How the Default Probability is Defined by the CreditRisk+Model?

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1 Iteratioal Joural of Global Eergy Marets ad Fiace, 28, Vol, No, 2-25 vailable olie at Sciece ad Educatio Publishig DOI:269/igefm---4 How the Default Probability is Defied by the CreditRis+Model? bdelader Derbali * Higher Istitute of Maagemet of Sousse, Uiversity of Sousse, Tuisia *Correspodig author: derbaliabdelader@outloofr bstract The aim of this paper is to ivestigate theoretically oe of the curret models of credit portfolio maagemet There are curretly three types of models to cosider the ris of credit portfolio: the structural models (Moody's KMV model ad CreditMetrics model also defied by the models of the value of the firm, reduced form models also defied by models with itesity models (the actuarial models ad the ecoometric models (the Macrofactors model The developmet of the three types of models is based o a theoretical basis developed by several researchers The evolutio of their default frequecies ad the sie of the loa portfolio are expressed as fuctios of macroecoomic ad microecoomic coditios as well as uobservable credit ris factors, which explaied by other factors We developed this paper to explai the differet characteristics of the CreditRis+ models The purpose of this model is to calculate the default probability of credit portfolio Keywords: ris maagemet, credit ris, default probability, structural models, KMV model, CreditRis+, credit portfolio view Cite This rticle: bdelader Derbali, How the Default Probability is Defied by the CreditRis+Model? Iteratioal Joural of Global Eergy Marets ad Fiace, vol, o (28: 2-25 doi: 269/igefm---4 Itroductio The problem of evaluatio of the failure probability of ay borrower was the ceter of the baers as soo as they bega to led some moey The quatitative modelig of the credit ris for a debtor is rather recet i fact Besides, the modelig of the credit ris associated with istrumets of a portfolio of credit such as, the loas, the pledges, the guaratees ad the by-products (who costitute a recet cocept certai umber of models were developed, icludig at the same time the applicatios of property developed for the iteral custom by the fiacial istitutios, ad the applicatios iteded for the sale or for the distributio [] The big fiacial istitutios recogie his ecessity, but there is a variety of approaches ad rival methods There are three types of models of credit portfolio i the course of use at preset [2]: The structural models: there are two models of maagemet of credit portfolio who are supplied i the literature: Moody's KMV model (Portfolio Model ad CreditMetrics model by JPMorga The Macro-factors model (Ecoometric model: The Credit Portfolio View model itroduces i 998 by Mcisey The actuarial models CSFP (Credit Suisse First Bosto: this model (CreditRis+ is developed i 997 The mai idea for this study is to aswer the questio follows: How the default probability is defied by the CreditRis+ model? The, the orgaiatio of this paper is as follows I sectio 2, we preset the CreditRis+ model ad we defie the forces ad the weaesses of this model Fially, we coclude i sectio 3 2 The Model CSFP: CREDIT RISK+ MRKET RISK Sice 99s, Credit Suisse First Bosto (CSFB has developed ew methods of ris maagemet I 993, the credit Swiss Group lauched, i parallel of a importat proect which aims at moderiig its credit ris maagemet ad by usig the expertise of CSFB, ew oe maagemet tool of the credit portfolio i the future I December, 996, Credit Suisse Group preseted the CreditRis+ model as beig a model of the credit portfolio maagemet The structural models preset a icoveiece cocerig the default These models suppose that the default caot have arise by surprise because the maret value of assets is supposed to follow a cotiuous process of distributio I this aliged, a process of Fish was used i the actuarial models the purpose of which is to model the upredictable character of the emergece of the default what is developed i the model CreditRis+ CreditRis+ is a model with itesity is which presets o hypothesis o the causes of failure of a compay It is model statistical of the default of credit ris which maes o claim about the causes of the default This approach is similar to that of the maagemet of the maret ris, i which o attempt is made model the causes of the movemets of maret prices This model does ot

22 Iteratioal Joural of Global Eergy Marets ad Fiace cosider the cosequeces of a deterioratio of the quality of the quality of the couterparty So, the umber of failures i a credit portfolio durig the

