Since the discussion above is not coming to an end, I am going to show a summary of my assumptions. If you don not agree, let me know where exactly you do not agree.
1) The return
i) the simple annual return of a 1-year period is:
$$
r_{1989}=value_{1989}/value_{1988}-1
$$
for
$$
value_{1988}=100, value_{1989}=110, value_{1990}=100
$$
we get
$$
r_{1989}=10\%, r_{1990}=-9,1\%
$$
ii) the simple annual return of a 2-year period is
$$
r^2_{1990}=value_{1990}/value_{1988}-1=0\%
$$
iii) the mean of the sample of returns is
$$
r_{mean}=(r_{1989}+r_{1990})/2 = 0.45\%
$$
iv) to calculate the end value we use the mean and take it to the power of the number of periods (2 in this case)
$$
endvalue = (r_{mean}+1)^2*startvalue=100.09
$$
But this is not our endvalue. We could easily employ this method to daily returns, calculate the mean of daily returns and annualize them. We could take the daily returns the number if periods required to to get the end value of that 2-year period. However to get a real annualized value we simply take the mean of daily returns and take it to the power of one year (annualizing). However, this procedure will still overestimate the true end value. As I showed above, it will overestimate the endvalue by a lot.
v) the log return is calculated for a 1-year-period as:
$$
logr_{1989}ln(value_{1989}/value_{1988})
$$
for a two year period
$$
logr_{1990}=ln(value_{1990}/value_{1988}) = ln(value_{1989}/value_{1988})+ln(value_{1990}/value_{1989})
$$
and the mean
$$
logr_{mean} = logr_{1990}/2
$$
in this example you get
$$
logr_{mean}=(0.95-0.95)=0\%
$$
which is the true return of that period.
To get from an annual average to the end value you multply by the number of periods (in this case 2)
$$
endvalue=(1+0\%*2)*startvalue=100
$$
Now we see the log method leads to the correct endvalue. However for a very specific one-year period, the simple return is the most correct return to get. For a period that is greater than 1, to get an annual return, you cannot annualize the simple daily returns, since the degree of overestimation is way too worse. If you can proof me wrong, I am glad to know. However the annualized mean of the daily logreturns underestimate the true return by a bit with greater sampling size
vi) To get an average return annual return from a two-period case you do this:
$$
r_{geometric\ mean}=(value_{1990}/value_{1988})^{(1/2)}-1=(100/100)^{1/2}-1=0\%
$$
2) the standard deviation (of one year)
The standard deviation is the average deviation around the mean. Mean is the keyword: Around which mean? Typically around the arithmetic, not the geometric mean. But you can show me a formula of a geometric standard deviation if you have one.
Now, I have shown that the mean of the simple return leads to very incorrect annual return, when taking the daily return and annualizing it. Again, if you use a very specific one year period, the simple return is the most correct choice. If you have more than one period and still want the average yearly standard deviation your compared mean return will be the geometric mean from vi) or the logreturn from v). Since the standard deviation is the average deviation from the arithmetic mean (of a sample (=here: of daily returns)), and since the simple daily return is wrong, I see only one return that is left: that is the log return. What sence would is make to say in the same sentence: "the geometric mean is 10% and the standard deviation of the daily log returns is 9% (while the standard deviation is the average deviation around its mean (the mean of log returns and not of the geometric mean))" It gets even worse when we report measures like the sharpe ratio or the VaR that a calculated using different approaches. Now, you can for sure use the standard deviation of the simple daily returns but then you should also report the annualized mean of the daily returns. And if you do not believe me, please try it out yourself: Using daily simple returns and trying to get to the endvalue will dramaticall overstimate the true return of an entire period.
3) reporting
This is what my initial post intented to ask: Might it be not the return itself that matters so much but rather on what purpose we are using it for? For statistcal purposes like regressions, t-statistics, annualization, key metrics like the sharpe ratio, the log return seems to me favourable. For Sales, I definitively would use the simple return.
final remarks
As a private investor that tries to optimize his performance, I am trying to apply carefull methods that rather underestimate return and rather overestimate risk. Because, if something goes wrong, I will still have a cusion. And if nothing goes wrong I can happily get more return than I expected. I have proven that the simple return is not only wrong, but even more wrong than the logreturn, especially if you consider risk of false decisions caused by too high expectations or wrong risk-return portfolios. If you do not think that the simple return is more wrong than the logreturn, I am fully open to hear other opinions but I'd like to have some proof of that.