Reading Absolute Return Distributions on Log-Log Charts
Summary
The document interprets a chart of S&P absolute returns sampled across time scales, described as using logarithmic axes and a power-law fit. It suggests the horizontal axis represents standardized absolute returns, while the vertical axis shows the fraction of observations exceeding each return threshold. In this interpretation, the plotted tail frequencies help reveal how often unusually large moves occur relative to smaller ones.
The post also raises uncertainty about how the original chart defines normalization and whether marker shapes distinguish positive from negative returns. It mentions comparing the empirical distribution with normal and Student-t distributions, but supplies no chart values or detailed derivation. The explanation is explicitly tentative: axis labels are missing, and the standardization method may depend on a time-varying volatility estimate. Readers should therefore treat the axes and marker interpretation as a proposed reading rather than a confirmed description of the source figure.
Key ideas
- A log-log plot can show the tail frequency of absolute standardized returns across return thresholds.
- The vertical measure is described as a cumulative exceedance fraction rather than a raw count.
- Standardization may use a volatility estimate that changes over time.
- The suggested reading of marker shapes and normalization remains uncertain because the chart lacks labels.
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Full text
# What does this absolute return distribution chart show?
# What does this absolute return distribution chart show?
I was reading some pages in Professional Automated Trading by Eugene Durenard when I came across this chart:
The caption says: "S&P Absolute Return Distribution: Log-Log Scale".
The brief description in the text says:
> [The] log-log scale shows the fat tails and a power-law fit of the absolute returns of the S&P index sampled at time scales from ticks to years.
The lack of a label on the y-axis, and the lack of units of both axes makes it difficult for me to understand the plot. I am also unsure what "absolute normalized move" means.
Could you explain what the chart is saying? What am I looking at?
## Answer by Kermittfrog (score 3)
https://quant.stackexchange.com/a/59808
at closer inspection of the axes I think that this is a plot of tail frequencies in basis 10. I think they
- sort the absolute normalized returns from low (0.0) to high ($31\approx 10^{1. 5...}$) and present the numbers in base 10.
- For each item under 1, present the empirical excess frequency, i.e. the frequency of observed absolute returns above the selected absolute return. They present this number in base 10.
- They then (seem to) select the symbol (triangle or circle) depending on the sign of the return. (to be verified,though.)
I have added a chart for comparison with a standard normal and a standardized student-$t$ distribution with 5 degrees of freedom below.
Nota bene: I forgot to add axis labels as well. The vertical axis is (de-)cumulative frequency ($k/N$ , not $k$). The horizontal axis is standardised return, e.g. standardised by (potentially time varying) standard deviation or such. I will add this to my answer as well.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.