University of Amsterdam
Amsterdam Business School
Master in International Finance
Integrating liquidity risk into a VaR model
An application to the Chinese stock market
Linxi Li
Student Number: 6487718
Supervisor: Prof. Peter Boswijk
10 September 2012
1
Abstract
As a noticeable risk in the financial world, liquidity risk has been taken into
consideration in recent risk measurements. Bangia at el. (1999) proposed the wellknown BDSS model to calculate the liquidity adjusted VaR, which is widely recognized
and has been improved in the following years. BDSS suffers from three main drawbacks.
First and foremost, it presumes that market price risk and liquidity risk are perfectly
correlated. Second, it captures a scaling factor and a “fat-tailed” correction factor with
historical empirical distributions. Moreover, the liquidity indicator in the BDSS model,
the bid-ask spread, is not available in the Chinese stock market, as it is an order-driven
market,. This paper aims to explore the correlation between two types of risks in the
Chinese stock market. First, I modify the original BDSS model to fit the situation in China.
Furthermore, GARCH family models are used to find the best estimation equation for
the volatility of all risk indicators. Then I conduct a regression to capture the correlation
factor ρ. After running a backtesting exercise on several models, the empirical results
indicate that by adding two types of risk directly will overestimate the total risk because
it overlooks their imperfect correlation.
2
Acknowledgements
My profound thanks and sincere gratitude go first and foremost to my responsible
supervisor, Professor Peter Boswijk, for his considerable suggestions, patience and
unwavering supports on my thesis. The completion of this thesis would be impossible
without his guidance. Second, I would like to extend my thanks to other teachers and
staff of the MIF, who have instructed and helped me a lot in last precious year. Then, I
want to thank my parents for their spiritual support when I was in bad times. Last, I
leave my special thanks to Jingwen Zhang, Di Guan and other dearest friends, who not
only stand by my side whenever is needed but also make my life in Amsterdam shining
and meaningful.
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Content
1. Introduction .................................................................................................................... 6
1.1 Background ............................................................................................................... 6
1.2 Motivation of this study ............................................................................................ 7
1.3 Dataset and methodology ........................................................................................ 8
1.4 Structure of the thesis .............................................................................................. 8
2. Literature review ............................................................................................................. 8
2.1 Traditional VaR .......................................................................................................... 8
2.2 Liquidity and liquidity risk ....................................................................................... 10
2.3 Liquidity-adjusted VaR ............................................................................................ 11
3. VaR ................................................................................................................................ 14
3.1 Basic VaR ................................................................................................................ 14
3.2 Various approaches to VaR ..................................................................................... 15
3.3 Liquidity adjusted VaR ............................................................................................ 19
3.4 Correlation between two types of risks .................................................................. 22
4. Empirical study .............................................................................................................. 24
4.1 Data ......................................................................................................................... 24
4.2 Market risk VaR ....................................................................................................... 24
4.3 Liquidity risk VaR ..................................................................................................... 28
4.4 Prior integration ...................................................................................................... 29
4.4 Correlation effect .................................................................................................... 30
4.5 Backtesting .............................................................................................................. 31
5. Conclusion ..................................................................................................................... 33
Appendices .................................................................................................................... 35
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Appendix 1: Histograms ............................................................................................ 35
Appendix 2a: Parameters of GARCH class models estimation – Returns ................. 36
Appendix 2b: Parameters of GARCH class models estimation – Adjusted spreads . 37
Appendix 2c: Parameters of GARCH class models estimation – Liquidity adjusted
returns ....................................................................................................................... 38
Appendix 3: Results of ARCH-LM tests ......................................................................... 39
Appendix 4a Backtesting- failure days .......................................................................... 40
Appendix 4b Backtesting- Kupiec Test under 95% confidence level ............................ 41
References .................................................................................................................... 42
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1. Introduction
1.1 Background
Nowadays, Value-at-Risk (VaR) is one of the most popular tools for quantifying market
risk across financial institutions worldwide. A fundamental assumption underlying this
traditional method is that the market price will not be impacted by actions of any
participant and no transaction costs would be incurred when transactions are settled at
current price within a fixed time period. In reality, however, the capital market is not as
liquid as we expect.
Liquidity risk contains two main parts: funding liquidity risk and market liquidity risk.
“The first one has received the most attention from financial institutions, including
banks, for its significance “(Ioan and Adrian, 2010). As to the market liquidity risk, it is
essentially contingent on the availability of a sufficient number of counterparties and
their intention to trade, gaining more attention in the latest years. It is the main focus in
this paper.
Investors, especially institutional investors who hold large positions, will inevitably
influence the asset price when they liquidate their assets, causing liquidity risk. In an
order-driven market, if the investors all follow a similar investment strategy, hold the
similar assets or display herd behavior, market liquidity risk will increase. In general,
investors are facing not only market capital risk, which is measured by fluctuations in
the market price, but also market liquidity risk. Furthermore, both exogenous and
endogenous liquidity risk exist in the Chinese market. The exogenous factors are related
to characteristics of the market microstructure. It is common to the whole market and
beyond the individual’s actions. In contrast, the endogenous factors are specific to the
traders holding large positions willing to unwind them. It varies across the market.
Hence, it is necessary to incorporate liquidity risk in the VaR model.
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1.2 Motivation of this study
A range of approaches addressing the liquidity factor has been suggested in the
academic literature. Under the VaR framework, prior research can be classified into two
group, one based on an optimal execution strategy model (e.g. Hisata and Yamai, 2000
and Shamroukh, 2000) and the other is based on a bid ask spread model (e.g. Bangia et
al. 1999, Le Saout, 2002 and Roy S. 2004). The first one focuses on the effect of large
block transactions on security prices. It measures the VaR, including the liquidity risk, by
seeking an optimal liquidation strategy. On the other hand, the bid ask spread model
treats the market risk and liquidity risk equally. These models are modified by adding
the observed bid ask spread, which is used as an indicator of liquidity risk, to the
traditional VaR model. A major drawback of the bid ask spread model is that it ignores
the dependence between market risk and liquidity risk. Empirically, the correlation
between them tends to increase during extreme events. By simply summing up two
types of risks, the model will overestimate the total risk.
Although the articles above make an improvement to financial risk measurement, they
are mostly studied in developed markets (quote-driven markets). The Chinese stock
market, a market considered as a very important emerging market, has no market
makers, so the previous models need adjustments to fit the situation. Here, an
appropriate liquidity adjusted VaR model for the Chinese stock market will be explored.
Moreover, by applying and backtesting different models, the imperfect correlation
between market risk and liquidity risk will be confirmed.
This paper makes the following three main contributions. First, I apply a more suitable
approach to model the liquidity adjusted VaR for the Chinese market, thereby filling the
gap of research on the Chinese market. Second, this study adds to the evidence on the
importance of liquidity risk in VaR measures. Third, this paper clarifies the existence of
the imperfect correlation between price and liquidity.
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1.3 Dataset and methodology
For the empirical analysis, I use the CSI300 index as study object. The CSI300 index,
covering almost 60% of the Chinese stock market capital, is considered highly
representative and investable. It is based on prices of 300 stocks over 7 years, which is
available from the CSMAR Database. In this paper, following the model initiated by
Bangia et al. (1999), I first present a special framework for the Chinese stock market to
integrate exogenous liquidity risk into a standard VaR model. I mainly address two issues
on the original BDSS model, one is to parametrically estimate a proper percentile of the
non-normal distribution with the Cornish-Fisher expansion, another is to modify the
liquidity VaR calculation suited to the Chinese order driven market with the trading
volume and turnover ratio. Then, I perform statistical analyses on the sample of 1640
observations using the software EViews 5.0 and Excel. The GARCH model class is applied
to estimate the volatility. Based on the former modified BDSS model, I incorporate the
liquidity cost into returns priorly and conduct a regression of the correlation factor to
check the correlation between market risk and liquidity risk. Finally, the backtesting
results provide necessary evidence to draw some meaningful conclusions.
1.4 Structure of the thesis
The remainder of this paper is organized as follows. Section 2 presents a review of
current thinking and practices regarding liquidity risk measures. Based on this prior
literature, the liquidity adjusted VaR model is developed. Section 3 introduces the
sample, model and methodology used in this paper. Section 4 reports and analyzes the
empirical results of the model. Section 5 concludes the paper and discusses some
limitations.
2. Literature review
2.1 Traditional VaR
In the history of VaR, J. P. Morgan, who published the methodology and gave free access
to estimates of the underlying parameters in 1994, contributed a lot to VaR
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development. Two years later, the Basel Capital Accord played a significant role as it
recommended the application of VaR as the risk supervision standard, which
undoubtedly upgraded the VaR method’s status in the world of risk management and
financial regulation. Since then, VaR has been in favor, and has been studies extensively
in the academic literature.
Beder (1995), through a careful study on eight common VaR methods over three
portfolios, finds that VaR varies dramatically, depending on the selection of parameters,
