Heavy-Tailed Distributions for Simple Asset Return Models
Summary
The document considers how to model returns on external assets in simple agent-based banking models, where the main subject is systemic risk rather than asset pricing. It questions the convenience of independent normally distributed returns, noting that Gaussian assumptions may understate the significance of extreme outcomes. It also raises a modeling concern about distributions with unbounded negative returns and the distortions that can follow from truncating their left tail.
The response lists several alternatives used to represent heavier tails: log-Cauchy, log-Gamma, Lévy, Burr, Weibull, and mixed normal distributions. The exchange does not compare these choices, establish which is most appropriate, or provide evidence that one fits a particular market. It names papers as further reading, but the discussion itself offers no survey or selection procedure. The practical takeaway is to treat return-distribution choice as a modeling assumption with consequences for tail behavior, rather than assuming normality is automatically suitable.
Key ideas
- Independent Gaussian returns are convenient but may be a poor assumption for financial tail risk.
- Truncating a distribution with a heavy negative tail can materially affect the resulting model.
- The response lists log-Cauchy, log-Gamma, Lévy, Burr, Weibull, and mixed normal distributions as alternatives.
- The document does not rank the listed distributions or recommend one for a particular application.
- Distribution choice matters even when asset returns are secondary to a model’s main research question.
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Full text
# Toy models of asset returns # Toy models of asset returns When making simple agent-based models of banking systems to look at global properties (say systemic risk) one of the basic decisions you have to make is how to model returns on external (to the banking network) assets. The goal is to have as simple (and general) of a model as possible, without it being fundamentally unreasonable. What are some standard approaches? If I was trying the first thing that comes to mind then I would assume that returns on different assets are independently and normally distributed. This is tempting because of the central-limit theorem and how easy Gaussians are to work with, but unfortunately it is also considered one of the most dangerous assumptions in finance. Instead, it seems that distributions with fatter tails are preferred. For instance this recent paper (which is written at a level of abstraction that I'd be interested in working at) uses student's t-distribution (with df = 1.5) to sample returns. However, I've been told by some quants that even this distribution doesn't have a heavy enough tail. Further, distributions with full support on $(-\infty, +\infty)$ simply don't make sense as models of returns for me. How could you possibly lose more money than your original investment? But if you simply truncate the distribution on the left though then the fat tail you cut off will result in a significant effect (as compared to truncating something like a Gaussian). What are typical distributions used to model returns in work that needs to discuss external assets but is not primarily focused on them? Is there a good survey paper discussing advantages and limitations of simple return models? ## Answer by Matt Wolf (score 5) https://quant.stackexchange.com/a/7394 Some of the used heavy-tail distributions are: - Log-Cauchy and Log-Gamma - Lévy - Burr and Weibull - Mixed normal Here two papers that cover some of them and others: - http://ect-pigorsch.mee.uni-bonn.de/data/research/papers/Financial_Economics,_Fat-tailed_Distributions.pdf - http://www.rff.org/RFF/Documents/RFF-DP-11-19-REV.pdf
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