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Revista Română de Statistică - Supliment nr. 9 / 2019 135
IMPACT OF USED METHODOLOGY IN RISK PREDICTION: VARIANCE-COVARIANCE VS.
HISTORICAL SIMULATION
Associate lecturer Dumitru-Cristian OANEA PhD ([email protected])
„Artifex” University of Bucharest
Abstract
The purpose of this article is to identify the diff erence between
applying the variance-covariance method (used by the RiskMetrics model)
and the historical simulation, in quantifying the risk. As a result of the diff erent
assumptions behind these models, we conclude that the risk is more correctly
or less correctly identifi ed.
The main analyzed hypotheses that infl uence the ability of the models
to quantify the risk are: the way of estimating the decay factor, the supposed
distribution of returns, as well as the methodological framework used.
Keywords: RiskMetrics, decay factor, variance-covariance, historical
simulation
JEL Classifi cation: D81, G32
Introduction
The economic literature has delimited three approaches to quantifying
Value at Risk, namely: the variance-covariance approach (advantage =
easy application; disadvantage = predetermined hypothesis regarding the
distribution of the set of returns), the historical simulation and the Monte
Carlo simulation.
For the practical analysis, we will use the daily values of the stock
exchange indices for the capital markets of Central and Eastern Europe: BET
(Romania), BUX (Hungary), PX (Czech Republic), SAX (Slovakia), SOFIX
(Bulgaria) and WIG (Poland). The sample consists of 6 capital markets from
Central and Eastern Europe, including only those countries that are members
of the European Union and from which Romania imported more than 1% of the total value of imports from 2011-2013, according to the data, obtained from the offi cial website of the National Institute of Statistics.
Research methodology, data, results and discussions
Variance-covariance method
We will start from using the methodology proposed by JP. Morgan in
risk estimation, namely RiskMetrics model, based on which we estimated 3
types of models, namely:
Romanian Statistical Review - Supplement nr. 9 / 2019136
►RM1 – 0=l 0.94 and 2 types of distribution for returns: Normal
distribution (RiskMetrics-1994) and t-Student distribution (RiskMetrics-2006);
► RM2 – λ is estimated based on error checking function, and 2
types of distribution for returns: Normal distribution (RiskMetrics-1994) and
t-Student distribution (RiskMetrics-2006);
► RM3 – λ is estimated based on square errors, and 2 types of
distribution for returns: Normal distribution (RiskMetrics-1994) and t-Student
distribution (RiskMetrics-2006);
A fi rst fi nding concerns the manner of infl uence of a mathematical
model in the estimation can refer to the existence of diff erences in the
estimation of the decay factor. The diff erences are presented in table 1. Easelly
we can see that the hypothesize regarding the distribution of returns, and also
the selected level of confi dence infl uence in the end the estimated value for
decay factor. In this respect, a direct proportional relationship is observed
between the decay factor value and selected level of confi dence.
Decay factor estimation
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Legend: The table presents the estimates of the decay factor for the entire sample. Bold values
show discrepancies greater than 4% between the two functions based on which we made the
estimation
The estimation of decay factor value based on the function of the
square errors is presented in fi gure 1.
Revista Română de Statistică - Supliment nr. 9 / 2019 137
Estimation of the decay factor based on the function of the squared
errors
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Performance of estimated Value at Risk based on the model RM1 Table 2
IndexNormal distribution t-Student distribution10% 5% 1% 10% 5% 1%
The unconditional coverage test (Kupiec)Crisis period: 2008 – 2009BET 0.57 2.40 6.76 6.87 5.62 1.72 BUX 0.13 0.27 0.05 8.89 1.90 0.03 PX 0.25 8.12 2.72 8.19 12.08 0.36 SAX 2.74 1.47 25.98 0.08 1.90 10.21 SOFIX 0.03 5.62 23.42 4.04 11.02 6.76 WIG 0.25 1.90 14.17 11.96 3.54 3.90
Post-crisis period: 2010 – 2016BET 7.58 0.59 17.01 0.06 0.29 5.73 BUX 1.07 0.29 7.80 5.65 2.50 0.48 PX 0.06 6.24 23.50 8.41 15.02 3.94 SAX 3.71 0.92 32.69 0.01 5.27 7.80 SOFIX 2.82 1.36 12.71 1.74 3.20 4.80 WIG 1.07 5.27 25.24 3.42 15.02 3.16
The conditional coverage test (Christoff ersen)Crisis period: 2008 – 2009BET 3.57 4.54 7.43 13.34 7.90 2.07 BUX 1.81 0.98 0.23 13.96 2.14 0.16 PX 2.69 9.66 4.59 9.57 13.87 3.41 SAX 4.78 1.66 27.76 1.05 2.15 11.08 SOFIX 4.14 6.66 27.77 10.17 12.00 11.37 WIG 2.69 6.25 15.26 17.89 6.75 4.40
Post-crisis period: 2010 – 2016BET 7.75 3.05 18.39 1.66 1.18 6.24 BUX 7.44 3.32 8.92 10.89 6.25 1.00 PX 4.50 12.25 26.42 15.69 21.71 4.61 SAX 10.01 2.48 32.79 8.81 8.59 8.17 SOFIX 6.69 11.67 29.98 3.06 11.35 7.98 WIG 11.60 15.57 27.97 9.53 25.08 6.84
Note: Critical values for 2
)1(c are 2.706 (90%), 3.841 (95%) and 6.635 (99%). Critical values
Romanian Statistical Review - Supplement nr. 9 / 2019138
for 2
)2(c are 4.605 (90%), 5.991 (95%) and 9.210 (99%). Bold values indicate that the model is
accepted (the probability of failure is equal to the selected confi dence level – α).
