Model CreditRisk+ : The Economic Perspective of Portfolio Credit Risk Part II
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1 Model CreditRis+ : The Ecoomic Perspective of Portfolio Credit Ris Part II Semiar: Portfolio Credit Ris Istructor: Rafael Weissbach Speaer: Kexu Li
2 geda Review to the modelig assumptios ad calibratio Sector aalysis ctive Portfolio Maagemet
3 Review to the modelig assumptios ad calibratio What did we do last wee? Closed-form aalytic model Oly probability of default PD is stochastic Each obligor has a defiite ow PD over oe year: P Default rate of each obligor follows the Beroulli distributio X ~ B1, p Small default rates of obligors
4 Review to the modelig assumptios ad calibratio What did we do last wee? Probability geeratig fuctio pgf for the expected umber of default evets Fz F z 1 + pz 1 Fz e μ z 1 e μ e μz 0 e μ! μ z Pr defaul ts μ e μ! The pgf for the portfolio losses distributio G j z pdefaultsz ν 0 0 j e μ μ! j z ν j e ν j j μ + μ z j Gz e μ Pz 1 FPz
5 geda Review to the modelig assumptios ad calibratio Sector aalysis ctive Portfolio Maagemet
6 Sector aalysis Sector aalysis: calculatig the variable PD Idea of sector aalysis: PDs are volatile over time small umber of bacgroud factors systematic factors, cocetratio ris as driver of variability of PDs llocatig all obligors ito a sigle sector, i which PDs move together Sectors bacgroud factors are mutually idepedet Specific factors idiosycratic factors capture uiquely the fortue of a obligor Basic Modelig: llocatig each obligor ito a sigle sector Geeral sector aalysis: variability of each obligor s PD is simultaeously iflueced by several bacgroud factors
7 Sector aalysis Sector aalysis: Notatio Obligor Sector S Radom variable of umber of defaults Mea value μ μ Stadard deviatio Probability of default p p Basic uit of exposure L L Exposure sizes i uits L Lν L L Expected Losses i uits X λ Lε j 1 λ j 1 1 x ν 1 j L ε j m j 1 j m
8 Properties of pdf withi oe sector dditivity of the first ad the secod momets: - mea value of oe sector: - stadard deviatio SD: Liear combiatio betwee mea ad SD: p ν ε μ μ ω 1 ν ε μ μ ν ε p p p p p p ω μ the ratio is accordig to historical data the same for obligors i the same sector: p ω The SD ca be geerated by estimatig the ratio ω Sector aalysis
9 Sector aalysis Excursus: Gamma Distributio I Writte Γ α, β Cotiuous two parameter distributio with the pdf: Px X x + dx 1 α β Γ α 2 2 μ αβ αβ e x α 1 X β dx I our sector model: α μ 2 2 / β / μ 2 Why gamma distributio: ctuarial method Sewess: the smaller the alpha, the closer the distributio to empirical data Mixture form of Poisso rated gamma distributio later
10 Sector aalysis Excursus: Gamma Distributio II α1 β2.0 α2 β2.0 α3 β2.0 α5 β0.1 α9 β0.5
11 Sector aalysis Excursus: Negative Biomial Distributio I Discrete two parameter distributio Describig the probability distributio of defaults adαsurvivals i a series of i.i.d. Beroulli trials with survival o the last trial Probability of defaults: p Probability of survival: 1-p pmf: α + α 1 1 p p pgf: g x z 1 p 1 pz α Why egative biomial distributio The most importat distributio i actuary beside Poisso distributio Sewess ppropriate to recurrece computatio i order to geerate portfolio losses distributio
12 Sector aalysis Excursus: Negative Biomial Distributio II Negative biomial distributio: target successes2, probability of default0.1 Probability mass Number of defaults util target successes
13 Sector aalysis Modelig with stochastic PD: default rate distributio I Key assumptio: Sector default rates are Gamma distributed with mea μad stadard deviatio 2 2 X ~ Γ α, β with α μ ad β / μ pgf of coditioal default rate: pgf of ucoditioal default rate: F z 0 x 0 P e F z x z 1 f x dx 2 / x z 1 [ x x] e defaults z z P defaults x f x dx 0 x 0 Poisso characteristic Gamma distributio Mixture model
14 Modelig with stochastic PD: default rate distributio II pgf for sigle sector : where Expadig i Taylor series: egative biomial distributio 1 p β β z dx x x e z F x x z x + Γ β β α β α α α α β z z F p 1 p 1 α z p p z F α α Sector aalysis
15 Modelig with stochastic PD: losses distributio Capture portfolio losses i pgf: usig exposure multiples Pgf of portfolio losses distributio: Easy calculatio through geeral recurrece relatio Example spreadsheet-based implemetatio m j v j j j z v p p z G z G α ε μ p β β 1 + x z x x dx x f e dx x f LossofL x P z z G v Sector aalysis v
16 geda Review to the modellig assumptios ad calibratio Sector aalysis ctive portfolio maagemet Geeral sector aalysis Pairwise correlatio Ris cotributio
