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Add variances before taking a square root
For this learning model, the annual active-return quantity is the sum of twelve monthly active-return quantities. If their variances are equal and their pairwise covariances are zero, the variance of that sum is twelve times the monthly variance. Its standard deviation is therefore the monthly standard deviation multiplied by √12.
Keep the percentage units consistent
Use 0.169% × √12 ≈ 0.5854%. In decimal form, the same calculation is 0.00169 × √12 ≈ 0.005854. Converting that decimal result back to a percentage gives the same answer.
| Calculation | Result |
|---|---|
| 0.169% × √12 | Approximately 0.5854% |
| 0.169% × 12 | 2.028%; linear scaling, not standard-deviation scaling |
| No scaling | 0.169%; remains a monthly statistic |
Identify the missing terms when returns are correlated
For monthly active returns X₁ through X₁₂, the variance of their sum also includes twice the sum of the covariances between different months. Positive covariance can increase the variance relative to the zero-covariance calculation; negative covariance can reduce it. The square-root rule is not an identity that holds regardless of the return process.
Equal variances and zero off-diagonal covariances reduce the right side to 12 times one monthly variance.
Separate scaling from forecasting
The calculation scales a statistic under the supplied assumptions. It is not a compounded-return calculation or a guarantee about next year’s active risk. A twelve-observation sample can also estimate periodic dispersion imprecisely.