Geometric return ≈ arithmetic return minus sigma^2 / 2.
The intuitive demonstration. Up 50% then down 50% leaves you at 75, not 100. Losses hurt compounding more than equivalent gains help it.
Why it matters practically. It is the reason leveraged ETFs decay relative to the multiple of their index over long horizons, and the reason volatility reduction can raise compound returns even without raising average returns.
It is also the mathematical basis for Kelly: maximising growth means maximising expected log wealth, which penalises variance.