A simple return measures percentage change over an interval, while a log return measures the natural logarithm of the price ratio over that interval. If an asset moves from positive price Pt-1 to Pt, the simple price return is Rt = Pt/Pt-1 − 1, and the log return is rt = ln(Pt/Pt-1) = ln(1 + Rt).
The formulas answer different compounding questions
Simple returns combine multiplicatively. If consecutive simple returns are R1 and R2, the two-period simple return is (1 + R1)(1 + R2) − 1.
Log returns add across consecutive periods: r1,2 = r1 + r2. That additive property is convenient in time-series analysis, but convenience does not make log returns universally preferable.
A numerical example
If price rises from 100 to 105, the simple return is 105/100 − 1 = 5%. The log return is ln(1.05), approximately 4.879%.
If price then falls from 105 back to 100, the second simple return is about −4.762%. Adding the two simple percentages would incorrectly suggest a positive result. The two log returns sum to zero because the ending price equals the starting price.
Simple returns map directly to ordinary wealth changes
If an investment worth 10,000 earns a simple return of 5% over the interval and there are no cash-flow complications, its value becomes 10,500. This direct interpretation makes simple returns natural for portfolio wealth calculations and reporting.
Log returns are convenient for repeated time aggregation
Because log returns add through time, daily log returns can be summed to obtain the log return over a longer interval when the same price series and boundaries are used. This property is useful in statistical modeling and continuous-compounding calculations.
The longer-period simple return can then be recovered as exp(sum of log returns) − 1.
The two measures are close only for small returns
For small percentage changes, ln(1 + R) is numerically close to R. The approximation becomes less accurate as the magnitude of the return increases. Researchers should therefore avoid treating the two as identical merely because they look similar in ordinary daily data.
Log returns require a valid logarithm
The price ratio inside the logarithm must be positive. Standard positive asset-price series normally satisfy that condition, but transformed variables, spreads or synthetic series can be zero or negative. In those cases, a log-return transformation may be undefined or inappropriate.
Cash flows and corporate actions must be handled consistently
A price-only formula ignores dividends and other distributions unless the series is already adjusted or a total-return construction is used. Splits and other corporate actions can also create false jumps when raw prices are used without adjustment.
The return definition should therefore state what price and cash-flow convention it uses.
Do not choose a return type because one produces better model results
The transformation should be selected before interpreting performance and should match the economic question. Switching between simple and log returns after seeing the output can create avoidable researcher discretion.
Financial Time Series provides the wider data context, while Stationarity and Volatility Modelling explain properties commonly studied after returns are constructed.
Simple and log returns are related transformations, not competing claims about the market. State the formula, data convention and purpose, then use the same definition consistently through the analysis.