data, assumptions and methodologies. Kupiec (1995) proposes a backtesting method to
verify the results of VaR calculations and develops the VaR confidence interval under
different holding periods. This method is widely used in the following empirical studies.
Angelidis et al. (2004) evaluate the performance of the ARCH and GARCH model family,
including the GARCH, TARCH and EGARCH models, for the construction of daily VaR on
perfectly diversified portfolios. Moreover, they make additional assumptions on
distributions and sample sizes to better illustrate the real situations. Their results
indicate that leptokurtic distributions perform better when conducting VaR forecasts
and that the forecast accuracy depends on the selection of the sample size.
On the basis of the original calculation method, some improvements have been
proposed by various researchers. Ho et al. (2000) model the tails of the return
distribution of Asian financial markets during the Asian crisis using an extreme value
approach. They show that the extreme value approach works better for VaR calculation
of markets characterized by high degrees of leptokurtosis. Duan et al. (2006) derive
limiting models with GARCH-Jump process, and find that the GARCH-Jump model
outperforms alternative models for financial return series.
As simple as it is, the popularity of VaR is accompanied by several drawbacks. Artzner et
al. (1999) criticize VaR on two aspects. First, VaR is defined to measure a quantileof the
return distribution, and it ignores the distribution of extreme losses beyond this quantile.
Second, VaR is not a coherent measurement and it is not always sub-additive, which
may cause problems in risk management applications. They propose the use of expected
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shortfall to solve the problem. By their definition, “Expected shortfall measures the
conditional expectation of loss given that the loss is beyond VaR”. Moreover, Follmer
and Schied (2001) also prove that ES is coherent. Thus, VaR, together with expected
shortfall have been two main measures to describe the loss distribution in real world
applications.
2.2 Liquidity and liquidity risk
According to the literature so far, liquidity concepts can be defined into three categories.
The first one is based on Cost Theory. Liquidity refers to the transaction cost, including
the currency cost and time cost. This theory, building on the loss from lack of liquidity,
was first proposed by Demsetz (1968) based on research of the relationship between
financial market liquidity and the cost of transacting. Amihud and Mendelson (1989) say
that “liquidity refers to the cost to complete the transaction in a certain period or the
time to find an ideal price”.
The second group holds the idea that liquidity is the capability of financial assets being
liquidated with a lower cost and faster speed. Black (1971) pointed out that a liquid
market is able to settle trades on any number of securities immediately at the current
market price. Massimb and Phelps (1994) say that “liquidity refers to a market’s ability
to provide immediate execution for a potential market order (immediacy) and the ability
to execute small market orders without large change in the market price (market
depth)”.
The third group describes liquidity according to multi-dimensional indicators. Garbad
(1985) and Schwahz (1991) consider that liquidity spans into following dimensions:
depth, breadth and resiliency.
Liquidity risk consists of two main parts: funding liquidity risk and market liquidity risk.
Market liquidity risk is concerned with the ease with which positions in the trading book
can be unwound, while funding liquidity risk is concerned with being able to meet cash
needs as they arise (Hull, 20.., 403-404).
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According to the concept of liquidity, several studies have focused on liquidity indicators.
Amihud and Mendelson (1986) employ the bid-ask spread relating to trading cost. So do
Bangia et al. (1999), which is widely used as liquidity proxy nowadays. Bekaert et al.
(2007) construct a liquidity measure by observing the proportion of zero daily returns
over the relevant month in an emerging market. Amihud (2002) defined illiquidity cost
as” the average ratio of daily absolute return to the dollar trading volume”. Another
original liquidity measure is the Amivest liquidity ratio, which is put to use in NASDAQ
and it is simply computable on a daily basis (Brunner, 1996, Elyasiani et al, 2000). Wyss
(2004) summarizes those liquidity measures into two groups, one-dimensional and
multi-dimensional ones. Multi-dimensional liquidity measures combine properties of
one-dimensional liquidity measures and thus represent more aspects. He mentions 4
indicators in the fractional form with spread in the numerator and volume in the
denominator. Xu, Feng and Wu (2004), Zhao and Mou (2009) are concerned about the
Chinese market and propose the adjusted liquidity indicator. They state that Chinese
stock market has experienced a rapid expansion, so adjusted by trading volume alone
could cause distortion. Their new indicator of liquidity is modified by the turnover ratio.
In this paper, I will select the proper indicator of liquidity in the Chinese stock market to
measure liquidity risk. The liquidity risk discussed here refers to market liquidity part.
2.3 Liquidity-adjusted VaR
Liquidity risk has gained attention in recent years. There are many studies in the related
literature of incorporating market liquidity risk in VaR models.
One way to deal with liquidity risk is based on the optimal execution strategy. Hisata and
Yamai (2000) propose an approach to quantify the market liquidity risk based on the
consideration of market impact with optimal execution strategy for liquidating the
trader’s own position. The paper also presents a specific model providing a closed form
solution for calculating liquidity-adjusted VaR, and tests the effectiveness of this
framework in the financial risk management through numerical empirical examples.
Tinga, Warachka and Zhao (2007) examined three parametric specifications that proxy
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for increasingly realistic market conditions. They find that in less liquid markets the
optimal strategies facilitate rapid liquidations and that volatility is stochastic when
market liquidity is unpredictable. All these models are used in markets with a market
maker.
The second approach for the integration of liquidity risk in VaR consists of modeling the
exogenous liquidity component. Bangia et al. (1999) state that liquidity risk could be an
additional source of market risk. Accordingly, they develop a simple liquidity risk
methodology by building a model consisting of a liquidity component, which is
measured by the bid-ask spread, and integrate it into the ‘standard’ VaR, showing that
in emerging markets, total risk could be distorted by up to 30% when ignoring the
liquidity factor. Angelidis and Benos (2006) estimate a trade volume dependent model
based on the components of the bid ask spread and add it to the ‘standard’ VaR
measure. With a similar methodology, Le Saout (2002) applies the liquidity adjusted VaR
model provided by Bangia et al. (1999) to the French stock market. The results show
that the exogenous liquidity risk can represent more than half of the market risk on
illiquid stocks. Moreover, Roy (2004) relates the model provided by Bangia et al. (1999)
to the Indian debt market. His study shows that liquidity risk plays a key role in the
aggregate risks absorbed by the financial institutions. All these studies model the market
risk and liquidity risk by simply summing up the two parts.
Meanwhile, Shamroukh (2000) argues that the existence of market and liquidity risk
factors is common. The paper also proposes a model where the liquidation of portfolio
is taken orderly if the liquidation period occupies the same time to holding period. He
says that “Market liquidity risk can be modeled by expressing the liquidation price as a
function of trade sizes, thus imposing a penalty on instantaneous unwinding of large
position”.
In the relatively recent study, Stange and Kaserer (2009) integrate liquidity risk
measured by the weighted spread into a VaR framework. They use a data set from
Deutsche Borse AG, which is considered to be the most representative for the German
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stock market, finding that the original price risk has increased by over 25% due to the
additional liquidity risk and that the common approach of simply adding VaR together
substantially overestimates total risk because of the (imperfect) correlation between
liquidity and price. In contrast, Wu (2009) estimates ‘standard’ VaR on the basis of
liquidity adjusted returns and she forms a skewed Student’s t AR-GJR model to capture
the asymmetric, non-normality and excess skewness of returns. The empirical evidence
supports her viewpoint that simply adding the two risk measures would underestimate
the risk. Surprisingly, the empirical results in these two papers are against each other.
Nonetheless the imperfect correlation between price and liquidity has been verified in
both studies. In another paper by Zhang et al. (2010), they model the risk with a GARCHEVT approach to deal with the time-varying heteroscedasticity and fat tail factors. The
result indicates that the correlation between liquidity risk and market risk is increasing
in the upper tail and lower tail and it is symmetric.
The relation between market returns and liquidity is also the subject of a large body of
research. Amihud (2002) shows that over time, expected market illiquidity has a positive
effect on ex ante excess stock return, stating that expected excess stock return
represents an illiquidity premium to some extent. Bekaert et al. (2007) provide a
comprehensive study on the liquidity impact on expected returns or valuations. They
show a negative relation based on emerging markets. More directly, Hameed et al.
(2008) report evidence that market liquidity declines during bear markets.
Ernst et al. (2008) propose a new and easily implementable parametric method to
adjust a Value-at-Risk risk measure for liquidity risk. To deal with non-normality in price
and liquidity cost data, they employ the Cornish-Fisher approximation, which modifies
the normal distribution scaling factor based on the skewness and kurtosis of the true
distribution. Then they test the modified L-VaR, as well as a standard specification by
Bangia et al. (1999) and provide evidence that the new methodology produces much
more accurate results than alternative empirical risk estimations.
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Following Bangia et al. (1999) and noticing the various drawbacks it contains, in this
paper I modify their model to fit the Chinese stock market situation with the most
appropriate approaches proposed in the literatures above, getting a new liquidityadjusted VaR model.
3. VaR
3.1 Basic VaR
VaR is defined as “a measure of the worst expected loss that a firm may suffer over a
period of time that has been specified by the users, under normal market conditions and
a specified level of confidence”. (Learning Curve, 2003)
Assume that an asset has initial value P0 and R is the return during the holding period.
Then the asset value in that certain time can be presented as P=P0 (1+R). Additionally,
assuming that returns have expected mean μ and volatility σ, under the selected level of
confidence, the asset relative VaR is calculated as,
VaRrelative  E  P   Pmin   P0 ( Rmin   )
If we do not consider the mean of asset value as the basis, then the absolute VaR is
calculated as,
VaRabsolute  P0  Pmin   P0 Rmin
Here, Pmin and Rmin are the minimum asset value and minimum asset return, respectively.
The most commonly used VaR models are under the assumption that asset returns
follow a normal distribution with zero mean. For instance, the Basel Accord usually
requires a one-day horizon VaR and assumes that one-day returns are normally
distributed, and then the 99% worst value can be calculated as,
( E  r   2.33 )
P99%  Pe
t
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Where E(r) and σ denote the first two moments of distribution of the asset returns, and
the multiple of 2.33 represents the corresponding percentile.
3.2 Various approaches to VaR
At the moment, the calculation of VaR can be classified into three types: the nonparametric approach, the semi-parametric approach and the parametric approach. The
non-parametric and semi-parametric approaches are aimed at getting the distribution
function of asset values or asset returns directly, whereas the parametric approach
starts with building a formula of volatility under some distributional assumptions.
3.2.1 Non-parametric approach to VaR

