# Is there any economic meaning to log returns?

I recently had a discussion with a friend regarding log returns and any economic meaning that could have as a measure in the sense of "assessing the benefit received for compensating work done in a previous time period."

He rather suggested that this was a mathematical construct beneficial in constructing stories such as historical var for the assessment of risk (owing to their additive nature.. $\log(a/b)+\log(b/a)$ causing no information loss).

Is there any economic meaning that could be ascribed to log or arithmetic returns?

• I think your friend was right, log return it is just a useful measure with no 'direct' economic interpretation. Arithmetic calculation of return is a simple measure which usually gives you a good estimate of the geometrically calculated return. – Giskard Dec 17 '15 at 9:34
• "assessing the benefit received for compensating work done in a previous time period" would refer to lagged returns, not log returns. Is there some typo or mis-hearing going on hear that's confused the issue? – EnergyNumbers Dec 17 '15 at 10:47

Take the 1st order Taylor series the function $log(x)$. around one: $$log(x) \approx log(1) + \frac{x-1}{1} = x-1$$
Therefore, for a random variable r that is close to zero: $$log(1+r) \approx r$$
Log returns are typically something like $log(P_{new}/P_{old})$: $$\Delta log(P_t) = log(P_{new}/P_{old}) = log((P_{old} + \Delta P)/P_{old}) = log(1 + \Delta P/P_{old}) \approx \Delta P/P_{old}$$ But $\Delta P/P_{old}$ is just the percent change in $P_{old}$. We've therefore shown that we can approximate the percent change in a variable $X$ with $\Delta log(X)$.
Using logs, or summarizing changes in terms of continuous compounding, has a number of advantages over looking at simple percent changes. For example, if your portfolio goes up by 50% (say from \$100 to \$150) and then declines by 50% (say from \$150 to \$75), you’re not back where you started. If you calculate your average percentage return (in this case, 0%), that’s not a particularly useful summary of the fact that you actually ended up 25% below where you started. By contrast, if your portfolio goes up in logarithmic terms by 0.5, and then falls in logarithmic terms by 0.5, you are exactly back where you started. The average log return on your portfolio is exactly the same number as the change in log price between the time you bought it and the time you sold it, divided by the number of years that you held it.
• Derivatives could also aid intuition even though economic data is discrete for obvious reason: $\frac{d\log(p)}{dt} = \frac{d\log(p)}{dp} \frac{dp}{dt} = \frac{1}{p} \frac{dp}{dt} = \frac{dp/dt}{p}$. Where $t$ can be thought of as time and $p$ as a price. – snoram Oct 11 '16 at 0:50