The question asks to calculate the Net Present Value (NPV) at the end of each branch in a decision tree, given an interest rate of 15%. I'll walk through the steps for each branch of the tree based on the scenario and decision outcomes provided.

1. Scenario: Patent Granted (p = 0.7)

• License the Technology ($25M):

  • You receive $25M, starting in Year 3, spread evenly over five years at $5M/year.
  • NPV calculation: $NPV = \frac{5}{(1+0.15)^3} + \frac{5}{(1+0.15)^4} + \frac{5}{(1+0.15)^5} + \frac{5}{(1+0.15)^6} + \frac{5}{(1+0.15)^7}$

• Develop and Market the Product (-$10M investment):

  • You need to invest $5M in Year 3 and $5M in Year 4.
  • Proceeds depend on demand levels starting from Year 5 and lasting until Year 9.

  For each demand level:

  • High Demand (p = 0.25): You receive $55M spread evenly over 5 years ($11M/year).

    NPV calculation: $NPV = \sum_{t=5}^{9} \frac{11}{(1+0.15)^t} - \frac{5}{(1+0.15)^3} - \frac{5}{(1+0.15)^4}$

  • Medium Demand (p = 0.55): You receive $33M spread evenly over 5 years ($6.6M/year).

    NPV calculation: $NPV = \sum_{t=5}^{9} \frac{6.6}{(1+0.15)^t} - \frac{5}{(1+0.15)^3} - \frac{5}{(1+0.15)^4}$

  • Low Demand (p = 0.20): You receive $15M spread evenly over 5 years ($3M/year).

    NPV calculation: $NPV = \sum_{t=5}^{9} \frac{3}{(1+0.15)^t} - \frac{5}{(1+0.15)^3} - \frac{5}{(1+0.15)^4}$

2. Scenario: No Patent (p = 0.3)

• End of development: A sunk cost of -$2M and no additional revenue.
• NPV: -2M (as there are no further cash flows in this branch).

I will now calculate the specific NPVs using these formulas.Here are the NPVs for each branch of the decision tree based on a 15% discount rate:

1. License the Technology:
   NPV = $12.67M

2. Develop and Market the Product:

   • High Demand (p = 0.25):
     NPV = $14.94M
   • Medium Demand (p = 0.55):
     NPV = $6.50M
   • Low Demand (p = 0.20):
     NPV = -$0.40M (negative, meaning a loss)

The decision path without a patent results in a sunk cost of -$2M (as noted earlier).

Would you like further details or calculations? Here are five related questions to extend this problem:

1. How does the probability of receiving a patent affect the expected value of each decision path?
2. What happens to the NPVs if the interest rate changes to 10% or 20%?
3. How would altering the demand probabilities affect the overall decision-making process?
4. Could there be a break-even interest rate for the low-demand scenario?
5. What would happen if the cost to develop and market increased or decreased?

Tip: Understanding NPV helps assess long-term profitability, especially in project decision-making under uncertainty.