Research Article Complexity Study of the Credit Risk of a...

Research ArticleComplexity Study of the Credit Risk of a Business Group

Yang Yang1 Jing Gu2 and Zongfang Zhou1

1College of Management and Economics University of Electronic Science and Technology of China Chengdu 610054 China2School of Economics Sichuan University Chengdu 610064 China

Correspondence should be addressed to Jing Gu gj0901scueducn

Received 25 March 2015 Accepted 31 May 2015

Academic Editor Taher Lotfi

Copyright copy 2015 Yang Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A business group is a complex system thus it is much more difficult to predict its credit risk than that of an individual companyThis study proposes an iterative model which describes the internal interactions and dynamic credit risk of a business group Theproposed model was analyzed from a complex dynamics perspective The simulation results based on this model show that chaoswill emerge in the credit risk of a business group due to the dynamic decision-making processes of its subsidiaries even if theinteractions in the business group are fairly simple The results of this study might explain some economic phenomena and theyalso provide insights into the credit risk of a business group

1 Introduction

With the continuous development of economic globaliza-tion business groups play increasingly important roles ineconomic activities and are prevalent around the world (eg[1 2]) Because of the scale advantages of business groupsattempts by banks to attract business groups by providingfavorable rates and flexible policies have been a tremendoussuccess in the past two decades Many banks have gainedsizable benefits from their business group customers (eg [34]) However the worldwide financial crisis in 2008 and sub-sequent bankruptcy of numerous business groups demon-strate that banks can never afford to ignore the importanceof the credit risk of business groupsThus the need for creditrisk control by banks with business group customers hasnever been as urgent as it is at present Compared with indi-vidual companies the credit risk of business groups is alwaysmuch more complex and often unpredictable in reality Butwhy should this be the caseWhy is it difficult to predict creditrisk for business groups What is the mechanism that under-lies the evolution of the credit risk of a business group Thisstudy attempts to answer these questions by constructing amathematical model conducting numerical simulations andperforming a dynamic system analysis We argue that chaoswill occur in the credit risk of a business group as time passes

In recent years increasingly close attention has been paidto the credit risk of business groups Siegel and Choudhury

[5] argued that business groups have a relatively low creditrisk compared with individual companies Subsidiaries withlimited access to intermediate funds in business groups canbenefit from the support of the rest of group when they suffernegative cash-flow shocks Gopalan et al [6] and Khanna andYafeh [7] found that the internal capital market in a businessgroup can decrease the level of credit risk for its subsidiariesThese conclusions are also consistent with the view ofMasuliset al [8] who stated that the financing advantages of businessgroups outweigh their disadvantages By contrast someresearchers have stated the opposite opinion where theysuggest that the credit risk of business groups is often higherthan that of individual companies because of tunneling (eg[9 10]) In addition other studies have demonstrated thatwhether the credit risk of subsidiaries affiliated with businessgroups is lower than that of individual companies dependson a variety of parameters (eg [11 12]) In general theinconsistent conclusions of previous studies suggest that thecredit risk of business groups is complex Chen and Zhou [13]developed a method to measure the credit risk of businessgroups based on a structural model Many studies haveanalyzed the default correlation using copula functions (eg[14ndash16]) Although these studies are significant they still failto explain the complexity of the credit risk of business groups

In addition many major economic events such as thecollapse of Barings Bank the failure of China Aviation Oil(Singapore) the Enron scandal and the bankruptcy petition

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 527854 8 pageshttpdxdoiorg1011552015527854

2 Discrete Dynamics in Nature and Society

of General Motors serve as reminders that the credit risk ofbusiness groups is sensitive and a butterfly effect may exist inthe credit risk of a business groupWe therefore constructeda newsvendor model which is a fairly simple method todescribe the interactions among the subsidiaries of a businessgroup Next we analyzed the complexity of the credit riskof the business group based on numerical simulations anda dynamic systems analysis of the model The results of thisstudy show that the credit risk of business groups leads tochaos even if the interactions within the business group arevery simple Our results also provide new insights into creditrisk management for business groups

The remainder of this paper is organized as follows In thenext section we construct amodel using discrete dynamics todescribe the credit risk of a business group and find the Nashequilibriumpoint of credit risk with a given set of parametersThe emergence of chaos in the credit risk of a business groupis demonstrated in Section 3 based on a numerical simulationand stability analysis Section 4 provides the conclusions ofthis study

2 Model Construction and Analysis

The model we constructed in this study was as conciseas possible This iterative model shows that credit risk forbusiness groups can be complex and unpredictable even ifthe interaction among their subsidiaries is described in a verysimple manner

21 Assumptions and Nomenclature Following Jarrow andTurnbull [17]Ha andTong [18] Chen andBell [19] andWangand Ma [20] the specific assumptions and notations of theproposed model are summarized as follows

(1) The business group in our model is assumed to com-prise 119899 subsidiaries which are independent decisionmakers with the goal of profit maximization

(2) The products of all the subsidiaries are perishableEach subsidiary faces a newsvendor-type demandmarketWe assume that the demand of any subsidiary119894

119889119894119905

follows the uniform distribution 119880[0 2119886119894119905]

where 119886119894119905represents the effort level selected by sub-

sidiary 119894 in period 119905(3) To describe the collaboration and interactions among

subsidiaries in the business group we simply assumethat the market price 119875

119905 which depends on the total

effort level is the same for all the subsidiaries

119875119905= 120572+120573119878

119905minus 120574119878

2119905 (1)

where 119878119905= sum119899

119894=1 119886119894119905(4) Each subsidiary 119894 has a nonlinear cost function 119862

119894119905

for their effort levelThe subsidiaries always make theoptimal effort level decision to obtain the maximummargin of profit in every period

119862119894119905

= 119898119894+ 119899119894119886119894119905

+ 1199021198941198862119894119905 (2)