Performance of estimated Value at Risk based on the model RM2Table 3
IndexNormal distribution t-Student distribution
10% 5% 1% 10% 5% 1%The unconditional coverage test (Kupiec)
Crisis period: 2008 – 2009BET 0.78 4.18 6.76 8.19 5.62 0.36 BUX 0.06 0.03 0.05 6.25 2.94 0.05 PX 0.39 10.01 10.21 7.52 14.30 1.72 SAX 0.93 4.18 23.42 0.00 4.18 14.17 SOFIX 4.04 14.30 23.42 18.30 20.55 6.76 WIG 0.18 0.12 14.17 4.55 1.47 0.93
Post-crisis period: 2010 – 2016BET 5.08 1.21 10.13 0.01 0.05 5.73 BUX 1.62 0.10 5.73 7.16 2.19 0.84 PX 0.02 7.27 34.67 7.57 14.28 11.39 SAX 1.07 2.19 17.01 0.30 6.24 2.45 SOFIX 2.63 10.83 14.09 16.61 18.15 3.94 WIG 1.42 5.74 23.50 1.18 8.39 1.29
The conditional coverage test (Christoff ersen)Crisis period: 2008 – 2009BET 7.89 5.61 7.43 13.70 6.66 0.59 BUX 3.11 0.19 0.23 9.26 3.66 0.23 PX 2.56 12.32 13.84 9.10 15.66 3.93 SAX 1.92 4.33 25.05 1.41 4.33 15.26 SOFIX 11.53 18.33 27.77 26.34 22.92 11.37 WIG 6.94 0.98 14.63 9.97 4.19 1.22
Post-crisis period: 2010 – 2016BET 5.33 1.88 10.39 0.47 0.81 6.24 BUX 7.61 6.95 6.71 13.26 7.69 1.41 PX 4.65 11.33 34.79 16.63 19.78 11.61 SAX 5.06 6.26 17.10 8.88 9.88 3.33 SOFIX 3.00 12.70 15.75 16.89 20.31 4.79 WIG 12.73 22.36 36.54 7.99 29.96 5.84
Note: Critical values for 2
)1(c are 2.706 (90%), 3.841 (95%) and 6.635 (99%). Critical values
for 2
)2(c are 4.605 (90%), 5.991 (95%) and 9.210 (99%). Bold values indicate that the model is
accepted (the probability of failure is equal to the selected confi dence level – α).
Finally, we checked the performance of the models used in the
estimation respectively RM1 (decay factor is 0.94), RM2 - decay factor is
estimated based on the error checking function and RM3 - decay factor is
estimated based on the squared error function for each index.
