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!!!

• the fat tail effects are extremely hard to modelize since it can drastically change from distribution to distribution(and sample to sample if you use monte carlo simulations )
– lays
Jul 14, 2021 at 17:04
• I agree with @lays. However, I think the answer they were looking for is (d). If you have a few observations (let's say 50) and you fit a a t-distribution with a low degrees of freedom parameter then this will change the VaR at 1/200 quite drastically (increase it) versus the normal. This could be considered a better fit of the fat tail of the distribution generating the observations. Jul 14, 2021 at 20:22
• Seems like consensus is (b) not (d) as expected shortfall is considered more sensitive to fat tails. Jul 15, 2021 at 7:11

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.

• How do you distinguish B and D? Both would seem to be true, is one more appropriate than the other? I agree that this is a terrible question. Jul 14, 2021 at 21:58
• I think the distinction is that ES is generally always non normal in it's generic definition, whereas if you google "value at risk of normal distribution" you will find plenty of examples where the assumption is that possible outcomes are normally distributed about the mean. For example parametric var on Investopedia. Wikipedia's Expected shortfall page states "ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution." Jul 14, 2021 at 22:02