1. Levon Kazaryan, Gregory Kantorovich Higher School of Economics
Taking into account the rate of convergence in CLT under Risk evaluation on financial markets
Levon Kazaryan, Gregory Kantorovich
Higher School of Economics
Higher School of Economics Moscow, 2015

2. 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
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3. Introduction
Consider Sj>0 index prices over n time intervals. Define Xj = 𝑗=1 𝑛 ln( 𝑆 𝑗 𝑆 𝑗−1 ) Logarithm of the index return over the whole period Yn = X1 + … + Xn From CLT with assumption that n is large, and a conclusion that cumulative distribution function (c.d.f.) Fn (t) of Yn coincides with c.d.f. Φ(t) of a normally distributed random variable. Decomposing Fn(t) = [Fn(t) – Φ(t)] + Φ(t) For example one of important cases is probability of six-standard-deviations loses on US stock market. CLT promised us Φ(-6σ) ~ But empirical research of historical stock returns shows that: Pr{ Y253 < -6σ } = F253 (-6σ) ~ 10-2= 1% So fatness ratio between CLT result and empirical research is really huge. Investors face six-standard-deviations loses 10 million times frequently.
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4. 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
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5. 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
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6. 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.
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7. Construction of G(n,t) tail estimates
G1-bounds (G1,n(t)) - combination of Berry-Esseen’s -type estimates with Chebyshev-type inequality
Berry-Esseen’ s-type inequality:
For c.d.f. Fn (t) of Yn there exists such normal Φn (t) and non-dependent on n constant C that for all t:
𝑠𝑢𝑝 𝑡 Fn t – Φ t ≤ 𝐶𝜌 𝑛 , so we can estimate Fn(t) as :
𝐹 𝑛 𝑡 = 𝐹 𝑛 𝑡 − 𝛷 𝑛 𝑡 + 𝛷 𝑛 t ≤ 𝐶𝜌 𝑛 + 𝛷 𝑛 t
Chebyshev-type one-sided inequalities for random variables:
F n t ≤ 1 1+ t 2
G2-bounds (G2,n(t)) - combination of G1-bounds with Nagaev-Nikulin-type inequality
Nagaev-Nikulin-type inequalities for sums of independent random variables:
| 𝐹 𝑛 𝑡 −Ф(𝑡)|≤ 𝐶(𝑡)𝜌 𝑛 (1+ |𝑡| 3 )
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8. 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
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9. Results of construction G bounds
G bounds of S&P500
1*σ
2*σ
3*σ
4*σ
5*σ
6*σ
7*σ
8*σ
9*σ
10*σ
H(t)
0.0951
0.0259
0.0096
0.0047
0.0023
0.0011
0.0007
0.0003
0.0001
Φ(t)
0.1587
0.0228
0.0013
0.0000
Ψ(Φ,t)
0.5991
1.1380
7.4015
1.47E+02
8.10E+03
1.16E+06
5.70E+08
5.13E+11
1.21E+15
1.20E+19
ΔKS
0.0406
CH(t)
0.5000
0.2000
0.1000
0.0588
0.0385
0.0270
0.0200
0.0154
0.0122
0.0099 KS
0.1993
0.0634
0.0419
0.0407
G1(t)
Ψ(G1,t)
0.4770
0.4090
0.2294
0.1144
0.0572
0.0280
0.0180
0.0079
0.0034
0.0022
NC(t)
9.0590
7.2512
6.0329
5.7370
NN(t)
1.4518
0.3165
0.0988
0.0327
0.0121
0.0052
0.0025
0.0008
0.0006
G2(t)
Ψ(G2,t)
0.1421
0.1925
0.2205
0.2909
0.2398
0.1782
0.1620
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10. Results of construction G bounds
G bounds of MICEX
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
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11. 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)
5.7E+08
5.13E+11
1.21E+15
1.20E+19
Ψ(G1,t)
0.2294
Ψ(G2,t)
Fatness of left tail MICEX
Fatness
1*σ
2*σ
3*σ
4*σ
5*σ
6*σ
7*σ
8*σ
9*σ
10*σ
Ψ(Φ,t)
1.42E+09
2.93E+12
1.61E+16
2.39E+20
Ψ(G1,t)
Ψ(G2,t)
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12. Results of logit model Results of logit model testing Coef. Std. Err.z
P>|z|
[95% Conf. Interval]
-1*σ X
0,68
0,495
const
-0,
1,929778
-0,49
0,621
-4,736332
2,828257
-2*σ
101228,70
,00
0,08
0,93
,00 ,0
0,22
1,44
0,15
0,88
-2,60
3,04
-3*σ
,40
276671,00
-0,82
0,41
,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
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13. 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
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14. Conclusion
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