Revista Română de Statistică - Supliment nr. 9 / 2019 139
Performance of estimated Value at Risk based on the model RM3Table 4
IndexNormal distribution t-Student distribution
10% 5% 1% 10% 5% 1%The unconditional coverage test (Kupiec)
Crisis period: 2008 – 2009BET 0.57 3.54 6.76 6.87 6.41 1.72 BUX 0.25 0.75 0.36 9.62 2.40 0.03 PX 1.29 11.02 5.25 8.19 14.30 0.05 SAX 2.29 1.47 23.42 0.48 1.90 14.17 SOFIX 0.06 7.24 28.63 7.52 9.05 6.76 WIG 0.01 1.08 12.13 10.37 4.18 2.72
Post-crisis period: 2010 – 2016BET 5.46 0.05 20.15 0.02 1.36 8.93 BUX 0.91 0.05 5.73 6.02 1.62 0.22 PX 0.73 7.82 34.67 16.02 15.02 10.13 SAX 4.03 0.73 20.15 0.01 3.20 6.73 SOFIX 0.06 3.20 20.15 6.39 6.24 4.80 WIG 0.51 5.27 27.04 4.31 15.02 3.94
The conditional coverage test (Christoff ersen)Crisis period: 2008 – 2009BET 4.90 5.18 7.86 13.34 8.42 2.07 BUX 1.70 1.23 0.59 14.26 2.59 0.16 PX 3.42 13.07 6.57 9.01 16.82 0.23 SAX 4.09 1.66 25.05 1.19 2.15 15.26 SOFIX 4.51 9.01 32.13 18.59 10.38 11.37 WIG 4.88 6.30 12.71 14.61 7.06 3.14
Post-crisis period: 2010 – 2016BET 5.68 1.74 21.28 1.06 1.81 11.27 BUX 7.00 3.68 6.71 11.00 7.66 0.69 PX 3.54 11.66 36.52 20.23 16.90 11.40 SAX 10.10 2.18 20.21 8.27 5.76 7.17 SOFIX 2.31 9.54 30.15 7.34 13.93 7.98 WIG 11.06 11.86 29.57 8.71 21.71 7.36
Note: Critical values for 2
)1(c are 2.706 (90%), 3.841 (95%) and 6.635 (99%). Critical values
for 2
)2(c are 4.605 (90%), 5.991 (95%) and 9.210 (99%). Bold values indicate that the model is
accepted (the probability of failure is equal to the selected confi dence level – α).
Analyzing the results presented in tables 2, 3 and 4, we fi nd that the best models that manage to capture the risk are represented by RM1 ( .0=l0.94) and RM2 (λ is estimated based on the error checking function).
The two models are accepted for over 50% of the indexes analyzed, both during the fi nancial crisis (2008 - 2009) and in the post-crisis period (2010 - 2016).
Historical simulation method
Using the second method of estimating individual risk, namely historical simulation, we estimated VaR for the same market indices analyzed.
Romanian Statistical Review - Supplement nr. 9 / 2019140
VaR for period 2008 – 2016
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In order to highlight the impact of the size of the estimation sample,
we estimate the Value at Risk in two ways: taking into account all available
historical values, and also based on the use of a fi xed sample, using the rolling window method. The results obtained for the Value at risk can be seen in fi gure 2
Revista Română de Statistică - Supliment nr. 9 / 2019 141
Performance of estimated Value at Risk based on historical simulation
method
Table 5
IndexFull sample Rolling window
10% 5% 1% 10% 5% 1%The unconditional coverage test (Kupiec)
Crisis period: 2008 – 2009BET 68.14 82.35 31.36 68.14 77.73 25.98BUX 48.54 48.33 8.42 48.54 46.44 12.13PX 42.64 35.76 16.32 56.37 40.96 18.58SAX 6.30 0.33 0.36 3.75 0.00 0.36SOFIX 57.99 39.19 14.17 69.89 42.76 16.32WIG 44.09 24.75 23.42 45.55 27.73 23.42
Post-crisis period: 2010 – 2016BET 64.73 53.93 10.90 87.23 45.64 13.29BUX 16.92 13.63 3.19 40.23 33.32 4.28PX 6.69 12.73 5.57 12.34 17.58 7.08SAX 4.01 1.89 1.55 7.57 1.62 3.19SOFIX 52.96 38.28 16.08 54.35 41.85 13.29WIG 37.93 17.58 0.97 24.88 18.67 3.19
The conditional coverage test (Christoff ersen)Crisis period: 2008 – 2009BET 73.52 87.81 37.49 74.57 82.78 33.30BUX 51.67 54.11 9.32 53.28 52.70 12.71PX 45.33 41.64 18.81 57.98 47.05 20.76SAX 12.33 0.76 0.59 7.99 3.04 0.59SOFIX 66.91 47.61 20.68 79.74 51.91 22.23WIG 51.44 30.52 27.77 52.36 34.32 27.77
Post-crisis period: 2010 – 2016BET 66.57 54.30 10.94 91.52 45.81 13.32BUX 26.25 22.90 3.34 55.85 39.03 4.40PX 19.79 35.99 5.67 28.49 35.96 12.08SAX 9.88 5.80 4.62 14.59 3.53 6.90SOFIX 74.03 57.73 16.10 70.21 62.78 13.32WIG 60.58 44.65 16.40 50.70 41.96 3.34
Note: Critical values for 2
)1(c are 2.706 (90%), 3.841 (95%) and 6.635 (99%). Critical values
for 2
)2(c are 4.605 (90%), 5.991 (95%) and 9.210 (99%). Bold values indicate that the model is
accepted (the probability of failure is equal to the selected confi dence level – α).
