You are interviewing for an internship with an investment firm. They ask you to come up with the price to charge on a security that makes growing annual payments for 10 years. The first payment is one year from today in the amount of $100 and it grows at 2% per year. The investment firm thinks potential investors have a required return of 9% per year. What is the highest price investors would be willing to pay for this security?

To determine the highest price investors would be willing to pay for this security, we need to find the **present value** of the growing payments. This type of problem involves a **growing annuity formula**, which is used when the payments grow at a constant rate over time.

### Variables:
- $ P_0 = 100 $ (initial payment one year from now)
- $ g = 2\% = 0.02 $ (growth rate of payments)
- $ r = 9\% = 0.09 $ (required return by investors)
- $ n = 10 $ (number of years)

The **present value** of a growing annuity is calculated using the following formula:

\[
PV = \frac{P_0}{r - g} \left[ 1 - \left( \frac{1 + g}{1 + r} \right)^n \right]
\]

### Step-by-step calculation:

1. **Substitute the known values into the formula:**

\[
PV = \frac{100}{0.09 - 0.02} \left[ 1 - \left( \frac{1.02}{1.09} \right)^{10} \right]
\]

2. **Simplify the difference between the required return and the growth rate:**

\[
PV = \frac{100}{0.07} \left[ 1 - \left( \frac{1.02}{1.09} \right)^{10} \right]
\]

3. **Calculate the fraction:**

\[
PV = 1428.57 \left[ 1 - \left( \frac{1.02}{1.09} \right)^{10} \right]
\]

4. **Compute the ratio $\left( \frac{1.02}{1.09} \right)$:**

\[
\frac{1.02}{1.09} = 0.93578
\]

5. **Raise this value to the power of 10:**

\[
0.93578^{10} = 0.52276
\]

6. **Subtract this value from 1:**

\[
1 - 0.52276 = 0.47724
\]

7. **Multiply by the factor $1428.57$:**

\[
PV = 1428.57 \times 0.47724 = 681.98
\]

### Final Answer:

The highest price that investors would be willing to pay for this security is **$681.98**.

---

Would you like more details or have any questions?  
Here are some follow-up questions to deepen understanding:

1. What would happen if the required return increased?
2. How does the length of the annuity affect the price of the security?
3. What is the impact if the growth rate is higher than the required return?
4. How would you calculate the price if payments were made semi-annually instead of annually?
5. How does this method differ from valuing a fixed annuity?

**Tip:** When the required return is higher than the growth rate, the present value calculation ensures that investors are compensated for the time value of money and risk.

Learn how to calculate the highest price investors would pay for a security with growing payments over 10 years, starting at $100 with a 2% growth rate and a required return of 9%. This step-by-step solution explains the use of the present value of growing annuity formula. Suitable for finance students.

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