Log Return

Log return is calculated as ln(price_today / price_yesterday). It differs from a simple percentage return, though the two nearly match for small moves and diverge for large ones: a +50% simple gain is about +40.5% as a log return, and a −50% simple loss is about −69.3%.

Risk math uses log returns for two properties. Time-additivity: multi-period log returns simply add up, which makes scaling volatility across days, weeks, or years mathematically clean — the basis for "annualizing." Symmetry: an equal-magnitude log gain and loss cancel exactly, avoiding the upward bias you get from averaging simple percentages (where a −50% move needs a +100% move to recover).

This is why standard deviation, Sharpe, and Sortino are built on daily log returns rather than simple percentage changes — log returns behave better statistically and annualize consistently (daily log-return standard deviation × √365 for 24/7 crypto).

Worked example: a move from $1.00 to $1.50 is a +50% simple return but a log return of ln(1.50) ≈ +40.5%; a move from $1.00 to $0.50 is a −50% simple return but ln(0.50) ≈ −69.3%. The two log returns do not cancel the way the simple percentages appear to — the same asymmetry that makes an 80% drawdown need a 400% gain to recover.