(5) During each period the business group has a fixedfinancial cost 119863 which is used to repay its externaldebt If the total profit of the business group is lowerthan119863 in any period the business group will defaultIn this study the probability of default is used todescribe the credit risk of the business group

The assumptions given above imply three intuitive con-clusions First increasing the effort level can improve thedemand market and increase the expected demand Thusthe expected demand will be higher when the level ofeffort selected by the subsidiary is also higher Howevera high effort level can lead to a higher cost Furthermorea very high level of effort may also decrease the profitof other subsidiaries because of the relatively low marketprice Second the market price is decided jointly by allthe subsidiaries of the business group which can affect theprofits of all the subsidiaries Third the uncertainty in ourmodel only originates from market demand The volatilityof demand directly affects the credit risk of the businessgroup The pattern of the business group we consider issimilar to a modularized monopoly structure which hasthe shape of horizontal integration and the character ofcompetition However if we interpret the variable 119886

119894119905as the

production decision making and interpret the formula (1)as the nonlinear inverse demand function of subsidiary 119894 inperiod 119905 we can probably explain these assumptions in thecontext of the supply chain

To simplify the model the influence on the subsidiariesdue to the actual controller and the internal tunnelingproblem of the business group are not considered Eachsubsidiary pursues its own profit maximization strategy anddoes not care about the overall welfare of the business groupeven though the decisions of each subsidiary affect the othersubsidiaries in the business group

22 Model Construction We define 120587119894119905as the profit of sub-

sidiary 119894 in period 119905

120587119894119905

= 119875119905119889119894119905

minus119862119894119905 (3)

According to the assumptions in the previous sectionthe subsidiaries of the business group are independent andpursue their own profit maximization strategies When thesubsidiaries decide their effort level they tend to increasetheir effort until the expected marginal revenue is equal tothe marginal cost Therefore

120597119864 [120587119894119905]

120597119886119894119905

= minus 1205741198782119905+120573119878119905+120573119886119894119905

minus 2120574119886119894119905119878119905minus 2119902119894119886119894119905

+120572

minus 119899119894

(4)

The optimal effort level might not be achieved in everyperiod because of the information asymmetry among thesubsidiaries and the bounded rationality of decision makersThe subsidiaries should adjust their effort level in period 119905+1according to their performance in period 119905 A strategy thatthe subsidiaries are highly likely to adopt in the dynamicdecision-making process is marginal adjustment to achieve

Discrete Dynamics in Nature and Society 3

the aim of profit maximization This strategy could be rep-resented by the following iterative equation

119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[

120597119864 [120587119894119905]

120597119886119894119905

] (5)

where 119896119894is the adjustment coefficient for subsidiary 119894 In

our model this coefficient reflects the flexibility of the sub-sidiariesThe subsidiaries with higher adjustment coefficientsmay respond more vigorously to market changes comparedwith the subsidiaries with relatively lower adjustment coeffi-cientsThe coefficient can also be regarded as ameasure of theindividual characteristics of different decision makers Thesubsidiaries that are controlled by radical decision makerstend to have higher adjustment coefficients whereas thesubsidiaries that are controlled by conservative decisionmakers tend to have lower adjustment coefficients Since theadjustment coefficient vector (1198961 1198962 119896119899) is closer to theoptimal decision when the information is more comprehen-sive and transparent it may also represent some features ofthe information structure of the business group

According to (4) and (5) the dynamic decision-makingprocess of subsidiary 119894 can be written as follows

119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[minus1205741198782119905+120573119878119905+120573119886119894119905

minus 2120574119886119894119905119878119905

minus 2119902119894119886119894119905

+120572minus 119899119894]

(6)

Hence the dynamic decision-making process of thebusiness group can be represented by an 119899-dimensional non-linear iterative system

1198861119905+1 = 1198861119905 + 11989611198861119905 [minus1205741198782119905+120573119878119905+1205731198861119905 minus 21205741198861119905119878119905

minus 211990211198861119905 +120572minus 1198991]

1198862119905+1 = 1198862119905 + 11989621198862119905 [minus1205741198782119905+120573119878119905+1205731198862119905 minus 21205741198862119905119878119905

minus 211990221198862119905 +120572minus 1198992]

119886119899119905+1 = 119886

119899119905+ 119896119899119886119899119905

[minus1205741198782119905+120573119878119905+120573119886119899119905

minus 2120574119886119899119905

119878119905

minus 2119902119899119886119899119905

+120572minus 119899119899]

(7)

The profit of subsidiary 119894 is 120587119894119905 thus the total profit of the

business group in period 119905 can be written as follows

120587119905=

119899

sum

119894=1120587119894119905 (8)

We also assume that 119889119894119905

are independent for differentsubsidiaries and we define

119889119905as follows

119889119905=

119899

sum

119894=1

119889119894119905 (9)

Consequently 119889119905follows the uniform distribution 119880[0

2119878119905]In this study the probability of default is used to measure

the credit risk of the business group According to the fifthassumption the default will only occur in a period when thetotal profit of the business group is less than its fixed financialcost Therefore the business group defaults if and only if thefollowing inequality is satisfied

120587119905minus

119899

sum

119894=1119862119894119905

lt 119863 (10)

Hence the probability of default can bewritten as follows

Pr( 119889119905lt

119863 + sum119899

119894=1 119862119894119905

119875119905

) (11)

Since 119889119905follows the uniform distribution the probability

of default in period 119905 can ultimately be written as follows

Pr119905(119886119905) =

119863 + sum119899

119894=1 119862119894119905

2119878119905119875119905

(12)

where 119886119905= (1198861119905 1198862119905 119886119899119905)

Specifically the financial cost of the business group maybe due to the liabilities of its subsidiaries and the constant119863 measures the total financial cost of all the subsidiariesin each period In reality the business group often movesmoney to subsidiaries that face the risk of default from othersubsidiaries based on its internal capital market thus robPeter to pay Paul Therefore we do not need to consider thedifferent debt structures among the subsidiaries and we onlyassume a fixed total financial cost to simplify the model inthis study

