The geometric mean return for a set of returns over multiple periods is calculated using the formula:

$\text{Geometric Mean Return} = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{1/n} - 1$

where $r_i$ represents the annual returns, and $n$ is the number of years.

Steps to Calculate:

1. Convert the annual returns into their decimal forms and add 1:

   • $1 + r_1 = 1 + 0.10 = 1.10$
   • $1 + r_2 = 1 + 0.20 = 1.20$
   • $1 + r_3 = 1 - 0.40 = 0.60$
   • $1 + r_4 = 1 - 0.30 = 0.70$

2. Multiply these values together: $(1.10) \times (1.20) \times (0.60) \times (0.70) = 0.5544$

3. Take the fourth root (since $n = 4$): $\text{Fourth Root of } 0.5544 = 0.9169$

4. Subtract 1: $0.9169 - 1 = -0.0831 \, \text{or } -8.31\%$

Final Answer:

The geometric mean return is approximately -8.31% per year.

Would you like a detailed step-by-step explanation for any part?

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Related Questions:

1. What is the arithmetic mean return for these same values?
2. How does the geometric mean differ from the arithmetic mean in financial contexts?
3. How would an additional year with a positive return affect the geometric mean?
4. Why is the geometric mean preferred for compound growth rates in investments?
5. What would the total return over the four years be, combining all the returns?

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Tip:

When analyzing stock performance, use the geometric mean to assess long-term growth because it accounts for compounding effects and varying returns.