17 ctive portfolio maagemet Effect of cocetratio ad correlatio o credit ris
18 ctive portfolio maagemet Potetial beefits i a typical ba portfolio Source: Oliver, Wyma & Compay1999 : Credit portfolio maagemet
19 geda Review to the modellig assumptios ad calibratio Sector aalysis ctive portfolio maagemet Geeral sector aalysis Pairwise correlatio Ris cotributio
20 ctive portfolio maagemet Geeral sector aalysis I Idea of geeral sector aalysis Replacig the cocept of a sector with that of a systematic factor Factor aalysis: factor loadig Sesitivity of every bacgroud factor to a sigle obligor θ Critical assumptio of CR+ : idepedet bacgroud factors Startig from the pgf of losses distributio G z x e 0 x z v 1 f x dx
21 ctive portfolio maagemet Geeral Sector alysis II Previously: x v ε ν x z 1 z 1 μ ν Now: The θ For a sector : x x μ z v 1 ε v 1 θ x μ Example spreadsheet-based implemetatio θ ε v θ x μ ε ν z ν 1 explais the ifluece of oe systematic factor to a sigle obligor Sum of all iflueces to a obligor is equal to 1 θ
22 geda Review to the modellig assumptios ad calibratio Sector aalysis ctive portfolio maagemet Geeral sector aalysis Pairwise correlatio Ris cotributio
23 ctive portfolio maagemet Pairwise correlatio Defie a idicator fuctio: I Correlatio coefficiet: 1 0 if obligor defaults i the time period otherwise ρ B ρ I,IB Default evet correlatio betwee distict obligors ad B: ρ B μ μ B 1 θ θ B μ If the obligors ad B have o sector i commo, the the correlatio betwee them will be zero No systematic factor affects them I geeral oe would expect default correlatios to typically be of the same order of magitude as default probabilities themselves. order: μ μ B 2 meas of I, IB are captured from μ, μb
24 ctive portfolio maagemet Correlatio extesio what happes if sectors are mutually depedat? Versio I : Bürgisser et al s model Itegratig correlatio Versio II: Factor loadig through pricipal axis trasformatio Versio III: Global bacgroud factor Uderestimatio of default losses from the model CreditRis+ CreditRis+ Versio I - III CreditRis+ Source: Buergisser 1999 Source: Giese1999
25 geda Review to the modellig assumptios ad calibratio Sector aalysis ctive portfolio maagemet Geeral sector aalysis Pairwise correlatio Ris cotributio
26 ctive portfolio maagemet Ris cotributio ad capital allocatio Key problem: who drives the ris ad how much? Ex ate margial ris cotributio Servig for ris-based pricig From ris measuremet to ris maagemet Ris cotributio i CR+: aalytic approximatio without simulatio
27 ctive portfolio maagemet Ris Cotributio Defiitios of Ris Cotributio: Margial effect of a sigle obligor o the stadard deviatio of the distributio of credit losses UL 2 E RC E RC E 2 E Margial effect of a sigle obligor o a give loss percetile of portfolio aggregate ris VaR VaR p VaR ε +ξrc PL E[L] + ξ VarL 1 p VaR
28 Calibratig ris cotributio o stadard deviatio Variace of loss distributio for the whole portfolio: RC for obligor o stadard deviatio: v ε μ ε + 2 E E RC θ ε μ μ 2 E 2 E E E RC ctive portfolio maagemet
29 ctive portfolio maagemet Portfolio maagemet usig Ris Cotributio The total ris cotributios for the idividual obligors is approximately equal to the ris of the etire portfolio RC E 2 E 2 Ris cotributios allow the effect of a potetial chage i the portfolio e.g. the removal of a exposure to be measured I geeral, a portfolio ca be effectively maaged by focusig o a relatively few obligors that represet a sigificat proportio of the ris but costitute a relatively small proportio of the absolute portfolio exposures Example spreadsheet-based implemetatio
30 Summary: Sector aalysis Modelig the loss distributio through probability geeratig fuctio, where the total probability of defaults for sectors follows a gamma distributio, default rate of obligors i sigle sector is Poisso-distributed ad the sectors are idepedet to each other. ctive portfolio maagemet usig CreditRis+ Geeral sector aalysis allows the portfolio to be allocated to sectors to reflect the degree of diversificatio ad cocetratio. Pairwise correlatio represets a aalytical form of portfolio maagemet i the level of sigle obligors Ris cotributio is widely used to reduce Icremetal Credit Reserve
31 Discussio: dvatages ad disadvatages of CreditRis+?
32 Literatures Bürgisser, P./Kurth,./Wager,./Wolf, M.1999: Itegratig Correlatios, i: Ris Magazie July 1999, S Credit Suisse First Bosto Iteratioal, CreditRis+, Credit Ris Maagemet Framewor, Giese, G Ehacig CreditRis+. Worig paper. Oliver, Wyma & Compay 1999: Credit portfolio maagemet, ERis.com
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