Log Returns vs Simple Returns: What Traders Must Know

A stock closes up 10 percent on Monday and down 10 percent on Tuesday. Add the two moves and you get zero, so the week looks like a wash. The account says otherwise: a 100 position is now worth 99, a one percent loss. That one-point gap is the whole reason a trader needs to be clear about log returns vs simple returns before comparing a single figure across tools.

Most charting packages, volatility estimators, regression scripts, and backtest engines report returns one way or the other, and they don’t announce which. Get the convention wrong and your volatility reading, your compounded track record, and your risk numbers drift away from what actually happened in the account. The math takes one line. The fallout can run through a whole backtest.

Log returns vs simple returns: the two formulas

A simple return is the percentage change from one price to the next. For a move from P0 to P1, it’s P1 divided by P0, minus 1. A move from 100 to 110 is 110/100 minus 1, which is 0.10, or 10 percent. That’s the plain percentage gain or loss, the number that lines up with your account statement.

A log return is the natural logarithm of the price ratio: the natural log of P1 divided by P0. The same 100-to-110 move is ln(1.10), which is 0.0953, about 9.53 percent. It describes the identical move. It’s just measuring it on a different scale.

For small moves the two nearly coincide. I run every new return series through the same quick check: a 0.5 percent day should read close to 0.4988 as a log return, near enough to ignore. Because ln(1 + r) is approximately r for small r, a quiet market makes the choice look academic. The gap only opens once moves get large.

A two-period example that shows the difference

Take a price that runs from 100 to 110, then falls from 110 to 99. Work the simple returns first. Day one is plus 10 percent. Day two is (99 minus 110) divided by 110, which is minus 0.10, or minus 10 percent. Adding those gives zero, so you’d call it a wash, yet the price ended at 99, a one percent loss over the two days. Simple returns don’t add across time. They compound: 1.10 times 0.90 is 0.99, and 0.99 minus 1 is that minus 1 percent.

Now the log returns. Day one is ln(1.10), plus 9.53 percent. Day two is ln(0.90), minus 10.54 percent. Add them and you get minus 1.005 percent. That sum equals ln(99/100) exactly, the log return of the whole two-day move. Log returns add across consecutive periods because a logarithm turns multiplication into addition. The compounding you had to do by hand for the simple returns is baked into the arithmetic. That single property, addition standing in for compounding, is what the rest of this comes back to. Convert that log sum back and you land exactly where you started: e to the minus 0.01005, minus 1, is minus 1 percent, the same loss the simple returns showed. Both conventions agree on the outcome; they’re just carrying it differently through time.

Why log returns add and simple returns multiply

That additivity is why quantitative work leans on log returns. A multi-period log return is the plain sum of the single-period log returns inside it, so rolling daily data up to weekly, monthly, or annual figures becomes ordinary addition. The property is what makes clean time aggregation possible without re-compounding at every step.

Additivity also drives the square-root-of-time rule. If daily log returns are roughly independent with a stable variance, variance adds across days, so the standard deviation grows with the square root of the number of days. That’s why a daily volatility gets annualized by multiplying by the square root of 252. Applied to simple returns the same scaling is only an approximation, which starts to matter once you’re estimating historical volatility over longer windows. Log returns also sit more symmetrically and closer to a normal shape, which is why regressions and price-path simulations are usually written in log space.

Where simple returns stay the right tool

Log returns win on the time axis and lose on the cross-section. A portfolio’s return over a single period is the weighted average of the simple returns of its holdings, with the weights summing to one. That clean weighting doesn’t hold for log returns, so building a one-period portfolio return by averaging the logs of each position is a quiet error that grows as you scale the book. If you want the percentage your holdings made today, the simple return is the honest measure.

Simple returns are also the number tied to cash. Put 10,000 into an unlevered position that gains 3 percent and you hold 10,300, exactly. The simple return maps one-to-one onto the money in the account, which is why the performance you’d show a reader or a client is almost always stated that way. Whichever convention you use, the figures are only as trustworthy as the underlying adjusted price series. A missed split or a dropped dividend corrupts both measures equally.

The downside isn’t symmetric

Here’s a difference that surprises people. A simple return has a floor. A price can fall to zero but no further, so the worst possible simple return is minus 100 percent. A log return doesn’t have a floor. As price approaches zero the log return runs toward negative infinity, unbounded on the downside. On the upside a doubling, a plus 100 percent simple return, is only ln(2), about plus 69.3 percent in log terms. Log space compresses the extremes.

The two scales also disagree about round trips. A rise from 100 to 110 is plus 10 percent simple, but the fall from 110 back to 100 is only minus 9.09 percent simple, an asymmetry that trips up anyone eyeballing percentage moves on a chart. In log terms both legs are 9.53 percent, equal and opposite, one more reason modelling work prefers them. Reading a chart, remember that the up move and the matching down move aren’t the same percentage.

Volatility drag and the equity curve

Mixing the conventions does real damage on an equity curve. Picture a path that gains 50 percent, then loses 50 percent: 100 to 150 to 75. The two simple returns average to zero, yet the account finished down 25 percent. The arithmetic mean of simple returns overstates what actually compounds, and the gap widens with volatility. That gap is volatility drag, and it runs to roughly half the variance of the returns. Ralph Vince built much of his money-management work on this geometry of compounding, and his treatment of it rewards a slow read (Ralph Vince).

Log returns catch the loss honestly, because ln(1.5) plus ln(0.5) is minus 0.288, the log return of that 75 ending value. It’s the same effect that pulls the mean path of a Monte Carlo equity curve away from its median, and the reason a geometric growth rate describes a system better than a straight average of its period returns. On my own equity-curve work I read the geometric figure first, because averaging period returns quietly flatters a volatile system, and an expectancy stated in R-multiples means little when the compounding underneath it is negative.

Check the convention before you trust the number

None of this decides whether a strategy’s any good. It decides whether two figures can even be compared. Before you line up two performance numbers, two volatility estimates, or two backtests, confirm they’re both on the same convention. A daily chart of quiet large-cap moves reads almost identically either way. A leveraged name that swings 30 percent in a week, a multi-year compounded record, or a simulation of thousands of paths won’t.

The conversion is exact and worth pinning above the desk:

• Log to simple: take e raised to the log return, then subtract 1.
• Simple to log: take the natural log of one plus the simple return.

Convert first, then combine. Adding a log return to a simple return, or averaging a mix of the two, produces a number that means nothing. Small daily moves forgive the sloppiness. Large moves, long horizons, and compounded results punish it. When two reports of the same system disagree and the logic is identical, the return convention is the first place I look.

Read the label, then the number

Choosing a return convention is bookkeeping, not edge. It won’t rescue a backtest built on survivorship-biased data, nor one leaking future information through look-ahead bias, nor a series mangled by a botched corporate-action adjustment, nor a fantasy fill no live order would ever get. Those errors survive any choice of formula. What the convention does settle is whether your numbers are comparable to each other at all, and that’s a low bar you can’t afford to miss.

So know which number you’re reading. Compound simple returns, add log returns, and never mix the two without converting first, from the opening line of the calculation to the last. Learn the pattern. Ride the trend. Keep the gains.

Educational content only. Not investment advice. Trading involves risk. You are responsible for your decisions.