23 Model Analysis Without loss of generality we considerthe special case of 119899 = 3 for simplicity Then the dynamicdecision-making process model can be constructed as thefollowing 3-dimensional iterative system

1199091015840= 119909+ 1198961119909 [minus120574119878

2119905+120573119878119905+120573119909minus 2120574119909119878

119905minus 21199021119909+120572

minus 1198991]

1199101015840= 119910+ 1198962119910 [minus120574119878

2119905+120573119878119905+120573119910minus 2120574119910119878

119905minus 21199022119910+120572

minus 1198992]

1199111015840= 119911+ 1198963119911 [minus120574119878

2119905+120573119878119905+120573119911minus 2120574119911119878

119905minus 21199023119911 + 120572

minus 1198993]

(13)

where 119909 = 1198861119905 1199091015840= 1198861119905+1 119910 = 1198862119905 119910

1015840= 1198862119905+1 119911 = 1198863119905 and

1199111015840= 1198863119905+1


Therefore the fixed points in our model satisfy the fol-lowing algebraic equations

1198961119909 [minus31205741199092minus 120574119910

2minus 120574119911

2minus 4120574119909119910minus 4120574119909119911 minus 2120574119910119911

+120573 (2119909+119910+ 119911) minus 21199021119909+120572minus 1198991] = 0

1198962119910 [minus31205741199102minus 120574119909

2minus 120574119911

2minus 4120574119910119911 minus 4120574119909119910minus 2120574119909119911

+120573 (2119910+119909+ 119911) minus 21199022119910+120572minus 1198992] = 0

1198963119911 [minus31205741199112 minus 1205741199092minus 120574119910

2minus 4120574119910119911 minus 4120574119909119911 minus 2120574119909119910

+120573 (2119911 + 119909+119910) minus 21199023119911 + 120572minus 1198993] = 0

(14)

In the dynamic decision-making process parameters120572 120573 120574119898

119894 119899119894 119902119894 119863 are relatively fixed whereas the adjustment

coefficients 119896119894 which depend on the specific characteristics

of the subsidiaries are not In order to make the studyconvenient we fix the parameters as follows

120572 = 5

120573 = 05

120574 = 1

1198981 = 02

1198991 = 05

1199021 = 01

1198982 = 03

1198992 = 04

1199022 = 02

1198983 = 04

1198993 = 03

1199023 = 03

119863 = 2

(15)

Then the algebraic equations (14) can be rewritten as fol-lows

1198961119909 [minus31199092minus119910

2minus 119911


+ 05 (2119909+119910+ 119911) minus 02119909+ 45] = 0

1198962119910 [minus31199102minus119909

2minus 119911

2minus 4119910119911minus 4119909119910minus 2119909119911

+ 05 (2119910+119909+ 119911) minus 04119910+ 46] = 0

1198963119911 [minus31199112 minus1199092minus119910


+ 05 (2119911 + 119909+119910) minus 06119911 + 47] = 0

(16)

There are six fixed points 1199081 (0 08374 08176)1199082 (06160 06095 06037) 1199083 (0 0 09416) 1199084 (0 0minus68386) 1199085 (0 0 0) and 1199086 (minus04405 minus05008 minus05693)According to the algebra we can easily show that the equilib-rium and the adjustment coefficients are independent In ourmodel the nonpositive equilibrium solutions are meaning-less Thus only 1199082 (06160 06095 06037) is the Nashequilibrium of the dynamic decision-making process At theequilibrium the probability of default is 4102 for thebusiness group

In order to find the stable region of the Nash equilibriumpoint we put 1199082 into the Jacobian matrix and obtain thefollowing

119869 =[

[

[

minus477341198961 minus270451198961 minus270451198961minus266801198962 minus483711198962 minus266801198962minus263561198963 minus263561198963 minus490491198963

]

]

]

(17)

Therefore the characteristic equation of the Jacobianmatrix can be written as follows

119891 (120582) = 1205823+119860120582

2+119861120582+119862 (18)

where

119860 = 477341198961 + 483711198962 + 490491198963

119861 = 15873811989611198962 + 16693711989621198963 + 16285111989611198963

119862 = 478501119896111989621198963

(19)

According to the Routh-Hurwitz criterion the necessaryand sufficient condition for the asymptotic stability of theequilibrium point can be written as follows

119891 (1) = 119860+119861+119862+ 1 gt 0minus 119891 (minus1) = 119860minus119861+119862minus 1 gt 0

1198622minus 1 lt 0

(1minus1198622)

2minus (119861minus119860119862)

2gt 0

(20)

Figure 1 shows the stability region of the Nash equilib-rium point represented by algebraic equations (20) If thevector of the adjustment coefficients (1198961 1198962 1198963) is in thisthree-dimensional region the Nash equilibrium point will bereached after a certain number of iterations

3 Simulation and Complexity Analysis ofthe Credit Risk

We conducted numerical simulations to understand theevolution of the credit risk in more depth The model weconstructed shows that the evolution of the credit risk isdriven by the dynamic decision-making processes of thesubsidiaries which are described by algebraic equations (13)Therefore we first focus on the dynamic decision-makingprocess and consider a general circumstance where 1198961 is freeand 1198962 1198963 are fixedWithout loss of generality we assume that1198962 = 03 and 1198963 = 02


0 01 02 03 04 05 06 07 08 09 1

Adjustment coefficient of

subsidiary 1


subsidiary 2

Adju

stmen

t coe

ffici

ent

of su

bsid

iary

3

0020406081

002040608

1

Figure 1 The stability region of the Nash equilibrium

0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1

Adjustment coefficient

Lyap

unov

char

acte

ristic

expo

nent

Figure 2 Lyapunov characteristic exponent of the dynamic deci-sion-making process

