Log Returns vs Simple Returns: What Each One Measures

Open almost any quantitative finance paper and returns are written as a logarithm. Open a broker statement or a TradingView percent-change readout and they are written as a percentage. Both are called "returns," both are computed from the same two prices, and on an ordinary trading day they agree to the second decimal place. The question of which one to use only becomes interesting when moves are large — and understanding why tells you something useful about how any volatility measure is built.

This article defines both, shows exactly where they part ways, and explains which convention sits underneath a rolling-volatility band.

Two formulas for the same price change

The simple return is the one everybody already knows: the change in price divided by the starting price. A move from 100 to 101 is a simple return of 1%. It answers the question a person with money in the market actually asks — by what fraction did my position change?

The log return takes the natural logarithm of the price ratio instead: ln(101 / 100), which is about 0.995%. It answers a slightly different question — at what continuously compounded rate did the price grow over the period?

The two are related exactly, not approximately: the log return equals ln(1 + simple return), and the simple return equals e raised to the log return, minus one. They are the same information in two units, the way a temperature can be written in Celsius or Fahrenheit. The useful question is not which one is "correct" but which one makes a particular calculation easier or more honest.

Where they agree, and where they don't

For small moves the two conventions are nearly indistinguishable. A 1% simple return is a 0.995% log return. A 2% simple return is a 1.98% log return. At the scale of a typical daily close on a liquid stock or index, the difference is a few thousandths of a percentage point — invisible on a chart and far below the noise in any volatility estimate.

The gap grows with the size of the move, and it grows asymmetrically. A 10% simple return is a 9.53% log return. A 50% simple return is a 40.5% log return. A −50% simple return is a −69.3% log return. (These are constructed arithmetic examples, not market data.) Log returns compress large gains and stretch large losses, which is exactly the behavior that makes them attractive for some purposes and awkward for others.

The practical rule that falls out of this: on daily or longer bars in liquid markets, the choice between conventions rarely changes a conclusion. On very long horizons, in thin or crisis-hit markets, or anywhere single-period moves run into double digits, it can.

The property quants care about: additivity over time

The reason log returns dominate time-series work is a single algebraic convenience. Multi-period log returns add. If a price goes from 100 to 110 and then to 121, the two log returns are each ln(1.1), and their sum is ln(1.21) — the log return over the whole stretch. Simple returns do not behave this way: 10% followed by 10% is 21%, not 20%, because the second gain compounds on the first.

Additivity makes a great deal of statistical machinery cleaner. Averaging over time, scaling a volatility estimate between bar sizes, and reasoning about the sum of many small moves all work more neatly when the quantity being summed actually sums — which is why the log convention is the default in most academic return series and statistics libraries.

Simple returns have the mirror-image advantage: they add across assets at a single point in time. The return on a portfolio is the weighted average of the simple returns of its holdings. The same statement is false for log returns, except as an approximation that worsens as moves get larger. So the two conventions divide the territory cleanly — log returns for one asset across time, simple returns for many assets at one time — and most of the confusion online comes from using one in the other's domain.

Symmetry, and why it matters for a volatility measure

A price that goes from 100 to 110 and back to 100 has gone nowhere. In simple returns the round trip is +10% followed by −9.09%: two unequal numbers whose average is positive even though the position is flat. In log returns it is +9.53% followed by −9.53%: two equal and opposite numbers that sum to zero. Log returns are symmetric around zero in a way that simple returns are not, and the asymmetry grows with the move — a 50% loss needs a 100% gain to recover.

A standard deviation is a measure of how far observations scatter around their mean. Feed it a series where up-moves and down-moves of the same real size are recorded with different magnitudes, and the scatter is very slightly misstated, with the mean pulled a little positive. On daily data the effect is tiny; on a series with frequent large moves it is not, and this is the second reason log returns are the conventional input when volatility is the object of study.

Which one a band on a chart is built from

Why a volatility band should be computed on returns at all, rather than on raw price levels, is a prior question with its own answer: price levels trend and drift, so their dispersion mostly records how far price has travelled rather than how much it moves. That argument is set out in what is rolling volatility and, in more depth, in return space vs price space. Once returns are the input, the remaining choice is which return.

Oisigma's own construction, as described on the How It Works page, samples a rolling window of recent percentage returns — the simple convention — takes their mean and standard deviation, and projects an expected range from the prior close. Simple returns are the natural unit for a tool whose output is drawn back onto a price chart: a range expressed as "within about 1% of the prior close" is read directly off the axis, and the projection back to price is a multiplication rather than an exponential. At daily and intraday bar sizes, where the conventions differ by thousandths of a point, the choice has essentially no effect on where the lines land.

What the convention does not change is the claim the band makes. It describes how wide recent moves have been and draws a range from that; it does not forecast direction, and a close outside it records that the move was unusual, not that anything is about to happen. The calibration record behind that range — how often the next close historically landed inside it — is on the Proof page and in the working paper, and those figures are historical: past behavior is not a guarantee of future results. The paper validates the range's calibration, not the profitability of any use of it; whether any such use delivers value after costs is an open question.

The takeaway

Log and simple returns are two notations for one fact. The simple return is the percentage change a position actually experienced; the log return is the continuously compounded rate that produced it. On typical daily moves they are the same number to three decimal places, and the choice is a matter of convenience: log returns add over time and treat gains and losses symmetrically, which suits time-series statistics; simple returns add across assets, which suits portfolios and anything read directly off a price axis. The mistake is not picking one — it is mixing them without noticing.

Frequently asked questions

Can a log return be less than −100%? Yes, and that is one of the places the two conventions behave very differently. A simple return is bounded below at −100%, because a price cannot fall further than to zero. A log return has no lower bound: a fall from 100 to 1 is a simple return of −99% but a log return of about −460%, and a fall all the way to zero is a log return of negative infinity. This is why log returns are awkward for instruments that can expire worthless or trade near zero, and why the convention needs special handling in those cases.

Do log returns make market returns normally distributed? No. Taking logarithms pulls in the right tail and stretches the left, which makes the distribution of daily returns a little more symmetric than the simple-return version, and that is often described loosely as "closer to normal." It does not remove the fat tails that real markets show in either convention — large moves remain far more frequent than a bell curve predicts — and the consequences of that for any band are discussed in is this price move normal.

Does the choice matter more on intraday charts or on daily charts? Less on intraday charts. Single-bar moves on a 15-minute or hourly chart are smaller than daily moves, and the gap between the two conventions shrinks with the square of the move, so on intraday bars the conventions are indistinguishable for any practical purpose. The difference becomes noticeable only as bar size and per-bar move size grow, which is the opposite of where most chart-based volatility tools are used. How a return-space range behaves at different bar resolutions is covered in does an expected range work on intraday charts.

Would building the band on log returns instead change its containment rate? Nothing on that has been published, so no figure can be quoted. What can be said from the arithmetic is that at the daily scale the paper measured, the two return series differ by thousandths of a percentage point on almost every bar, so the standard deviation computed from either would be very nearly the same and the band drawn from it would move by a hair. The published containment figures on the Proof page were measured on the percentage-return construction described on How It Works, and are historical measurements only.

If the distinction has mostly been an abstraction until now, the quickest way to make it concrete is to watch a return-based range drawn live on a chart you already follow. BTM draws a calibrated expected range on any TradingView chart, and you can start a free 30-day trial to see how the width tracks the size of recent moves.