In this post, I continue trying to fit the daily log returns of TASI index using heavy-tailed distributions. In the previous post, I used Pareto distribution to model TASI index’s left tail. In this post, I use Student t distribution.

Recently, Student t distribution has been used widely by financial engineers as models for heavy-tailed distribution such as the distribution of financial markets.The t-distribution is characterized by *degrees of freedom v *(shape parameter) in addition to mean and variance. Both the kurtosis and weight of the tails increase as the *v degrees of freedom* decreases. When *v **≤ 4,* the tail weight is so high that the kurtosis is infinite. Thus, *v degrees of freedom* is also interpreted as the *tail index α.*

The following figure contains Quantile-Quantile plots of the normal and t probability with 1, 2, 4, 8, and 15 degrees of freedom. The data used is TASI Index daily log returns from Jan 7, 2007 to Feb 2, 2016.

As shown, none of the plots looks exactly linear, but the t plot with 2 degrees of freedom is almost straight through the bulk of the data. There are approximately two returns in the left tail and one in the right tail that are not along the line, but these can be ignored considering the sample size of 2270. Nevertheless, it is worthwhile to keep in mind that the historical data have more extreme outliers than a t-distribution.

Conclusion: Extreme returns from TASI index are more likely than the probability given by a t-distribtuion with 2 degrees of freedom.

Thank you.

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## Author: Thamir K. AlHashemi

Practitioner in the fields of quantitative finance and risk management. A lifetime learner. Reading is the only hobby that I haven't failed in ;)
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