The Lyapunov characteristic exponent of a dynamicalsystem is a commonly used quantity which characterizesthe rate of separation of infinitesimally close trajectories Thedynamic decision-making process in our model is a three-dimensional dynamic system The Lyapunov characteristicexponent in each dimension of the dynamic decision-makingprocess is shown in Figure 2

Figure 2 shows that the dynamic decision-making pro-cess is stable and the subsidiaries tend to select effort levelsthat are asymptotically equal to the Nash equilibrium pointwhen the adjustment coefficient is 1198961 lt 02231 As theadjustment coefficient increases the maximum Lyapunovcharacteristic exponent is always initially less than zero Nextit equals zero which shows that the dynamic system under-goes a period of doubling bifurcation where the oscillationis periodic When 1198961 gt 02865 the maximum Lyapunovcharacteristic exponent becomes greater than zero whichindicates that the dynamic system has become chaotic Atthis moment the decisions made by the subsidiaries becomemore complex Since the credit risk of the business groupdepends on these decisions the credit risk may also becomeunpredictable

To obtain a better understanding of the complex creditrisk the bifurcation shown in Figure 3 is used to describe thedynamic decision-making process and the evolution of thecredit risk of the business group Figures 3(a) 3(b) and 3(c)show the dynamic stability characteristics of decisionmakingby the subsidiaries in the business group The bifurcationsoccur as the adjustment coefficient increases Finally thesystem is chaotic Figure 3(d) shows the bifurcation of thecredit risk According to Figure 3(d) the first bifurcationoccurs at 1198961 = 02231 where the probability of default is5621 the second bifurcation occurs at 1198961 = 02581 andthe third occurs at 1198961 = 02655 thus chaos emerges

We also show the chaotic attractors of the dynamic systemin Figure 4 Figures 4(a) 4(b) and 4(c) show the two-dimensional chaotic attractors of the dynamic systembetween the credit risk and effort levels of the three sub-sidiaries Figure 4(d) shows the three-dimensional chaoticattractor for the effort levels selected by three subsidiaries inthe business group

Figures 3 and 4 show that the credit risk of the businessgroup will lead to chaos even if the interactions among thesubsidiaries are described in a very simple manner This mayexplain why the credit risk of a business group is alwayscomplex and sometimes unpredictable To illustrate the creditrisk process we use the sequence diagram of the credit riskFigure 5 shows the changes in the credit risk over time at both1198961 = 024 and 1198961 = 034 We can conclude from this figurethat the credit risk of the business group changes periodicallyand that it can be predictedwithout difficulty when 1198961 = 024However the orbit of the credit risk becomes complex as 1198961increases It is safe to state that the chaos orbit of the credit riskwill traverse all of the value sections within a specific time

Next we illustrate the butterfly effect of the credit riskThe chaotic system is sensitive to its initial conditions thuswe can expect that trivial differences in the initial condi-tions of the subsidiaries will result in departures from theevolutionary trajectories We simulated the evolution of thecredit risk of the business group with the initial conditions(03 02 03) and (03001 02 03) We also calculated thedifference between the credit risks with the two initialconditions over the time as shown in Figure 6 It can be seenthat the difference between the trajectories is very small andnegligible at the initial stage However these two adjacenttrajectories separate after about 20 iterations The differenceincreases over time and the adjacent trajectories are ledinto different domains of attraction This figure may explainwhy there are many different credit risk distributions inhomogeneous business groups

In fact if we define 1198961 = 1198962 = 1198963 = 119896 in our model thenas 119896 increases the credit risk of the business group will reachits first stable point at 4102 Next it will circulate betweenbiperiodic points which means that there are two Nashequilibrium points in this stage If we continue to increase119896 the quadric-periodic phenomenon and octoperiodicphenomenon will occur in turn According to Li and Yorke[21] the credit risk is in chaos at the quadric-periodic stageAt this time even a tiny change in the business group canlead to the credit risk in very different directions



Effor

t lev

el of

subs

idia

ry 1

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(a)


Effor

t lev

el of

subs

idia

ry 2

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(b)

0 005 01 015 02 025 03Adjustment coefficient

Effor

t lev

el of

subs

idia

ry 3

0

01

02

03

04

05

06

07

08

09

1

(c)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(d)

Figure 3 Bifurcations of the decision-making process and the credit risk

4 Conclusion

Credit risk of a business group is complicated and difficultto predict compared with that of an individual companyThe available literature has pointed out some of the mainreasons for the complexity of credit risk of a business groupHowever previous studies have not produced a formal frame-work that explains the mechanism behind this phenomenonIn this paper we proposed an iterative model to describe theinternal interactions and dynamic decision-making processof a business groupThe credit risk of the business group wasthen characterized in each iteration by default probabilityThe subsequent stability analysis and Lyapunov characteristicexponents derived by numerical simulations have at lastrevealed the complexity of the credit risk of a business groupFirst the credit risk of a business group might be led intochaos as time passes even if the internal interactions of thebusiness group were described in a fairly simple mannerSecond the flexibility of the subsidiaries characteristics of

decision makers and information structure in a businessgroup can all affect the Nash equilibrium of its credit riskLastly the credit risk of a business group appeared to beacutely sensitive to initial conditions and the butterfly effectwas found in the evolution of the credit risk of a businessgroup

In this study we have two particular contributions Firstwe proposed an iterative model which was constructed tobe as concise as possible to capture the main characteristicsof the credit risk of a business group The analysis andsimulations of this model essentially revealed the complexityof the credit risk of a business group and to some extentexplained the observed phenomenon that the credit risk ofa business group is more difficult to predict than that of anindividual company Second our results provided importanttheoretical insights for understanding the complexity of thecredit risk of a business group which can benefit furtherresearch in this area In addition this study also providedpractical implications for risk control of a business group


0 02 04 06 08 1Effort level of subsidiary 1

Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(a)

