Suppose a certain company stock has its share price rise by the following percentages during each of the last 5 years: 4%, 6%, 6%, 8%, 13%. Find the average annual gain over the last five years (i., e., the geometric mean).
Round your answer to two decimal places answer= %

To find the average annual gain using the geometric mean, we need to take into account the compounded nature of growth over multiple periods. Here are the steps:

1. **Convert the percentages to growth factors** by adding 1 to each percentage as a decimal:
   - 4% growth → \(1 + 0.04 = 1.04\)
   - 6% growth → \(1 + 0.06 = 1.06\)
   - 6% growth → \(1 + 0.06 = 1.06\)
   - 8% growth → \(1 + 0.08 = 1.08\)
   - 13% growth → \(1 + 0.13 = 1.13\)

2. **Multiply all growth factors** together to find the compounded growth factor over five years:
   \[
   1.04 \times 1.06 \times 1.06 \times 1.08 \times 1.13
   \]

3. **Take the 5th root** of the result to find the geometric mean, since we’re averaging over five years.

4. **Subtract 1 and convert to a percentage** to get the average annual gain.

Let's calculate:

\[
\text{Geometric mean} = \left(1.04 \times 1.06 \times 1.06 \times 1.08 \times 1.13\right)^{\frac{1}{5}} - 1
\] 

Calculating, we get approximately:

\[
\text{Geometric mean} \approx 0.0731 = 7.31\%
\]

So, the average annual gain over the last five years is approximately **7.31%**. 

Would you like further clarification on any step?

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### Related Questions:
1. How does the geometric mean differ from the arithmetic mean in financial contexts?
2. What would happen to the geometric mean if one of the years had a negative return?
3. How does the geometric mean help in understanding compounded growth over time?
4. Can the geometric mean be used to calculate average losses, and how?
5. What is the impact of varying percentage changes on the overall average gain?

#### Tip:
The geometric mean is especially useful in finance for calculating average growth rates, as it accounts for compounding effects over time.

Suppose a certain company stock has its share price rise by the following percentages during each of the last 5 years: 4%, 6%, 6%, 8%, 13%. Find the average annual gain over the last five years (i.e., the geometric mean). Round your answer to two decimal places answer= %

Discover how to find the average annual gain of a company's stock price over five years using the geometric mean. Given growth rates of 4%, 6%, 6%, 8%, and 13%, this guide breaks down each step in calculating compounded growth, with a final result of approximately 7.31%. Ideal for students in Grades 10-12 learning about financial growth and statistics.

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