B Tables and Figures - Tail index estimation: quantile driven threshold selection

B.1 Tables

Table 4: Hall expansion parameters values Stable Student-t Fr´echet α (1, 2) (2, ∞) (2, ∞) β α 2 α A _π1Γ (α) sin απ₂ √1 απ Γ(α+1₂ ) Γ(α₂) α (α−1)/2 ₁ B −1 2 Γ(2α) sin(απ) Γ(α) sin(απ₂ ) − α2 2 α+1 α+2 1 2

Table 5: Estimates of k∗ under different methods for the four families of distributions

α KS dif KS rat TH 5% Eye-Ball Drees Du Bo

Student-t 2 509.89 5.32 281.00 500.00 19.67 1036.73 968.18 3 343.13 9.63 132.00 500.00 35.21 841.59 895.65 4 227.99 13.92 78.00 500.00 51.48 754.68 859.72 5 164.88 17.93 53.00 500.00 69.02 708.41 837.61 6 140.07 20.25 40.00 500.00 84.55 677.95 823.24 Stable 1.1 240.82 2.43 817.00 500.00 8.00 1481.88 1180.98 1.3 172.74 2.97 292.00 500.00 10.14 1466.24 1218.68 1.5 137.18 3.54 146.00 500.00 12.66 1376.89 1214.89 1.7 200.45 4.34 74.00 500.00 18.88 1176.03 1153.69 1.9 667.03 4.13 27.00 500.00 108.09 861.59 1061.44 Frechet 2 217.71 5.39 928.00 500.00 19.26 1500.70 1305.65 3 231.47 9.58 928.00 500.00 34.99 1501.00 1304.65 4 226.54 14.53 928.00 500.00 51.35 1501.00 1305.28 5 227.16 19.49 928.00 500.00 67.51 1501.00 1303.90 6 229.31 25.70 928.00 500.00 84.04 1501.00 1304.10 ARCH 2.30 290.39 8.53 500.00 31.32 1131.36 1244.62 2.68 300.24 10.29 500.00 36.21 1036.93 1244.78 3.17 290.97 12.06 500.00 42.90 947.32 1245.28 3.82 246.72 14.90 500.00 52.81 864.97 1246.05 4.73 202.79 17.84 500.00 64.75 791.26 1247.14

This table depicts the mean for the estimated k∗ for the different methodologies. The samples are drawn from four different heavy tailed distribution families. The samples are drawn from the Student-t, Symmetric Stable, Frechet distribution and ARCH process. The different methods are stated in the first row. KS dif is the Kolmogorov-Smirnov distance metric in (10). The KS rat is the Kolmogorov-Smirnov distance in (12). TH is based on the theoretically derived optimal k from minimizing the mse for specific parametric distributions, presented in Equation (17) in the Appendix. The automated Eye-Ball method in (6) is the heuristic method aimed at finding the first stable region in the Hill Plot. For the column Drees the k∗ is determined by the methodology described by Drees and Kaufmann (1998). Du Bo is the Double Bootstrap procedure by Danielsson et. al. (2001). Here α indicates the corresponding theoretical tail exponent for the particular distribution which the sample is drawn from. The sample size is n = 10, 000 for 10, 000 repetitions.

Table 6: Different statistics for ˆα in the left tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo mean 3.40 3.99 2.70 3.19 2.41 1.98 median 3.35 3.78 2.72 3.19 2.32 2.05 st. dev. 0.81 5.01 0.58 0.65 0.94 0.53 min 0.27 0.49 0.16 0.16 0.48 0.19 max 7.79 427.86 15.04 8.52 53.00 10.09 skewness 0.40 55.10 0.71 0.09 18.34 -0.58 kurtosis 3.07 3835.31 19.53 4.56 717.86 11.60

This table presents the results of applying the six different methods to estimate α for left tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from the 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The six different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). Different statistics are calculated for the distribution of ˆα. The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 7: Different statistics for ˆα in the right tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo mean 2.97 3.58 2.37 2.87 2.22 1.77 median 2.90 3.40 2.39 2.87 2.13 1.83 st. dev. 0.81 3.91 0.52 0.63 0.76 0.55 min 0.54 0.32 0.12 0.11 0.32 0.12 max 7.48 357.56 7.22 7.09 45.42 34.91 skewness 0.42 53.65 -0.22 0.15 17.52 12.51 kurtosis 3.01 4196.10 6.79 4.11 779.80 740.25

This table presents the results of applying the six different methods to estimate α for right tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from the 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The six different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). Different statistics are calculated for the distribution of ˆα. The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 8: The median absolute difference of the estimates of α in the left tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 0.00 0.22 0.70 0.46 1.02 1.40 KS rat 0.22 0.00 1.14 0.77 1.45 1.80 5% 0.70 1.14 0.00 0.50 0.40 0.68 Eye-Ball 0.46 0.77 0.50 0.00 0.86 1.19 Drees 1.02 1.45 0.40 0.86 0.00 0.27 Du Bo 1.40 1.80 0.68 1.19 0.27 0.00