Effort level of subsidiary 2

Cred

it ris

k

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

(b)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(c)

002

0406

081

0

05

10

02

04

06

08

Effort level of subsidiary 1Effort level of subsidiary 2

Effor

t lev

el of

subs

idia

ry 3

(d)

Figure 4 Chaotic attractors of the dynamic system

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1

Iterations (k1 = 024)

(a)

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(b)

Figure 5 Sequence diagram showing the changes in the credit risk of the business group


0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03

Figure 6 Sensitivity of the credit risk to the initial conditions

It suggested that an appropriate internal control mechanisminformation sharing and decision process management canhelp to decrease the credit risk of a business group Althoughthe numerical simulation results in this study are based onthe special case of a business group with three subsidiariesthey can easily be extended to a general circumstance involv-ing a business group with more subsidiaries

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was funded by the National Natural Sci-ence Foundation of China (Approval no 71271043 and no71401116) and the Specialized Research Fund for DoctoralProgram ofHigher Education (Approval no 20110185110021)

References

[1] H V Almeida and D Wolfenzon ldquoA theory of pyramidalownership and family business groupsrdquo Journal of Finance vol61 no 6 pp 2637ndash2680 2006

[2] T Khanna and K Palepu ldquoIs group affiliation profitable inemerging markets An analysis of diversified Indian businessgroupsrdquoThe Journal of Finance vol 55 no 2 pp 867ndash891 2000

[3] M Deloof and M Jegers ldquoTrade credit corporate groups andthe financing of Belgian firmsrdquo Journal of Business Finance andAccounting vol 26 no 7-8 pp 945ndash966 1999

[4] I Mevorach ldquoAppropriate treatment of corporate groups ininsolvency a universal viewrdquo European Business OrganizationLaw Review vol 8 no 2 pp 179ndash194 2007

[5] J Siegel and P Choudhury ldquoA re-examination of tunnelingand business groups new data and new methodsrdquo Review ofFinancial Studies vol 25 no 6 pp 1763ndash1798 2012

[6] R Gopalan V Nanda and A Seru ldquoAffiliated firms and finan-cial support evidence from Indian business groupsrdquo Journal ofFinancial Economics vol 86 no 3 pp 759ndash795 2007

[7] T Khanna and Y Yafeh ldquoBusiness groups and risk sharingaround the worldrdquo Journal of Business vol 78 no 1 pp 301ndash340 2005

[8] R W Masulis P K Pham and J Zein ldquoFamily business grouparound the world costs and benefits of pyramidsrdquo Review ofFinancial Studies vol 24 pp 3556ndash3600 2011

[9] G Jiang CMC Lee andH Yue ldquoTunneling through intercor-porate loans the China experiencerdquo Journal of Financial Eco-nomics vol 98 no 1 pp 1ndash20 2010

[10] S Johnson R La Porta F Lopez-de-Silanes and A ShleiferldquoTunnelingrdquoThe American Economic Review vol 90 no 2 pp22ndash27 2000

[11] V Atanasov B Black C Ciccotello and S Gyoshev ldquoHowdoes law affect finance An examination of equity tunneling inBulgariardquo Journal of Financial Economics vol 96 no 1 pp 155ndash173 2010

[12] F Urzua ldquoToo few dividends Groupsrsquo tunneling through chairand board compensationrdquo Journal of Corporate Finance vol 15no 2 pp 245ndash256 2009

[13] L Chen and Z F Zhou ldquoThe research on measure defaultcorrelation of related corporations controlled by an enterprisegrouprdquoChinese Journal ofManagement Science vol 5 no 18 pp159ndash164 2010

[14] T Ane and C Kharoubi ldquoDependence structure and riskmeasurerdquo The Journal of Business vol 76 no 3 pp 411ndash4382003

[15] R Frey and A J McNeil ldquoVaR and expected shortfall in port-folios of dependent credit risks conceptual and practicalinsightsrdquo Journal of Banking and Finance vol 26 no 7 pp 1317ndash1334 2002

[16] G N F Weiszlig ldquoAre copula-gof-tests of any practical useEmpirical evidence for stocks commodities and FX futuresrdquoThe Quarterly Review of Economics and Finance vol 51 no 2pp 173ndash188 2011

[17] R A Jarrow and S M Turnbull ldquoPricing derivatives onfinancial securities subject to credit riskrdquo Journal of Finance vol50 pp 53ndash86 1995

[18] A Y Ha and S L Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[19] J Chen and P C Bell ldquoThe impact of customer returns onsupply chain decisions under various channel interactionsrdquoAnnals of Operations Research vol 206 no 1 pp 59ndash74 2013

[20] G H Wang and J H Ma ldquoModeling and complexity study ofoutput game amongmultiple oligopolistic manufacturers in thesupply chain systemrdquo International Journal of Bifurcation andChaos vol 23 no 3 Article ID 1350038 11 pages 2013

[21] T Y Li and J A Yorke ldquoPeriod three implies chaosrdquo TheAmerican Mathematical Monthly vol 82 no 10 pp 985ndash9921975

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


of General Motors serve as reminders that the credit risk ofbusiness groups is sensitive and a butterfly effect may exist inthe credit risk of a business groupWe therefore constructeda newsvendor model which is a fairly simple method todescribe the interactions among the subsidiaries of a businessgroup Next we analyzed the complexity of the credit riskof the business group based on numerical simulations anda dynamic systems analysis of the model The results of thisstudy show that the credit risk of business groups leads tochaos even if the interactions within the business group arevery simple Our results also provide new insights into creditrisk management for business groups

The remainder of this paper is organized as follows In thenext section we construct amodel using discrete dynamics todescribe the credit risk of a business group and find the Nashequilibriumpoint of credit risk with a given set of parametersThe emergence of chaos in the credit risk of a business groupis demonstrated in Section 3 based on a numerical simulationand stability analysis Section 4 provides the conclusions ofthis study