Historical simulation
Among all the ways to implement the VaR calculation, this is the simplest and easiest.
The basic idea behind the assumption is that the change in market value or return is
stationary through time. Putting it slightly differently, we could use the historical data to
forecast the future value.
Despite the undeniable advantage of simplicity, historical simulation, as a full valuation
suffers from several obvious drawbacks. In the real world of finance, markets change
from time to time and never follow a single pattern. Moreover, a large number of data
points is essential and the result of VaR might be volatile.
There are various extensions to the basic historical simulation approach. The weighted
historical approach combines the advantages from Riskmetrics and the ordinary
historical approach. It allows the weights of the observations to decrease exponentially
as the observations become older, making the most recent observations more
important. Another way is the volatility updating procedure, which combines the
advantages from the GARCH approach by taking account of fluctuations in the volatility
over the period.

Monte Carlo simulation
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The Monte Carlo simulation process is implemented with computer techniques. First,
repeat simulations of a random process to determine the asset value. After thousands
of times, the distribution of the simulations will converge to the true distribution of the
asset price, from which VaR estimates can be calculated.
Monte Carlo simulation can simulate large amount of scenarios, including different
behaviors and distributions, thus it is able to deal with issues such as non-linearity, timevarying volatility and fat tails. However, it overly depends on the stochastic process and
the selected historical data. Another potential risk lies in the model building process.
The data series are simulated based on an estimated model, which may be distorted.
3.2.2 Parametric approach
Based on historical data in the sample period, the parametric approach estimates the
parameters of the distribution function or the density function of returns, from which
the VaR is calculated. This is the most commonly used approach in constructing a VaR
model and is the main approach in this paper.

