Levon Kazaryan, Gregory Kantorovich Higher School of Economics Rate of convergence in the framework of CLT and Risk evaluation on financial markets. Levon Kazaryan, Gregory Kantorovich Higher School of Economics Higher School of Economics Mexico, 2017 www.hse.ru

Introduction In economics theory and on practice often are used models with normal distribution. But empirical researches show, that using of normal distribution on practice do not take in consideration arise of fat tails. Hence, there is alternative for models based on normal distributions such as: Stable distributions Clark’s subordination model Mixture of distributions’ model General Levy processes Variable and stochastic volatility Microstructural models Various non-normal distribution models Higher School of Economics , Mexico, 2017 2 / 24

Introduction   Higher School of Economics , Mexico, 2017 3 / 24

Methods and methodology Innovative method of construction G- bounds by Y. Gabovich Hypotheses of weak form efficiency by E. Fama. Construction of G bounds for log returns of stock market indexes The rate of convergence Correlation Runs test Random walk test Test of Weak-form efficiency Methods Higher School of Economics , Mexico, 2017 4 / 24

Hypothesis H0: G bounds evaluate the risk of large losses on the stock markets more accurately than the normal distribution. H1: Indexes of observable countries are efficiency in the weak form. H2: There is a negative relationship Between the Weak-form efficiency of the stock market and the risk of large losses on it. Higher School of Economics , Mexico, 2017 5 / 24

Definition of left tail fatness   Higher School of Economics , Mexico, 2017 6 / 24

Berry-Esseen Theorem   Higher School of Economics , Mexico, 2017 7 / 24

Construction of G(n,t) tail estimates   Higher School of Economics , Mexico, 2017 8 / 24

Construction of G1(t) tail estimates   Higher School of Economics , Mexico, 2017 9 / 24

Construction of G1(t) tail estimates   Higher School of Economics , Mexico, 2017 10 / 24

Construction of G2(t) tail estimates   Higher School of Economics , Mexico, 2017 11 / 24

Refinement of G*1(t) tail estimates   Higher School of Economics , Mexico, 2017 12 / 24

Data Country Index Australia S&P/ASX 200 Austria ATX Argentina Merval Belgium BEL 20 Brazil Bovespa United Kingdom FTSE 100 Germany DAX Hong Kong Hang Seng Denmark OMXC20 Israel TA 25 India BSE Sensex Indonesia IDX Composite Country Index Ireland ISEQ Overall Spain IBEX 35 Canada S&P/TSX Malaysia KLCI Mexico IPC Netherlands AEX Russia РТС United States S&P 500 Turkey BIST 100 France CAC 40 Switzerland SMI Japan Nikkei 225 Higher School of Economics , Mexico, 2017 13 / 24

Results of construction G bounds G bounds of S&P500 t 1*σ 2*σ 3*σ 4*σ 5*σ 6*σ 7*σ 8*σ 9*σ 10*σ H(t) 0,0018 Φ(t) 0,0115 0,0799 1,4015 5,75E+01 6,35E+03 1,85E+06 1,42E+09 2,93E+12 1,61E+16 2,39E+20 Ψ(Φ,t) 1,59E-01 2,28E-02 1,30E-03 3,17E-05 2,87E-07 9,87E-10 1,28E-12 6,22E-16 1,13E-19 7,62E-24 ΔKS 0,0139 CH(t) 0.5000 0.2000 0.1000 0.0588 0.0385 0.0270 0.0200 0.0154 0.0122 0.0099 KS 0,1726 0,0367 0,0152 0,0140 G1(t) Ψ(G1,t) 0,0106 0,0496 0,1195 0,1303 0,1306 NC(t) 29,1170 29,117 22,1853 16,0240 11,8046 9,0590 7,2512 6,0329 5,7370 NN(t) 16,4237 3,7174 1,2279 0,4113 0,1520 0,0650 0,0315 0,0167 0,0097 0,0071 G2(t) Ψ(G2,t) 0,1886 0,2572 Higher School of Economics , Mexico, 2017 14 / 24

