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Nonparametric factor analytic risk measurement in Common Stocks in
Korean Financial Firms:an empirical perspective
SeunghoBaek
University of Chicago
Joseph D.Cursio
Illinois Institute of Technology
SeungYoun Cha1
University of Chicago
Keywords Risk management, Value at Risk, Expected Shortfall, Full valuation, Monte Carlo simulation,
Principal Component Analysis, Itos lemma, Nonparametric method, Systemic Risk, Systematic Risk
JEL Classification: G11
1Corresponding Author: SeungYoun Cha, Financial Mathematics, University of Chicago,
Chicago, IL 60637Phone: 773-834-4385, Fax: 773-702-9787, email: [email protected]
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Abstract
The purpose of this research is to measure risk in common stocks in Korean financial firms by industrial
clusters applying a nonparametric methodology, which is Monte Carlo simulation, but also to identify the
most critical factor explaining the volatility of stocks in financial firms and in each sector of financial
firms (banks, insurance companies, and investment and security trading companies). The study suggests
that the stock returns of Korean firms are covariated because of this parallel shift factor. The result shows
similar VaRsand ESs for each industry when using a factor analytic approach.
KeywordsRisk management, Value at Risk, Expected Shortfall, Full valuation, Monte Carlo simulation,
Principal Component Analysis, Itos lemma, Nonparametric method, Systemic Risk, Systematic Risk
JEL Classification: G11
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1.Introduction
Financial institutions are regarded as the most important monetary resources in each countrys
economy. Sullivan and Sheffrin(2003) definethe financial sectoras the system that allows the transfer of
money between investors and borrowers.However, it is complicatedly interconnected with financial
intermediates, financial products, investors and markets. In this respect, the financial sector is highly
regulated and monitored by governments and even rated by credit rating agencies such as Fitch, Standard
and Poors, and Moodys. Despite their regulatory efforts, critical disasters have come from the financial
industry. In particular, the 2007 worldwide financial meltdown was triggered by U.S. financial institutions,
due to their speculative investment strategy and improper risk management of fixed income products,
including subprime mortgage. Large financial institutions such Lehman Brothers, Wachovia, Merrill
Lynch, Washington Mutual, Bank of America, J.P Morgan, Citigroup, and AIG held sizable toxic assets,
which were unable to be liquidated, and thus faced critical liquidity issues. Some banks survived only
because of government bail-outs, at substantial cost to tax-payers, while other banks such as Lehman
Brothers declared bankruptcy. As a result, the U.S. and global markets entered into a severe economic
downturn and the global economy is still suffering from the aftermath of the financial collapse. From the
recent financial crisis, risk management failure or insolvencyof an individual financial firm could result in
the disaster of entire economic system. Because of this spillover effect of shocks from one institution to
another, measuring an individual firms risk and the systematic risk over the entire financial system is
critical in risk management.
To measure the level ofrisk, Value at Risk(VaR) has emerged as the standard benchmark for
quantifying downside risk. VaR estimatesthe potential loss amount of a firms assets over a given period.
Many banks and financial institutions have disclosed VaR information in their quarterly and annual
financial reportsnot only to satisfy regulation policybut also to show their competitiveness of asset
management. From VaR reports, risk managers in financial firms and government financial supervisors
monitor individual firm and sector-wide risk exposure and identify the level of capital adequacy for their
business and for the entire financial industry. According to Jorions(2002) study of the relation between
the trading VaR disclosure of U.S. commercial banks and the variability of trading revenue, the result
suggests that VaR disclosures are informative.
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Generally there are two major approaches to VaR, local valuation and full valuation methods. The
local valuation method is an analytical procedurewhich assumes that the distribution of returns is known.
In other words, providedwith a parametric probability density function (e.g. normal density), thismethod
values portfolio risk using a linear combination of parameters such as mean and standard deviation. The
full valuation method is a simulation, or scenario-based, approach. There are two prominent empirical
techniques in the full valuationmethod, historical simulation and Monte Carlo(MC) simulation. Both
historical and MC simulations are more useful than the local valuation method in that they 1)better reflect
extreme events, 2) the non-constant correlation between underlying assets, and 3) fat-tails for measuring
risk. Unlike the local valuation method, the full valuation method is not a linear approach but instead
computesVaR using the ranking of percentage and does not assume the shape of the distribution of asset
returns. As long as information of underlying asset is properly contained, this method would be efficient.
However, the main disadvantage ofthe full valuation methodis that it does not consider any future
volatility higher than the peakhistorical volatility,because historical simulation depends on the historical
events over a specified time frame. On the other hand, Monte Carlo simulation is more flexible than
historical simulation because it considers extreme events and fat-tail issues when applying stochastic
process regardless of sample size.When applying the MC methodto generate various scenarios ofasset
prices at t+1, both Cholesky decomposition and eigenvalue decomposition (EVD) have been used.
Especially, EVD is a part of principal component analysis(PCA) and thus PCA can be applicable as
adecomposition technique, which is a useful multivariate statistical technique in portfolio risk
management that explains the covariance structure of time seriesand reduces the number of risk factors
that affects the movement of portfolio values. Thus, we can identify how individual variables affect a
portfolio movement because with a few of principal components we can reproduce the covariance
structure of asset price along with individual variable effect.
Generally, PCA can be performed by eigenvalue decomposition (EVD) and singular value
decomposition(SVD).By applying EVD-based PCAJamshidian and Zhu(1997) first compute VaR of
multivariate currencies and interest rate assets employing the MC method while Frye(1997),
Loretan(1997)study PCA applicability in the context of parametric based VaR.Under the nonparametric
framework,this study considerssingular value decomposition (SVD), first suggested by Stewart (1993), as
well as eigenvalue decomposition. SVD has been widely used in digital imaging processing in
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engineering and signal processing, but today many investment banks and hedge funds are employing this
method to derive asset price of financial derivatives and to extract invisible risk factorsin interest rate
modeling. Since the eigenvalue approach to perform PCA tends to show lack of precisionfrom a
numerical analysis point of view, SVD widely is regarded as a numerically better approach than
eigenvalue method.
In the meantime, despite of its popularity and usefulness among practitioners, several studies warn of
undesirable aspects of using VaR when measuring risk. In their post-mortem of the market crisis in the
fall of 1999, the BIS Committee on the Global Financial System(1999) states responses among the
interviewees as to whether the magnitude of mature market turbulence was within or above their VaR
limits. A large majority of interviewees admitted that last autumns events were in the tails of the
distribution and that therefore their VaR models were useless for measuring and monitoring market risk.
Also, Beder(1995) says that VaR is not a sufficient method to control risk because VaRis extremely
dependent on parameters, data, assumptions, and choice of methodology. Likewise, the standard VaR
method is heavily dependent on the assumption of normality. In other words, when the distribution is
either positively or negatively skewed or either leptokurtic or platykurtic, VaR cannot measure downside
risk properly.
To overcome this problem, Artzner et al(1997, 1999) suggest another risk measurement that explains
extreme loss of risk, alternatively called Expected Shortfall(ES) or Conditional Value at Risk(CVaR). This
risk measure is more sensitive to the tail loss than that of VaR in that ES (or CVaR) isdefined as expected
loss amount beyondVaR. Empirically, Yamai and Yoshida(2005) suggest ES is a better risk measure in
terms of tail risk through the simulation of credit portfolio and foreign exchange under market stress if ES
obtain enough large sample size.
From this consideration, this studyattempts to compute VaR and ES for a portfolio of financial firms
by acknowledging the fact that asset prices in these financial firms are highly correlated, and
appliesMonte Carlo(MC) based PCA factor analytic approach. The conventional MC method focuses on
calculating VaR based on either Cholesky decomposition or eigenvalue decomposition.Although the
conventional MC method ismore desirable than historical simulation, it is also more time consuming to
obtain risk measurement values because they compute covariance matrix on the basis of all variables.
However, by using the PCA dimension reduction method; this study suggests a more efficient way to
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quantify risk. In addition, considering the problem of tail risk, and the numerical issue of inaccuracy of
EVD, this research suggests using factor analytic VaR and ES calculations applying SVD under the
framework of Monte Carlo simulation as well. This study investigates whether the applicability of this
research model through an empirical test identifyingasingle principal risk factorfor a portfolio of stock
pricesof South Korea financial firms. Analyzing an equity portfolio of financial institutions gives us an
understanding of how an individual asset price movement can affect the change of the entire
portfolio.This study examines how one principal factor can explain how the equity asset price changes in
an individual firm can affect the movement of the entire financial sector.