VaR with normal distribution
We can simplify the VaR calculation under the normal distribution. The advantage of the
normal assumption is that it is easy to transform the VaR from different confidence
levels and thus to compare among institutions. Meanwhile, another assumption of zero
mean returns is usually made. From empirical results, large data-sets of daily equity
returns suggest that mean returns are around zero, albeit a little bit positive (Dowd,
1998). For example, assume that the returns of an asset are normally distributed with
mean μ and variance σ2, then VaR can be calculated as,
VaR=-P0Zασ
Where P0 refers to the initial asset value, α is significant level, Zα denotes the α
percentile of the standardized normal distribution which is negative and easy to
transform under different confidence levels, ensuring the final VaR result is positive.
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As a convenient way to forecast VaR, the assumption of normality has a number of
pitfalls. Substantial empirical evidences on returns show the distribution to display fat
tails and leptokurtosis, which may result in distorted estimation of the VaR. So other
distributions have been suggested as a better description of returns, e.g. the student t
distribution and the generalized error distribution (GED).
Another shortcoming underlying the simple VaR model is that it presumes a constant
volatility σ throughout the observation period. However, a stylized fact of stock returns
is volatility clustering, which is defined as large price variations followed by large price
variations (Cont, 2001). Su et al. (1997) suggest that volatility shows high persistence
and is predictable. To solve the problem, ARCH and GARCH models are introduced,
which gives access to predicting time-varying volatility with historical data.

Autoregressive Conditional Heteroskedasticity (ARCH) model
The ARCH model, introduced by Engle in 1982, is described as ARCH (q) where q is the
number of lagged values on r2 used in the model, stated as the order of the ARCH
process. The ARCH model can be shown as,
p
 t2 = + i t2-i
i =1
where εt = rt – E(r). For the conditional variance to be positive, the parameters must
satisfy ω>1 and β≥0 for all i=1 to p. To ensure stationarity of σt+1 2, the parameters are
also required to satisfy the constrains that,
p
  <1
i
i =1

GARCH
The difficulties with the ARCH process, as noted by Bollerslev (1986) are a totally free lag
distribution, where the required number of lags could be high, which could lead to a
violation of the non-negativity constraints. He presented an extension of the ARCH
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model denoted as GARCH (p, q) where p represents the order of the GARCH elements
and q represents the order of the ARCH elements. The model looks like this,
p
q
i =1
j =1
 t2 = + i t2-i +  j t2-j
(1)
For stationarity and the variance to be mean reverting, it is required that Σiαi+βj<1.

TARCH and E-GARCH
The original GARCH model has some limitations. Several improvements have been made;
some of those capture the leverage effect on the financial market. The leverage effect
refers to the situation that a large negative shock is expected to increase volatility more
than a large positive shock. In order to describe the asymmetry on the news impact,
Zakonian (1990) proposed the threshold GARCH (TARCH or GJR GARCH) model. The
model looks like this,
 t2 = + t2-1 + t2-1dt -1 +  t2-1
(2)
dt =1 if  t <0，bad news

Where dt is a dummy variable 
, and t has a symmetric
t
dt =0 if  t  0, good news
distribution. To guarantee the positive of  t2 , the constraints are ω>0, α≥0, β≥0, γ≥0. For
1
stationary, the constraint is that 0< + +  <1 .
2
Another way to measure the asymmetry is EGARCH, proposed by Nelson (1991). In
contrast to the GARCH model, no restrictions need to be imposed on the parameters,
since the logarithmic transformation ensures that the forecast of the variance is nonnegative. The model looks like this,
ln  t2 = +
 t -1

