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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.

Reconsider the research-and-development decision in Figure 3.32. If you decide to continue the project, you will have to come up with the $2 million this year (Year 1). Then there will be a year of waiting (Year 2) before you know if the patent is granted. If you decide to license the technology, you will receive the $25 million distributed as $5 million per year beginning in Year 3. On the other hand, if you decide to sell the product directly, you will have to invest $5 million in each of Years 3 and 4 (to make up the total investment of $10 million). Your net proceeds from selling the product, then, would be evenly distributed over Years 5 through 9. Assuming an interest rate of 15%, calculate the NPV at the end of each branch of the decision tree.

Explore the detailed calculation of Net Present Value (NPV) in a research-and-development decision scenario. This problem involves discounting cash flows at a 15% interest rate, probability assessments, and evaluating different strategic outcomes. Ideal for undergraduate finance and economics students.

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