logreturn

In finance, log return, or continuously compounded return, measures the percentage change in price using natural logarithms. It is defined for a price series P_t as r_t = ln(P_t / P_{t-1}). It can also be written as r_t = ln P_t − ln P_{t-1}.

If dividends are not paid out of price, total return can be used by adjusting the price series to include dividends or by using a total return index. In that case, the log return reflects the combined effect of price change and distributions.

Properties include time additivity: the sum of log returns over multiple periods equals the log of the cumulative price ratio, r_{t0+1} + ... + r_t = ln(P_t / P_{t0}). This makes log returns convenient for multi-period analyses. For small changes, log returns approximate simple returns, since r_t ≈ R_t = (P_t − P_{t-1}) / P_{t-1}. Under geometric Brownian motion, log prices are normally distributed, so log returns are often modeled as convenient, sometimes assumed iid or normally distributed for analytical tractability.

Uses include asset pricing, risk management, and portfolio optimization. The additive property of log returns simplifies aggregation over time and in likelihood-based estimation. They are standard inputs to statistical models of returns and are used to test hypotheses about mean and volatility.

Limitations include the need for positive prices, sensitivity to delistings or zero-price events, and the fact that log returns can be less intuitive for investors than simple percentage changes. Dividends, splits, and other corporate actions must be accounted for to use true total returns.