+ t -1 + ln  t2-1
 t -1
 t -1
18
(3)
Where γ indicates the asymmetric effect. With γ<0, the leverage effect could be showed
in the model.
In this paper, in order to find out a suitable model for VaR, I will try to incorporate
leverage effects and fat tails with the returns’ volatility estimated by a GARCH model
under a non-normal distribution.
3.2.3 Semi-parametric approach
The traditional parametric and non-parametric approaches are precise in estimating the
part of the distribution where data are sufficient, whereas VaR is more concerned about
the part of the distribution with less data. Therefore several drawbacks are maintained
in those approaches, especially for forecasting. The semi-parametric approach is
developed to fix this problem, which includes tail estimation based on Extreme Value
Theory and conventional autoregressive VaR (Engle and Manganelli, 2004). In this paper,
I mainly focus on the parametric approach, so no further details about semi-parametric
methods are given here. Future research may follow this newly arisen approach to
improve the liquidity adjusted VaR.
3.3 Liquidity adjusted VaR
In the preceding subsection, appropriate models to estimate the volatility have been
discussed. From the distribution of returns and the volatility estimation, VaR can be
calculated.
3.3.1 BDSS model
As mentioned before, exogenous liquidity risk is caused by market characteristics, which
influence every participant equally. In a liquid market, a large volume of trading happens,
and the bid-ask spread offered by the market maker should be small and stable.
Measuring the liquidity risk with the bid-ask spread and including the spread into VaR
model, Bangia et al. (1999) built the famous BDSS model, which consists of two parts:
one measures the market risk caused by the devaluation of capital and the other one
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measures the liquidity risk caused by the willingness to liquidate current assets,
presented as PVaR and COL (cost of liquidity) respectively. The model looks like this,
1
LAdj -VaR =PVaR+COL=Pt (1-eE(rt )-Z t )+ Pt (S +a )
2
rt = ln (
Pt
)
Pt -1
Where Pt refers to the mid-price at time t, rt refers to the log return, E[rt] and  t are
the first two moments of the distribution of r, most of time E(r)=0. Z refers to the α
percentile of the mid log return distribution. BDSS simplifies Z =2.33 as the multiple for
standard deviation under normal distribution. S denotes to the average relative spread,
 refers to the standard deviation of the relative spread. a is the non-Gaussian
distribution scaling factor ranging from 2 to 4.5 according to empirical analysis.
Additionally, the BDSS model regards the possible situation that asset returns deviate
far from normality. They designed a correction factor θ to describe the fat tailed or
leptokurtic distribution.
 =1.0+ ln( /3)
Here κ denotes kurtosis, the fourth moment of return distribution. For instance, κ=3 and
θ=1 if the returns follow normal distribution. And  refers to a constant whose value
depends on the tail probability. BDSS derive the constant  by running a regression of
the right hand side of equation with historical VaR.
3.3.2 Modified BDSS model
Following the model initiated by Bangia et al. (1999), I present a special framework for
Chinese stock market to integrate exogenous liquidity risk into a standard VaR model. I
mainly mend two parameters on the original BDSS model, one is to estimate a proper
percentile of the non-normal distribution with the Cornish-Fisher expansion, another is
20
to modify the liquidity VaR calculation suited to Chinese order driven market with the
trading volume and turnover ratio.
My modified model looks like this,
LAdj  VaR=VaRmarket +VaRliquidity  Pt (1  e  z * t ) 
Pt 
1
Pt (V  zV V )
2
P0  Ph  Pl  Pc
S
P  Pl
,S h
,V =
,
tr*Vo
Pt
4
(4)
Where P0 ,Ph ,Pl ,Pc refer to the opening price, highest price, lowest price and closing
price of daily trade respectively. ( ) denote the modified parameters which will be
introduced below.
The BDSS model is applied to a market with market makers, who provide the bid-ask
spread S. However, the Chinese stock market is an order-driven market and the bid-ask
spread is replaced by the daily spread between the highest and lowest price in the open
order, S=
Ph  Pl
. Then I follow the Amihud (2002) illiquidity measure by dividing the
Pt
trading volume to the spread. Last, in order to eliminate the effect of Chinese stock
market rapid expansion on the trading volume, following Xu, Feng and Wu (2004), I
modify the liquidity indicator with turnover ratio and the final expression of liquidity
indicator looks like this,
V=
S
tr*Vo
Here tr and Vo refer to the turnover ratio and Yuan trading volume (in 1 billion),
respectively. Yuan volume in one billion means that I assume the position for each
trader is 1 billion yuan. Large amount as it may look like, the fictitious investment
portfolio in our later empirical analysis only contains the CSI300 index, which covers 60%
of total market capital. So the large trading position is necessary and reasonable. This V
21
ratio illustrates the rate of spread change caused by unit trading volume adjusted with
turnover ratio. Zhao and Mou (2009) also use this ratio to measure the liquidity risk. In
the empirical analysis, I retain the model without turnover ratio adjustment to compare
with the model above. The Model with turnover ratio adjustment outperforms other
model, supporting my integration of this parameter.
Since the scaling factor and correction factor in BDSS are both derived from historical
methods, which suffer the same drawbacks as mentioned in the previous section, I
modify these two parameters with the Cornish-fisher expansion developed by Cornish
and Fisher (1937). Johnson (1978) says that “we obtain explicit polynomial expansions
for standardized percentiles of a general distribution in terms of its standardized
moments and the corresponding percentiles of the standard normal distribution”. As we
see in the analysis later, returns and modified spreads do not follow Gaussian
distributions, especially the spreads deviates from the normal distribution. The proper
application of an estimated correction factor is necessary. The expansion for the
approximate α-percentile of a standardized random variable is calculated as,
1
1
1
z  z + （z2 -1）* + ( z 3  3z )*   (2 z 3  5 z)*  2
6
24
36
(5)
Here z is the α percentile of the standard normal distribution, γ refers to the skewness
and κ refers to the excess-kurtosis of the random variable.
3.4 Correlation between two types of risks
To simplify the calculations, the BDSS model implicitly assumes perfect correlation
between prices and liquidity cost. However if they are other than perfectly related, BDSS
will wrongly estimate the total risk. Also, most empirical evidence (e.g. Amihud 2002,
Bekaert et al. 2007, Hameed et al., 2010) show that illiquid securities should have higher
expected returns. Hence, it may be inaccurate in calculating liquidity adjusted VaR if we
omit the relation between market risk and liquidity risk.
Stange and Kaserer (2009) apply a decomposition of total risk and define the correlation
factor κ as residual of
22
VaRtotal  VaRmarket  VaRliquidity   *VaRliquidity
(6)
Where κ≤0 measures the tail correlation factor between mid-price return and liquidity
cost (κ=0 in case of perfect correlation). Later I will follow the same framework to verify
the existence of imperfect correlation between two types of risks.
The total VaR on the left hand side of this equation (6) is calculated from the liquidity
adjusted return. Similar to the adjustment in the liquidity indicator of modified BDSS
model before, based on the paper of Amihud (2002), I integrate the turnover ratio to
the daily spread as well, the liquidity cost looks like this,
V=
S
tr*Vo
where S and Vo denote the daily spread and trading volume respectively. Hence the
object of liquidity adjusted return looks like this,
LArt =rti -Vt i .
VaRtotal (q)=1-exp (LArt )
On the right hand side of equation (6),
VaRprice (q)=1-exp (rt ) ,
VaRliquidity (q)=1-exp(Vt )
Then, by regressing equation (6) with 3 VaR series above, I can obtain the correlation
factor κ. If κ =0, the two risks will be perfectly correlated. If κ <0, then adding two types
of risks will overestimate the total risk.
In Stange and Kaserer’s empirical results, on average, 60% of the liquidity cost risk is
diversified away, verifying the non-perfect tail correlation. I will apply this approach in
my following analysis.
23
4. Empirical study
4.1 Data
This paper will use the CSI300 index ranging from 4/April/2005 to 30/December/2011,
making nearly 7 years of 1640 observations. The data are from the CSMAR China stock
market Trading Database, which is widely used by Chinese academe and financial
companies. Besides, a dataset of turnover ratios is provided by the China Securities
Index Co. LTD, who compiles the CSI300 index.
The CSI300 index, covering almost 60% of the Chinese stock market value, is considered
highly representative and investable. Initiated in April 2005, it is based on 300 stocks.
With these most actively traded constituent stocks, CSI300 reflects the situation of the
main part of market investment activity. Returns of the CSI300 index are available on
trading days (5 days per week without holidays).
All estimations of econometric models in this paper are implemented with EViews 5 and
Excel.
4.2 Market risk VaR
4.2.1 Statistics description
The returns on the CSI300 index are calculated on a continuously compounded or logreturn basis, as follows
rt = ln (
Pt
)
Pt -1
where rt represents the return at time t and Pt represents the index at the given time.
Table 1 presents descriptive statistics for the returns. As expected, the mean of the
returns is around zero, while the skewness (0.002272) and kurtosis (6.093776) show
that returns do not follow the normal distribution, but have a peaked and fat tailed
distribution. (See also appendix 1)
24
series
observations
Mean
maximum
minimum
Std. Dev.
skewness
kurtosis
Returns
1639
0.000521
0.107272
-0.079636
0.016400
0.002272
6.093776
Table 1 Descriptive statistics of CSI300 index returns
Returns of CSI300
0.15
0.1
0.05
0
-0.05
-0.1
Date
2006-07-05
2007-09-24
2008-12-17
2010-03-15
2011-06-09
Figure 1 Line graph of CSI300 index returns
Figure 1 above shows how daily returns on the CSI300 index have changed throughout
the period 04/04/2005 to 30/12/2011, ranging from about -10% to 10%. This fits the
price limits policy in China, which has been imposed by Chinese government since 1996.
Any stock on the exchange is not allowed to fluctuate more than the daily price spread
of 10% (Ji, 2009 and Feng, 2002). From the graph, volatility clustering is evident, as some
periods show high volatility while others show low volatility. As can be seen in the figure,
the most recent financial crisis starting in 2008 have led to higher variances.
4.2.2 Autocorrelation
At the beginning of the process, it is good to check how the sample data behaves by
constructing correlograms for returns. The correlogram in table 2 left side shows the
correlations between the returns and the lagged returns referred to as autocorrelation,
a lag of one would in our sample show the correlation between the daily returns and the
daily returns one day back in time. A lag of two would go two days back in time etc.
25
After estimating several AR (q) equations of returns (with constant term included), I
conclude to start with AR (3) with the smallest AIC.
A correlogram of the squared returns is used to see how the autocorrelations of the
variances are. Table 2 right side shows the correlogram for the squared returns. It is
interesting to see that the correlations tend to decline with increasing lag lengths. This is
a clear signal for volatility clustering and therefore heteroskedasticity since with
volatility clustering, correlations are expected to decline when lags move further away in
time. I decide to start with GARCH (1,1).
Returns
Squared Returns
lags
AC
PAC
P-value
AC
PAC
P-value
1
2
3
4
0.291
-0.062
0.085
0.087
0.291
-0.16
0.171
-0.008
0.00
0.00
0.00
0.00
0.220
0.137
0.136
0.177
0.220
0.093
0.094
0.129
79.568
110.30
140.80
192.52
5
0.001
0.001
0.00
0.111
0.036
212.88
Table 2 Correlograms of CSI300 index (squared) returns
4.2.3 Test of normality
From table 1, there is clear evidence that the returns have a leptokurtic distribution.
Figure 3 gives additional evidence of this. The quantile-quantile plot is a graphical
technique for determining if two datasets come from populations with a common
distribution (definition in e-Handbook of Statistical Methods). If two series follow the
same population distribution, the points should fall approximately along the reference
line. In EViews, the vertical axis refers to the quantile from the normal distribution,
while the horizontal axis refers to the returns distribution. The departure indicates a
non-normal distribution.
In order to model more adequately the thickness of the tails, I will use two different
distributional assumptions for the standardized residuals: Student’s-t and the
Generalized Error Distribution (GED).
26
Theoretical Quantile-Quantile
8
6
Normal Quantile
4
2
0
-2
-4
-6
-.12
-.08
-.04
.00
.04
.08
.12
CSI300
Figure 2 Q-Q plot on CSI300 index returns
4.2.4 GARCH model selection and scaling factor
There will be two selection criteria used to see which GARCH model is considered most
fitting on the time-series. These are Akaike Information Criterion (AIC) and Schwartz
Bayesian Criterion (SBC or SC). The preferred model will show the lowest values of these
criteria. Results supporting the strongest model based on both AIC and SC is TARCH (1, 1)
with t-distribution. TARCH model reflects the asymmetry news impact in the CSI300
returns series.
The results of ARCH-LM tests on the residuals of six GARCH models suggest that no
more heteroscedasticity exists. All models give a good fit of the time-varying
characteristics of returns. See details in appendix 2a and 3.
According to the Cornish-Fisher expansion, the original scaling factor, which
corresponding to normal distribution, could be improved to fit the real distribution.
Adjusted scaling factors are 1.5916 and 3.0612 under the 95% and 99% confidence level,
respectively.
4.2.5 VaR calculation
Finally, the estimated parameters, plugged in Equation (2), which are used to estimate
the market returns volatility, look like this,
27
 t2 =1.45E-06+0.048752 t2-1 +0.029414 t2-1dt -1 +0.932733 t2-1
Meanwhile, the market risk VaR (under 99% confidence level) is calculated as,
VaR market =Pt (1  e3.0612* t )
4.3 Liquidity risk VaR
4.3.1 Statistics description
The variable used to capture the liquidity component is the modified spread, since the
Chinese stock market is order-driven and the “bid-ask spread” employs the spread
between the highest and lowest price in the open order. When it comes to the liquidity
variable, I use the turnover ratio and trading volume to modify the bid-ask spread S in
BDSS, V=
S
, where tr and Vo refer to the turnover ratio and trading volume.
tr*Vo
Moreover, I keep the modified spread without turnover ratio division V=
S
as a
Vo
comparison to V
The statistics of V are in table 3 below, no negative value appears in the liquidity
indicators. Compared to returns, the density becomes less normal with more right tails
(high skewness). So the normal distribution is also inappropriate for the spread. Instead,
I consider the t-distribution or GED. (See also appendix 1.) However, since these two
distributions are also symmetric, it is not clear that they perform better than normal. In
this paper, for the sake of consistency, I simply use them to capture the leptokurtosis of
spreads, but future studies can work on this topic to obtain more suitable distributions.
series
observations
mean
maximum
minimum
Std. Dev.
skewness
kurtosis
M. spread
1640
0.000623
98bp
12.6bp
0.001007
4.210972
26.40166
Table 3 Descriptive statistics on CSI300 index modified spreads
4.3.2 Autocorrelation and test of normality
The same rationale is used in testing the autocorrelation and normality to spread terms
as to return in the preceding subsection. Judging from the correlogram as well as the AIC
28
for each AR (q) equations, AR (4) is chosen finally. Likewise Q-Q plot for spread supports
the conclusion of non-Gaussian distribution.
4.3.3 GARCH model selection and VaR calculation
Based on AIC and SC, the EGARCH (1, 1) with GED distribution gives the best fit,
indicating asymmetry in spread. In fact, although all the parameters are significant from
the model, the fitting effect of final model is not satisfactory enough because of the
unique distribution of the modified spread (see appendix 1). Future study may focus on
exploring a more proper model to describe the spreads. The liquidity risk VaR is
calculated as,

