How Return Aggregation Relates to Heavy-Tail Exponents

Summary

The document asks whether the power-law tail exponent of log returns should remain the same when returns are aggregated from daily to longer horizons. It notes the theoretical argument that a regularly varying tail can preserve its exponent under aggregation, while recognizing that applying this directly to annual returns may seem strong, especially when annual observations are scarce for estimating tail risk.

As a check, the author reports comparing right-tail log-log plots of cumulative returns across horizons, using a large historical sample of stocks and grouping observations by volatility decile and cohort. The plotted slopes appear similar across periods, which is presented as supporting evidence for a common exponent. This is an empirical observation rather than a definitive result: the document focuses on right tails, does not provide numerical estimates or uncertainty measures, and does not establish that the same behavior holds for left tails or all assets and regimes.

Key ideas

• Regularly varying return tails may retain their exponent under temporal aggregation.
• Sparse long-horizon data makes direct estimation of annual tail risk difficult.
• The author reports similar right-tail log-log slopes across return horizons in a historical stock sample.
• The evidence is visual and does not establish equivalence across left tails, assets, or market regimes.

Tags

# Are the tail exponents of daily, monthly, and annual log returns the same?


# Are the tail exponents of daily, monthly, and annual log returns the same?












Mathematically it should be the same, $\nu$ doesn't change if we aggregate x.

$$Pr(X>x)∼Cx^{−ν}$$

Yet, it feels a bit extreme to assume that annual and daily log returns have same $\nu$.

I need to know $\nu$ for annual log returns to calculate tail risk, but don't know how to estimate it - there's too little data. And using same $\nu$ as for daily feels a bit extreme.

Assuming that daily, monthly and annual log returns described by $\text{SkewStudentT}$ (possibly with different tail exponent for left and right tails).

UPDATE:

Indeed it seems it is. Log Log plot of right tails for $log S_T/S_0$ for $T \in [1, ..., 730]$ The slope seems to be the same for different periods.

Data: 250 stocks 1972-2025 years. Color - volatility decile, multiple lines of same color for periods > 30 - different cohort (coherent class, to avoid overlapping bias).

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.