Annualize refers to converting a short-term number, such as an investment return or interest rate, into an annual rate.

Note:

• annual volatility: computed on several years e.g. 2020, 2021, 2022.

• annualized volatility: computed on one year based on monthly or daily returns.

In practice, we use the annualized volatility because we don’t have the annual returns of a large sample of years.

Let us say we want to compute the annualized standard deviation based on the monthly returns. When returns are continuous:

\[\begin{align*} var(r_{2020}) &= var(r_{jan, 2020}+...+r_{dec, 2020}) \quad \text{Reminder: when returns are continuous: } r_{0,2} = r_{0,1}+r_{1,2}\\ &= var(\sum_{i=1}^{12}r_{i, 2020}) \\ &= \sum_{i=1}^{12}var(r_{i, 2020}) \quad \text{assuming independent returns}\\ &= 12var(r_{monthly}) \\ \end{align*}\]

Thus, we have:

\[\sigma_{annualized} = \sqrt{12}\sigma_{monthly}\]

Similarly, assuming trading happens all day long:

\[\sigma_{annualized} = \sqrt{365 * var_{daily}(x)} = \sqrt{365} \sigma_{daily}\]