To calculate the present value (PV) of the drug's profits over the 17-year period, we can use the concept of a growing annuity, which applies to cash flows that increase at a constant growth rate.

The formula for the present value of a growing annuity is:

$PV = \dfrac{C_1}{r - g} \left( 1 - \left( \dfrac{1 + g}{1 + r} \right)^T \right)$

Where:

• $C_1$ is the profit in the first year ($2 million),
• $r$ is the discount rate or interest rate (10%, or 0.10),
• $g$ is the growth rate (5%, or 0.05),
• $T$ is the number of years of growth (17 years).

Step 1: Plugging in the values:

$PV = \dfrac{2}{0.10 - 0.05} \left( 1 - \left( \dfrac{1 + 0.05}{1 + 0.10} \right)^{17} \right)$

$PV = \dfrac{2}{0.05} \left( 1 - \left( \dfrac{1.05}{1.10} \right)^{17} \right)$

Step 2: Simplifying the terms:

$PV = 40 \times \left( 1 - \left( \dfrac{1.05}{1.10} \right)^{17} \right)$

$\dfrac{1.05}{1.10} = 0.954545$

So,

$PV = 40 \times \left( 1 - (0.954545)^{17} \right)$

Now, calculate $0.954545^{17}$:

$0.954545^{17} \approx 0.45459$

Step 3: Final calculation:

$PV = 40 \times \left( 1 - 0.45459 \right)$

$PV = 40 \times 0.54541 = 21.8164$

Thus, the present value of the new drug's profits over the 17-year period is approximately $21.82 million.

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Would you like to see the details on any step, or do you have further questions?

Related Questions:

1. How would the present value change if the growth rate were 3% instead of 5%?
2. What happens to the present value if the interest rate decreases to 8%?
3. How can inflation affect the real value of the drug's profits over time?
4. How would the result differ if the patent duration were only 10 years?
5. What would the future value of the drug's profits be after 17 years?

Tip:

When calculating the present value of any cash flow, always ensure that the growth rate $g$ is less than the discount rate $r$ for the formula to work correctly.