Comparing Arithmetic, Log, and Geometric Average Returns
Summary
The document explains why averaging simple daily percentage returns can misrepresent compounded performance. A loss followed by a gain of the same percentage does not restore the starting value, even though the arithmetic average of those two returns is zero. The question arose because one return series had a lower average daily return but a higher cumulative return.
The accepted response recommends log returns, whose period values add to the total log return. Another answer explains that averaging log growth and transforming back gives a geometric average, the constant per-period growth rate consistent with the compounded ending value. It cautions that individual log returns may be less intuitive than simple returns and are not themselves the same as the realized percentage change. The example illustrates the distinction, but the discussion does not provide a broader performance-measurement framework or address fees, cash flows, or statistical comparisons across strategies.
Key ideas
- Simple returns compound multiplicatively, so their arithmetic average need not reflect cumulative performance.
- Log returns add across periods and correspond to the total log change over the full interval.
- Transforming the average log return back gives the geometric average growth rate.
- The geometric rate matches the constant periodic return that produces the same final value.
- Log returns can be less intuitive than simple percentage changes when interpreting individual periods.
Tags
Full text
# How can I measure returns such that the average is useful?
# How can I measure returns such that the average is useful?
If I measure daily returns by simple percent change, a -50% day then a +50% day (or vice versa) results in a true -25% total change, but the average makes it look like you would expect a total 0% change.
Is there a way to measure returns differently that eliminates this effect? For context I have two series of returns and the one with a lower average daily return ends up with a higher total return and I suspect this phenomenon is to blame.
Currently I measure daily return by: (priceDay2-priceDay1)/priceDay1
## Answer by Newquant (score 8, accepted)
https://quant.stackexchange.com/a/71287
Take the log return between days.
## Answer by AKdemy (score 6)
https://quant.stackexchange.com/a/71290
What does not work with the geometric mean?
The geometric mean is computed with the following formula: $${\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}$$
which is equivalent to the arithmetic mean in logscale (see Wikipedia):
$${\displaystyle \exp {\left({{\frac {1}{n}}\sum \limits _{i=1}^{n}\ln x_{i}}\right)}}$$
A quick implementation in Julia looks like this.
```
geom = ((1-0.5)*(1+0.5))^(1/2)-1
100*(1+geom)^2
(1+geom)^2-1
exp(1/2*(log(0.5)+log(1.5)))-1
```
The inaccuracy is due to decimal precision of floating point math, see for example this answer.
While I think taking logs is generally useful, I do not really think using log returns is particularly meaningful (easy to interpret) in this example. While it is true that the sum of log returns in each period corresponds to the log return over the entire period, I am not sure what you can do with this result, unless you transform it back into the geometric mean as shown above.
If you simply use the log returns, you get the following values:
However, as you wrote, the true change is $-25\%$ and not $\approx -28.76\%$. Also, each period change ($-69.31\%$ and $+40.54\%$) is far from the actual change of $\pm 50\%$.
On the other hand, the geometric mean provides you with the constant growth rate (return) that yields the correct final amount.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.