 t -1

2
+0.471427 t -1 +0.997855 ln  t2-1
ln  t =-0.050326-0.098420

 t -1
 t -1

1

VaRliquidity = 2 Pt (S  4.2337 S )
See more details on estimated results in appendix 2b and 3.
4.4 Prior integration
4.4.1 Statistics description
The variable used to calculate the total VaR is the liquidity adjusted return, where the
adjusted returns are equal to returns minus liquidity cost:
LArt =rti -Vti
Since the adjusted return is a reflection of the market risk and the liquidity risk
simultaneously, it is not required to put additional terms into the formula. Moreover, I
keep the liquidity adjusted return expression without turnover ratio adjustment as a
comparison to LArt.
The statistics of LArt are in table 4 below; the mean of the returns is around zero (0.0001), while the skewness (0.018988) is a little bit larger than the original returns, and
29
the kurtosis (5.927795) shows that the distribution is peaked and fat tailed. (See also
appendix 1)
series
observations
mean
maximum
minimum
Std. Dev.
skewness
kurtosis
Adj. returns
1639
-0.00010
0.107202
-0.079904
0.016593
0.018988
5.927795
Table 4 Descriptive statistics on CSI300 index liquidity adjusted returns
4.3.2 Autocorrelation and test of normality
Judging from the correlogram as well as the AIC for each AR (q) equation, AR (3) is
chosen finally. Not surprisingly, the LAr time series, which is the differences between the
returns and the illiquidity costs, also exhibits volatility clustering.
4.3.3 GARCH model selection and VaR calculation
Based on AIC and SC, the TGARCH (1, 1) with t distribution is preferred. The modified
scaling factor is 3.0343. Total VaR is calculated as,
2
2
2

