APJFS Paper Docx Format Ver22_final Version

7/30/2019 APJFS Paper Docx Format Ver22_final Version

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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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References

Adrian, T., and M. Brunnermeier, 2007, Hedge fund tail risk,Presented at the NY Fed on February 14.

At-Sahalia, Y., A.W. Lo, 2000, Nonparametric risk management and implied risk aversion,Journal of

Econometrics 94 (1/2), 9-51.

Alexander, S., 2007, Application of Monte Carlo simulation methods in risk management,Journal of

Business Economics and Management7 (3), 165168.

Artzner, P., Delbaen, F., Eber, J.M., and D. Heath, 1997, Thinking coherently,Risk10 (11), 6871.

Artzner, P., Delbaen, F., Eber, J.M., and D. Heath, 1999, Coherent measures of risk,Mathematical

Finance 9 (3), 203228.

Baek, S. H., Baek, S. Y., and Seung Y. Cha, 2011, A study on common risk factors in Korean bank

stocks,The Korean Journal of financial management28 (4), 147171.

Basak, S., and A. Shapiro, 2001, Value-at-risk based risk management: Optimal policies and asset

prices,The Review of Financial Studies 14 (2), 371405.

Basle Committee on Banking Supervision, 1996a, Amendment to the Capital Accord to incorporate

market risk.

Basle Committee on Banking Supervision, 1996b, Supervisory Framework for the use of Backtesting

in conjunction with the Internal Models Approach to Market Risk Capital Requirements.

Beder, T. S., 1995, VAR: Seductive but dangerous,Financial Analysts Journal51(5), 1224.

Berkowitz, J., 2001,Testing density forecasts, with applications to risk management,Journal of Business

and Economic Statistics 19, 465474.

BIS Committee on the Global Financial System, 1999,A Review of Financial Market Events in Autumn

1998, October.


25/78

25

Frey, R., and A. J. McNeil, 2002,VaR and expected shortfall in portfolios of dependent credit risks:

Conceptual and practical insights,Journal of Banking and Finance 26 (7), 13171334.

Frye, J., 1997, Principals of risk: Finding value at risk through factor based interest rate scenarios,

London, Risk Publications.

Holton, G., 2003, Value at Risk: Theory and Practice, 1stedn.(Academic Press)

Jamshidian, F., and Y. Zhu, 1997, Scenario simulation: theory and methodology,Finance and Stochastics

1(1), 43-67.

Jorion, P., 2006, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd

edn.(McGraw-Hill)

Jorion, P. and G. Zhang, 2007, Good and bad credit contagion: Evidence from credit default swap,

Journal of Financial Economics 84(3), 860-883

Jorion, P. and G. Zhang, 2009, Credit contagion from counterparty risk, Journal of Finance64(5), 2053-

2087

JP Morgan, 1996,Riskmetrics, New York, December 17, 4th

edn., JP Morgan.

Jorion, P., 2002, How Informative Are Value-at-Risk Disclosures?,The Accounting Review 77(4), 911-

931.

Kaufman, G.G., 1994, Bank contagion: A review of the theory and evidence,Journal of Financial

Services Research 8, 123150.

Krein, A., Merkoulovitch, L., Rosen, D., and Michael Zerbs, 1998, Principal component analysis in quasi

montecarlosimulation, Algo Research Quarterly 1(2), 21-30.

Kyle, A. S., and W. Xiong, 2001, Contagion as a wealth effect,Journal of Finance 56, 14011440.

Lintner, John (1965). The valuation of risk assets and the selection of risky investments in stock

portfolios and capital budgets,Review of Economics and Statistics, 47 (1), 13-37.

Loretan, M., 1997, Generating market risk scenarios using principal components analysis: methodological

and practical considerations, Federal Reserve Board.

Martinez-Jaramillo, S., Perez, O. M., Embriz, F. A., and F. L. G. Dey, 2010, Systemic risk, financial

contagion and financial fragility,Journal of Economic Dynamics and Control34(11), 2538-2374.

Mossin, Jan., 1966, Equilibrium in a Capital Asset Market,Econometrica, 34(4), 768783.


26/78

26

Nijskens, R. and W. Wagner, 2011, Credit risk transfer activities and system risk : How banks become

less risky individually but posed greater risks to the financial system at the same time, Journal of

Banking and Finance 35(11), 1391-1398.

Rockafellar, R.T., and S.Uryasev, 2002, Conditional value-at-risk for general loss distributions,Journal of

Banking and Finance 26 (7), 14431471.

Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk,

Journal of Finance, 19 (3), 425-442

Suh, S., 2011, Measuring systemic risk of Korean banking sector,Journal of Money and Finance 25(2),

57-81.

Sullivan, A., Sheffrin, S. M., 2003, Economics: Principles in action, Upper Saddle River, New Jersey

07458, Pearson Prentice Hall, pp. 551.

Stewart ,G. W., 1993, On the early history of the singular value decomposition,SIAM Review 35, pp 551

566.

Szego, G., 2002, Measures of risk,Journal of Banking and Finance 26, 12531272.

Tsay, R. S., 2003, Analysis of financial time series, 2nd

edn.(Wiley Series)

Viale, A.M, Kolari, J.W., and Donald R. Fraser, 2009, Common risk factors in bank stocks, Journal of

Banking and Finance 33(3), 464-472.

Wong, W. K., 2008,Backtesting trading risk of commercial banks using expected shortfall,Journal of

Banking and Finance 32 (7), 14041415.

Yamai, Yasuhiro, and ToshinaoYoshiba, 2005, Value-at-risk versus expected shortfall: A practical

perspective,Journal of Banking &Finance29, pp. 997-1015.


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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


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


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


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


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


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


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



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


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


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


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



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


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


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


-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


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


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


-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

-0.05

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


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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

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