There arefew papers explaining risk of financial industry. Viale et al(2009) and Baek et al
(2011)attempt to identify systematic risk of banking industry employing CAPM, ICAPM, and Fama-
French(1992,1993)s factor model. However, unlike Baek et al (2011)s systematic approach identifying
the risks of banking industry, this paper attempt to identify systemic risk
The paper proceeds as follows. Section 1 describes this research purpose and motivation. Section 2
introduces the different methods of measuring risk, VaR and ES. Section 3 describes scenario generation
applying decomposing methods and explains eigenvalue decomposition and single value decomposition
to perform principalcomponent analysis. Section 4 illustrates the results of empirical tests and discusses
the implications in risk management. Section 5 concludes this paper.
2. Measuring Risk
For measuring risk using VaR, two approaches can be applied. The first application is local valuation
which is a method to quantify risk by using parameters (e.g. mean and standard deviation) from a specific
distribution, generally a normal distribution as written in equation (1).
VaR(X) = ( + ) (1)where is the inverse of a corresponding cumulative normal density function q. Although its methodcomputesVaRwith this simple analytical form employing normal distribution, there are a couple of
drawbacks. The major drawbacks of this parametric approach areusing a linear combination of mean and
variance(covariance) and using the assumption of normality. With only with mean and standard variance
as parameters, it is unlikely to get precise information for the distribution.In other words, in the case
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thatthe realization of asset price does not show a symmetrical bell-shape curve, the distribution of
financial assets does not capture a plausible loss in advance underasymmetric conditions. If the normality
and symmetric conditions are violated, it is highly likely to underestimate or overestimate the loss of asset.
The second approach is full valuation.Jorion(2006) states that it measures risk by fully repricing the
portfolio over a range of scenarios.Normally such repricing is performed by employing risk factor that is
generated by historical simulation and Monte Carlo simulation. Unlike the local valuation approach, this
method does not assume any particular distribution of assetsnor considers constant correlated asset
pricing movements. Given the distribution of scenarios, VaR is computed by percentile ranking.
VaR(X) = inf{x: P(X x) q} = F(q) (2)The historical approach uses past time series data under the assumption that previous events would
happen again in the future. But when we look at equity market collapses in U.Smarket and Korean stock
market in figure 1, it could not have predicteddownside risk appropriately only with past historical price
in that something that did not show up in the previous data would happen in the future. In this sense, it
does not reflect extreme even case because of its backward looking perspective.
On the other hand, Monte Carlo simulation is used togeneraterisk factor by applying stochastic
process. Geometric Brownian motion explains stock assets while either the Vasicek or Cox-Ingersoll-Ross
model is used to explain the evolution of interest rates. Since it considers extreme events using a large set
of correlated assets changes, it is anefficient method.
In the meantime, Alexander(2007) summarizes VaR is applied to risk limitation, proper return
calculation, and capital adequacy calculations. However, Artzner et al(1997, 1999), Rockafellar and
Uryasev(2002), Yamai and Yoshida(2005), Wong(2008) point out that VaRtend to disregard tail event so
extreme shortfall (ES), or conditional VaR would be an alternative measure to consider expected loss
beyond VaR.Mathematically, this is written as
ES(X) = EXX > VaR(X) = 11 q x() dq(3)
whereX refers to a loss amount, and q is quantile.It means that ES represents theaverage loss amount
given that the loss is greater than quantile at a1 % confidence level. From this consideration, thisstudy measures VaR as well as ES applying MC simulation to consider tail risk.
3. Monte Carlo Simulation
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For our empirical tests, we conduct the research in accordance with JP MorgansRiskmetricstechnical
document(1996). Firstly using the covariance structure of asset return movement, we generate a set of
scenariosforfuture price changeson the basis of lognormal model by employing decomposition methods.
Thengiven generated future prices, portfolio valuesare calculated. Finally, from the distribution of profit
and loss, we calculate Value at Risk and expectedshortfall for each set of scenarios.
3.1 Scenario Generation
The stock price scenarios that we generate are based on a lognormal return model. By using Its
lemma and lognormal model, we can model the price of assets in the time oftas
P = Pexp (ty) (4)where P0 is current asset price, Pt is the price at time t, is volatility estimate within time periods, andy is
a standard normal distribution. To prove the equation, we can start the derivation from Itsformula. For
simplification, we suppose there are two risk factors then the process generating the returns for each risk
factor can be written as
()() = + () , = 1, . . , , = 1, , (5)where
() = , ~ (6)and
()
, ()
= (7)
where is the correlation between returns of risk factori and risk factorj.
From ItsLemma, the process followed by lnPis
,() = () = 12 + () (8)so that
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() () = 12 + (9)or equivalently
() = () (10)Since this study assumes driftless returns for simplicity, it regards drift term in equation (10) as zero.
Accordingly equation (8) can be written as equation (11).
,() = () () = () = (11)So, it is summarized as in equation (12).
() = () (12)Thus, on the basis of equation (1), we can generate future prices at time tconsidering various scenarios.
3.2Decomposition
It is very simple to simulate normal random variable if we assume all the values of assets are
independent. However, in real markets, the assumption of asset independence does not hold in that assets
are to some extent correlated with each other. In this respect, it is more desirable to account for
correlation via the variance-covariance matrix in the simulation model.In this study, we considerthree
methods in decomposing this variance-covariance matrix: by Choleskydecomposition, by eigenvalue
decomposition(EVD), andbysingular value decomposition(SVD), when generating random variables in
which correlation isinvolved. Moreover, this study pays attention to the fact that both eigenvalue and
singular valuedecompositionssuggest principal components.In other words, these two methods can
provide otherwise invisible risk factors for financial firms. We attempt to identify these common risk
factors that explain the volatility of asset price for clusters of financial firms such as banks, insurance
companies, and securities trading companies.
Of decomposing methods, Cholesky decomposition is the easiest and simplest.If there is a
symmetrical matrix satisfying a semi-positive definite, the Cholesky method can decompose the matrix
into a lower triangular and an upper triangular matrix. In other words,
= = = (13)where
refers a variance-covariance matrix,
is a lower triangular matrix, and
Uis a upper
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triangularmatrix. In equation (13), the U matrix is the transpose of the L matrix and the matrix is thetranspose of U matrix. Cholesky decomposition can be described as a combination of the L and U
matrices. For convenience, we defined Cholesky decomposition as a combination of the Cholesky matrix
and its transpose.
= (14)Through theCholesky decomposition methodology, we can generate random variables considering the
impact of correlation between assets.
Alternatively, the other decomposition methods areeigenvalue decomposition and singular value
decomposition. Eigenvalue decomposition is another way of decomposing variance-covariance matrix.
Basic eigenvalue decomposition is
= (15)whereis a matrix which consist of eigenvectors and is a diagonal matrix. The diagonal matrix can bedissembled into two matrices as shown the below.
= .
.
=
0
0
0
0
(16)
Also,
= . (17)Then equation (17) can be rewritten as
= .. = (18)Particularly through eignevalues in equation (16), we can identify the explicability of each principal
component variance and the desirable number of principal components.Another decomposition method is
singular value decomposition (SVD), a matrix factorization introduced by Stewart(1993). While EVD
requires a square matrix to decompose, SVD can be applied to any matrices which are not square. In this
sense, SVD is more general and flexible approach that EVD. However, our study focuses on decomposing
the variance-covariance matrix so we only consider a square matrix.
SVD is
= = (19)And let
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= . (20)Since = = , we can have a similar equation as shown in equation (21).
= .. = (21)
Like the eigenvalue decomposition method, SVD is also able to extract a most desirable factor because
we can identify on how much percentage of variance can be explained by each factor. Thus, this study
examines whether factor analytic simulation is also applicable to quantify the riskiness of every industry.
Both EVD and SVD methodologies extract important factors with respect of the explicability of
variance. The most desirable aspects when we use these factor analytic methodsare dimension reduction
and the identification of risk. For example,suppose that there is Mby Msized covariance matrix. Then
PCA extractsMnumbers of factors. As we use all M factors, we can perfectly approximate the original
covariance matrix. Obviously, the fewer factors we use, the lowerthe quality of estimation we perform.
However, EVD or SVD based PCA approaches can accurately approximate the original data structure
with few of factors. That is, through the PCA we are able toobtain a high quality approximation of the
estimated covariance matrix with just a few principal factors and identify key aspects of the covariance
structure. Based on theseprincipal risk factors, we can quantify risk as well.
The singular value of the square matrix
is defined as the square root of the eigenvalues of
.
Since eigenvectors are orthogonal, which is = , is orthonormal as in equation (18). InSVD, = and Uis equal to . Then we derive a mathematical form as in equation(22)because = .
= .. = .. = (22)
3.3 Covariance Estimation by PCA
On the basis of the fact that reduced principal components can approximate the covariance matrix,
this research attempts to applyboth EVD and SVD to estimate the variance-covariance structure.
Generally speaking, the covariance can be estimated with a linear combination of eigenvector and
diagonal covariance matrix of principal components.
