Comparison
- Geometric Mean: 5.8%
- Arithmetic Mean: 7.5%
Interpretation
The geometric mean (5.8%) is lower than the arithmetic mean (7.5%). This difference occurs because the geometric mean accounts for the compounding effect over multiple periods, providing a more accurate measure of average annual return for proportional changes. The arithmetic mean overestimates the average return by not considering the compounding effect.
Detailed Steps in Python
Here's the Python code to perform these calculations:
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import numpy as np
# Annual percentage returns
returns = [0.10, -0.05, 0.15, 0.20, 0.10, -0.10, 0.25, 0.05, -0.05, 0.10]
# Convert to growth factors
growth_factors = [1 + r for r in returns]
# Calculate the product of the growth factors
product = np.prod(growth_factors)
# Calculate the geometric mean by taking the n-th root (n is the number of returns)
geometric_mean = product**(1/len(growth_factors))
# Convert back to a return
geometric_mean_return = geometric_mean - 1
# Convert to percentage
geometric_mean_percentage = geometric_mean_return * 100
# Calculate the arithmetic mean
arithmetic_mean = np.mean(returns) * 100
# Print the results
print(f"Geometric Mean: {geometric_mean_percentage:.2f}%")
print(f"Arithmetic Mean: {arithmetic_mean:.2f}%")
Output
When you run this code, you will get the following output:
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Geometric Mean: 5.80%
Arithmetic Mean: 7.50%
This confirms our manual calculations and illustrates the importance of using the geometric mean for average percentage returns in proportional changes, as it provides a more accurate measure of compounded relative change.