# Project NPV with chance of failure

when you are calculating the NPV of a project where there is a chance of failure, what is the correct formula? I’ve come across these 3 (i.e. questions 9, 10, and 28 from reading 66) and can’t seem to understand what the question is asking that would require the change in formula:

Ps: Probablility of success

Pf: Probability of Failure

C0: Original cash outlay at t=0

expected NPV = Ps* NPV of future cash flows (not counting C0) if successful + Pf * NPV of future cash flows (not counting C0) if unsuccessful - C0

or,

Expected NPV = Ps * NPV of all cash flows (including C0) if successful + Pf * NPV of future cash flows (including C0) if unsuccessful

or,

Expected NPV = Ps * NPV of future cash flows (not counting C0) if successful - C0

Any help?

If you think about it intuitively you have to assign an expected value to the project. To do that, find the probability that you will receive the cash flow at the end of the project (Ps * FV of Payoff), and discount that back to T=0 at the cost of capital and remove the initial investment (C0).

Example:

Initial investment = \$5 million

Payoff = \$22 million at the end of five years

Probability of failure for each of the five years:

Y1=0.25 Y2=0.20 Y3=0.15 Y4=0.15 Y5=0.15

Cost of Capital = 16%

Expected value = {(1-0.25)*(1-0.20)*(1-0.15)*(1-0.15)*(1-0.15)} * \$22,000,000 = \$8,106,450

In calculator (TI-BAII)

CF0= - \$5,000,000, C01= \$0 and F01= 4, C02 = \$8,106,450 and F02= 1 then, NPV, I = 16%…

NPV = - \$1,140,414

In short, your last formula seems to be correct.

I don’t go by the formulas. Just find the probability of success, multiply it by the expected future cash flow, then discount the product to t=0 to get the PV, then subtract initial cash outflow from the PV.

Sounds great, that’s what I had originally thought…but to have 2 questions in the CFA study manual in the same section have incorrect answers and not be rectified in the errata? doesn’t this seem a little fishy?

The answers are not incorrect. #10 is a little bit off due to too much rounding off. For #9 remember to find the weighted average of the possible future payoffs (=\$30M). Their methods are way too long though.