Log Return

Most people measure returns as simple percentage changes. Log returns are an alternative used widely in finance and statistics because they have useful mathematical properties.

For example, if a share rises from 100 to 110, the simple return is 10%, while the log return is about 9.53%. If it then falls back from 110 to 100, the simple return is about −9.09%, while the log return is about −9.53%. The two log returns add up to zero, matching the fact that the share ended where it started.

This additivity is the main advantage: log returns over several periods can simply be added, while simple returns must be compounded by multiplying. Log returns are also symmetric for rises and falls, and they fit naturally with models that assume prices follow a lognormal distribution.

For small changes, log and simple returns are very close. For large changes, they differ noticeably. Simple returns are better for adding up the returns of different assets in a portfolio at one point in time, so the choice depends on the task.