How to annualize returns and volatility

Annualized volatility uses √T scaling and annualized mean return uses T scaling under IID or Brownian increment assumptions.

Annualization converts a return or volatility measured over one horizon into a comparable value for another horizon, usually one year. The familiar square root of time rule says that volatility over T periods equals one period volatility multiplied by the square root of T. A related linear rule says that expected return over T periods equals one period expected return multiplied by T. These rules are exact only under specific conditions.

The mechanism comes from summing period by period returns under independence. If simple excess returns r1, r2, ..., rT are independent and identically distributed with mean μ and variance σ2, then the T period cumulative arithmetic return has expected value Tμ and variance Tσ2. Taking the square root of the variance gives a standard deviation of σ√T. This is the source of square root of time volatility scaling. In continuous time, the same scaling appears in Brownian motion, where increments over disjoint intervals are independent and variance grows proportionally with elapsed time. The Sonar fundamentals material presents annualized return and annualized volatility using these standard scaling conventions, with volatility scaling by the square root of the number of periods and return scaling by the number of periods when period returns are treated as IID increments.

Drafted with AI assistance from cited sources. Reviewed and approved by Sonar Sciences Quant & Research Team.