For the estimation of covariance using EVD based PCA, which is the traditional PCA method, we can
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explain it using a simple example. Suppose we have a random vector which represents stock returns offinancial firms and principal components, P, of . From principal component analysis, we can obtain
ieigenvectors,
e, of covariance of
.
eused to represent a change of movement of
and the
elements of each column of e represents factor loadings that able to explain each individual elementcontribution to the fluctuation of Z. Then Z is written as a combination of principal components and
eigenvectors in equation 23.
= Pe + Pe + + Pe = Pe = eP (23)Then we can obtain the estimate ofZ,, with a linear combination of a few principal components and
eigenvector. For example, if we use two principal components, the estimate of Z is described in equation
24.
= Pe + Pe (24)Also the general form of equation with the reduced principal components is expressed in algebraic form
as
= (25)where refers reduced eigenvectors and is reduced principal factors.
The vector of principal components P has a covariance matrixof, which is the same matrix as shownin equation 16. Then the covariance matrix of Z is = (26)
and the estimated covariance ofZis
= (27)where means reduced diagonal matrix.For SVD based PCA, the estimated covariance is calculated as
= (28)Base on this applicability of PCA, this research studies considers factor analytic scenario analysis to
obtain the information of individual fluctuation of asset returns and the efficiency of computation time.
3.4Building Asset Price Simulator
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Given these three decomposition methods, we set up the realization of simulated return from each
method as described in table 1.On the basis of historical volatility, the scenario prices at time tfor each
method can be generated as shown in the third column.Looking closely at the equations, this is a simple
asset pricing concept using continuous compounded return and thus we can obtain the future prices from
Monte Carlo simulation, and then value an amount of profit and loss for a portfolio. In other words, given
generated future prices from these equations, we can obtain future prices in accordance with stochastic
price series and can compute the distribution of profit and loss(P/L) between current price and future
price at time t. Mathematically, we can express the value of P/L at time, V, as in equation(29) and thereturn of P/L at time, R, in equation (30).
V = P P (29)R = P PP ln (PP) (30)
Once the distribution of P/L is given, we simply compute maximum loss amount at a certain confidence
level and expected loss amount beyond VaR.
4. Empirical Test
4.1 Data and Summary Statistics
The data we use in this study is the daily price ofstocks in Korean financial companies trading in
Korea Exchange (KRX). As of May 31, 2012, there are 30 companies listed in KRX financial sector. In
this research, to reflect the huge market collapse in 2008 in the covariance structure of our simulation, we
collect the price data which contains the period when KOPSI index had declined from 1888.88 on May 16,
2008 to 968.97 on October 29, 2009, as depicted in figure 1 - b) and 1 - c), because of the subprime
mortgage collapse in the U.S., as shown in figure 1- a).For our empirical tests, we refined the research
data.To avoid the violation of semi-positive definite condition for Cholesky decomposition, we only
select 26 financial firms. In case of firms that do not have a complete time series data within this period,
we removed those firms in our analysis. So anincomplete time series data within this period (for the firms
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Samsung lifeInsurance Co., Daehanlife Insurance Co., Kiwoom Security Co., and Samsung Card Co.) are
taken out of the analysis. The stock price information is retrieved from KISVALUE database during the
period between February15, 2006 and June 31, 2011.
With the data consisting of 26 companies: 10 banks or bank holding companies, six insurance
companies, and the remaining 10 investment and security trading companies,table 2exhibitsthe simple
statistics of stock returns for each financial firm. Average stock returns for all finance firms are
approximatelyequal to zero. In terms of standard deviation, the investment and securities sector and
insurance sectors are more volatile than that of banking sector. More specifically, the standard deviations
for banks are less than 3.00% except for Woori and Hana Bank Holdings while the standard deviation for
investment and securities companies and insurance corporations are greater than 3.00% but for Samsung
Securities Co. Ltd. and Samsung Fire & Marine Insurance Co.Ltd, and Korean Reinsurance Co. The
distributions of stock returns for five companies(JeonbukBank, Woori Bank Holdings, Samsung Fire
&Marine Insurance Co.Ltd, SK Securities Co., Ltd., SamsungSecurities Co., Ltd.)are positively skewed,
whereas that of the remaining companies shows negative skewness. For excess kurtosis, thedistributions
for every stock are leptokurtic, i.e.values are greater than 3.On the basis of these descriptive statistics, the
shape of stock returns is not well approximated by the normal distribution,as a parametricVaRassumes. It
is obvious that the risk measurement would be inaccurate if we employ a simple parametric VaR
methodology which depends on Gaussian distribution.
4.2 Analysis of Principal Factors
Before we run MC simulation, we analyze principal components for all financial firms in general and
independently for each financial sector. By applying principal component analysis(PCA) to all financial
firms, the financial industry has three most explicable principalcomponents or factors in describing
covariance movements such as parallel movement, movement by banks and insurances versus investment
& trading, movement by business sectors as shown in figure 2.
Table 3reports the eigenvalues of the principal components of the covariance of daily stock returns in
financial firms. Each eigenvalue exhibits the explicability of each component.Thistable indicates that
about53% of variance can be explained by the first principal factor, 8.5% by the second factor, and 6.2 %
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by the third one. The most critical factor that is able to explain the variance in financial firms is parallel
shift or market movement. In other words, if there is either negative or positive exogenous and indigenous
impact to financial firms, all financial firms stock prices would react moving toward the same direction
as shown in figure 2-a). In this research, we regard this factor as a key risk factor in our analysis that
canbe described asprocyclical condition and risk contagion.
In figure 2-a), the first factor can describe the procyclical figure in Korean financial firms because
procyclical condition describes how the values of assets in market correlated with market fluctuation.
Also the risk contagion as suggested by Kaufman(1994) can be explained by this factor because risk
contagion explains the relation between the movement of individual firm and entire firms.For each kind
of individual effect, we can explain using factor loading values from PCA. The higher values of factor
loading indicate the higher impact on the entire market movement whereas the lower factor loading imply
the less impact on the entire sector.More specifically the graph a) in figure 2 depicts a bar chart of each
factor loading for financial companies.2
Since five companies (SK Securities Co. (0.25), Daewoo
Securities Co. (0.25), Hyundai Securities Co .(0.26), Hanwha Securities Co. (0.26), and Dongyang
Securities Co. (0.29)) have higher factor loadings, their asset movements affect substantially to a portfolio
of entire financial market movements comparedto two banks such as Cheju Bank (0.07) and Jeonbuk
Bank (0.10).
Figure 2-b) and 2-c) describe that business areas can be classified by other principal factors. The
second factor indicates that it can explain the changes in variance-covariance structure of financial firms
2Each value of factor loading is as follows: Cheju Bank (0.07), Jeonbuk Bank (0.10), Samsung
Marine and Fire (0.11), Korea Reinsurance (0.12), Korea Exchange Bank (0.14), Daegu Bank Group
(0.16), Shinhan Bank (0.16), Busan Bank (0.16), Koomin Bank (0.18), Industrial Bank of Korea (0.18),
Hana Bank (0.18), LIG (0.19), Meritz Insurance (0.19), Hankook Investment Holding (0.19), Samsung
Securities Co. (0.19), Hyundai Marine and Fire (0.20), Woori Bank (0.20), Daeshin Securities Co. (0.22),
Bongbu Fire and Insurance (0.22), Woori Investment Co. (0.22), Mirae Asset Securities Co. (0.23), SK
Securities Co. (0.25), Daewoo Securities Co. (0.25), Hyundai Securities Co .(0.26), Hanwha Securities
Co. (0.26), and Dongyang Securities Co. (0.29).
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in terms of insurance versus noninsurance businesses.Since 2003 Korea banks have started
bancassurance(or also known as Bank Insurance Model), a composite word of bank and assurance
which describes the partnership between a bank and an insurance business.Through this business model,
an insurance companycan sell its products via the banks sales channel and bank and insurance company
share the sales commission. And the other factor as shown in figure 2-c) explains the movement by each
industrial sector(Banks, Insurances, and Investment & Trading).Tsay(2003) names this factor as industrial
movement.However, it could also be classified as the change of the movement of asset returns in terms of
security trading or financial instruments trading business versus non-trading business.
Even if the percentage of variance explained for the remaining two factors isrelativelylower than that
of the first factor (14.7% versus 53%), they suggest that there are co-movements by each business area. It
is possible to use the change of asset price with these two other factors. However, unlike the other two
factors, the first factor commonly appears in both all financial firms and each group of financial firms,
this research determinesthatthe one-factorapproach is better than three-factor approachesin that this
research focuses on the risk aspects with respect to each business area and examine whether there is a
common risk factor that explains each business area.
As seen table 4, the percentage of variance explained by first principal factor in each industrial sector
is more than 62%. One interesting figure is that the first principal component in every financial industry
indicates parallel shift or market movement.
