Log Versus Simple Returns in Reporting

A simple return is the one an investor actually feels: start with $100, end with $110, that's a 10% return, full stop. A log return is the natural logarithm of the price ratio — for the same move, $\ln(110/100) \approx 0.0953$, or 9.53%. The two are close for small moves and diverge more the bigger the move is, and the reason both exist is that they're useful for different jobs.

Log returns are additive across time: the log return over ten days is just the sum of the ten daily log returns, which makes them convenient for statistical modeling and for combining return series computationally. Simple returns are the ones that actually compound the way an investor's account balance does, and multi-period simple returns must be multiplied, not added, to get a correct cumulative figure.

A common mistake is summing daily log returns over a year and quoting the sum directly as "annual return." It's close for calm markets but can be meaningfully wrong after a volatile year, which is exactly when accurate reporting matters most. The safe rule is to use log returns internally for any calculation, and convert to simple returns only at the final step before a number reaches a client or a performance report.