 t =0.0000015+0.050113* t -1 +0.026417* t -1*( t -1 <0)+0.932331* t -1


VaRtotal =Pt (1-exp(-3.0343* t ))
See more details on estimated results in appendix 2c and 3.
4.4 Correlation effect
Following the Stange and Kaserer (2009), I conduct the regression on the parameter κ as
residual of
VaRtotal  VaRmarket  VaRliquidity   *VaRliquidity
κ
Results are in line with former researches, i.e. correlation factors are negative ( =-0.94
κ
under 95% confidence level and =-0.99 under 99% confidence level ), verifying the
imperfect tail correlation between market risk and liquidity risk and that adding two
types of risk substantially overestimates total risk. Moreover, κ is quite small, indicating
that a large proportion of liquidity risk is eliminated due to the correlation in Chinese
30
stock market, which further implies a tiny liquidity impact based on Stange and Kaserer
(2009).
Table 5 below provides two examples of one-day liquidity adjusted VaR given α=5% and
1%, respectively. Comparing VaRtotal with Modified BDSS model ( VaRmarket  VaRliquidity ),
we could see that VaR (LAr) is smaller than the summation, which indicates that the
simply adding method overestimates the risk as I state earlier.
α=5%
α=1%
VaRtotal
61.6659
115.9327
VaRmarket  VaRliquidity
71.6993
138.2944
Table 5 Comparison of two methods on one-day VaR (11/10/2010)
4.5 Backtesting
Backtesting is simply a historical test of the accuracy of the VaR model. The most
commonly used test of a VaR model is to count the VaR failure days when the losses in
the asset positions exceed the VaR estimates. A good VaR estimation implies that the
number of failure days is on average the same as the confidence level indicates. In
reality, it is rarely the case that we observe the exact amount of failure days expected to
happen. However, models with less days of failure are in general considered to be
better intuitively.
Appendix 4a generates the results of backtesting from different testing windows and
different confidence levels on all the models in this paper. Compared with the expected
failure days, the VaR estimates from the modified BDSS model, together with the VaR of
the liquidity adjusted returns have no (or only one) failures. However, when looking
closely at the details, the modified BDSS model, although it seems to offer the best
results, overestimates the risk in fact, in that failure days are much fewer than the
expected number. In general, the VaR based on liquidity adjusted returns has a
remarkable performance.
Alternatively, define the hit sequence as,
31
p