Figure 3 describes that this common risk factor exists among commercial banks and bank holding
companies, insurance companies, and investment & trading companies and that each factor loading is the
indicator of individual firms risk contribution to the entire market movement. The bar charts factor
loadings for banks, insurance companies, and investment & trading corporations are depicted in graph a),
b), and c) respectively.For banks, the entire portfolio of risk in banks equity assets are dependent on the
variation of Woori and Hana banks whilethe risks of Cheju Bank and Jeonbuk Bank are relatively lower,
(0.10 and 0.18 respectively), thanthe values forHana Bank and Woori, (0.39 and 0.41 respectively). Graph
b) displays factor loadings for the insurance companies. Samsung Marine and Fire (0.24) and Korea
Reinsurance (0.27)have less impact to the change in asset return in insurance equities while Meritz
Insurance (0.41), LIG (0.45), Hyundai Marine and Fire (0.49), and Dongbu Marine and Fire
(0.51)havesubstantial impact to the volatility of the industry. Graph c) shows individual firms risk factor
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values for investment and trading companies. Among the investment and trading firms, it appearsthat the
equity of Hankook Investment Holding is less risky in that it has the lowest factor loading (0.29).
To identify whether this factor is able to reflect changes in asset prices over the course of MC scenario
generation, we examine whether this factor can appropriately approximate the original covariance
structure. Unless the estimated covariance matrix from this factor approximates the original covariance
matrix, this risk factor would be no longer applicable because it induces false values of VaR and ES
through MC simulation. In this sense, we determine that this diagnosis step is important to maintain our
coherent analysis. Also, to compare EVD-based PCA and SVD-based PCA, we compute two estimated
covariance structures. Because SVD is regarded as a more precise method than EVD to conduct PCA at
the perspective of numerical analysis, and because PCA can possibly generate the loss of precision of
square of form of matrix3, i.e, we also check whether there is discrepancy between two methods.
Tables5 through 8 suggest the results of the estimated covariance matrices for all financial firms and
those firms within each industrial cluster.Table 9 summarizes the difference between original covariance
and estimated covariance matrices. This result suggests that regardless ofthe choice of EVS and SVD
approach, using only one factor can approximate the original covariance matrix. For all financial firms,
the maximum difference between estimated and original covariance is only 0.06% (or 0.0006) for both
EVA and SVD, which is approximatelyzero. In terms of banks, insurance companies, investment &
trading companies, the maximum values are 0.03%, 0.04%, and 0.05% for the EVD factor approach, and
0.04%, 0.05%, 0.05% for the SVD factor approach.From these results, we can infer that with only one
factor,the parallel shift factor, we can explain volatilities of asset prices of financial firms.
4.2.1 Monte Carlo Risk Measurement
Employing the Choleskyand PCA methods, we generate the evolution of asset price changes with
respect to scenarios.Based on the generated scenarios prices, we revalue equal weighted portfolios
consisting of all financial firms, an industrial sector of only banks, an industrial sector of only insurance
companies, and an industrial sector of only investment and trading securities companies.By employing
3Usually, Lauchli matrix is well known for this problem.
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equations 29 and 30 for each scenario, we compute the profit and loss of an equally weighted portfolio for
each scenario, and obtain empirical distributions of profit and loss for the entire financial firms and for
each industrial sector across the scenarios. On the basis of these P/L distributions, we obtain each VaR
and ES for each method. When we run the simulation, we assume that the evolution of price of equation
is driftless and t = 1 when we examine 1 day 99 percent VaR and ES.Figure 4 shows the Monte Carlo simulation for financial firms using nonfactor analytic approach.
Panel A shows the distribution of profit and loss from an equally weighted portfolio of twenty six
financial institutions using different number of Monte Carlo simulations using Cholesky decomposition
for the estimated variance-covariance matrix. Panel B isthe distribution of profit and loss for eigenvalue
decomposition and panel C describes the case of singular value decomposition. The first chart uses one
thousand simulations, the second chart uses five thousand simulations, and the third chart uses ten
thousand simulations.When analyzing panels A, B, and C, the results show that regardless of the
decomposition method, as the number of scenarios increases, each empirical distribution displays nearly
identical values and therefore we can have similarVaR and ES values.
On the other hand, figure 5shows the Monte Carlo simulation for financial firms using the factor
analytic approach. Panel A shows the distribution of profit and loss from an equally weighted portfolio of
twenty six financial institutions using different number of Monte Carlo simulations using the eigenvalue-
based analytic approach for the estimated variance-covariance matrix. The first chart uses one thousand
simulations, the second chart uses five thousand simulations, and the third chart uses ten thousand
simulations. Similarly, panel B shows the distribution of profit and loss from an equally weighted
portfolio of twenty six financial institutions using different number of Monte Carlo simulations using the
singular value-based analytic approach for the estimated variance-covariance matrix. The first chart uses
one thousand simulations, the second chart uses five thousand simulations, and the third chart uses ten
thousand simulations. Like nonfactor analytic method, these two methods suggest that as the size of
simulation grows the distributions from each method show similarity.
Most of all, figure 5 shows that distributions from nonfactor analytic method s and factor analytic
methods do not differ(comparing with figure 4 and fig 5) if the number of simulation is sufficiently large.
In other words, it implies that these factor analytic methods are very useful in measuring risk with
accuracy and efficiency because factor analytic approaches employ the reduced number of dimensions. To
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clarify this observation, we examine all five methods (Cholesky, EVD, SVD, EVD factor analytic, and
SVD factor analytic methods) by each financial industry.
Figures6 through 10 depict profit and loss distributions generated from estimated variance-covariance
matrices using the Cholesky, EVD, SVD, EVD factor analytic, and SVD factor analytic methods. In each
figure, Panel A shows the distribution of profit and loss from an equally weighted portfolio ten banks
using different number of Monte Carlo simulations by Cholesky decomposition for the estimated
variance-covariance matrix. The first chart uses one thousand simulations, the second, uses five thousand
simulations, and the third chart uses ten thousand simulations.Panel B shows the distribution of profit and
loss from an equally weighted portfolio six insurance companies using different number of Monte Carlo
simulations using Cholesky decomposition for the estimated variance-covariance matrix. The first chart
uses one thousand simulations, the second, uses five thousand simulations, and the third chart uses ten
thousand simulations. Similarly, panel C shows the distribution of profit and loss from an equally
weighted portfolio the ten investment & securities companies using different number of Monte Carlo
simulations using Cholesky decomposition for the estimated variance-covariance matrix. The first chart
uses one thousand simulations, the second, uses five thousand simulations, and the third chart uses ten
thousand simulations. As shown these figures, as the number of scenarios increase, we obtain amore
smooth shape for the distribution of profit and loss.
Table 10 shows simple means and standard deviations of the values of profit and loss from MC
simulation over the number of simulationsforbanks, insurance companies, and trading firms. Overall, the
simple statistics from Panels A, B, C, and D suggest that the means and standard deviations obtained by
each Cholesky, EVD, SVD, EVD factor, and SVD factor approach do not vary with respect to simulation
frequencies and by decomposing methods. Therefore the table indicates both VaR and ES for each
decomposing method would be coherent regardless of the number of simulations. However, even if MC
based risk measurement does not care about the shape of distribution, VaR and ES would be accurately
obtained as the number of simulation is sufficiently enoughbecause figures 4 through 10 describe the
distributions of profit and loss become reliable when the number of simulation is more than 5,000.
Table 11 summarizes the Monte Carlo simulation based VaR and ES by the different decomposition
methods. In panel A of table 11for all financial firms, it shows that maximum amount of loss ranges
between 4.5% and 5.1% amount within 99% confidence level when the number of scenarios is equal to
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1,000. SVD based factor analytic VaRsseem to be underestimated. However, when N is equal to 5,000
and 10,000, the values range from 4.9% to 5.1% respectively, and the range of values by decomposition
methods is narrowed. Also when N is equal to 1,000, the expected loss amount exceeding VaRs are a
range of 5.1% through5.7% whereas ES ranges from 5.7% to 5.8% as N is 5,000 and 10,000, respectively.
In Panel A of table 11, the range of 1 day 99% VaR is 0.45% and of ES is 0.51% for N=1,000 while the
difference between maximum and minimumis VaR and ES are0.19% for VaR and 0.12% for ES as
N=5,000 and 0.12%for VaR and 0.16% as N = 10,000. For insurance companies, panel C and D suggests
that as N is getting large, the range of VaR and ES is reduced in similar as shown in panel A and B.
