The geometric mean return of a set of returns is calculated using the formula:

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

Where:

• $R_i$ is the return in each year, expressed as a decimal.
• $n$ is the number of years.

Step-by-Step Solution:

1. Convert annual returns from percentages to decimals:

   • Year 1: $10\% = 0.10$
   • Year 2: $20\% = 0.20$
   • Year 3: $-40\% = -0.40$
   • Year 4: $-30\% = -0.30$

2. Add 1 to each return to calculate the growth factors:

   • Year 1: $1 + 0.10 = 1.10$
   • Year 2: $1 + 0.20 = 1.20$
   • Year 3: $1 - 0.40 = 0.60$
   • Year 4: $1 - 0.30 = 0.70$

3. Multiply the growth factors: $\text{Product} = 1.10 \times 1.20 \times 0.60 \times 0.70 = 0.5544$

4. Take the 4th root of the product (since $n = 4$): $G = \left( 0.5544 \right)^{1/4} - 1$

5. Compute:

   • $0.5544^{1/4} \approx 0.8665$
   • Subtract 1: $G = 0.8665 - 1 = -0.1335$

6. Convert to percentage: $G = -13.35\%$

Final Answer:

The geometric mean return is -13.35% per year.

Would you like more details or have any questions? 😊

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Related Questions:

1. What is the difference between geometric and arithmetic mean returns?
2. How does compounding affect the geometric mean return?
3. Can geometric mean return be negative even if some yearly returns are positive?
4. What are the advantages of using geometric mean return in finance?
5. How would the geometric mean change if all returns were positive?

Tip: The geometric mean is crucial in finance because it accounts for compounding effects over multiple periods, giving a more accurate representation of long-term growth.