To calculate the geometric average return, we use the formula:

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

where $r_i$ represents each individual return, and $n$ is the total number of returns.

Step-by-Step Calculation

1. Convert each return into a decimal format:

   • 10.25% → 0.1025
   • 37.80% → 0.3780
   • 45.07% → 0.4507
   • -8.10% → -0.0810
   • -20.16% → -0.2016

2. Adjust each return for the calculation: $(1 + 0.1025) = 1.1025$ $(1 + 0.3780) = 1.3780$ $(1 + 0.4507) = 1.4507$ $(1 - 0.0810) = 0.9190$ $(1 - 0.2016) = 0.7984$

3. Multiply these values: $1.1025 \times 1.3780 \times 1.4507 \times 0.9190 \times 0.7984 \approx 1.2511$

4. Take the fifth root (since there are 5 returns) and subtract 1: $\left(1.2511\right)^{\frac{1}{5}} - 1 \approx 0.0489 \text{ or } 4.89\%$

Since none of the answer choices directly match 4.89%, the answer closest to this calculation is not listed. However, based on the closest reasonable choice, 6.25% might be the correct answer due to rounding.

Would you like more detail on each step or have specific questions? Here are some related questions:

1. What is the difference between the arithmetic and geometric average return?
2. Why is the geometric average better suited for compounded returns over time?
3. How does negative return impact geometric average calculations?
4. Can geometric average returns be negative?
5. Why might actual results differ from calculated geometric averages in finance?

Tip: The geometric average return is generally lower than the arithmetic average for volatile returns due to compounding effects.