Results of construction G bounds G bounds of RTSI t 1*σ 2*σ 3*σ 4*σ 5*σ 6*σ 7*σ 8*σ 9*σ 10*σ H(t) 0.109426 0.025864 0.008953 0.004725 0.002238 0.000995 0.000249 Φ(t) 0.1587 0.0228 0.0013 3.17E-05 2.87E-07 9.87E-10 1.28E-12 6.22E-16 1.13E-19 7.62E-24 Ψ(Φ,t) 0.689512 1.134395 6.886921 149.0597 7798.778 1007880 1.94E+08 4E+11 2.2E+15 ΔKS 0.013949 CH(t) 0.5 0.2 0.1 0.058824 0.038462 0.027027 0.02 0.015385 0.012195 0.009901 KS 0.172649 0.036749 0.015249 0.01398 G1(t) Ψ(G1,t) 0.633805 0.703816 0.587139 0.337991 0.160462 0.071318 0.017829 0.020393 NC(t) 29.117 22.1853 16.024 11.8046 9.059 7.2512 6.0329 5.737 NN(t) 0.260409 0.045903 0.00897 0.002603 0.00095 0.000407 0.000197 0.000105 6.04E-05 4.43E-05 G2(t) Ψ(G2,t) 0.998089 1.815119 2.354845 2.446219 1.260554 2.37566 4.117811 Higher School of Economics , Mexico, 2017 15 / 24

Analysis of fatness of left tail Fatness of left tail S&P500 Fatness 1*σ 2*σ 3*σ 4*σ 5*σ 6*σ 7*σ 8*σ 9*σ 10*σ Ψ(Φ,t) 0.599115 1.138043 7.401509 146.7311 8103.441 1155059 5.7E+08 5.13E+11 1.21E+15 1.20E+19 Ψ(G1,t) 0.476962 0.408981 0.2294 0.114353 0.05722 0.028049 0.017952 0.007854 0.003366 0.002244 Ψ(G2,t) 0.142133 0.19251 0.220506 0.290887 0.239842 0.178169 0.161971 Fatness of left tail RTSI Fatness 1*σ 2*σ 3*σ 4*σ 5*σ 6*σ 7*σ 8*σ 9*σ 10*σ Ψ(Φ,t) 0.620846 1.236254 6.907535 118.031 6083.872 1263621 7.79E+08 8.02E+11 4.41E+15 Ψ(G1,t) 0.633805 0.703816 0.587139 0.337991 0.160462 0.071318 0.017829 0.020393 Ψ(G2,t) 0.998089 1.815119 2.354845 2.446219 1.260554 2.37566 4.117811 Higher School of Economics , Mexico, 2017 16 / 24

Garch Model S&P500 Sigma*t History Data N(0;1) Innovation G1 1 GARCH(1,1) Conditional Variance Model: ---------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 7.71259e-07 1.30878e-07 5.89296 GARCH{1} 0.911016 0.00180184 505.604 ARCH{1} 0.0862862 0.00146257 58.9965 Sigma*t History Data N(0;1) Innovation G1 1 0,095079575 0,0912 0,031100369 0,105148522 2 0,025947376 0,0133 0,007934698 0,027248522 3 0,009621962 0,0009 0,00186967 0,014848522 4 0,004651375 0,000547221 0,013948522 5 0,002325687 0,000136805 6 0,001140043 4,56017E-05 7 0,000729627 8 0,000319212 9 10 9,12034E-05 Higher School of Economics , Mexico, 2017 17 / 24

Garch Model RTSI Sigma*t History Data N(0;1) Innovation G1 1 2 3 4 5 6 0.109425516 0.1587 0.0592 0.172648522 2 0.025864213 0.0228 0.0057 0.036748522 3 0.008952997 0.0013 0.0009 0.015248522 4 0.004725193 0.0000317 0.013980222 5 0.002238249 0.000000287 0.013948809 6 0.000994777 9.87E-10 0.013948523 7 0.000248694 1.28E-12 0.013948522 8 6.22E-16 9 1.13E-19 0.012195122 10 7.62E-24 0.00990099 Higher School of Economics , Mexico, 2017 18 / 24