1, if R t +1 <-VaRt +1,
It +1 = 
p

0, if R t +1  -VaRt +1,
T
Consider N =  I t , where N refers to the observed number of failure days in the test
t =1
period. As argued in Kupiec (1995), “the failure number follows a binomial distribution,
N ~ B(T , p) and consequently the appropriate likelihood ratio statistic, under the null
hypothesis that the expected exception frequency
N
 p , is:”
T
N T -N N N
) ( )
T
T
)
T -N N
(1-p) p
(1LR =2ln (
The null hypothesis for this test is that Pr (It+1=1) =p on average. If the null hypothesis is
rejected, the model is deemed inaccurate.
The critical values of the χ2 distribution with degrees of freedom 1 and significant level
of 5% and 1% are 3.841 and 6.63, respectively. Appendix 4b summarizes the results on
this test. The entire LR statistic in the modified BDSS VaR is larger than the critical value,
leading to the rejection of null hypothesis. This is another proof of the overestimation
by adding up two types of risks directly. Meanwhile, under the more strict 99%
confidence level, the results reject the null hypothesis in each cell. So no outcomes are
presented in the appendix.
To sum up the two approaches on backtesting, I could say that the VaR estimated from
the liquidity adjusted returns work better on the CSI300 index. Although the modified
BDSS model is able to maintain the failure days under a certain level, it overestimates
the risk, which may lead to a higher cost in risk management. As the correlation
between market risk and liquidity risk is imperfect, we need to consider their co-effect
when calculating the total VaR in the future.
32
5. Conclusion
In this paper, following the framework in Bangia et al. ‘s (1999) BDSS model, I try to
measure the Chinese stock market VaR by splitting risk into two parts, the market risk
and the liquidity risk. Unlike most of the developed market, the Chinese stock market
has no market maker. So the liquidity variables cannot be derived from the market bidask spread. Based on several liquidity measures researches, I modify the liquidity
variables with turnover ratio and trading volume to get around this problem. Moreover,
the non-normal distribution of both price returns and spreads is settled by incorporating
the Cornish-Fisher expansion.
In order to estimate the volatility of CSI300 returns, I conduct autocorrelation tests and
model the time series with the GARCH family. My study shows that the extended
versions of GARCH work better to eliminate the volatility clustering effect in the index
returns, under all distributional assumptions and both confidence levels. As expected,
the leptokurtic distributions provide better estimators of VaR since they perform better
in the low probability regions which VaR tries to measure. The final models applied to
estimate the returns and spreads series are selected by AIC and SC.
However, former studies show that adding the two risk measures directly would
overestimate the total VaR. After incorporating liquidity adjusted returns to model the
VaR directly, where liquidity adjusted returns equal to the original returns minus
liquidity cost, the correlation has been verified through a regression on the correlation
factor. My empirical results stand in line with the overestimation conclusion.
In the last section, judging from the backtesting results on VaR models, I point out that
the VaR estimated from the liquidity adjusted returns fits CSI300 better. As the
correlation between market risk and liquidity risk is imperfect, we need to consider their
co-effect when calculating the total VaR in the future.
Several venues are still open for future research. Since I mainly focus on the simple
parametric approach to calculate VaR, other possible approaches are not discussed
33
within this paper. Additionally, more appropriate model to describe the spreads
distribution can be considered. Third, there is no guarantee that the model in this paper
will work the same throughout the whole Chinese stock market. It is hoped that my
work will help gain further progress in integrating liquidity risk into a risk model for the
Chinese stock market and facilitate communication among worldwide financial
institutions about these risk issues.
34
Appendices
Appendix 1: Histograms
Returns
600
500
400
300
200
100
0
Adjusted spreads (in bp)
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 More
Liquidity adjusted returns
500
400
300
200
100
0
-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01
0
35
0.01 0.02 0.03 0.04 0.05 0.06 0.07 More
Appendix 2a: Parameters of GARCH class models estimation – Returns
These three tables (2a, 2b and 2c) summarize the results of estimated parameters in GARCH model class. All the parameters are
significant at 95% confidence level. Column 8 and 9 are Akaike Information Criterion (AIC) and Schwartz Bayesian Criterion (SBC or
SC). The preferred model will show the lowest value of these criteria. Results in 2a support that, based on both AIC and SC, the
strongest model is TARCH with t-distribution, whereas 2b suggests the EGARCH with GED distribution and 2c recommends the
TARCH with t-distribution. (*) denotes the standard error of the corresponding estimated parameter.
Model
Distribution
Student’s t
GARCH
GED
Student’s t
TARCH
GED
Student’s t
EGARCH
GED
ω
α
β
1.2E-06
(5.87E-07)
1.22E-06
(5.64E-07)
1.45E-06
(6.38E-07)
1.55E-06
(6.28E-07)
-0.231703
(0.051748)
-0.243878
(0.050876)
0.059519
(0.0111)
0.059087
(0.010594)
0.048752
(0.013233)
0.047333
(0.012423)
0.159075
(0.025715)
0.159173
(0.024301)
0.937243
(0.01077)
0.937217
(0.010415)
0.932733
(0.011217)
0.932050
(0.010803)
0.986976
(0.004954)
0.985606
(0.004965)
Estimated Parameters
γ
DOF.
7.111593
0.029414
(0.016686)
0.031786
(0.015313)
-0.026889
(0.013187)
-0.030270
(0.012355)
36
7.187139
7.046284
AIC
SC
DW
-5.79239
-5.76929
2.065092
-5.784802
-5.761697
2.065142
-5.792930
-5.766524
2.066242
-5.785830
-5.759424
2.067532
-5.786626
-5.760220
2.071427
-5.779754
-5.753348
2.071782
Appendix 2b: Parameters of GARCH class models estimation – Adjusted spreads
Model
Distribution
Student’s t
GARCH
GED
Student’s t
TARCH
GED
Student’s t
EGARCH
GED
ω
α
1.17E-10
(4.95E-11)
1.01E-10
(4.52E-11)
1.13E-10
(2.68E-11)
9.29E-11
(2.36E-11)
-0.036613
(0.19394)
-0.050326
(0.020035)
0.347946
(0.050041)
0.307053
(0.039089)
0.377061
(0.041736)
0.348614
(0.034073)
-0.106548
(0.014164)
-0.098420
(0.013331)
Estimated Parameters
Β
γ
0.750715
(0.021844)
0.762876
(0.021983)
0.860097
(0.012405)
0.871160
(0.011706)
0.998528
(0.000978)
0.997855
(0.001006)
37
-0.556687
(0.054974)
-0.526928
(0.045632)
0.490970
(0.028052)
0.471427
(0.025179)
DOF.
AIC
SC
DW
4.060250
-14.65008
-14.62038
2.142855
-14.64669
-14.61698
2.188864
-14.72709
-14.69408
2.186467
-14.71978
-14.68678
2.210193
-14.76520
-14.76219
2.203603
-14.78242
-14.74941
2.209037
5.124936
5.975551
Appendix 2c: Parameters of GARCH class models estimation – Liquidity adjusted returns
Model
Distribution
Student’s t
GARCH
GED
Student’s t
TARCH
GED
Student’s t
EGARCH
GED
ω
α
1.28E-06
(6.32E-07)
1.34E-06
(6.06E-07)
1.5E-06
(6.82E-07)
1.62E-06
(6.66E-07)
-0.227995
(0.052434)
-0.239875
(0.051158)
0.059900
(0.011233)
0.058911
(0.010646)
0.050113
(0.013645)
0.048447
(0.012796)
0.158768
(0.025764)
0.157515
(0.024263)
Estimated Parameters
Β
γ
0.936514
(0.010996)
0.936741
(0.010628)
0.932331
(0.011454)
0.931965
(0.011015)
0.987453
(0.005023)
0.986009
(0.004977)
38
0.026147
(0.016495)
0.027715
(0.015078)
-0.023086
(0.012991)
-0.025902
(0.012178)
DOF.
AIC
SC
DW
7.127492
-5.765898
-5.742793
2.067056
-5.758545
-5.735440
2.067972
-5.766081
-5.739675
2.067225
-5.759061
-5.732656
2.068825
-5.758971
-5.732565
2.075507
-5.752069
-5.725663
2.078054
7.178906
6.995226
Appendix 3: Results of ARCH-LM tests
This table summarizes the results of ARCH-LM tests on the residual series of each GARCH models. They are tests on returns (column
1), adjusted spreads (column 2) and liquidity adjusted spreads (column 3). All the p-values in the tests exceed 5% significant level,
indicating no more volatility clustering in the series. (*) denotes the p-value of the corresponding statistic value.
(1)
Returns
Model
Distribution
Student's t
GARCH
GED
Student's t
TARCH
GED
Student's t
EGARCH
GED
F-statistic
0.266404
(0.931531)
0.275491
(0.926689)
0.258022
(0.935877)
0.275521
(0.926673)
0.265293
(0.932114)
0.297386
(0.914494)
LM
1.335841
(0.931201)
1.38137
(0.92634)
1.293844
(0.935564)
1.381519
(0.926324)
1.330277
(0.931786)
1.491055
(0.9141)
(2)
Adjusted spreads
F-statistic
LM
1.011942
5.06263
(0.409023)
(0.408285)
0.085862
4.297615
(0.508267)
(0.507409)
1.627969
8.129181
(0.149379)
(0.149258)
1.437832
7.183922
(0.2076)
(0.207317)
1.746508
8.717932
(0.12089)
(0.120858)
1.814724
9.056553
(0.106823)
(0.10683)
39
(3)
Liquidity adjusted returns
F-statistic
LM
0.268648
1.347086
(0.930348)
(0.930013)
0.282633
1.41715
(0.922792)
(0.922428)
0.291808
1.463115
(0.917669)
(0.917287)
0.309261
1.550539
(0.907584)
(0.907165)
0.291463
1.461388
(0.917864)
(0.917482)
0.326311
1.635936
(0.897323)
(0.89687)
Appendix 4a Backtesting- failure days
This table shows the results of backtesting on different VaR models. They are the VaR of the original returns only (column 2), the
liquidity adjusted VaR based on the modified BDSS model with turnover ratio division (column 3)/ without turnover ratio division
(column 4), and the VaR of the liquidity adjusted returns with turnover ratio division (column 5)/ without turnover ratio (column 6).
The red numbers are observation days beyond the expected days, indicating the fact that model does not work well.
Dates
Windows
(1)
Expected failure
days
(2)
Returns
(3)
Modified BDSS
(tr*Vo)
(4)
Modified BDSS
(Vo)
(5)
adj. returns
(tr*Vo）
(6)
adj. liquidity
(Vo)
95%
99%
95%
99%
95%
99%
95%
99%
95%
99%
95%
99%
2005-2011
1639
81.95
16.39
86
8
49
2
64
6
81
8
89
8
2006-2011
1459
72.95
14.59
75
8
43
2
62
6
73
8
78
8
2007-2011
1218
60.9
12.18
64
8
41
2
57
6
64
8
69
8
2008-2011
976
48.8
9.76
49
7
29
2
42
6
49
7
54
7
2009-2011
730
36.5
7.3
32
5
21
1
28
4
32
5
35
5
2010-2011
486
24.3
4.86
25
2
14
1
21
1
25
2
28
2
40
Appendix 4b Backtesting- Kupiec Test under 95% confidence level
This table shows the results of the backtesting Kupiec test on different VaR models. They are the VaR of the original returns only
(column 1), the liquidity adjusted VaR based on the modified BDSS model with turnover ratio division (column 2)/ without turnover
ratio division (column 3), and the VaR of liquidity adjusted returns with turnover ratio division (column 4)/ without turnover ratio
(column 5). The critical value of the LR test is 3.84. The red numbers in column 2 show that modified BDSS model with turnover ratio
division rejects the entire null hypothesis. Under this approach, the remaining four approaches perform well.
(1)
Dates
Windows
Returns
(2)
Modified BDSS
(tr*Vo)
(3)
Modified BDSS
(Vo)
(4)
adj. returns
(tr*Vo）
(5)
adj. liquidity
(Vo)
p
LR statistic
p
LR statistic
p
LR statistic
p
LR statistic
p
LR statistic
2005-2011
1639
0.0525
0.2075
0.0299
16.1921
0.0390
4.4612
0.0494
0.0116
0.0543
0.6218
2006-2011
1459
0.0514
0.0601
0.0295
15.0852
0.0425
1.8189
0.0500
0.0000
0.0535
0.3602
2007-2011
1218
0.0525
0.1635
0.0337
7.6961
0.0468
0.2684
0.0525
0.1635
0.0567
1.0894
2008-2011
976
0.0502
0.0009
0.0297
9.8346
0.0430
1.0447
0.0502
0.0009
0.0553
0.5646
2009-2011
730
0.0438
0.6082
0.0288
8.1267
0.0384
2.2577
0.0438
0.6082
0.0479
0.0657
2010-2011
486
0.0514
0.0210
0.0288
5.3884
0.0432
0.4935
0.0514
0.0210
0.0576
0.5665
41
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