2 22 Iteratioal Joural of Global Eergy Marets ad Fiace cosider the cosequeces of a deterioratio of the quality of the quality of the couterparty So, the umber of failures i a credit portfolio durig the give period ustifies itself by a process of Fish CreditRis+ uses a methodology based o techiques ad quatitative methods The preset model is based o a actuarial calculatio to determie ad preset the distributio of the losses of a credit portfolio The CreditRis+ presets four hypotheses: Every idividual credit presets oly two possible states: failures or o failures The default probability of a idividual credit is low The default probability for a big group of borrowers is very low The umber of default over a period is idepedet from that of ay other period By basig itself o these hypotheses, the probability distributio of the umber X of defaults over a give period (oe moth or oe year for example ca be represeted by usig the law of Fish of average µ ad of stadard deviatio μ: μ P( X! Where, µ is the average of the umber of default a year μ P Where, P idicate the default probability of the obligor The aual umber of the defaults,, is a stochastic variable of average µ ad a stadard deviatio μ ccordig to CreditRis+, the calculatio of the distributio of the losses requires the use of a approach by bods; that is issued i a portfolio are grouped ad collected by edge of exposure The process of determiatio of the distributio of the losses of a portfolio is costituted by three stages: The determiatio of the geerative fuctio of probability for every bod The diversio of the geerative fuctio of probability for the whole portfolio The determiatio of the distributio of the losses for the whole portfolio The distributio of the losses of default for a portfolio is diverted i two stages as the watch represets it below (Figure Util here, we suppose that the distributio of fish allows movig closer to the distributio of the umber of the evets of defect The, we should expect that the stadard deviatio of the default rate is approximately equal to the square root of the average I case of defect of a obligor, the couterparty icurs a loss equal to the quatity possessed by the obligor less a quatity of restorig I CreditRis+ the exposure for every obligor is adusted by the rate plaed by restorig, to calculate the loss of default These adusted exposures are exogeous i the model, ad are idepedet of the maret ris ad miimie the ris To divert the distributio of loss for a diversified portfolio, the losses are divided ito bads with the level of the exposure i every bad To aalye the distributio of the resultat losses of the whole portfolio, presetig us the default probability expressed by the fuctio defied i terms of variables auxiliary by respectig itself the followig approach of the formulatio of the geerative fuctio: We cosidered X a whole ad positive radom variable The geerative fuctio of X is the whole series: G( PX ( Where P(X is the probability that the radom variable X taes the value to obtai P(X from the geerative fuctio G(, we use the followig formula: d G P( X (! d I that case, the geerative fuctio associated amog default X arise amog all the bods of a portfolio is give by the expressio below: F( P( X μ exp ( μ (! Figure CreditRis+ ris measuremet framewor [2]

3 Iteratioal Joural of Global Eergy Marets ad Fiace 23 This fuctio ca be writte as follows: ( ( F F Where F ( idicate the geerative fuctio of a portfolio costituted by a sigle bod of the issuer So, every portfolio cosists of m idetical bod of exposure of idicatios (, 2, m Every bod is characteried by: ε μ * ε Thus implies that: μ With, ε idicate the expected average loss expressed i multiple of a stadard exposure L, μ idicate the expected umber of defaults which is a ow value ad θ idicate the exposure expressed i multiple of L i the bad I that case, the iputs of the model to be developed are: the idividual exposure L ad the probability of default P for the issuer (debtor The, the loss hoped for the debtor is expressed as follows: λ L *P λ ε L The expressio above is obtaied whe the expected loss is expressed i uits of L So, the expected loss ε for the bod is give the as follows: ε ε I this perspective, the expected umber of defects μ for each of the idicated bod is the give by: μ ε ε ε Thus, the umber of waited defects total µ for them m bod is expressed as follows m m ε μ μ The expressio of the geerative fuctio of the icluded losses is obtaied by: G ( P ( gregate?losses * L ( m G G ( G ( ( PV Where V represets the amout of the losses of the bod ad P(V idicates the probability of the loss Furthermore, we have: μ ( ( PV P X Thus we obtai: d! μ μ e G (?? exp( μ + μ! The, if we put: m m G ( exp( μ + μ m P(? μ μ m ε m ε The, the geerative fuctio of the icluded losses ca be writte i the followig way: ( ( ( ( ( G exp μ P F( P Where from, we ca obtai the distributio of the losses of the total portfolio of a amout (*L as follows: d G (?! d Lad us ote i that case that, ca be calculated i cotiuous by basig itself o the followig formula ad uder the hypothesis accordig to which µ is costat Where from we obtai: m ε ( exp( G μ exp ε? The CreditRis+ model cosiders that every sector is drive by a simple fudametal factor This factor explais the variability of the rate of average defect measured for this sector The fudametal factor iflueces the rate of defects plaed i the cocered sector which is modeled by a radom variable of average µ ad of stadard deviatio μ idicated for every sector The stadard deviatio reflects the degree to which, i all the probability of default, the obligors i the portfolio are exposed are more or less that their levels of the average By cotiuig this aalysis, the model CreditRis+ bases o the hypothesis that µ is costat So, by basig itself o the distributio of Fish of parameter µ the probability of failures are uderestimated I that case, it is ecessary to tae ito accout the existece of a average umber of variable failures I this aligmet, the parameter µ is cosidered as beig a stochastic variable ad depeds o characteristics

