You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 1616 years. You expect that the​ drug's profits will be ​$22 million in its first year and that this amount will grow at a rate of 2 %2% per year for the next 1616 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 7 %7% per​ year?

To find the present value (PV) of the drug’s profits over the next 16 years, we can apply the **Present Value of a Growing Annuity** formula, which accounts for the growth in profits and the discount rate (interest rate).

### Given:
- $ C_1 = 22 $ million (the first year's profit)
- $ g = 2\% $ or 0.02 (annual growth rate of profits)
- $ r = 7\% $ or 0.07 (annual interest rate)
- $ n = 16 $ years (duration of the patent)

### Formula for the Present Value of a Growing Annuity:

\[
PV = \frac{C_1}{r - g} \times \left[ 1 - \left( \frac{1 + g}{1 + r} \right)^n \right]
\]

Where:
- $ C_1 $ is the profit in the first year,
- $ r $ is the discount rate,
- $ g $ is the growth rate of profits,
- $ n $ is the number of years the patent will last.

### Plugging in the values:

\[
PV = \frac{22}{0.07 - 0.02} \times \left[ 1 - \left( \frac{1.02}{1.07} \right)^{16} \right]
\]

### Step-by-step Calculation:
1. $ r - g = 0.07 - 0.02 = 0.05 $
2. $ \frac{22}{0.05} = 440 $
3. $ \frac{1.02}{1.07} = 0.953271 $
4. $ 0.953271^{16} \approx 0.478594 $
5. $ 1 - 0.478594 = 0.521406 $
6. Now calculate the present value:
   \[
   PV = 440 \times 0.521406 \approx 229.4186
   \]

### Conclusion:
The present value of the drug's profits over the next 16 years is approximately **$229.42 million**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. How would changes in the growth rate affect the present value?
2. What if the interest rate changes to 5% instead of 7%?
3. How does the duration of the patent (number of years) influence the PV calculation?
4. How would the profits change if there was no growth in the profits (i.e., g = 0%)?
5. What are the risks of patent expiration for the pharmaceutical company?

**Tip:** Always ensure to adjust the growth and interest rates for inflation when evaluating long-term investments like drug patents.

You work for a pharmaceutical company that has developed a new drug. The patent on the drug will last 16 years. You expect that the drug's profits will be $22 million in its first year and that this amount will grow at a rate of 2% per year for the next 16 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 7% per year?

Find out how to calculate the present value of drug profits over 16 years, accounting for a 2% annual growth and a 7% discount rate. This step-by-step guide walks you through the Present Value of a Growing Annuity formula. Ideal for advanced high school or college students.

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