Standard deviation: Annual to Monthly

Quick question. Came across a VAR question earlier that provided annualized returns and s.d. and asked for the monthly VAR. To go from annual to monthly return, you divided by 12. But to go from annual to monthly s.d., you divided by the square root of 12. Does anyone know why this is the case? I did a quick search and could only find that that’s how you do it, but never an explanation of why. Thanks.

(monthly SD) ^ 2 = monthly variance = annual variance/12

monthly SD = (annual variance/12) ^ (1/2)

=(annual variance ^ (1/2) ) / (12 ^ 0.5)

=annual SD / (12 ^ 0.5)


Assuming independence, variances are additive, so annual variance = 12 × monthly variance.

Then follow breadmaker’s derivation, above.

So when convering Sharpe ratio from monthly to annually be basically increase it by 12^0.5?

A way of gaming the Sharp is to increase the length of measurment periods…

annual to monthly adjustments for var:

e(x) /12

st dev / radic 12

when it comes to sharp ratio, annual to monthly will increase the st dev and will understate the SR. Usually problems are associated with overstating the sharp ratio ie weekly to monthly - > st dv decreases - > sharp ratio increases.