Estimating Extreme Return Risk with Heavy-Tailed Distributions

Summary

The document considers how to estimate a high percentile of absolute daily returns when the available sample is short and the desired tail quantile lies beyond what the observations can reliably show. Because the estimate requires extrapolation, it recommends making an explicit distributional assumption and names the truncated Lévy flight distribution as one possible heavy-tailed model for capturing higher kurtosis. It points readers toward introductory and research material, but supplies no fitting procedure, parameter estimation guidance, or empirical comparison with alternative distributions.

It also recommends examining Expected Shortfall, also called Conditional Value at Risk, alongside Value at Risk. VaR identifies a loss threshold at a chosen percentile, while Expected Shortfall describes the average loss conditional on exceeding that threshold. This gives a broader view of tail severity, though the document does not discuss how either measure behaves with a limited sample or how to validate the chosen model. The proposed distribution is a recommendation, not a universal standard.

Key ideas

• Estimating an extreme return percentile from a short sample requires extrapolating beyond observed data.
• A distributional assumption is necessary, and the answer suggests a truncated Lévy flight model for heavy tails.
• Expected Shortfall complements Value at Risk by measuring average losses beyond the selected threshold.
• The document does not provide a model fitting method or compare the recommendation against alternatives.

Tags

# What distribution should I apply to estimate the likelihood of extreme returns?


# What distribution should I apply to estimate the likelihood of extreme returns?












Say I have a limited sample, a month of daily returns, and I want to estimate the 99.5th percentile of the distribution of absolute daily returns.

Because the estimate will require extrapolation, I will need to make a distributional assumption. Is there a standard approach to take here if I want to include the higher kurtosis in my estimate?

## Answer by Tal Fishman (score 7, accepted)

https://quant.stackexchange.com/a/3840

What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at Morningstar has written a few papers on this topic, and you can find more by googling him and this topic. Here is a less technical, more introductory piece on the topic.

Also, you should consider other risk measures, most notably Expected Shortfall (ES), also known as Conditional-Value-at-Risk (CVaR), which examines not just what the 99.5% percentile is, but what the conditional expected return is beyond that point. I also recommend you read this MSCI piece on the topic.

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.