Expected NPV

CFA Level I Quant
#1

Hi,

kindly help to to explain the below calculus for expected NPV.

An investment of 2.5 million in a venture capital has the following probablilities of failure over the next 4 years: 60%, 55%, 45%, 30%.

If the cash projects succeeds at the end of the 4th year the investor can exit with 13.5 million. Given the risk of the investment the discount rate is 40%. Should the investment undertaken?

Solution: No, expected NPV -2.26 mln.

The probability of success over the 4 years is the product of probabililies of the 4 years:

Prob(S) = (1-60%) * (1-55%) * (1-45%) * (1.30%) = 0.069

Prob(F) = 1 - 0.0696 = 0.931

Prob(F) is the probability of ONE failure over the next 4 years. PV of CF after 4 years: 13.3/1.4^4 = 3.51 mln

Expected NPV = 0.069 * 3.51 + 0.931 * 0 - 2.5 = -2.26 < 0 - project should be turned down

#2

For the investment to succeed, it has to succeed in the first year, and succeed in the second year, and in the third year, and in the fourth.

The probability of succeeding in the first year is 1 – P(F1) = 1 – 0.6 = 0.4.

The probability of succeeding in the second year given that it succeeded in the first year is 1 – P(F2) = 1 – 0.55 = 0.45. Thus, the probability of surviving the first two years is P(S1,2) = P(S1) × P(S2|S1) = 0.4 × 0.45 = 0.18.

The probability of succeeding in the third year given that it succeeded in the second year is 1 – P(F3) = 1 – 0.45 = 0.55. Thus, the probability of surviving the first three years is P(S1,2,3) = P(S1) × P(S2|S1) × P(S3|S2) = 0.4 × 0.45 × 0.55 = 0.099.

The probability of succeeding in the fourth year given that it succeeded in the third year is 1 – P(F4) = 1 – 0.3 = 0.7. Thus, the probability of surviving the first four years is P(S1,2,3,4) = P(S1) × P(S2|S1) × P(S3|S2) × P(S4|S3) = 0.4 × 0.45 × 0.55 × 0.3 = 0.0693. If the project survives four years, the payoff is $13.5 million.

The probability of not surviving the first four years (i.e., failing in year 1 or in year 2 or in year3 or in year 4) is 1 – P(S1,2,3,4) = 1 – 0.0693 = 0.9307. If the project fails anytime in the first four years, the payoff is $0.

The expected value of the payoff is 0.0693 × $13.5 million + 0.9307 × $0 = $935,550. The present value of the expected payoff is $935,550 / 1.4^4 = $243,531.

Thus, the expected net present value is -$2,500,000 + $243,531 = -$2,256,469.

#3

cheers my friend, thank you very much :slight_smile:

#4

You’re most welcome.