A stock's daily returns have a standard deviation of 1.4%. Annualise it on 252 trading days to the nearest tenth of a percent, and state the approximation you used for sqrt(252).

A stock's daily returns have a standard deviation of 1.4%. Annualise it on 252 trading days to the nearest tenth of a percent, and state the approximation you used for sqrt(252).

Approach: Scale by the square root of the number of periods, and get sqrt(252) from the nearest perfect square with a first order correction.

22.2%. Volatility scales with the square root of time, so the annual sigma is the daily sigma times sqrt(252). Get sqrt(252) from 16^2 = 256: the shortfall of 4 divided by 2 * 16 = 32 is 0.125, so sqrt(252) is 15.875 against the true 15.8745. Then 1.4% * 15.875 = 22.225%. That rounds to 22.2%. The desk shortcut is to multiply the daily move by 16 and trim 0.8%, since 16 overstates sqrt(252) by 0.79%. Running it the other way is the more common trading question, because a 20% annual volatility over 252 trading days is a 1.25% daily standard deviation.

Follow-up: The same stock trades 6 hours a day. What is the standard deviation of a one hour return, and what assumption does that answer rely on?

Key concepts: square root of time, annualisation, trading days.