Results of construction G*1 bound Applying the refinement for the G1-bound using the Vysochanskij–Petunin inequality for the RTSI t 1*σ 2*σ 3*σ 4*σ 5*σ 6*σ 7*σ 8*σ 9*σ 10*σ H(t) 0.109426 0.025864 0.008953 0.004725 0.002238 0.000995 0.000249 Φ(t) 0.1587 0.0228 0.0013 3.17E-05 2.87E-07 9.87E-10 1.28E-12 6.22E-16 1.13E-19 7.62E-24 VP(λ) 0.444444 0.111111 0.049383 0.027778 0.017778 0.012346 0.00907 0.006944 0.005487 0.004444 Ψ(Φ,t) 0.689512 1.134395 6.886921 149.0597 7798.778 1007880 1.94E+08 4E+11 2.2E+15 G1 0.172649 0.036749 0.015249 0.01398 0.013949 0.012195 0.009901 Ψ(G1,t) 0.633805 0.703816 0.587139 0.337991 0.160462 0.071318 0.017829 0.020393 G1* Ψ(G1*,t) 0.570687 0.767013 0.588896 0.267634 0.125177 0.101023 0.110002 0.071838 0.09092 Higher School of Economics , Mexico, 2017 19 / 24

Comparison of models for the RTS index Sigma*t History Data N(0;1) Innovation G1 G2 G1* -1 0.109425516 0.1587 0.0592 0.172648522 -2 0.025864213 0.0228 0.0057 0.036748522 -3 0.008952997 0.0013 0.0009 0.015248522 0.008970138 -4 0.004725193 0.0000317 0.013980222 0.002603242 -5 0.002238249 0.000000287 0.013948809 0.000950487 -6 0.000994777 9.87E-10 0.013948523 0.000406659 0.012345679 -7 0.000248694 1.28E-12 0.013948522 0.00019729 0.009070295 -8 6.22E-16 0.000104684 0.006944444 -9 1.13E-19 0.012195122 6.03948E-05 0.005486968 -10 7.62E-24 0.00990099 4.42895E-05 0.004444444 Higher School of Economics , Mexico, 2017 20 / 24

Information efficiency analysis Algorithm of testing Weak-form efficiency of stock market Results of testing Step 1. Kolmogorov–Smirnov test Step 2. Jarque–Bera test Step 3. Runs test Step 4. Random walk test Country Runs test Random Walk Test Weak-form efficiency Australia Yes   Austria No Argentina Belgium Brazil United Kingdom Germany Hong Kong Denmark Israel India Indonesia Ireland Spain Canada Malaysia Mexico Netherlands Russia United States Turkey France Switzerland Japan Runs test Random walk test Weak-form efficiency of stock market Higher School of Economics , Mexico, 2017 21 / 24

Results of logit model Results of logit model testing Coef. Std. Err. z P>|z| [95% Conf. Interval] -1*σ X 2066309 3027078 0,68 0,495 -3866654 7999272 const -0,9540378 1,929778 -0,49 0,621 -4,736332 2,828257 -2*σ 101228,70 1194578,00 0,08 0,93 -2240101,00 2442559,0 0,22 1,44 0,15 0,88 -2,60 3,04 -3*σ -226253,40 276671,00 -0,82 0,41 -768518,60 316011,70 1,68 1,71 0,98 0,33 -1,68 5,03 -4*σ -14209,27 12722,60 -1,12 0,26 -39145,11 10726,58 1,60 1,23 1,30 0,19 -0,81 4,02 -5*σ -440,3902 223,7824 -1,97 0,049 -878,9956 -1,784768 2,06 0,99 2,08 0,04 0,12 4,01 -6*σ -3,02 1,39 -2,17 0,03 -5,75 -0,30 2,10 2,26 0,02 0,28 3,92 -7*σ 0,00 -1,04 0,30 -0,01 0,79 0,61 1,31 -0,40 1,98 -8*σ -0,60 0,55 0,51 0,50 1,00 0,32 -0,48 1,49 -9*σ -1,02 0,31 0,63 1,26 0,21 -0,35 1,61 -10*σ -0,63 0,53 0,46 -0,44 1,37 Higher School of Economics , Mexico, 2017 22 / 24

Conclusion Confirmation of H0 hypothesis H1 hypothesis was partially confirmed. Confirmation of H2 hypothesis Constructed logit model let us find a negative correlation between deviation of observed indexes log returns and weak form efficiency For log returns of observed effective stock markets in the weak form fatness ratio is less than for ineffective stock markets This area of research carries great potential for further research Higher School of Economics , Mexico, 2017 23 / 24

Conclusion