $279 in a new factory. Cash flows two years from now depend on the economy. At full capacity, those cash flows produce the following pattern. If there are two good years, the cash flow is $814. If one good year is followed by one bad year, the cash flow is $581. If one bad year is followed by a good year, the cash flow is the same. That is the pattern recombines. Finally, if two bad years occur, the cash flow is $-255. A market portfolio produces a risk neutral risk probability of 0.7 for upward transitions and the annual risk free rate is 8%. What is the value of an option to scale production back to 50% at no cost if unfavorable conditions occur in year 1?

$\pi$ = .7

1 - $\pi$ = .3

$r_{f}$ = .08

Solve using the Two-Period Binomial Valuation

Scenario 1 (Full Production)
$V_{u}$ = (($\pi$*uu) + ((1 - $\pi$)*ud)) / (1 + $r_{f}$)
$V_{u}$ = ((.7*814) + (.3*581))/1.08
$V_{u}$ = 688.98

$V_{d}$ = (($\pi$*du) + ((1 - $\pi$)*dd)) / (1 + $r_{f}$)
$V_{d}$ = ((.7*581) + (.3*-255))/1.08
$V_{d}$ = 305.74

$V_{0}$ = (($\pi$*u) + ((1 - $\pi$)*d)) / (1 + $r_{f}$)
$V_{0}$ = ((.7*688.98) + (.3*305.74))/1.08
$V_{0}$ = 531.49

Two-Period Binomial Tree:
Yr0………Yr1…….Yr2
……………………….814
…………688.98
531.49…………….581
…………305.74
………………………-255

Scenario 2 (50% Production in Unfavorable Conditions)

$V_{u}$ = 688.98 (No change since there are favorable conditions in Year 1)

$V_{d}$ = ((.7*290.50) + (.3*-127.50))/1.08 (Scale back production to 50% in Year 1)
$V_{d}$ = 152.87

$V_{0}$ = ((.7*688.98) + (.3*152.87))/1.08
$V_{0}$ = 489.03

One-Period Binomial Tree:
Yr0……….Yr1……….Yr2
………………………….814
………….688.98
489.03……………….581
………………………..290.50
………….152.87
……………………….-127.50

Value of the Option = Value at Full Production - Value at 50% Production
Value of the Option = 531.49 - 489.03
Value of the Option = 42.46