Why Cumulative Return Distributions Can Have Two Modes

Summary

The document asks why cumulative returns can show two peaks when ordinary, period-by-period returns often have one. Its explanation is that the shape can reflect distinct market regimes: a long sample may combine bull and bear markets, crises, and shifts in investor behavior, each with a different return profile. Pooling observations across these regimes can produce a bimodal distribution even if returns within an individual regime are unimodal.

The answer is a qualitative explanation rather than a statistical derivation or empirical demonstration. It does not specify a test for detecting regimes, define the return sampling procedure, or establish that cumulative returns are generally bimodal. The distribution will depend on the asset and the time window, and cumulative returns have different stationarity properties from regular returns. The note also raises whether cumulative returns appear in established financial models, but does not answer that part of the question.

Key ideas

• Combining observations from distinct market regimes can produce a multimodal return distribution.
• Bull and bear markets may have different return behavior and can contribute separate peaks.
• Returns within a single regime may be unimodal even when the pooled sample is not.
• The explanation depends on the asset and sample period and is not a formal test.

Tags

# Why do cumulative returns have a bimodal distribution?


# Why do cumulative returns have a bimodal distribution?












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?

## Answer by John (score 2, accepted)

https://quant.stackexchange.com/a/55616

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.

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.