<p>Take the 1st order Taylor series the function $log(x)$. around one:
$$ log(x) \approx log(1) + \frac{x-1}{1} = x-1$$</p>

<p>Therefore, for a random variable r that is close to zero:
$$ log(1+r) \approx r$$</p>

<p>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 $\Delta log(X)$.</p>

<p>Why bother? Mostly because log differences can be summed but percent changes cannot be:</p>

<blockquote>
  <p>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.</p>
</blockquote>

<p><a href="http://econbrowser.com/archives/2014/02/use-of-logarithms-in-economics" rel="nofollow">Jame Hamilton: Use of logarithms in economics</a> </p>

CC BY-SA 3.0
2016-10-11T00:50:05.810Z
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
CC BY-SA 3.0
2015-12-17T16:54:15.940Z
2016-10-10T01:23:53.837Z
2016-10-10T01:23:53.837Z
BKay
<p>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."</p>

<p>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).</p>

<p>Is there any economic meaning that could be ascribed to log or arithmetic returns?</p>

CC BY-SA 3.0
2015-12-17T09:34:46.307Z
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
CC BY-SA 3.0
2015-12-17T10:47:29.010Z
"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?
410 gone
<mathematical-economics>
Is there any economic meaning to log returns?
user6731
# Is there any economic meaning to log returns?

(Asked by: username_0 on 2015-12-17)

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?

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: 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.  
> **username_2**: "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?  

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## 1 answers

### Answer by username_3 on 2015-12-17. Score: 13

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 $\Delta log(X)$.

Why bother? Mostly because log differences can be summed but percent changes cannot be:

> 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.

[Jame Hamilton: Use of logarithms in economics](http://econbrowser.com/archives/2014/02/use-of-logarithms-in-economics)

> **Comments:**
> **username_4**: 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.  

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