How Aggregating Short-Horizon Returns Affects Tail Risk

Summary

The document asks whether returns that are highly skewed or fat-tailed over short intervals become more normal when compounded over a longer period. It describes an example of accumulating weekly yields across a year and observes that, when yields are independent and identically distributed, the sum of those yields may be approximated by a normal distribution under the central limit theorem. The compounded return is presented as being dominated by this sum.

Selling out-of-the-money covered calls with strikes adjusted to keep the theoretical call probability roughly constant is offered as an example of a strategy with very fat-tailed short-horizon returns. The author suggests its annual yield may look nearly normal and well behaved. The document provides no data, derivation, or empirical test for that claim. Its conclusion depends on the stated independence and identical-distribution assumption and should not be taken to mean that compounding removes tail risk in general; the behavior of the strategy's compounded returns is left unexamined.

Key ideas

• The document approximates compounded returns using the sum of periodic yields.
• Under an independent and identically distributed yield assumption, the central limit theorem motivates a normal approximation for longer-horizon sums.
• Selling out-of-the-money covered calls is given as an example of a strategy with fat-tailed short-term returns.
• The claim that annual returns become nearly normal is an intuition in the document, not a result supported by data or analysis.

Tags

# Does tail risk disappear in the long horizon in any rolling over strategy with shorter frequency?


# Does tail risk disappear in the long horizon in any rolling over strategy with shorter frequency?












Say I am investing to gain weekly yields ${y_{i}}$ for over a year, gaining the overall yield:

$\prod_{n=1}^{52} (1+y_i) -1$

The most dominant term in the above product is the sum of all yields, which by CLT should now converge to a normal distribution even if short horizon returns are themselves 'fat tailed'. (Assuming yields are iid).

These yields could correspond to successively selling an OTM covered calls (with strikes adjusted so that the theoretical probability of call is approximately constant), which is extremely fat tailed in the short term, but if I look at the yearly yield, it is almost normal and very well behaved.

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.