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Regular returns (log-differenced prices) have statistical distributions that are bell-shaped and unimodal (one mode/peak) despite being non-normal and fat-tailed.

Cumulative returns, on the other hand, computed from regular returns as $[\prod (1+r)] -1$, are bi-modal (with two modes/peaks). Is there a reason for cumulative return distributions having this shape?

And in spite of cumulative returns being non-stationary unlike regular returns, are they used in any well-known financial models at all?

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depending on the time period and asset considered, it comes from different market regimes.

e.g. if you consider a long enough period, you will clearly have distinct bull and a bear markets with crisis and changes in investors behaviors, with very different returns, there is an interesting article there.

all in all if you consider just 1 regime, would expect the returns to be monomodal.

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  • $\begingroup$ if this is the reason, then wouldn't more than two market regimes be possible (tri-, quad- modal cumulative return distributions)? if I currently only see two modes, would elongating the historical dataset another two decades surely cause more modes to appear? $\endgroup$ – develarist Jul 13 at 2:45
  • $\begingroup$ yes the longer the time period the more modes, in the article linked, the time series spans over a hundred years $\endgroup$ – John Jul 13 at 3:11
  • $\begingroup$ in spite of cumulative returns being non-stationary unlike regular returns, are they used in any well-known financial models at all? If cumulative returns appear in market regime models, is that all they're good for? $\endgroup$ – develarist Jul 13 at 3:12

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