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.

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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.