Using Student’s t Distributions to Model Fat-Tailed Returns
Summary
The document addresses whether a Student’s t distribution can represent the fat tails often observed in stock returns and how that choice relates to risk measures. It concludes that the distribution’s degrees of freedom can be used to model heavier tails than a normal distribution, and that this feature matters directly when estimating expected shortfall, which depends on the shape of the loss tail.
The answer cites prior studies comparing Student’s t distributions with empirical stock returns and describes examples of their use in risk estimation. It suggests expected shortfall as the clearest answer to the exam question, while allowing that value at risk can also be affected depending on its distributional assumptions. The response emphasizes that the correct multiple-choice answer depends on the course framing; it does not establish that the t distribution fits every asset or captures all relevant tail behavior.
Key ideas
- A Student’s t distribution can model heavier tails than a normal distribution.
- Its degrees of freedom control the degree of tail heaviness.
- Expected shortfall is sensitive to the modeled tail shape.
- Value at risk can also change when the assumed return distribution changes.
- The t distribution is a useful candidate model, but its fit and suitability depend on the data and application.
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Full text
# Fat tailed can be estimated through a t-distributions? # Fat tailed can be estimated through a t-distributions? I have a simple question that makes me doubt a bit. In a multiple choise exam I ecountered this question: "if the stocks returns are not normally distributed, the fat tail effect can be estimated trhough a t-distribution?" a)no, since data are usually normally distributed b)yes, since it affect ES c)no since the t-distribution is estimating badly the fat tails effects d)yes, since it affects the VaR. My answer was C, exam grade was quite good, but I could not access the correction. I hope C is the right answer Thank you!!! ## Answer by AKdemy (score 4) https://quant.stackexchange.com/a/66035 B is the correct choice. I honestly would wish multiple choice would not even exist. It is the worst way of testing knowledge in my opinion. Without knowing the details of what was taught, I would say choosing C is definitely the wrong answer. The df in t-student can be used to estimate/model fat tails. According to Fat Tails in Financial Return Distributions Revisited, `P. D. Praetz, The Distribution of Share Price Changes, Journal of Business 45(1) (1972) 49-5519`, and `R. Blattberg, N. Gonedes, A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices, Journal of Business 47 (1974) 244-280.` showed that Student’s t distribution has similar distributional properties to those observed for actual returns. ` A. Peiro, The Distribution of Stock Returns: International Evidence, Applied Financial Economics 4 (1994) 431-439` presented evidence that Student’s t distribution in stock markets such as those in the United States, Japan, United Kingdom, Germany and France is very close to the empirical distribution of returns. `G. Zumbach, A Gentle Introduction to the RM2006 Methodology, Technology Paper, RiskMetrics (2006).` showed the usefulness of the risk estimation model based on Student’s t distribution with five degrees of freedom, using the return data of FTSE 100. Now, the paper also lists reasons why it may not be ideal to use t-student for such purposes. However, if the exam would discuss such nuances, I would expect it to not use multiple choice. What is correct? Most likely B, potentially B and D. Frequently, VaR assumes returns are normally distributed. Now obviously there are more general version of VaR like TVaR but expected shortfall would directly relate to the shape of the distribution. Long story short, without knowing the course material, I would certainly select B.
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