Additionally, in table 11 when closely looking at VaR and ES by industrial segments, a portfolio of
investment and security trading companies seems to be highly exposed to system risk and the portfolio
fluctuation of asset prices in investment and trading securities is highly affected by the change of the asset
prices of Dongyang, Hanwha, and Hyundai while the change of the portfolio prices is less affected by the
risk of Samsung securities companies and HankookInvestment Holdings. The maximum loss amount of
an equity portfolio of investment and trading companies within 99% confidence level and the expected
shortfall amount are on average 6.61% and 7.58% respectively, which are the highest values among
financial sectors. The maximum loss amount of an equity portfolio of banks within 99% confidence level
and the expected shortfall amount are on average 4.83% and 5.51%. Average 1-day 99% VaR for a
portfolio value of insurance companies is 5.58% and the expect loss amount beyond the maximum loss
amount with 1% significance level is6.39%. The reason why VaR and ES of the banks are relatively lower
than that of other sectors is since 1997 Asian crisis, the Korean government has supported banking system
deregulating policies to make their business competitive and has improved monitored capital adequacies
on regular basis,and banks themselves have innovated their business channels and diversified asset
classes to hedge against system risk.
Moreover, table 11 indicates that there is no significant difference in VaR and ES computation
between factor analytic approaches and conventional methods when the number of scenarios is
sufficiently large enough.Also, the result of table11 suggests that both EVD and SVD factor analytic
methods are robust, as the number of scenarios is increased from 1,000 to 5,000 to 10,000, both VaR and
ES from factor analytic methods are similar with those in conventional risk measure over banks,
insurance companies, and investment & trading companies.
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As seen in figures 11 and 12, left tail of the distribution generated by each decomposing method has
similar tail distribution. Also, it implies that we can obtain accuracy in measuring VaR and ES with only
one significant risk factor when running MC based scenario analysis, if we apply a factor analytic method.
5. Conclusion
This research examines the efficiency of nonparametric factor analytic approaches in measuring risk
in common stocks of Korean financial firms from the risk management perspective.Theoretically and
practically, this paper suggests that with a few risk factors we can obtain an estimated covariance matrix
withsufficient accuracy and we are therefore able to compute downside risk measures;VaR and ES,by
applying MC nonparametric scenario analysis.
Through a range of scenario analyses employing stochastic process (i.e., Its lemma), the results
show that there is no significant difference between both EVD and SVD factor analytic methods with
only one risk factor and nonfactor analytic methods measuring both VaR and ES. To check the robustness
of these results, this paper examines whether this approach is consistent in computing risk measures. In
the results, the values of VaRs and ESsare coherent in each case when the number of scenarios is 1,000,
5,000, and 10,000 and by each group of financial industries.
Additionally, this paper depicts factor analytic methods can identify how much the entire portfolio
risk can be affected by individual firm risk through factor loadings. From this perspective, this paper
introduces that our factor analytic approach can examine the risk for the entire financial system and for
each industry group; and that the results indicate that the fluctuation of Korea financial sector has been
dominated by trading and investment firms and insurance companies whose companies are highly
sensitive to market state, while the price movement of the financial sector has been less affected by banks.
The reason why the volatilities of banks are less than that of the other industries is that the Korean
government and policy makers have focused on developing, monitoring, and regulating the risk system of
banks and hence banks could hedge the risk relatively appropriately than for other industries. Thus, these
results imply that to reduce the volatility of entire Korean financial sector, both insurance andinvestment
& trading industries need more risk management activities.
In summary, this paper suggests factor analytic risk identification considering the risk contagion effect
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and tail risk by applying MC simulation. In conventional MC approaches, the issue of computation time
is a main shortcoming, butthis paper shows that our factor analytic methodcan reduce computational time
to obtain the estimated covariance matrix for MC simulation with sufficient accuracy. Furthermore, this
paper suggests through the principal risk factor, government and financial supervisors are able to track
risk transfer, or the risk impact from anindividual firm level to each financial segment or the entire
financial sector.
Out research presents how factor analytic method can explain systemic risk of financial firms.
Although the approach focuses systemic risk, this method can be applied to identify systematic risk as
well. In other words, the classic market equilibrium model sketches the market with a single systematic
risk factor. From our empirical analysis, the results identifies that the most explicable principal risk factor
to understand the fluctuation of equity assets of financial firms is the parallel shift or market movement.
The major finding of this paper could apply to the fundamental Capital Asset Pricing Model of Sharpe
(1964), Lintner (1965) and Mossin(1966) as the result shows that a single major factor could explain the
market sufficiently. Further research will cover this in great details.
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Table 1 Monte Carlo pricing simulation by decomposing methods
Methods Realization of Returns Simulation for Asset Price
Cholesky Decomposition = (, , , ) = = Eigenvalue Decomposition = (, , , ) = = Singular Value Decomposition = (, , , ) = = EVD based PCA = (, , , ) = = SVD based PCA = (, , , ) = =
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Table 2 Simple descriptive statistics for financial firms
This table uses daily stock returns from February 15, 2006 through June 31, 2011 for the twenty sixfinancial institutions with complete trading history in the Korean Exchange (KRX). Four other financial
institutions were excluded because of incomplete pricing.
Financial
SectorsCompanies Mean(%)
Std.
Deviation(%)Skewness
Excess
Kurtosis
Banks
Korea Exchange Bank -0.028 2.622 -0.005 8.766
Checju Bank -0.018 2.009 -0.323 16.505
Jeonbuk Banks -0.010 2.100 0.266 8.856
Industrial Bank of Korea 0.008 2.829 -0.412 8.586
Shinhan Bank Holdings 0.020 2.546 -0.140 7.957
Woori Bank Holdings -0.022 3.222 0.017 8.902
Hana Bank Holdings -0.007 3.197 -0.435 9.437
KB Holdings -0.029 2.855 -0.420 8.934BS Holdings 0.017 2.702 -0.350 6.587
DBG Holdings 0.008 2.760 -0.173 7.002
Insurances
MERITZ Fire &
Marine Insurance Co.,Ltd0.090 3.141 -0.216 7.685
Samsung Fire &
Marine Insurance Co.,Ltd0.062 2.238 -0.008 4.791
Hyundai Fire &
Marine Insurance Co.,Ltd0.074 3.318 0.041 5.438
LIG Insurance 0.051 3.179 -0.203 5.108
Korean Reinsurance Co. 0.027 2.849 -0.016 6.734
Dongbu Fire Insurance Co, Ltd. 0.089 3.497 -0.439 7.314
Investment
andSecurities
SK Securities Co., Ltd. 0.031 3.776 0.196 7.532
Hyundai Securities Co., Ltd. -0.005 3.502 -0.053 6.561
Dongyang Securities Co., Ltd. -0.023 3.856 -0.062 5.933
Hanwha Securities Co., Ltd. -0.028 3.734 -0.072 7.116
Daeshin Securities Co., Ltd. -0.030 3.012 -0.041 7.913
Woori Investment & Securities
Co., Ltd-0.005 3.059 -0.153 7.388
Daewoo Securities Co., Ltd 0.011 3.307 -0.139 7.007
Samsung Securities Co., Ltd. 0.040 2.716 0.181 6.793
Korea Investment & SecuritiesCo., Ltd.
0.009 3.165 -0.090 5.383
Mirae Asset Financial Group -0.018 3.386 -0.060 8.029
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Table 3 Principal components for the covariance of financial firms
This reports the eigenvalues of the principal components of the covariance of daily stock returns from
February 15, 2006 through June 31, 2011for the twenty six financial institutions with complete tradinghistory in the Korean Exchange (KRX). Four other financial institutions were excluded because of
incomplete pricing.
Component Eigenvalue Percent of Eigenvalue Cumulative Percent
PC1 0.01 53.06 53.06
PC2 0.00 8.47 61.53
PC3 0.00 6.19 67.72
PC4 0.00 2.72 70.44
PC5 0.00 2.59 73.03
PC6 0.00 2.18 75.22
PC7 0.00 2.04 77.26
PC8 0.00 1.93 79.19
PC9 0.00 1.88 81.07
PC10 0.00 1.69 82.76
PC11 0.00 1.59 84.34
PC12 0.00 1.52 85.86
PC13 0.00 1.44 87.30
PC14 0.00 1.33 88.63
PC15 0.00 1.26 89.89
PC16 0.00 1.21 91.10
PC17 0.00 1.18 92.28
PC18 0.00 1.13 93.41
PC19 0.00 1.07 94.48
PC20 0.00 0.95 95.43
PC21 0.00 0.95 96.38
PC22 0.00 0.89 97.27
PC23 0.00 0.87 98.14
PC24 0.00 0.73 98.87
PC25 0.00 0.62 99.49
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Table 4Percent of Eigenvalues for the Principal components for the covariance of each
financial sector
The first row reports the percent of eigenvalues of the principal components of the covariance of daily
stock returns from February 15, 2006 through June 31, 2011 for the ten banks with complete tradinghistory in the Korean Exchange (KRX).
The second row reports the percent of eigenvalues for the six insurance companies with complete
trading history over the same time period, and the third row reports the percent of eigenvalues for the ten
investment & securities companies with complete trading history over the same time period.
