The Sqrt(t) Volatility Band Is Wrong in Both Directions; Here's What to Compute Instead

If you have ever drawn a forecast band around a price series, you have probably reached for the same formula everyone reaches for: take a volatility estimate, multiply by the square root of the horizon, and call the result your interval. sigma * sqrt(t) is the closed-form move; it is one line of code, and for fat-tailed financial series it is wrong in a way that is worth understanding precisely, because the error does not even keep the same sign as you change the horizon. This is a short engineering write-up on what breaks, how we measured it, and the handful of things you actually have to com