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
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 , Moscow, 2015
2 / 15
Introduction
Consider Sj>0 index prices over n time intervals.
Define Xj =
𝑆𝑗
𝑛
ln(
𝑗=1
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) ~ 10-9.
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.
Higher School of Economics , Moscow, 2015
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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
Higher School of Economics ,
Moscow, 2015
Country
Ireland
Spain
Canada
Malaysia
Mexico
Netherlands
Russia
United States
Turkey
France
Switzerland
Japan
Index
ISEQ Overall
IBEX 35
S&P/TSX
KLCI
IPC
AEX
РТС
S&P 500
BIST 100
CAC 40
SMI
Nikkei 225
4/ 15
Methods and methodology
Methodology
Innovative method of construction G- bounds by Y. Gabovich
Hypotheses of weak form efficiency by E. Fama.
Methods
Construction of G
bounds for log
returns of stock
market indexes
The rate of
convergence
Higher School of Economics , Moscow, 2015
Correlation
• Runs test
• Random walk
test
Test of Weakform efficiency
5 / 18
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 , Moscow, 2015
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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:
𝑠𝑢𝑝𝑡 Fn t – Φ t
≤
𝐶𝜌
𝑛
, so we can estimate Fn(t) as :
𝐹𝑛 𝑡 = 𝐹𝑛 𝑡 − 𝛷𝑛 𝑡
+ 𝛷𝑛 t ≤
𝐶𝜌
𝑛
+ 𝛷𝑛 t
Chebyshev-type one-sided inequalities for random variables:
1
Fn t ≤
1 + t2
G2-bounds (G2,n(t)) - combination of G1-bounds with Nagaev-Nikulin-type inequality
Nagaev-Nikulin-type inequalities for sums of independent random variables:
𝐶(𝑡)𝜌
|𝐹𝑛 𝑡 − Ф(𝑡)| ≤
𝑛(1 + |𝑡|3 )
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Information efficiency analysis
Algorithm of testing Weak-form
efficiency of stock market
Step 1. Kolmogorov–Smirnov test
Step 2. Jarque–Bera test
Step 3. Runs test
Step 4. Random walk test
Runs test
Weak-form
efficiency of stock
market
Random walk
test
Higher School of Economics , Moscow, 2015
Results of testing
Country
Runs test
Random Walk Test
Yes
Yes
Austria
No
Yes
Argentina
Yes
Yes
Belgium
Yes
No
Brazil
Yes
Yes
United Kingdom
Yes
Yes
Germany
Yes
Yes
Hong Kong
No
Yes
Denmark
Yes
Yes
Israel
Yes
Yes
India
No
No
Indonesia
No
No
Ireland
Yes
No
Spain
Yes
Yes
Canada
No
Yes
Malaysia
No
Yes
Mexico
No
No
Netherlands
No
Yes
Russia
No
No
United States
Yes
Yes
Turkey
Yes
Yes
France
Yes
Yes
Switzerland
Yes
Yes
Japan
Yes
Yes
Australia
Weak-form efficiency













8/ 15
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
H(t)
0.0951
0.0259
0.0096
0.0047
0.0023
0.0011
0.0007
0.0003
0.0001
0.0001
Φ(t)
0.1587
0.0228
0.0013
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
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
0.0406
0.0406
0.0406
0.0406
0.0406
0.0406
0.0406
0.0406
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
0.0406
0.0406
0.0406
0.0406
0.0406
0.0406
G1(t)
0.1993
0.0634
0.0419
0.0407
0.0406
0.0406
0.0406
0.0406
0.0406
0.0406
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)
29.1170
29.1170
16.0240 11.8046
9.0590
7.2512
6.0329
5.7370
NN(t)
1.4518
0.3165
0.0988
0.0327
0.0121
0.0052
0.0025
