Why the Mean of Log Returns Is Zero When Prices End Where They Began

Summary

The document considers a price series that begins and ends at the same level and asks whether its arithmetic mean of log returns must be zero. The answer says yes, extending the claim to ordinary returns as well. The intuition is that the net change across the full interval is zero, though the intermediate path may fluctuate.

The exchange is very short and supplies no derivation or discussion of return conventions, sampling intervals, or how software might produce a different result. In particular, the claim concerns the arithmetic average of period-by-period returns, not the geometric average; for log returns, summing period changes telescopes to the log of the ending-to-starting price ratio. Readers investigating conflicting calculations should check how returns were formed and which observations were included.

Key ideas

• If a price series starts and ends at the same level, the sum of its log returns across the interval is zero.
• The arithmetic mean of those log returns is therefore zero when calculated over the same set of periods.
• Intermediate price fluctuations do not change the telescoping sum of log returns.
• When calculations disagree, check return definitions and the observations included in the sample.

Tags

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


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












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.

## Answer by charlie090 (score 0, accepted)

https://quant.stackexchange.com/a/51767

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.