Question

Your firm is thinking of expanding. If you invest​ today, the expansion will generate $ 11 million in FCF at the end of the​ year, and will have a continuation value of either $ 147 million​ (if the economy​ improves) or $ 55 million​ (if the economy does not​ improve). If you wait until next year to​ invest, you will lose the opportunity to make $ 11 million in FCF but you will know the continuation value of the investment in the following year​ (that is, in a year from now you will know what the investment continuation value will be in the following​ year). Suppose the​ risk-free rate is 5 %​, and the​ risk-neutral probability that the economy improves is 42 %. Assume the cost of expanding is the same this year or next year.

a. If the cost of expanding is $ 78 ​million, should you do so​ today, or wait until next year to​ decide?

b. At what cost of expanding would there be no difference between expanding now and​ waiting? To what profitability index does this​ correspond?

Answer 1

a) If expansion is done today

Expected continuation value after a year = Probability weighted continuation value

= probability that economy improves * continuation value in case economy improves + (1- probability that economy does not improve) * continuation value in case economy does not improve

=0.42* $147 million + 0.58* $55 million

=$93.64 million

So NPV = -78 + (11+93.64)/1.05

=$21.657143 million

=$21,657,143

In case expansion is done next year, it will be done only if the economy improves (42%)

So, NPV after one year = -78+147 = $69 million in case economy improves

or 0 otherwise

Expected Value of NPV after one year = 0.42*69+0.58*0 = $28.98 million

Value of NPV today = $28.98/1.05 = $27.60 million

As the NPV of waiting is more than the NPV of expanding today

It is better to Wait for one year for expanding.

b) IF the expansion cost were $X million for indifference

NPV of expanding today =  -X + (11+93.64)/1.05

=$(99.657143-X) million

For Waiting

So, NPV after one year = -X+147 = $(147-X) million in case economy improves

or 0 otherwise

Expected Value of NPV after one year = 0.42*(147-X)+0.58*0 = $(61.74 - 0.42X ) million

Value of NPV today = $(61.74 - 0.42X )/1.05

For Indifference between expansion today and waiting

(61.74-0.42X)/1.05 = 99.657143 -X

=> 61.74 - 0.42X = 104.64 - 1.05X

=> 0.63X = 42.90

X = $68.095238 million or $68,095,238

At this point, PV of benefit = (11+93.64)/1.05 =$99.657143 million

So, Profitability Index = PV of Benefits/PV of cost = 99.657143/68.095238 = 1.4635