0.0013
0.0008
0.0006
G2(t)
0.1993
0.0634
0.0419
0.0327
0.0121
0.0052
0.0025
0.0013
0.0008
0.0006
Ψ(G2,t)
0.4770
0.4090
0.2294
0.1421
0.1925
0.2205
0.2909
0.2398
0.1782
0.1620
Higher School of Economics , Moscow, 2015
29.1170 22.1853
10*σ
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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
H(t)
0,0018
0,0018
0,0018
0,0018
0,0018
0,0018
0,0018
0,0018
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)
10*σ
0,0018
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
0,0139
0,0139
0,0139
0,0139
0,0139
0,0139
0,0139
0,0139
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
0,0139
0,0139
0,0139
0,0139
0,0139
0,0139
G1(t)
0,1726
0,0367
0,0152
0,0140
0,0139
0,0139
0,0139
0,0139
0,0139
0,0139
Ψ(G1,t)
0,0106
0,0496
0,1195
0,1303
0,1306
0,1306
0,1306
0,1306
0,1306
0,1306
NC(t)
29,1170
29,117
29,1170
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)
0,1726
0,0367
0,0152
0,0140
0,0139
0,0139
0,0139
0,0139
0,0097
0,0071
Ψ(G2,t)
0,0106
0,0496
0,1195
0,1303
0,1306
0,1306
0,1306
0,1306
0,1886
0,2572
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10 / 15
Analysis of fatness of left tail
Fatness of left tail S&P500
Fatness
Ψ(Φ,t)
Ψ(G1,t)
Ψ(G2,t)
1*σ
0.599115
0.476962
0.476962
2*σ
1.138043
0.408981
0.408981
3*σ
7.401509
0.2294
0.2294
4*σ
146.7311
0.114353
0.142133
5*σ
8103.441
0.05722
0.19251
6*σ
1155059
0.028049
0.220506
7*σ
5.7E+08
0.017952
0.290887
8*σ
5.13E+11
0.007854
0.239842
9*σ
1.21E+15
0.003366
0.178169
10*σ
1.20E+19
0.002244
0.161971
Fatness of left tail MICEX
Fatness
Ψ(Φ,t)
Ψ(G1,t)
Ψ(G2,t)
1*σ
0.01148
0.010553
0.010553
2*σ
0.079908
0.049578
0.049578
Higher School of Economics , Moscow, 2015
3*σ
1.401468
0.119481
0.119481
4*σ
57.47345
0.13032
0.13032
5*σ
6348.113
0.130614
0.130614
6*σ
1845905
0.130617
0.130617
7*σ
1.42E+09
0.130617
0.130617
8*σ
2.93E+12
0.130617
0.130617
9*σ
1.61E+16
0.130617
0.18864
10*σ
2.39E+20
0.130617
0.257236
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Results of logit model
Results of logit model testing
Coef.
-1*σ
-2*σ
-3*σ
-4*σ
-5*σ
-6*σ
-7*σ
-8*σ
-9*σ
-10*σ
X
const
X
const
X
const
X
const
X
const
X
const
X
const
X
const
X
const
X
const
Higher School of Economics , Moscow, 2015
2066309
-0,9540378
101228,70
0,22
-226253,40
1,68
-14209,27
1,60
-440,3902
2,06
-3,02
2,10
0,00
0,79
0,00
0,51
0,00
0,63
0,00
0,46
Std. Err.
z
3027078
1,929778
1194578,00
1,44
276671,00
1,71
12722,60
1,23
223,7824
0,99
1,39
0,93
0,00
0,61
0,00
0,50
0,00
0,50
0,00
0,46
P>|z|
0,68
-0,49
0,08
0,15
-0,82
0,98
-1,12
1,30
-1,97
2,08
-2,17
2,26
-1,04
1,31
-0,60
1,00
-1,02
1,26
-0,63
1,00
0,495
0,621
0,93
0,88
0,41
0,33
0,26
0,19
0,049
0,04
0,03
0,02
0,30
0,19
0,55
0,32
0,31
0,21
0,53
0,32
[95% Conf. Interval]
-3866654
7999272
-4,736332
2,828257
-2240101,00
2442559,0
-2,60
3,04
-768518,60
316011,70
-1,68
5,03
-39145,11
10726,58
-0,81
4,02
-878,9956
-1,784768
0,12
4,01
-5,75
-0,30
0,28
3,92
-0,01
0,00
-0,40
1,98
0,00
0,00
-0,48
1,49
0,00
0,00
-0,35
1,61
0,00
0,00
-0,44
1,37
12 / 15
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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13 / 15
Conclusion
14 / 15