Log Returns
The natural log of the price ratio between two periods. Log returns add up across time, which makes them the default unit for quantitative analysis.
Log Return = ln(Pₜ / Pₜ₋₁) Simple Return = eʳ − 1
Log returns measure the change from one price to the next as ln(Pₜ / Pₜ₋₁) instead of the simple percentage (Pₜ − Pₜ₋₁) / Pₜ₋₁. For small moves the two are almost identical; for large moves they diverge.
The practical advantage is additivity. The log return over a week is the sum of the daily log returns, so multi-period returns, averages and standard deviations can be computed with ordinary sums. Simple returns compound by multiplication instead, and a +10% move followed by a −10% move does not bring you back to the start.
Log returns are also symmetric: a rise from 100 to 110 and a fall from 110 back to 100 have the same magnitude with opposite signs. Convert back to a percentage with eʳ − 1 before reporting results to anyone who thinks in simple returns.
On the desk
A stock moves 100 → 110 → 100. Simple returns: +10.00% then −9.09%, which do not sum to zero. Log returns: ln(1.10) = +0.0953 and ln(100/110) = −0.0953, which sum to exactly 0 — the correct net change.