Business Sectors PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10
Banks 62.65 6.79 5.49 4.78 4.15 4.08 3.75 3.26 2.96 2.09
Insurances 64.23 10.63 8.23 6.63 5.64 4.64
Investment &
Securities Co.71.31 6.02 4.87 4.04 3.38 3.02 2.31 2.16 1.76 1.14
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Table 5 Covariance Matrix for Korean financial firms: Percentage of Variance and Covariance
Panel A displays the historic variance-covariance of daily stock returns from February 15, 2006 through June 31, 2011 for the twenty six financial institutions with
complete trading history in the Korean Exchange (KRX). Four other financial institutions were excluded because of incomplete pricing.Panel B displays the estimated variance-covariance of daily stock returns from February 15, 2006 through June 31, 2011 for the twenty six financial institutions
with complete trading history in the Korean Exchange (KRX). Four other financial institutions were excluded because of incomplete pricing. This estimation is using
theParallel Shift Factor from eigenvalue decomposition (EVD).
Panel C displays the historic variance-covariance of daily stock returns from February 15, 2006 through June 31, 2011 for the twenty six financial institutions with
complete trading history in the Korean Exchange (KRX). Four other financial institutions were excluded because of incomplete pricing. This estimation is using the
Parallel Shift Factor fromsingular value decomposition (SVD).In the table, X1 through X10 are the variables of bank industry, X11 through X15 are the variable of insurance industry, and X16 through X26 are the variables ofinvestment and trading securities industry. The list of variable are as follows: Korean Exchange Bank(X1), Cheju Bank(X2), Jeonbuk Bank(X3), Industrial Bank of
Korea(X4), Shinhan Bank(X5), Woori Bank(X6), Hana Bank(X7), Kookmin Bank(X8), Busan Bank(X9), Daegu Bank Group(X10), Meritz Insurance(X11), Samsung
Marine and Fire(X12), Hyndai Marine and Fire(X13), LIG(X14), Korea Reinsurance(X15), Dongbu Marine and Fire(X16), SK Securities Co.(X17), Hyundai
Securities Co.(X18), Dongyang Securities Co.(X19), Hanwha Securities Co.(X20), Daeshin Securities Co.(X21), Woori Investment(X22), Daewoo SecuritiesCo.(X23), Samsung Securities Co.(X24), Hankook Investment Holdings (X25), Mirae Asset Securities Co.(X26).
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Panel A: Original Covariance Matrix
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26
x10.07
x20.01 0.04
x30.02 0.01 0.04
x40.04 0.02 0.03 0.08
x50.04 0.01 0.03 0.05 0.06
x60.05 0.02 0.03 0.06 0.06 0.10
x70.05 0.01 0.03 0.06 0.06 0.07 0.10
x80.04 0.01 0.03 0.06 0.06 0.06 0.06 0.08
x90.04 0.01 0.03 0.05 0.04 0.05 0.05 0.05 0.07
x100.04 0.01 0.02 0.05 0.04 0.05 0.05 0.05 0.05 0.08
x110.03 0.02 0.02 0.04 0.03 0.04 0.04 0.04 0.04 0.03 0.10
x120.02 0.01 0.01 0.02 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.05
x130.03 0.02 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.06 0.04 0.11
x140.03 0.02 0.03 0.04 0.04 0.04 0.04 0.04 0.03 0.03 0.06 0.03 0.07 0.10
x150.02 0.01 0.02 0.03 0.02 0.03 0.02 0.02 0.02 0.02 0.03 0.02 0.04 0.03 0.08
x160.04 0.02 0.03 0.05 0.04 0.06 0.05 0.05 0.04 0.04 0.06 0.04 0.08 0.07 0.04 0.12
x170.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.04 0.06 0.14
x180.04 0.02 0.03 0.05 0.04 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.04 0.06 0.09 0.12
x190.04 0.02 0.04 0.06 0.05 0.07 0.05 0.05 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.10 0.10 0.15
x200.04 0.02 0.03 0.05 0.04 0.05 0.04 0.04 0.04 0.04 0.06 0.03 0.06 0.05 0.04 0.07 0.10 0.09 0.11 0.14
x21 0.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.03 0.06 0.07 0.08 0.09 0.08 0.09
x220.04 0.02 0.03 0.05 0.04 0.06 0.05 0.05 0.04 0.04 0.05 0.03 0.05 0.04 0.03 0.05 0.07 0.08 0.09 0.07 0.07 0.09
x230.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.03 0.06 0.08 0.10 0.10 0.09 0.08 0.09 0.11
x240.03 0.02 0.02 0.04 0.03 0.04 0.03 0.04 0.03 0.03 0.04 0.03 0.04 0.04 0.03 0.05 0.06 0.07 0.08 0.07 0.06 0.06 0.07 0.07
x250.03 0.01 0.02 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.04 0.03 0.05 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05 0.10
x260.03 0.02 0.03 0.05 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.06 0.05 0.04 0.06 0.07 0.08 0.09 0.08 0.07 0.06 0.07 0.06 0.06 0.11
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Panel B: Estimated covariance matrixusing Parallel Shift Factor fromEVD
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26
x10.03
x20.01 0.01
x30.02 0.01 0.01
x40.03 0.02 0.02 0.04
x50.03 0.01 0.02 0.04 0.03
x60.04 0.02 0.03 0.05 0.04 0.05
x70.03 0.02 0.02 0.04 0.04 0.05 0.04
x80.03 0.01 0.02 0.04 0.04 0.05 0.04 0.04
x90.03 0.01 0.02 0.04 0.03 0.04 0.04 0.04 0.03
x100.03 0.01 0.02 0.04 0.03 0.04 0.04 0.04 0.03 0.03
x110.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05
x120.02 0.01 0.02 0.03 0.02 0.03 0.03 0.03 0.02 0.02 0.03 0.02
x130.04 0.02 0.03 0.05 0.04 0.05 0.05 0.05 0.04 0.04 0.05 0.03 0.05
x140.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.04
x150.02 0.01 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.02
x160.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.04 0.06
x170.04 0.02 0.03 0.06 0.05 0.06 0.06 0.06 0.05 0.05 0.06 0.04 0.06 0.06 0.04 0.07 0.08
x180.05 0.02 0.04 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.09
x190.05 0.02 0.04 0.07 0.06 0.08 0.07 0.07 0.06 0.06 0.07 0.04 0.08 0.07 0.05 0.08 0.09 0.10 0.11
x200.05 0.02 0.04 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.09 0.10 0.09
x210.04 0.02 0.03 0.05 0.04 0.06 0.05 0.05 0.04 0.04 0.05 0.03 0.06 0.05 0.03 0.06 0.07 0.07 0.08 0.07 0.06
x220.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.04 0.06 0.07 0.07 0.08 0.07 0.06 0.06
x230.05 0.02 0.03 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.08 0.09 0.08 0.07 0.07 0.08
x240.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.03 0.05 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05
x250.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.03 0.05 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05 0.05
x260.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.06 0.03 0.06 0.06 0.04 0.07 0.07 0.08 0.09 0.08 0.06 0.07 0.08 0.06 0.06 0.07
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Panel C: Estimated covariance matrixusing Parallel Shift Factor fromSVD
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26
x10.03
x20.01 0.01
x30.02 0.01 0.01
x40.03 0.02 0.02 0.04
x50.03 0.01 0.02 0.04 0.03
x60.04 0.02 0.03 0.05 0.04 0.05
x70.03 0.02 0.02 0.04 0.04 0.05 0.04
x80.03 0.01 0.02 0.04 0.04 0.05 0.04 0.04
x90.03 0.01 0.02 0.04 0.03 0.04 0.04 0.04 0.03
x100.03 0.01 0.02 0.04 0.03 0.04 0.04 0.04 0.03 0.03
x110.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05
x120.02 0.01 0.02 0.03 0.02 0.03 0.03 0.03 0.02 0.02 0.03 0.02
x130.04 0.02 0.03 0.05 0.04 0.05 0.05 0.05 0.04 0.04 0.05 0.03 0.05
x140.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.04
x150.02 0.01 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.02
x160.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.04 0.06
x170.04 0.02 0.03 0.06 0.05 0.06 0.06 0.06 0.05 0.05 0.06 0.04 0.06 0.06 0.04 0.07 0.08
x180.05 0.02 0.04 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.09
x190.05 0.02 0.04 0.07 0.06 0.08 0.07 0.07 0.06 0.06 0.07 0.04 0.08 0.07 0.05 0.08 0.09 0.10 0.11
x200.05 0.02 0.04 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.09 0.10 0.09
x210.04 0.02 0.03 0.05 0.04 0.06 0.05 0.05 0.04 0.04 0.05 0.03 0.06 0.05 0.03 0.06 0.07 0.07 0.08 0.07 0.06
x220.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.05 0.03 0.06 0.05 0.04 0.06 0.07 0.07 0.08 0.07 0.06 0.06
x230.05 0.02 0.03 0.06 0.05 0.07 0.06 0.06 0.05 0.05 0.06 0.04 0.07 0.06 0.04 0.07 0.08 0.08 0.09 0.08 0.07 0.07 0.08
x240.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.03 0.05 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05
x250.03 0.02 0.03 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.05 0.03 0.05 0.05 0.03 0.05 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05 0.05
x260.04 0.02 0.03 0.05 0.05 0.06 0.05 0.05 0.05 0.05 0.06 0.03 0.06 0.06 0.04 0.07 0.07 0.08 0.09 0.08 0.06 0.07 0.08 0.06 0.06 0.07
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Table 6Covariance approximation with the principal factor:
Commercial Banks and Bank holding companies
Panel A displays the historic variance-covariance of daily stock returns from February 15, 2006through June 31, 2011 for the ten banks with complete trading history in the Korean Exchange (KRX).