The effi ciency of the estimates is tested using the two tests well known
in the economic literature, namely: the unconditional coverage test and the
conditional coverage test.
The results obtained are summarized in table 5, based on which we
can see that the Value at Risk manages to better capture the risk on the capital
market, only if we consider the confi dence level of 1%, and especially for the post-crisis period.
Romanian Statistical Review - Supplement nr. 9 / 2019142
Comparison of variance-covariance methodology and historical
simulation
The purpose of this article was to estimate the individual risk existing
on the capital market, using for this purpose the indices of the capital markets
of Bulgaria, the Czech Republic, Poland, Romania, Slovakia and Hungary,
these being the most important capital markets in Central and Estern Europe.
When the variance-covariance method is applied in the risk estimation,
it can be observed that the models are mainly accepted in case of a t-Student
distribution of the returns, as well as of using 1% signifi cance level, which can be observed in the results presented in table 6.
The degree of acceptance of the variance-covariance models
Table 6
ModelNormal distribution t-Student distribution
10% 5% 1% 10% 5% 1%The unconditional coverage test (Kupiec)
Crisis period: 2008 – 2009RM1 83.3% 66.7% 33.3% 16.7% 50.0% 66.7%RM2 83.3% 33.3% 16.7% 16.7% 33.3% 66.7%RM3 100.0% 66.7% 33.3% 16.7% 33.3% 66.7%
Post-crisis period: 2010 – 2016RM1 50.0% 66.7% 0.0% 50.0% 50.0% 83.3%RM2 83.3% 50.0% 16.7% 50.0% 33.3% 83.3%RM3 66.7% 66.7% 16.7% 33.3% 50.0% 50.0%
The conditional coverage test (Christoff ersen)Crisis period: 2008 – 2009RM1 83.3% 50.0% 50.0% 16.7% 33.3% 66.7%RM2 50.0% 66.7% 33.3% 16.7% 50.0% 66.7%RM3 66.7% 50.0% 50.0% 16.7% 33.3% 66.7%
Post-crisis period: 2010 – 2016RM1 16.7% 50.0% 16.7% 33.3% 16.7% 100.0%RM2 16.7% 16.7% 16.7% 16.7% 16.7% 83.3%RM3 33.3% 50.0% 16.7% 16.7% 33.3% 66.7%
Table 7 presents the test results in the case of historical simulation. Based on these results, the importance of choosing the level of confi dence in risk estimation on capital markets is emphasized once again, as the only acceptable performance areas of historical estimation are the choice of a statistical signifi cance threshold of 1%, regardless of the estimation method: complete sample or rolling window. Comparing tables 6 and 7 we conclude that the variance-covariance methodology is more useful in estimating the risk on the capital market, generating better results compared to the historical simulation. In the same time, investors must also be aware of the impact that the assumption regarding the distribution of returns, as well as the selection of the degree of signifi cance have on the correctness of estimating the Value at Risk.
Revista Română de Statistică - Supliment nr. 9 / 2019 143
The degree of acceptance of the historical simulation models
Table 7
Model
Full sample Rolling window
10% 5% 1% 10% 5% 1%
The unconditional coverage test (Kupiec)
Crisis period: 2008 – 2009
Degree of acceptance 0.0% 16.7% 16.7% 0.0% 16.7% 16.7%
Post-crisis period: 2010 – 2016
Degree of acceptance 0.0% 16.7% 66.7% 0.0% 16.7% 50.0%
The conditional coverage test (Christoff ersen)
Crisis period: 2008 – 2009
Degree of acceptance 0.0% 16.7% 16.7% 0.0% 16.7% 16.7%
Post-crisis period: 2010 – 2016
Degree of acceptance 0.0% 16.7% 50.0% 0.0% 16.7% 50.0%
Conclusions
This research aims to identify the impact of main mathematical assumptions of the models used in the estimation of the Value at risk, using two main estimation methods: the variance-covariance approach and the historical simulation. We have shown the impact on the results of the estimation of the following main hypotheses: the chosen confi dence level, the assumed distribution of the returns, the way of estimating the decay factor in the RiskMetrics model, as well as the size of the selected sample. In this regards, it can be observed that the models are accepted especially in the case of the assumption of a t-Student distribution of the used returns, as well as of the threshold of 1% statistical signifi cance.
The choice of the statistical confi dence level in the estimation of the
risk is of particular importance in the estimation, since the only acceptable
performance areas of the historical estimation are represented by the choice of
a statistical signifi cance threshold of 1%.
Finally, it can be observed that the variance-covariance methodology
is more useful in estimating the risk on the capital market, generating better
results compared to the historical simulation.
Romanian Statistical Review - Supplement nr. 9 / 2019144
References
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