Log Returns as Continuously Compounded Growth
Summary
The document explains how log returns differ from simple returns using an asset that doubles in price. A simple return measures the gain against the starting price; the log return describes the constant rate that, applied across arbitrarily small intervals, compounds to the same ending price. Dividing the log return across progressively finer intervals illustrates why it approaches the observed total price change.
It also lists practical properties: log returns add across time, scale linearly with interval length, convert to and from simple returns, and treat upside and downside moves symmetrically in log space. The explanation is conceptual rather than a treatment of statistical assumptions or use in a particular trading strategy. Its symmetry statement refers to equal positive and negative percentage log moves, which do not leave an investor at the same price when interpreted as ordinary simple returns.
Key ideas
- A simple return measures the price change relative to the initial price.
- A log return is the continuously compounded rate corresponding to the observed price ratio.
- Log returns can be summed across consecutive periods.
- Log returns scale linearly with time and can be converted to simple returns.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.