Log Returns vs. Arithmetic Returns
If you gain +10% in January and lose −10% in February, do the two months add up to 0%? It's confusing with arithmetic returns, but with log returns the math becomes much cleaner.
Two kinds of returns
The 'simple return (arithmetic return)' we usually use is (ending price − starting price) ÷ starting price. If ₩100 becomes ₩110, that's +10%.
Log return is a bit different. It's ln(ending price ÷ starting price), that is, the value with the natural logarithm applied. For ₩100 → ₩110, it's ln(110/100) ≈ 0.0953, about 9.53%. The values are similar but subtly different.
The key to log returns: they can just be added
The biggest advantage of log returns is that they 'can be added across time (additivity).' When stringing together multiple periods, simply summing each period's log return gives the log return for the entire period.
Simple returns, by contrast, can't be added—they must be multiplied. For +10% in January and −10% in February, (1+0.10)×(1−0.10) − 1 = −1%, so the two-month total is not 0% but −1%. With log returns, ln(1.1) + ln(0.9) ≈ 0.0953 − 0.1054 = −0.0101, which likewise comes out to about −1% simply by adding.
Arithmetic returns are convenient 'when combining multiple assets at a single point in time,' and log returns are convenient 'when linking one asset across multiple periods.' They serve different purposes.
Why practice and research use log returns
Log returns are interpreted as 'continuous compounding.' They're good for comparing assets without worrying about the compounding frequency, and statistically they're treated as closer to a normal distribution, so they're often used in risk models and backtests.
That said, the smaller the return, the more nearly identical log and simple returns become. At the +1% level, the difference between the two is negligible below the decimal point. The gap widens with large moves like +30% or −40%.
Points to watch
Log returns are convenient, but they don't intuitively show 'the money I actually made.' Expressing a −50% loss as a log gives ln(0.5) ≈ −0.693, so it looks like −69.3%, but what actually vanished from your wallet is half.
So when finally stating 'how much did I make,' it's safer to convert back to simple (cumulative) returns for reporting. You can remember it as: use logs in the calculation process, and simple returns when reporting results.
Preguntas frecuentes
Q. Is a log return always smaller than a simple return?
In the positive (+) return range, the log return is a bit smaller than the simple return, and in the negative (−) range, its absolute value is larger. For example, +10% is about +9.53% as a log, and −10% is about −10.54% as a log. Because of this asymmetry, log returns effectively weight the burden of losses slightly more heavily.
Q. So which one should I use?
When viewing long-term performance as 'how much did it become cumulatively,' the simple (total) return is intuitive. When analyzing by stringing together monthly or yearly returns, or calculating volatility, log returns are convenient. If your purpose is an individual investor checking performance, simple returns and CAGR are enough.
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📋 Los resultados se basan en datos históricos; las rentabilidades pasadas no garantizan rentabilidades futuras.
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