The extractData() dataset analyzed with K2, K3, K4 distributions.

Summary This video is the third example of the forthcoming paper 'A family of distributions tailored to skewed and fat tails' by Patrice Kiener. Four parameters K2, K3 and K4 distributions mentionned in the paper are used to analyze the distribution of the daily log-returns of eight financial assets (Gold bullion, Société Générale, Vivendi, EURUSD exchange rate, VIX, CAC40, DJIA, SP500) during the period January 2007 - December 2013. The data come from the extractData() dataset available in the R package FatTailsR. The video displays the plots of the empirical distributions and their estimates as well as the plot of the logit of the distribution function, which is used for the parameter estimation, over the whole period and for each year. The main results are: (1) The eight assets are perfectly described by K2, K3 and K4 distributions which exhibit high flexibility and excellent adjustment capabilities to the data. (2) The Value-at-Risk (VaR) and the Expected Shortfall (ES) are easily calculated as the formulas have closed forms. (3) Almost all distributions exhibit skewed and fat tails. (4) Over the whole period (seven years = 1818 points), the tail parameter k (k in latin, kappa in greek) varies in the range [2.8, 6]. The lowest value corresponds to DJIA, SP500 indices while the largest value corresponds to EURUSD exchange rate. (5) On a yearly period (about 260 points per year), the fluctuations of the k parameter are in the range [2.5, 10]. The asymmetry of the distribution, measured by parameters d (delta = distortion) or e (epsilon = eccentricity), is much more pronounced. For further information, please contact patrice.kiener@inmodelia.com or dutangc@gmail.com.