To find arithmetic and geometric returns, you use different calculation methods, each providing a distinct perspective on investment performance. The arithmetic return gives a simple average, while the geometric return offers a more accurate reflection of compounded growth.
Arithmetic Return
The arithmetic return, often called the average return, is calculated by summing the returns over a period and dividing by the number of periods.
Formula:
Arithmetic Return = (Sum of Returns) / (Number of Periods)
Example:
Suppose an investment yields the following annual returns: 10%, 20%, and -5%. The arithmetic return would be calculated as follows:
(10% + 20% + (-5%)) / 3 = 8.33%
In summary, you sum all the periodic returns and then divide by the number of periods to arrive at an average. This average represents the arithmetic mean of the returns.
Geometric Return
The geometric return, also known as the time-weighted return, measures the actual rate of return on an investment over a given period, taking into account the effects of compounding. It's generally more useful than the arithmetic return for evaluating investment performance, especially over longer periods.
Formula:
Geometric Return = [(1 + Return1) (1 + Return2) ... * (1 + Returnn)](1/n) - 1
Where:
- Return1, Return2, ..., Returnn are the returns for each period.
- n is the number of periods.
Example:
Using the same annual returns as above (10%, 20%, and -5%), the geometric return is calculated as follows:
[(1 + 0.10) (1 + 0.20) (1 + (-0.05))](1/3) - 1 = (1.10 1.20 0.95)(1/3) - 1 = (1.254)(1/3) - 1 ≈ 1.077 - 1 = 0.077 or 7.7%
Steps for Calculation:
- Add 1 to each return (expressed as a decimal).
- Multiply all the results together.
- Raise the product to the power of (1/n), where n is the number of periods. This is equivalent to finding the nth root of the product.
- Subtract 1 from the result to get the geometric return.
Key Differences and When to Use Each:
| Feature | Arithmetic Return | Geometric Return |
|---|---|---|
| Calculation | Simple average | Compounded average |
| Use Case | Short-term, indicative return | Long-term, realistic performance measurement |
| Sensitivity | Less sensitive to volatility | More sensitive to volatility |
| Accuracy | Overestimates long-term performance in volatile markets | Provides a more accurate representation of growth |
In summary, calculate arithmetic returns by averaging periodic returns, whereas geometric returns involve compounding individual period returns and calculating the n-th root to give the annualised compounded return. The geometric return is typically preferred for evaluating investment performance, especially over longer periods, as it accurately reflects the actual growth achieved.