2 Model Construction and Analysis

The model we constructed in this study was as conciseas possible This iterative model shows that credit risk forbusiness groups can be complex and unpredictable even ifthe interaction among their subsidiaries is described in a verysimple manner

21 Assumptions and Nomenclature Following Jarrow andTurnbull [17]Ha andTong [18] Chen andBell [19] andWangand Ma [20] the specific assumptions and notations of theproposed model are summarized as follows

(1) The business group in our model is assumed to com-prise 119899 subsidiaries which are independent decisionmakers with the goal of profit maximization

(2) The products of all the subsidiaries are perishableEach subsidiary faces a newsvendor-type demandmarketWe assume that the demand of any subsidiary119894

119889119894119905

follows the uniform distribution 119880[0 2119886119894119905]

where 119886119894119905represents the effort level selected by sub-

sidiary 119894 in period 119905(3) To describe the collaboration and interactions among

subsidiaries in the business group we simply assumethat the market price 119875

119905 which depends on the total

effort level is the same for all the subsidiaries

119875119905= 120572+120573119878

119905minus 120574119878

2119905 (1)

where 119878119905= sum119899

119894=1 119886119894119905(4) Each subsidiary 119894 has a nonlinear cost function 119862

119894119905

for their effort levelThe subsidiaries always make theoptimal effort level decision to obtain the maximummargin of profit in every period

119862119894119905

= 119898119894+ 119899119894119886119894119905

+ 1199021198941198862119894119905 (2)

(5) During each period the business group has a fixedfinancial cost 119863 which is used to repay its externaldebt If the total profit of the business group is lowerthan119863 in any period the business group will defaultIn this study the probability of default is used todescribe the credit risk of the business group

The assumptions given above imply three intuitive con-clusions First increasing the effort level can improve thedemand market and increase the expected demand Thusthe expected demand will be higher when the level ofeffort selected by the subsidiary is also higher Howevera high effort level can lead to a higher cost Furthermorea very high level of effort may also decrease the profitof other subsidiaries because of the relatively low marketprice Second the market price is decided jointly by allthe subsidiaries of the business group which can affect theprofits of all the subsidiaries Third the uncertainty in ourmodel only originates from market demand The volatilityof demand directly affects the credit risk of the businessgroup The pattern of the business group we consider issimilar to a modularized monopoly structure which hasthe shape of horizontal integration and the character ofcompetition However if we interpret the variable 119886

119894119905as the

production decision making and interpret the formula (1)as the nonlinear inverse demand function of subsidiary 119894 inperiod 119905 we can probably explain these assumptions in thecontext of the supply chain

To simplify the model the influence on the subsidiariesdue to the actual controller and the internal tunnelingproblem of the business group are not considered Eachsubsidiary pursues its own profit maximization strategy anddoes not care about the overall welfare of the business groupeven though the decisions of each subsidiary affect the othersubsidiaries in the business group

22 Model Construction We define 120587119894119905as the profit of sub-

sidiary 119894 in period 119905

120587119894119905

= 119875119905119889119894119905

minus119862119894119905 (3)

According to the assumptions in the previous sectionthe subsidiaries of the business group are independent andpursue their own profit maximization strategies When thesubsidiaries decide their effort level they tend to increasetheir effort until the expected marginal revenue is equal tothe marginal cost Therefore

120597119864 [120587119894119905]

120597119886119894119905

= minus 1205741198782119905+120573119878119905+120573119886119894119905

minus 2120574119886119894119905119878119905minus 2119902119894119886119894119905

+120572

minus 119899119894

(4)

The optimal effort level might not be achieved in everyperiod because of the information asymmetry among thesubsidiaries and the bounded rationality of decision makersThe subsidiaries should adjust their effort level in period 119905+1according to their performance in period 119905 A strategy thatthe subsidiaries are highly likely to adopt in the dynamicdecision-making process is marginal adjustment to achieve



119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[

120597119864 [120587119894119905]

120597119886119894119905

] (5)




119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[minus1205741198782119905+120573119878119905+120573119886119894119905

minus 2120574119886119894119905119878119905

minus 2119902119894119886119894119905

+120572minus 119899119894]

(6)


1198861119905+1 = 1198861119905 + 11989611198861119905 [minus1205741198782119905+120573119878119905+1205731198861119905 minus 21205741198861119905119878119905

minus 211990211198861119905 +120572minus 1198991]

1198862119905+1 = 1198862119905 + 11989621198862119905 [minus1205741198782119905+120573119878119905+1205731198862119905 minus 21205741198862119905119878119905

minus 211990221198862119905 +120572minus 1198992]

119886119899119905+1 = 119886

119899119905+ 119896119899119886119899119905

[minus1205741198782119905+120573119878119905+120573119886119899119905

minus 2120574119886119899119905

119878119905

minus 2119902119899119886119899119905

+120572minus 119899119899]

(7)



120587119905=

119899

sum

119894=1120587119894119905 (8)




119889119905=

119899

sum

119894=1

119889119894119905 (9)




120587119905minus

119899

sum

119894=1119862119894119905

lt 119863 (10)


Pr( 119889119905lt

119863 + sum119899

119894=1 119862119894119905

119875119905

) (11)



Pr119905(119886119905) =

119863 + sum119899

119894=1 119862119894119905

2119878119905119875119905

(12)

where 119886119905= (1198861119905 1198862119905 119886119899119905)



1199091015840= 119909+ 1198961119909 [minus120574119878

2119905+120573119878119905+120573119909minus 2120574119909119878

119905minus 21199021119909+120572

minus 1198991]

1199101015840= 119910+ 1198962119910 [minus120574119878

2119905+120573119878119905+120573119910minus 2120574119910119878

119905minus 21199022119910+120572

minus 1198992]

1199111015840= 119911+ 1198963119911 [minus120574119878

2119905+120573119878119905+120573119911minus 2120574119911119878

119905minus 21199023119911 + 120572

minus 1198993]

