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To solve the problem, we need to calculate the geometric mean rate of return for each of the three stock exchanges (A, B, and C) over the four years from 2009 to 2012. The geometric mean is particularly useful in finance because it accounts for compounding returns.

### Geometric Mean Formula

The geometric mean \( G \) of a set of \( n \) values \( x_1, x_2, \dots, x_n \) is given by:

\[
G = \left(\prod_{i=1}^{n} (1 + x_i)\right)^{\frac{1}{n}} - 1
\]

Where \( x_i \) is the rate of return in decimal form for year \( i \).

### Steps:
1. Convert the percentage returns to their decimal form by dividing by 100.
2. Apply the geometric mean formula for each stock exchange.
3. Subtract 1 at the end to convert back to the mean rate of return.

Let's go through the calculations for each stock.

### Stock A:
Given returns: 18.27%, 10.00%, 4.87%, 7.57%

Convert to decimals:
- \( 1.1827 \)
- \( 1.1000 \)
- \( 1.0487 \)
- \( 1.0757 \)

Geometric mean:
\[
G_A = \left(1.1827 \times 1.1000 \times 1.0487 \times 1.0757\right)^{\frac{1}{4}} - 1
\]

### Stock B:
Given returns: 23.11%, 12.18%, 0.00%, 13.23%

Convert to decimals:
- \( 1.2311 \)
- \( 1.1218 \)
- \( 1.0000 \)
- \( 1.1323 \)

Geometric mean:
\[
G_B = \left(1.2311 \times 1.1218 \times 1.0000 \times 1.1323\right)^{\frac{1}{4}} - 1
\]

### Stock C:
Given returns: 43.41%, 16.14%, -2.15%, 15.77%

Convert to decimals:
- \( 1.4341 \)
- \( 1.1614 \)
- \( 0.9785 \) (since -2.15% becomes \(1 - 0.0215\))
- \( 1.1577 \)

Geometric mean:
\[
G_C = \left(1.4341 \times 1.1614 \times 0.9785 \times 1.1577\right)^{\frac{1}{4}} - 1
\]

I'll now calculate these values.The geometric mean rate of return for each of the three stock exchanges over the four-year period is as follows:

- **Stock A**: 10.07%
- **Stock B**: 11.83%
- **Stock C**: 17.20%

These percentages represent the average compounded annual growth rate for each stock exchange during the given period.

Would you like more details or have any other questions? 

Here are some related questions:

1. How does the geometric mean differ from the arithmetic mean in finance?
2. What is the significance of negative returns in calculating the geometric mean?
3. How can you interpret the geometric mean in the context of investment returns?
4. What are some limitations of using the geometric mean for financial data?
5. How does the geometric mean impact long-term investment strategies?
6. What is the relationship between volatility and geometric mean returns?
7. Can the geometric mean be used to compare different investment portfolios?
8. How would the geometric mean change if one year had a significantly larger return?

**Tip:** The geometric mean is always less than or equal to the arithmetic mean, and it is particularly useful for comparing investment returns over multiple periods.

Learn how to calculate the geometric mean rate of return for three stock exchanges (A, B, and C) over the four-year period from 2009 to 2012. This problem involves converting percentage returns to decimals and applying the geometric mean formula. Discover the compounded annual growth rates for each stock exchange.

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