This table presents the results of applying the six different methods to estimate α for left tail of stock returns. The data is from the CRSP database that contains all individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The matrix presents the median absolute difference between the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cutt-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 9: The median absolute difference of the estimates of α in the right tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 0.00 0.28 0.59 0.42 0.74 1.16 KS rat 0.28 0.00 1.12 0.78 1.25 1.64 5% 0.59 1.12 0.00 0.50 0.30 0.59 Eye-Ball 0.42 0.78 0.50 0.00 0.73 1.10 Drees 0.74 1.25 0.30 0.73 0.00 0.31 Du Bo 1.16 1.64 0.59 1.10 0.31 0.00

This table presents the results of applying the six different methods to estimate α for right tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12- 31 for NYSE, AMEX, NASDAQ and NYSE Arca. The matrix presents the median absolute difference between the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cutt-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 10: The correlation matrix of ˆα in the left tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 1.00 0.10 0.38 0.49 0.24 0.24 KS rat 0.10 1.00 0.05 0.06 0.04 0.03 5% 0.38 0.05 1.00 0.68 0.33 0.65 Eye-Ball 0.49 0.06 0.68 1.00 0.26 0.49 Drees 0.24 0.04 0.33 0.26 1.00 0.24 Du Bo 0.24 0.03 0.65 0.49 0.24 1.00

This table presents the correlation matrix for the estimates of α by applying the six different methods for left tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 11: The correlation matrix of ˆα in the right tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 1.00 0.13 0.41 0.55 0.32 0.26 KS rat 0.13 1.00 0.05 0.06 0.05 0.04 5% 0.41 0.05 1.00 0.72 0.39 0.62 Eye-Ball 0.55 0.06 0.72 1.00 0.37 0.45 Drees 0.32 0.05 0.39 0.37 1.00 0.25 Du Bo 0.26 0.04 0.62 0.45 0.25 1.00

This table presents the correlation matrix for the estimates of α by applying the six different methods for right tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 12: The mean absolute difference of the estimates of k∗ in the left tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 0.00 80.51 126.78 78.18 224.47 360.98 KS rat 80.51 0.00 172.30 34.54 294.60 439.86 5% 126.78 172.30 0.00 142.01 140.18 267.66 Eye-Ball 78.18 34.54 142.01 0.00 266.68 408.87 Drees 224.47 294.60 140.18 266.68 0.00 168.47 Du Bo 360.98 439.86 267.66 408.87 168.47 0.00

This table presents the results of applying the six different methods to estimate α for left tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The matrix presents the mean absolute difference between the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 13: The mean absolute difference of the estimates of k∗ in the right tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 0.00 92.13 128.39 85.37 195.12 349.54 KS rat 92.13 0.00 174.16 28.10 266.57 438.43 5% 128.39 174.16 0.00 149.29 120.49 264.38 Eye-Ball 85.37 28.10 149.29 0.00 243.99 413.43 Drees 195.12 266.57 120.49 243.99 0.00 191.04 Du Bo 349.54 438.43 264.38 413.43 191.04 0.00

This table presents the results of applying the six different methods to estimate k∗ _{for right tail of stock returns. The data is from the CRSP database that}

contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The matrix presents the mean absolute difference between the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 14: The correlation matrix of ˆα (k∗) in the left tail for the six different methods

This table presents the correlation matrix for the estimates of α by applying the six different methods for left tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013- 12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye- Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 15: The correlation matrix of ˆα (k∗) in the right tail for the six different methods

This table presents the correlation matrix for the estimates of α by applying the six different methods for right tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013- 12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye- Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 16: The correlation matrix of k∗ in the left tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 1.00 -0.04 -0.00 -0.10 0.19 -0.11 KS rat -0.04 1.00 -0.01 0.17 0.16 -0.27 5% -0.00 -0.01 1.00 0.00 0.01 0.01 Eye-Ball -0.10 0.17 0.00 1.00 -0.08 -0.26 Drees 0.19 0.16 0.01 -0.08 1.00 -0.29 Du Bo -0.11 -0.27 0.01 -0.26 -0.29 1.00

This table presents the correlation matrix for the estimates of k∗ by applying the six different methods for left tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

Table 17: The correlation matrix of k∗ in the right tail for the six different methods

KS dif KS rat 5% Eye-Ball Drees Du Bo KS dif 1.00 -0.13 -0.01 -0.11 0.21 -0.06 KS rat -0.13 1.00 0.02 0.20 0.13 -0.16 5% -0.01 0.02 1.00 0.00 -0.00 0.02 Eye-Ball -0.11 0.20 0.00 1.00 -0.06 -0.22 Drees 0.21 0.13 -0.00 -0.06 1.00 -0.10 Du Bo -0.06 -0.16 0.02 -0.22 -0.10 1.00

This table presents the correlation matrix for the estimates of k∗ _{by applying}

the six different methods for right tail of stock returns. The data is from the CRSP database that contains all the individual stocks data from 1925-12-31 to 2013-12-31 for NYSE, AMEX, NASDAQ and NYSE Arca. The different methods are the KS distance metric, KS ratio method, 5% threshold, automated Eye-Ball method, the iterative method by Drees and Kaufmann (1998) and the Double Bootstrap by Danielsson et. al. (2001). The stocks for which one of the methods has ˆα > 1, 000 are excluded. The maximum k is cut-off at 15% of the total sample size. There are 17,918 companies which are included in the analysis.