(13)

where 119909 = 1198861119905 1199091015840= 1198861119905+1 119910 = 1198862119905 119910

1015840= 1198862119905+1 119911 = 1198863119905 and

1199111015840= 1198863119905+1



1198961119909 [minus31205741199092minus 120574119910

2minus 120574119911


+120573 (2119909+119910+ 119911) minus 21199021119909+120572minus 1198991] = 0

1198962119910 [minus31205741199102minus 120574119909

2minus 120574119911


+120573 (2119910+119909+ 119911) minus 21199022119910+120572minus 1198992] = 0



+120573 (2119911 + 119909+119910) minus 21199023119911 + 120572minus 1198993] = 0

(14)





120572 = 5

120573 = 05

120574 = 1

1198981 = 02

1198991 = 05

1199021 = 01

1198982 = 03

1198992 = 04

1199022 = 02

1198983 = 04

1198993 = 03

1199023 = 03

119863 = 2

(15)


1198961119909 [minus31199092minus119910

2minus 119911


+ 05 (2119909+119910+ 119911) minus 02119909+ 45] = 0

1198962119910 [minus31199102minus119909

2minus 119911


+ 05 (2119910+119909+ 119911) minus 04119910+ 46] = 0



+ 05 (2119911 + 119909+119910) minus 06119911 + 47] = 0

(16)



119869 =[

[

[


]

]

]

(17)


119891 (120582) = 1205823+119860120582

2+119861120582+119862 (18)

where

119860 = 477341198961 + 483711198962 + 490491198963

119861 = 15873811989611198962 + 16693711989621198963 + 16285111989611198963

119862 = 478501119896111989621198963

(19)



1198622minus 1 lt 0

(1minus1198622)

2minus (119861minus119860119862)

2gt 0

(20)





0 01 02 03 04 05 06 07 08 09 1


subsidiary 1


subsidiary 2

Adju

stmen

t coe

ffici

ent

of su

bsid

iary

3

0020406081

002040608

1


0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1


Lyap

unov

char

acte

ristic

expo

nent











Effor

t lev

el of

subs

idia

ry 1

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(a)


Effor

t lev

el of

subs

idia

ry 2

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(b)


Effor

t lev

el of

subs

idia

ry 3

0

01

02

03

04

05

06

07

08

09

1

(c)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(d)


4 Conclusion






Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(a)


Cred

it ris

k

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

(b)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(c)

002

0406

081

0

05

10

02

04

06

08


Effor

t lev

el of

subs

idia

ry 3

(d)


0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(a)

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(b)



0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[

120597119864 [120587119894119905]

120597119886119894119905

] (5)




119886119894119905+1 = 119886

119894119905+ 119896119894119886119894119905

[minus1205741198782119905+120573119878119905+120573119886119894119905

minus 2120574119886119894119905119878119905

minus 2119902119894119886119894119905

+120572minus 119899119894]

(6)


1198861119905+1 = 1198861119905 + 11989611198861119905 [minus1205741198782119905+120573119878119905+1205731198861119905 minus 21205741198861119905119878119905

minus 211990211198861119905 +120572minus 1198991]

1198862119905+1 = 1198862119905 + 11989621198862119905 [minus1205741198782119905+120573119878119905+1205731198862119905 minus 21205741198862119905119878119905

minus 211990221198862119905 +120572minus 1198992]

119886119899119905+1 = 119886

119899119905+ 119896119899119886119899119905

[minus1205741198782119905+120573119878119905+120573119886119899119905

minus 2120574119886119899119905

119878119905

minus 2119902119899119886119899119905

+120572minus 119899119899]

(7)



120587119905=

119899

sum

119894=1120587119894119905 (8)




119889119905=

119899

sum

119894=1

119889119894119905 (9)




120587119905minus

119899

sum

119894=1119862119894119905

lt 119863 (10)


Pr( 119889119905lt

119863 + sum119899

119894=1 119862119894119905

119875119905

) (11)



Pr119905(119886119905) =

119863 + sum119899

119894=1 119862119894119905

2119878119905119875119905

(12)

where 119886119905= (1198861119905 1198862119905 119886119899119905)



1199091015840= 119909+ 1198961119909 [minus120574119878

2119905+120573119878119905+120573119909minus 2120574119909119878

119905minus 21199021119909+120572

minus 1198991]

1199101015840= 119910+ 1198962119910 [minus120574119878

2119905+120573119878119905+120573119910minus 2120574119910119878

119905minus 21199022119910+120572

minus 1198992]

1199111015840= 119911+ 1198963119911 [minus120574119878

2119905+120573119878119905+120573119911minus 2120574119911119878

119905minus 21199023119911 + 120572

minus 1198993]

(13)

where 119909 = 1198861119905 1199091015840= 1198861119905+1 119910 = 1198862119905 119910

1015840= 1198862119905+1 119911 = 1198863119905 and

1199111015840= 1198863119905+1



1198961119909 [minus31205741199092minus 120574119910

2minus 120574119911


+120573 (2119909+119910+ 119911) minus 21199021119909+120572minus 1198991] = 0

1198962119910 [minus31205741199102minus 120574119909

2minus 120574119911


+120573 (2119910+119909+ 119911) minus 21199022119910+120572minus 1198992] = 0



+120573 (2119911 + 119909+119910) minus 21199023119911 + 120572minus 1198993] = 0

(14)





120572 = 5

120573 = 05

120574 = 1

1198981 = 02

1198991 = 05

1199021 = 01

1198982 = 03

1198992 = 04

1199022 = 02

1198983 = 04

1198993 = 03

1199023 = 03

119863 = 2

(15)


1198961119909 [minus31199092minus119910

2minus 119911


+ 05 (2119909+119910+ 119911) minus 02119909+ 45] = 0

1198962119910 [minus31199102minus119909

2minus 119911


+ 05 (2119910+119909+ 119911) minus 04119910+ 46] = 0



+ 05 (2119911 + 119909+119910) minus 06119911 + 47] = 0

(16)



