A log return is the natural logarithm of the ratio between a price at time t and its price one period earlier, written as ln(P_t / P_{t-1}).

A log return is the natural logarithm of the ratio between a price at time t and its price one period earlier, written as ln(P_t / P_{t-1}). This transforms multiplicative price changes into additive return increments.

The key property is additivity across time. If one period return is ln(P_2 / P_1) and the next is ln(P_3 / P_2), then their sum is ln(P_2 / P_1) + ln(P_3 / P_2) = ln((P_2 / P_1) x (P_3 / P_2)) = ln(P_3 / P_1). The intermediate price cancels inside the ratio, so the summed log returns equal the log return over the full span.

A worked example makes this concrete. If price moves from 100 to 110, the first log return is ln(110 / 100) = ln(1.10). If price then moves from 110 to 121, the second log return is ln(121 / 110) = ln(1.10). Adding them gives ln(1.10) + ln(1.10) = ln(1.21), and the full-period log return from 100 to 121 is ln(121 / 100) = ln(1.21). The two one-period log returns therefore add exactly to the total log return.

This is simpler than arithmetic returns when combining periods. In the same example, the arithmetic return from 100 to 110 is (110 - 100) / 100 = 0.10, and from 110 to 121 it is (121 - 110) / 110 = 0.10. Adding them gives 0.20, but the full arithmetic return from 100 to 121 is (121 - 100) / 100 = 0.21. Arithmetic returns do not add across time because price changes compound multiplicatively, while log returns convert that compounding into addition.

That additive structure is why log returns are widely used in statistical work on asset performance. Summing period-by-period log returns produces the horizon return in log terms, which makes aggregation, modeling, and comparison across time more direct. Sonar’s fundamentals research identifies log returns as a core concept for rigorous quantitative analysis because this property aligns the mathematics of compounded prices with tools that operate naturally on sums.

Covered in depth in the Strategy research fundamentals pillar hub.