quick question: arithmetic mean of log returns that starts and ends with the same price in a time series

say a stock time series starts at t0 price 100 fluctuates in between the time series and ends at tx 100. arithmetic mean of log returns is 0, correct?

I ask this bc R seems to suggest otherwise or result in conflicting results.

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    $\begingroup$ Does the asset in question pay dividends or not? If not and you start at $P_0$ and end up at $P_n=P_0$ then indeed the sum of log returns will be zero and so will the mean of log returns. If there are dividends then you may have made a positive return. $\endgroup$ – noob2 Mar 21 at 22:10
  • $\begingroup$ Hi: Note that the comment of @noob2 holds for log returns also. In fact, I don't think anything that the OP said hinges on the returns being arithmetic. $\endgroup$ – mark leeds Mar 21 at 23:13
  • $\begingroup$ Thanks guys for the quick mental check $\endgroup$ – charlie090 Mar 22 at 12:39

In a arithmetic mean,

When P0 = Pt = end, then mean0 ~ t of returns = 0. Including log returns. For that matter, any returns.

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