119869 =[

[

[


]

]

]

(17)


119891 (120582) = 1205823+119860120582

2+119861120582+119862 (18)

where

119860 = 477341198961 + 483711198962 + 490491198963

119861 = 15873811989611198962 + 16693711989621198963 + 16285111989611198963

119862 = 478501119896111989621198963

(19)



1198622minus 1 lt 0

(1minus1198622)

2minus (119861minus119860119862)

2gt 0

(20)





0 01 02 03 04 05 06 07 08 09 1


subsidiary 1


subsidiary 2

Adju

stmen

t coe

ffici

ent

of su

bsid

iary

3

0020406081

002040608

1


0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1


Lyap

unov

char

acte

ristic

expo

nent











Effor

t lev

el of

subs

idia

ry 1

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(a)


Effor

t lev

el of

subs

idia

ry 2

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(b)


Effor

t lev

el of

subs

idia

ry 3

0

01

02

03

04

05

06

07

08

09

1

(c)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(d)


4 Conclusion






Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(a)


Cred

it ris

k

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1

(b)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(c)

002

0406

081

0

05

10

02

04

06

08


Effor

t lev

el of

subs

idia

ry 3

(d)


0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(a)

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(b)



0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198961119909 [minus31205741199092minus 120574119910

2minus 120574119911


+120573 (2119909+119910+ 119911) minus 21199021119909+120572minus 1198991] = 0

1198962119910 [minus31205741199102minus 120574119909

2minus 120574119911


+120573 (2119910+119909+ 119911) minus 21199022119910+120572minus 1198992] = 0



+120573 (2119911 + 119909+119910) minus 21199023119911 + 120572minus 1198993] = 0

(14)





120572 = 5

120573 = 05

120574 = 1

1198981 = 02

1198991 = 05

1199021 = 01

1198982 = 03

1198992 = 04

1199022 = 02

1198983 = 04

1198993 = 03

1199023 = 03

119863 = 2

(15)


1198961119909 [minus31199092minus119910

2minus 119911


+ 05 (2119909+119910+ 119911) minus 02119909+ 45] = 0

1198962119910 [minus31199102minus119909

2minus 119911


+ 05 (2119910+119909+ 119911) minus 04119910+ 46] = 0



+ 05 (2119911 + 119909+119910) minus 06119911 + 47] = 0

(16)



119869 =[

[

[


]

]

]

(17)


119891 (120582) = 1205823+119860120582

2+119861120582+119862 (18)

where

119860 = 477341198961 + 483711198962 + 490491198963

119861 = 15873811989611198962 + 16693711989621198963 + 16285111989611198963

119862 = 478501119896111989621198963

(19)



1198622minus 1 lt 0

(1minus1198622)

2minus (119861minus119860119862)

2gt 0

(20)





0 01 02 03 04 05 06 07 08 09 1


subsidiary 1


subsidiary 2

Adju

stmen

t coe

ffici

ent

of su

bsid

iary

3

0020406081

002040608

1


0 005 01 015 02 025 03minus2

minus15

minus1

minus05

0

05

1


Lyap

unov

char

acte

ristic

expo

nent











Effor

t lev

el of

subs

idia

ry 1

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(a)


Effor

t lev

el of

subs

idia

ry 2

0 005 01 015 02 025 030

01

02

03

04

05

06

07

08

09

1

(b)


Effor

t lev

el of

subs

idia

ry 3

0

01

02

03

04

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08

09

1

(c)


Cred

it ris

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0

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(d)


4 Conclusion






Cred

it ris

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(a)


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it ris

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0 02 04 06 08 10

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1

(b)


Cred

it ris

k

0

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06

07

08

09

1

(c)

002

0406

081

0

05

10

02

04

06

08


Effor

t lev

el of

subs

idia

ry 3

(d)


0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(a)

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(b)



0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 1


subsidiary 1


subsidiary 2

Adju

stmen

t coe

ffici

ent

of su

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1


0 005 01 015 02 025 03minus2

minus15

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minus05

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1


Lyap

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char

acte

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expo

nent











Effor

t lev

el of

subs

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ry 1

0 005 01 015 02 025 030

01

02

03

04

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06

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1

(a)


Effor

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subs

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0 005 01 015 02 025 030

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02

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(b)


Effor

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(c)


Cred

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(d)


4 Conclusion






Cred

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1

(a)


Cred

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0 02 04 06 08 10

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(b)


Cred

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Effor

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(d)


0 50 100 150 2000

Cred

it ris

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02

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1


(a)

0 50 100 150 2000

Cred

it ris

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Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Effor

t lev

el of

subs

idia

ry 1

0 005 01 015 02 025 030

01

02

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(a)


Effor

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(b)


Effor

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ry 3

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02

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08

09

1

(c)


Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(d)


4 Conclusion






Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(a)


Cred

it ris

k

0 02 04 06 08 10

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02

03

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1

(b)


Cred

it ris

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(c)

002

0406

081

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05

10

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04

06

08


Effor

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ry 3

(d)


0 50 100 150 2000

Cred

it ris

k

01

02

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06

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1


(a)

0 50 100 150 2000

Cred

it ris

k

01

02

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04

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06

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08

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1


(b)



0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

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01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Cred

it ris

k

0

01

02

03

04

05

06

07

08

09

1

(a)


Cred

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0 02 04 06 08 10

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(b)


Cred

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1

(c)

002

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0

05

10

02

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08


Effor

t lev

el of

subs

idia

ry 3

(d)


0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(a)

0 50 100 150 2000

Cred

it ris

k

01

02

03

04

05

06

07

08

09

1


(b)



0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100Iterations

Diff

eren

ce o

f cre

dit r

isk

minus04

minus03

minus02

minus01

0

01

02

03





Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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