Enter the returns (Ri) as a comma-separated list (and optionally the mean return and total number of returns) into the calculator to determine the variance of returns. Use consistent units (e.g., enter 10 for 10% for all values, or 0.10 for 10% for all values).
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Variance Of Returns Formula
Variance of returns measures how widely a series of periodic returns is spread around its average return. In finance, it is a core volatility metric used to compare the consistency of assets, strategies, and portfolios. A larger variance means returns are more dispersed; a smaller variance means returns are clustered more tightly around the mean.
V = \frac{\sum_{i=1}^{N}(R_i - R_m)^2}{N}This calculator uses the population variance form, which divides by the total number of returns.
Variable Definitions
| Term | Meaning |
|---|---|
| Variance of returns | The average of the squared deviations from the mean return. |
| Individual return | One observed return from the series being analyzed. |
| Mean return | The arithmetic average of all returns in the set. |
| Total number of returns | The number of observations included in the calculation. |
How to Calculate Variance of Returns
- List all returns for the same asset, portfolio, or strategy over matching time periods.
- Find the mean return.
- Subtract the mean from each return to get each deviation.
- Square each deviation so that positive and negative deviations do not cancel out.
- Add the squared deviations together.
- Divide that total by the number of returns.
If you still need the average return, use:
R_m = \frac{\sum_{i=1}^{N}R_i}{N}Why Variance Matters
- Risk measurement: Variance is a direct measure of return dispersion and is widely used as a volatility proxy.
- Comparison: Two investments can have the same average return but very different variability.
- Portfolio analysis: Variance is a building block for covariance, correlation, and portfolio risk calculations.
- Strategy evaluation: Lower variance often indicates more stable period-to-period performance.
Interpreting the Result
- A higher variance means returns fluctuate more around the average.
- A lower variance means returns are more consistent.
- A variance of zero means every observed return is identical.
- Variance is expressed in squared return units, so it is often converted into standard deviation for easier interpretation.
Standard deviation is the square root of variance:
\sigma = \sqrt{V}If returns are entered as percentages, the variance is in percentage-squared terms. If returns are entered as decimals, the variance is in decimal-squared terms. Use one format consistently throughout the calculation.
Example
Suppose five periodic returns are 10%, 15%, 12%, 8%, and 9%.
R_m = \frac{10 + 15 + 12 + 8 + 9}{5} = 10.8V = \frac{(10 - 10.8)^2 + (15 - 10.8)^2 + (12 - 10.8)^2 + (8 - 10.8)^2 + (9 - 10.8)^2}{5} = 5.36\sigma = \sqrt{5.36} \approx 2.315That result means the return series has moderate dispersion around the 10.8% mean, with a standard deviation of about 2.315 percentage points.
Population Variance vs. Sample Variance
The formula above is appropriate when the return set being analyzed is treated as the full population for that period. If the returns are only a sample and you want the sample variance, divide by one less than the number of observations instead.
s^2 = \frac{\sum_{i=1}^{N}(R_i - \bar{R})^2}{N - 1}Common Mistakes to Avoid
- Mixing decimal returns and percentage returns in the same calculation.
- Using returns from different time intervals, such as mixing monthly and annual observations.
- Rounding the mean too early, which can slightly distort the final variance.
- Assuming high variance is always bad; it indicates volatility, not necessarily poor performance.
- Confusing variance with standard deviation; standard deviation is usually easier to interpret because it returns to the original units.
Practical Uses
- Evaluating how stable a stock or fund has been over time.
- Comparing multiple investments with similar average returns.
- Estimating risk before position sizing or allocation decisions.
- Supporting broader portfolio metrics such as diversification and volatility analysis.