Panel B displays the estimated variance-covariance of daily stock returns from February 15, 2006
through June 31, 2011 for the ten banks companies with complete trading history in the Korean Exchange
(KRX). This estimation is using theParallel Shift Factor fromeigenvalue decomposition (EVD).
Panel C displays the historic variance-covariance of daily stock returns from May February 15, 2006
through June 31, 2011 for the ten banks with complete trading history in the Korean Exchange (KRX).This estimation is using the Parallel Shift Factor fromsingular value decomposition (SVD).
Panel A: Originalcovariance matrix
Banks KEB Cheju Jeonbuk IBK SHB Woori Hana KB BS DBG
KEB 0.07Cheju 0.01 0.04
Jeonbuk 0.02 0.01 0.04
Ibk 0.04 0.02 0.03 0.08
SHB 0.04 0.01 0.03 0.05 0.06
Woori 0.05 0.02 0.03 0.06 0.06 0.10
Hana 0.05 0.01 0.03 0.06 0.06 0.07 0.10
KB 0.04 0.01 0.03 0.06 0.06 0.06 0.06 0.08
BS 0.04 0.01 0.03 0.05 0.04 0.05 0.05 0.05 0.07
DBG 0.04 0.01 0.02 0.05 0.04 0.05 0.05 0.05 0.05 0.08
Panel B: Estimated covariance matrix using Parallel Shift Factor from EVD
Banks KEB Cheju Jeonbuk IBK SHB Woori Hana KB BS DBG
KEB 0.04
Cheju 0.01 0.01
Jeonbuk 0.02 0.01 0.02
Ibk 0.05 0.02 0.03 0.06
SHB 0.04 0.01 0.02 0.05 0.05
Woori 0.05 0.02 0.03 0.07 0.06 0.08Hana 0.05 0.01 0.03 0.06 0.06 0.08 0.08
KB 0.05 0.01 0.03 0.06 0.05 0.07 0.07 0.06
BS 0.04 0.02 0.03 0.05 0.04 0.05 0.05 0.05 0.06
DBG 0.04 0.02 0.03 0.05 0.04 0.05 0.04 0.05 0.06 0.06
Panel C: Estimated covariance matrixusing Parallel Shift Factorfrom SVD
Banks KEB Cheju Jeonbuk IBK SHB Woori Hana KB BS DBG
KEB 0.04
Cheju 0.01 0.00
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Jeonbuk 0.02 0.01 0.01
Ibk 0.05 0.02 0.03 0.06
SHB 0.04 0.01 0.03 0.05 0.05
Woori 0.05 0.02 0.03 0.07 0.06 0.08
Hana 0.05 0.02 0.03 0.06 0.06 0.07 0.07
KB 0.05 0.02 0.03 0.06 0.05 0.07 0.07 0.06
BS 0.04 0.01 0.03 0.05 0.05 0.06 0.06 0.05 0.05
DBG 0.04 0.01 0.03 0.05 0.05 0.06 0.06 0.05 0.04 0.04
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Table 7Covariance approximation with the principal factor: Insurance Companies
Panel A displays the historic variance-covariance of daily stock returns from February 15, 2006
through June 31, 2011 for the six insurance companies with complete trading history in the Korean
exchange (KRX).
Panel B displays the estimated variance-covariance of daily stock returns from February 15, 2006through June 31, 2011 for the six insurance companies with complete trading history in the Korean
exchange (KRX). This estimation is using theParallel Shift Factor fromeigenvalue decomposition
(EVD).
Panel C displays the historic variance-covariance of daily stock returns from February 15, 2006
through June 31, 2011 for the six insurance companies with complete trading history in the Korean
Exchange (KRX). This estimation is using the Parallel Shift Factor fromsingular value decomposition
(SVD).
Panel A: Original covariance matrix
Companies Meritz Samsung Hyundai LIG KoreaRe DB
Meritz 0.10
Samsung 0.03 0.05
Hyundai 0.06 0.04 0.11
LIG 0.06 0.03 0.07 0.10
KoreaRe 0.03 0.02 0.04 0.03 0.08
DB 0.06 0.04 0.08 0.07 0.04 0.12
Panel B: Estimated covariance matrix using Parallel Shift Factorfrom EVD
Companies Meritz Samsung Hyundai LIG KoreaRe DB
Meritz 0.06
Samsung 0.04 0.02
Hyundai 0.07 0.04 0.09
LIG 0.07 0.04 0.08 0.07
KoreaLaseIns 0.03 0.03 0.04 0.03 0.08
DB 0.08 0.04 0.09 0.08 0.04 0.09
Panel C: Estimated covariance matrix using Parallel Shift Factorfrom SVD
CompaniesMeritz Samsung Hyundai LIG KoreaRe DB
Meritz 0.06
Samsung 0.04 0.02
Hyundai 0.07 0.04 0.09
LIG 0.07 0.04 0.08 0.07
KoreaLaseIns 0.03 0.03 0.04 0.03 0.08
DB 0.08 0.04 0.09 0.08 0.04 0.09
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Table 8Covariance approximation with the principal factor:
Investment and Securities Trading Companies
Panel A displays the historic variance-covariance of daily stock returns from February 15, 2006
through June 31, 2011 for the ten investment & securities companies with complete trading history in theKorean Exchange (KRX).
Panel B displays the estimated variance-covariance of daily stock returns from February 15, 2006
through June 31, 2011 for the ten investment & securities companies with complete trading history in the
Korean Exchange (KRX). This estimation is using theParallel Shift Factor fromeigenvalue
decomposition (EVD).
Panel C displays the historic variance-covariance of daily stock returns from February 15, 2006through June 31, 2011 for the ten investment & securities companies with complete trading history in the
Korean Exchange (KRX). This estimation is using the Parallel Shift Factor fromsingular value
decomposition (SVD).
Panel A: Original covariance matrix
Companies SK HD DY HW DS Woori DW SS HK Mirae
SK 0.14
HD 0.09 0.12
DY 0.10 0.10 0.15
HW 0.10 0.09 0.11 0.14
DS 0.07 0.08 0.09 0.08 0.09
Woori 0.07 0.08 0.09 0.07 0.07 0.09
DW 0.08 0.10 0.10 0.09 0.08 0.09 0.11
SS 0.06 0.07 0.08 0.07 0.06 0.06 0.07 0.07
HK 0.06 0.06 0.07 0.06 0.05 0.06 0.06 0.05 0.10
Mirae 0.07 0.08 0.09 0.08 0.07 0.06 0.07 0.06 0.06 0.07
Panel B: Estimated covariance matrix using Parallel Shift Factorfrom EVD
Companies SK HD DY HW DS Woori DW SS HK Mirae
SK 0.09
HD 0.10 0.10
DY 0.11 0.11 0.12
HW 0.10 0.10 0.11 0.10
DS 0.08 0.08 0.09 0.08 0.07
Woori 0.08 0.08 0.09 0.08 0.07 0.07
DW 0.09 0.09 0.10 0.10 0.08 0.08 0.09
SS 0.07 0.07 0.08 0.07 0.06 0.06 0.07 0.05
HK 0.07 0.07 0.08 0.07 0.06 0.06 0.07 0.05 0.05
Mirae 0.08 0.08 0.09 0.09 0.07 0.07 0.08 0.06 0.06 0.07
Panel C: Estimated covariance matrix using Parallel Shift Factorfrom SVD
Companies SK HD DY HW DS Woori DW SS HK Mirae
SK 0.09
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HD 0.10 0.10
DY 0.11 0.11 0.12
HW 0.10 0.10 0.11 0.10
DS0.08 0.08 0.09 0.08 0.07
Woori 0.08 0.08 0.09 0.08 0.07 0.07
DW 0.09 0.09 0.10 0.10 0.08 0.08 0.09
SS 0.07 0.07 0.08 0.07 0.06 0.06 0.07 0.05
HK 0.07 0.07 0.08 0.07 0.06 0.06 0.07 0.05 0.05
Mirae 0.08 0.08 0.09 0.09 0.07 0.07 0.08 0.06 0.06 0.07
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Table 9The percentage difference between original covariance and estimated covariance
This reports summarized statistics of the difference of individual components of the variance-
covariance matrix using a different number of Monte Carlo simulations. The first section reports the mean
of the individual components of the variance-covariance matrix. The second section reports the standarddeviation of the individual components. The first row uses Cholesky decomposition for the estimated
variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses singular value
decomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses factor analytic
singular value decomposition.
