# 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?

at closer inspection of the axes I think that this is a plot of tail frequencies in basis 10. I think they

1. sort the absolute normalized returns from low (0.0) to high ($$31\approx 10^{1. 5...}$$) and present the numbers in base 10.
2. 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.
3. 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.

• If I have understood your answer correctly: the horizontal axis represents returns and these returns have been normalized in some way to make them comparable (to make 1 minute returns comparable with 1 year returns). The vertical axis represents the number of times that the return occurs.
– Flux
Commented Dec 8, 2020 at 7:56
• 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. Commented Dec 8, 2020 at 8:31