4 24 Iteratioal Joural of Global Eergy Marets ad Fiace of the sector I fact, ad accordig to the CreditRis+ model, a sector is cosidered as beig a sad of credits the rates of failure of which are subected to the same iflueces I the CreditRis+ model, every portfolio is divided ito sectors idicated by with K I particular, for every sector, we itroduce oe radom variable x which represets the average umber of defaults i this sector The average umber of the defects is equal i µ So, the hope of x for the sector is oted µ ad its stadard deviatio is equal i σ I this frame µ is calculated as follows: m ( ( ε μ ( I the case that µ is o costat; the geerative fuctio of the umber of defaults is give by: d ( ( F F F ( P ( defaults f ( x dx x x ( e f ( x dx x Where f(x idicates the desity of the variablex The cotiuatio of the calculatios is coditioed by the presece of a ature of distributio give ix I the CreditRis+ model, the choice is fixed to a distributio Gamma Г of average µ ad of stadard deviatioσ Thus we obtai: x β x ( e x F ( e x β Г ( Where the Gamma fuctio writte as follows: x ( Г e X dx x For every sector, we have two parameters of Gamma fuctio to be estimated adβ 2 2 μ 2, σ β σ μ By substitutig ad by basig itself o the defiitio of the Gamma fuctio, we obtai the: x β x ( e x F ( e x β Г ( Γ F ( Γ( ( + β β β β ( + fter this simplificatio, the geeratig fuctio of the distributio of the probabilities of default for the sector K is give by the followig expressio: F ( p p p? + β β fter the determiatio of the umber of defaults i a portfolio, oe goes i what follows to preset the geeratig fuctio of the losses icorporated i a portfolio fuctios writte is the followig ; So: G ( p( gregate losses * L? G( G (? F( P (? Where the polyomial fuctio P ( is writte as follows: ( ( m ( ε ( P ( ( m ( ε ( m ( ( ( ε μ ( Oe ca deduce the expressio from the geeratig fuctio G( which is writte i the followig way: ( ( ( p m ( ε G µ p ( I this respect, we ca deduct the distributio of the losses of portfolios from the which is give by: G ( Z So, i case G( verify the followig relatio:

5 Iteratioal Joural of Global Eergy Marets ad Fiace 25 ' G( ( G ( B ( Where ( ad B( are two polyomials of the followig shape: r ( a + + a r ( s B b + + b s Thus, the coefficiets ( verify the relatio of followig recurrece: mi( r, a i i i + b (, ( + mi s bl( l l l This relatio is applied owig that G( verify the followig coditio: ( p ( ( m ' ε ( G µ G ( ( ( p m ( ε ( μ Geerally, the CreditRis+ model is based o mathematical techiques i the modelig of the distributio of the losses i the field of the baig activities ad of the isurace The behavior of commo default of the borrowers is icorporated by treatig the rate of default as beig a commo radom variable for multiple borrowers So, the borrowers are assiged amog the sectors amog which each has a rate of average default ad a volatility of rate of default The volatility of rate of default is the stadard deviatio which would be observed o a portfolio of ifiitely diversified homogeeous credit The forces ad the weaesses relative to the CreditRis+ model are preseted i the table below [3]: Table The forces ad the weaesses relative to the CreditRis+ model The forces The use of a miimum of data sice the distributio of the losses depeds oly o oe reduced umber of parameters This characteristic maes it possible the CreditRis+ model to reduce ad miimie the ris of errors due to the ucertaity of the parameters The CreditRis+ model uses models based o closed formulas what allows him a fast executio of calculatios Source: Hamisultae [3] 3 Coclusio I this study, we expose a theoretical approach s cocerig the model of maagemet of credit portfolio by the actuarial models CSFP (Credit Suisse First Bosto However, structural models are based o optio theory ad capital structure the compay O ecoometric models, they li the probability fault of the compay to the state of the ecoomy The probability of failure depeds i these models of macroecoomic factors such as uemploymet, the rate of icrease GDP, the iterest rate The weaesses The CreditRis + model do ot tae ito accout the earigs or the loss of value of the portfolio provoed by chages of Ratig The iterest rates are supposed costat The used techiques of calculatio are ot simple ad are ot ecessarily accessible to every user of the model log-term Moreover, i the CreditRis+ models, the probability of default varies over time Refereces [] Koyluoglu, H, ad Hicma, (998 Recocilable differeces, Ris, (: [2] Crouhy, M, Galai, D ad Mar, R (2 comparative aalysis of curret credit ris models Joural of Baig & Fiace, (24: 59-7 [3] Hamisultae, H (28 Modèles de gestio du risque de crédit Ivestmet System R&D, Documet

CreditRisk + Download document from CSFB web site:

CreditRisk + Download document from CSFB web site: CreditRis + Dowload documet from CSFB web site: http://www.csfb.com/creditris/ Features of CreditRis+ pplies a actuarial sciece framewor to the derivatio of the loss distributio of a bod/loa portfolio.