B.2 Figures

Figure 6: Simulations Brownian motion for Symmetric Stable parameters

These two figures show the simulations for the limit function in (11). The parameters are for the Symmetric Stable distribution, which are found in Table 4 of the Appendix. The value for α for the different lines is stated in the legend. Here T is 1,500, therefore, the interval between w(si) − w(si+1) is normally distributed with mean 0 and variance 1_/_k_{. The path of the Brownian motion is simulated 1,000 times. The top figure shows the}

average number of order statistics at which the largest absolute distance is found for a given k. The bottom graph depicts the average distance found for the largest deviation at a given k. The top and bottom graphs are related by the fact that the bottom graph depicts the distances found at the ith_{observation found in the top graph.}

Figure 7: Simulations Brownian motion for Fr´echet parameters

These two figures show the simulations for the limit function in (11). The parameters are for the Fr´echet distribution, which are found in Table 4 of the Appendix. The value for α for the different lines is stated in the legend. Here T is 1,500, therefore, the interval between w(si) − w(si+1) is normally distributed with mean 0 and variance 1/k. The

path of the Brownian motion is simulated 1,000 times. The top figure shows the average number of order statistics at which the largest absolute distance is found for a given k. The bottom graph depicts the average distance found for the largest deviation at a given k. The top and bottom graphs are related by the fact that the bottom graph depicts the distances found at the ith observation found in the top graph.

Figure 8: Optimal ˆα for quantile metrics (Symmetric Stable distribution)

This Figure depicts simulation results of the average optimally chosen α(k) for a given level of T . Here T is the number of extreme order statistics over which the metric is optimized. In the upper left graph this is done for the KS distance metric for different Symmetric Stable distributions with degrees of freedom α. This is also done for the mean squared distance, mean absolute distance and the criteria used by Dietrich et al. (2002). The simulation experiment has 10,000 iterations for sample size n=10,000.

Figure 9: Optimal ˆα for quantile metrics (Fr´echet distribution)

This Figure depicts simulation results of the average optimally chosen α(k) for a given level of T . Here T is the number of extreme order statistics over which the metric is optimized. In the upper left graph this is done for the KS distance metric for different Fr´echet distributions with degrees of freedom α. This is also done for the mean squared distance, mean absolute distance and the criteria used by Dietrich et al. (2002). The simulation experiment has 10,000 iterations for sample size n=10,000.

Figure 10: Optimal k for quantile metrics (Symmetric Stable distribution)

This Figure depicts simulation results of the average optimally chosen k for a given level of T . Here T is the number of extreme order statistics over which the metric is optimized. In the upper left graph this is done for the KS distance metric for different Symmetric Stable distributions with degrees of freedom α. This is also done for the mean squared distance, mean absolute distance and the criteria used by Dietrich et al. (2002). The simulation experiment has 10,000 iterations for sample size n = 10, 000.

Figure 11: Optimal k for quantile metrics (Fr´echet distribution)

This Figure depicts simulation results of the average optimally chosen k for a given level of T . Here T is the number of extreme order statistics over which the metric is optimized. In the upper left graph this is done for the KS distance metric for different Fr´echet distributions with degrees of freedom α. This is also done for the mean squared distance, mean absolute distance and the criteria used by Dietrich et al. (2002). The simulation experiment has 10,000 iterations for sample size n = 10, 000.

Figure 12: Shape of the Hill Plot for different distributions

The Figure depicts the Hill estimates for the Student-t(3), Symmetric Stable(1.7), and Fr´echet(3) distribution against the number of order statistics used in the estimation. These graphs are constructed by taking the average Hill estimates over 500 simulations.

Figure 13: The shape of the Hill plot for the Student-t distribution

The Figure depicts the Hill estimates for the Student-t distribution family against the number of order statistics used in the estimation. This is done for different corresponding tail indices. These graphs are constructed by taking the average Hill estimates over 500 simulations.

Figure 14: The shape of the Hill plot for the Symmetric Stable distribution

The Figure depicts the Hill estimates for the Symmetric Stable distribution family against the number of order statistics used in the estimation. This is done for different corresponding tail indices. These graphs are constructed by taking the average Hill estimates over 500 simulations.

Figure 15: The shape of the Hill plot for the Fr´echet distribution

The Figure depicts the Hill estimates for the Fr´echet distribution family against the number of order statistics used in the estimation. This is done for different corresponding tail indices. These graphs are constructed by taking the average Hill estimates over 500 simulations.

Figure 16: The shape of the Hill plot for the ARCH stochastic process

The Figure depicts the Hill estimates for the ARCH(1) process against the number of order statistics used in the estimation. This is done for different corresponding tail indices. These graphs are constructed by taking the average Hill estimates over 500 simulations.

Figure 17: Optimal ˆα for quantile metrics (Symmetric Stable distribution)

Figure 18: Optimal ˆα for quantile metrics (Fr´echet distribution)

This Figure depicts simulation results of the average optimally chosen α(k) for a given

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