CompaniesEVD PCA SVD PCA
Max Min Max Min
All Firms 0.06 0.00 0.06 0.00
Banks 0.03 0.00 0.04 0.00
Insurances 0.04 0.00 0.05 0.00Investment and Securities 0.05 0.00 0.05 0.00
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Table 10 Simple statistics with respect to number of simulation
This reports summarized statistics of the difference of individual components of the variance-
covariance matrix using a different number of Monte Carlo simulations. The first section reports the mean
of the individual components of the variance-covariance matrix. The second section reports the standarddeviation of the individual components. The first row uses Cholesky decomposition for the estimated
variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses singular value
decomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses factor analytic
singular value decomposition.
Panel A: All Financial Firms
DecompositionMean Standard Deviation
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky 0.0009 0.0002 0.0002 0.0214 0.0215 0.0215
Eigenvalue 0.0004 0.0001 -0.0003 0.0222 0.0215 0.0217
Singular Value 0.0012 0.0003 0.0001 0.0209 0.0216 0.0217
Factor Analytic
Eigenvalue0.0006 0.0002 0.0000 0.0215 0.0215 0.0215
Factor Analytic
Singular Value0.0008 0.0003 0.0001 0.0211 0.0214 0.0214
Panel B: Commercial Banks and Bank holding companies
DecompositionMean Standard Deviation
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky 0.0001 0.0002 0.0001 0.0207 0.0204 0.0206
Eigenvalue 0.0001 0.0002 -0.0001 0.0204 0.0206 0.0206
Singular Value -0.0011 -0.0001 0.0005 0.0212 0.0211 0.0206
Factor AnalyticEigenvalue
0.0018 0.0001 -0.0003 0.0205 0.0204 0.0205
Factor Analytic
Singular Value0.0001 0.0002 0.0001 0.0207 0.0204 0.0206
Panel C: Insurance Companies
DecompositionMean Standard Deviation
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky 0.0009 0.0003 0.0000 0.0234 0.0237 0.0240
Eigenvalue -0.0012 -0.0005 -0.0003 0.0236 0.0239 0.0237
Singular Value -0.0001 0.0004 0.0003 0.0235 0.0240 0.0239
Factor Analytic
Eigenvalue0.0004 -0.0001 -0.0001 0.0233 0.0241 0.0238
Factor Analytic
Singular Value0.0003 -0.0003 0.0000 0.0232 0.0239 0.0239
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Panel D: Investment and Securities Trading Companies
DecompositionMean Standard Deviation
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky 0.0006 0.0005 0.0005 0.0283 0.0282 0.0281
Eigenvalue -0.0003 -0.0005 0.0001 0.0287 0.0277 0.0287
Singular Value -0.0011 0.0000 -0.0002 0.0281 0.0283 0.0281
Factor Analytic
Eigenvalue0.0011 -0.0002 -0.0002 0.0277 0.0283 0.0282
Factor Analytic
Singular Value-0.0004 -0.0002 0.0005 0.0290 0.0282 0.0282
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Table 11 Monte Carlo pricing simulation by decomposing methods
Panel A shows the one day Value at Risk at ninety-nine percent confidence (VaR) and corresponding
Expected Shortfall (ES) for an equally weighted portfolio of twenty six financial institutions using
different number of Monte Carlo simulations. The first row uses Cholesky decomposition for theestimated variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses
singular value decomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses
factor analytic singular value decomposition.
Panel B shows the one day Value at Risk at ninety-nine percent confidence (VaR) and corresponding
Expected Shortfall (ES) for an equally weighted portfolio of the ten banks using different number of
Monte Carlo simulations. The first row uses Cholesky decomposition for the estimated variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses singular value
decomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses factor analytic
singular value decomposition.
Panel C shows the one day Value at Risk at ninety-nine percent confidence (VaR) and corresponding
Expected Shortfall (ES) for an equally weighted portfolio of the six insurance companies using a different
number of Monte Carlo simulations. The first row uses Cholesky decomposition for the estimated
variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses singular valuedecomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses factor analytic
singular value decomposition.
Panel D shows the one day Value at Risk at ninety-nine percent confidence (VaR) and corresponding
Expected Shortfall (ES) for an equally weighted portfolio of the ten investment & securities companies
using a different number of Monte Carlo simulations. The first row uses Cholesky decomposition for theestimated variance-covariance matrix, the second row uses eigenvalue decomposition, the third uses
singular value decomposition, the fourth uses factor analytic eigenvalue decomposition, and the fifth uses
factor analytic singular value decomposition.
Panel A: All Financial Firms
Decomposition1 day 99% VaR Expected Shortfall
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky -0.0492 -0.0513 -0.0501 -0.0570 -0.0579 -0.0568
Eigenvalue -0.0491 -0.0493 -0.0513 -0.0566 -0.0575 -0.0576
Singular Value -0.0468 -0.0502 -0.0504 -0.0517 -0.0575 -0.0580
Factor Analytic
Eigenvalue-0.0501 -0.0492 -0.0491 -0.0571 -0.0577 -0.0565
Factor Analytic
Singular Value-0.0453 -0.0489 -0.0504 -0.0553 -0.0576 -0.0572
Panel B: Commercial Banks and Bank holding companies
Decomposition1 day 99% VaR Expected Shortfall
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky -0.0509 -0.0492 -0.0481 -0.0580 -0.0549 -0.0554
Eigenvalue -0.0464 -0.0484 -0.0483 -0.0515 -0.0549 -0.0549
Singular Value -0.0501 -0.0484 -0.0490 -0.0551 -0.0551 -0.0554
Factor AnalyticEigenvalue
-0.0486 -0.0473 -0.0475 -0.0563 -0.0540 -0.0541
Factor Analytic
Singular Value
-0.0483 -0.0487 -0.0488 -0.0529 -0.0561 -0.0557
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Panel C: Insurance Companies
Decomposition1 day 99% VaR Expected Shortfall
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky -0.0495 -0.0550 -0.0561 -0.0536 -0.0611 -0.0647
Eigenvalue -0.0553 -0.0558 -0.0558 -0.0628 -0.0642 -0.0640
Singular Value -0.0515 -0.0569 -0.0556 -0.0576 -0.0648 -0.0631
Factor Analytic
Eigenvalue-0.0558 -0.0551 -0.0550 -0.0645 -0.0638 -0.0641
Factor Analytic
Singular Value-0.0578 -0.0545 -0.0563 -0.0627 -0.0627 -0.0636
Panel D: Investment and Securities Trading Companies
Decomposition1 day 99% VaR Expected Shortfall
N=1,000 N=5,000 N=10,000 N=1,000 N=5,000 N=10,000
Cholesky -0.0680 -0.0670 -0.0652 -0.0768 -0.0759 -0.0753
Eigenvalue -0.0672 -0.0669 -0.0677 -0.0840 -0.0734 -0.0770
Singular Value -0.0653 -0.0638 -0.0652 -0.0804 -0.0739 -0.0748
Factor AnalyticEigenvalue
-0.0657 -0.0648 -0.0659 -0.0721 -0.0726 -0.0767
Factor Analytic
Singular Value-0.0721 -0.0653 -0.0663 -0.0800 -0.0750 -0.0750
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Figure 1US and K
(a) CRSP market ind
(b) KO
(c) The
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.15
-0.1
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0
0.05
0.1
0.15
-0.2
0
0.2
0.4
0.6
0.8
1
orean Market Indexes (Feb 16, 2006 to June 31
ex(NYSE/AMEX/NASDAQ/ARCA) Value weight
PI Composite Index Returns by adjusted prices
Cumulative Returns of KOSPI Composite Index
50
,2011 )
d return
-
7/30/2019 APJFS Paper Docx Format Ver22_final Version
51/78
51
-
7/30/2019 APJFS Paper Docx Format Ver22_final Version
52/78
52
Figure 2 Factor loading for financial firms
The graph a) in figure 2 depicts a bar chart of each factor loading for financial companies. Each value
of factor loading is as follows: Cheju Bank (0.07), Jeonbuk Bank(0.10), Samsung Marine and Fire