Log returns vs simple returns: when each convention is appropriate

Two common conventions exist for measuring period returns. Simple returns—the percentage change in price—are the convention most investors encounter in fund factsheets and platform reports. Log returns—the natural logarithm of the price ratio—are the convention most academic researchers and quantitative practitioners use. The two are nearly equal at small returns but diverge at larger ones, and the choice affects how returns aggregate across time and across assets.

What log and simple returns are

The simple return over a period is r_t = (P_t − P_{t-1}) / P_{t-1}—the percentage change in price. It is the figure most commonly used in retail and consumer-facing performance reports, because it answers the intuitive question "by what percent did the price change?".

The log return over the same period is r_t^log = ln(P_t / P_{t-1}) = ln(1 + r_t). It is the natural logarithm of the price ratio, and it has a different mathematical structure than the simple return. For small returns (r < ~5%), log and simple returns are nearly identical; the difference grows with the size of the return.

The relationship between the two is exact: r_t^log = ln(1 + r_t), or equivalently r_t = e^(r_t^log) − 1. They contain the same information; the difference is in how the information is organised.

How they differ in practice

The most consequential difference is how returns aggregate across time. Log returns are additive: the log return over multiple periods is the sum of the single-period log returns. Simple returns are multiplicative: the cumulative simple return is the product of (1 + simple return) terms minus one. The additivity of log returns makes them more convenient for many statistical operations—moments of sums of independent log returns equal the sums of their moments, which makes annualisation and aggregation straightforward.

Cross-sectional aggregation works the opposite way. The simple return on a portfolio is the weighted average of the simple returns on its components: r_portfolio = Σ w_i × r_i. Log returns do not have this property—the log return on a portfolio is not the weighted average of component log returns, except as an approximation that fails as returns get larger.

The implication is that simple returns are the right choice for portfolio-level reporting (where cross-sectional aggregation matters), and log returns are the right choice for time-series statistics (where time aggregation matters). Both conventions appear in practice, and the literature uses one or the other depending on the context.

For small returns, the choice rarely matters. A 1% simple return corresponds to a 0.995% log return—a difference of 0.5 basis points that is invisible in most practical applications. For large returns, the gap is meaningful: a 50% simple return is a 40.5% log return; a −50% simple return is a −69.3% log return. Comparing returns expressed in different conventions without converting can produce material errors.

What the evidence shows

The empirical use of log returns is strongest in academic finance and in quantitative trading contexts. The Black-Scholes option-pricing model assumes log-normally distributed returns, which is most cleanly expressed in log-return space. Time-series momentum strategies (Moskowitz, Ooi & Pedersen, 2012) typically operate on log returns, partly because the additive aggregation makes signal computation cleaner across multiple horizons.

The empirical use of simple returns dominates retail-facing performance reporting. Fund factsheets, advisor reports, and consumer-grade investment platforms typically present cumulative and annualised simple returns, because the multiplicative aggregation is what investors are accustomed to (a 10% return on USD 100,000 is USD 10,000 of additional capital, regardless of how the percentage is mathematically defined).

Statistical properties of the two conventions differ in expected ways. Log returns are typically closer to normally distributed than simple returns, particularly over short horizons—the log transformation pulls in the right tail of positive returns and stretches the left tail of negative ones. For volatility estimation, the standard deviation of log returns is the conventional input, and the resulting volatility estimate is approximately equal to the standard deviation of simple returns when both are small.

Limitations and trade-offs

The two conventions cannot be mixed without conversion. A portfolio of assets reported in log returns cannot be aggregated by simply weighting the log returns; the conversion to simple returns, weighting, and (optionally) re-conversion to log returns is required for accurate calculation. Mixing conventions in a single calculation is one of the more common sources of error in performance analysis.

Log returns also do not handle negative values gracefully. A 100% loss (price falls to zero) corresponds to a log return of negative infinity, which is mathematically meaningful but practically awkward. For instruments that can lose all their value—distressed stocks, single options, leveraged products with margin calls—the log return convention requires special handling around zero and below.

For most retail purposes, the simple-return convention is sufficient and is what platform reports use. For quantitative analysis, time-series modelling, and option pricing, log returns are the appropriate choice. The two should not be confused, and the convention being used should be stated explicitly in any performance comparison or statistical claim.

Log vs simple returns in pfolio

pfolio's analytics use simple returns by default for cumulative return and CAGR calculation. The choice of return convention matters most over short windows and for high-volatility series; for long-horizon, moderate-volatility series the difference is minimal. Underlying price data is exposed for users who want to compute log returns externally.

Related articles

• Cumulative return: how to measure total portfolio growth over time
• Geometric vs arithmetic mean return: when each measure is the right one to use
• CAGR explained: what compound annual growth rate means for your portfolio
• Annualisation in investing: how monthly and daily figures are scaled to annual rates