You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 17 years. You expect that the​ drug's profits will be $2 million on in its first year and that this amount will grow at a rate of 5% per year for the next 17 years. Once the patent​ expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 10% per​ year? 

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.

You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 17 years. You expect that the drug's profits will be $2 million in its first year and that this amount will grow at a rate of 5% per year for the next 17 years. Once the patent expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 10% per year?

Calculate the present value of a new drug's profits using the growing annuity formula over a 17-year patent period, considering a 5% growth rate and 10% interest rate. Learn the detailed steps involved in discounting cash flows for this pharmaceutical scenario.

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