Fat tails break the ordinary intuitions of risk. Variance estimated from quiet periods understates danger, correlation between assets rises exactly when diversification is needed, and a stop-loss does not cap your loss when the market gaps through it overnight.
They also break statistics. Estimators that depend on the variance converge slowly, sharpe-ratio comparisons become unstable, and a monte-carlo-simulation that draws from a fitted normal will systematically understate drawdown.
The design response is structural rather than statistical: size so that a move several times larger than anything in your sample is survivable, prefer instruments without embedded short-gamma, and never treat a stop as a guaranteed exit price.
Related: kurtosis, tail-risk, normal-distribution, stress-test