Credit Name Concentration Risk: Granularity Adjustment Approximation

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1 Journal of Fnancal Rsk Management, 06, 5, ISS Onlne: ISS Prnt: Credt ame Concentraton Rsk: Granularty Adjustment Approxmaton Badreddne Slme Ecole atonale de la Statstque et de l Admnstraton Economque (ESAE, Pars, France How to cte ths paper: Slme, B. (06. Credt ame Concentraton Rsk: Granularty Adjustment Approxmaton. Journal of Fnancal Rsk Management, 5, Receved: October 8, 06 Accepted: December 3, 06 Publshed: December 6, 06 Copyrght 06 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY Open Access Abstract Durng the last subprme mortgage crss, the concentraton rsk ssue has become ncreasngly mportant n the world of fnance. Ths rsk s defned as the loss that we can ncur from a large exposton of a sngle name counterparty, a sector or a product. Ths paper represents some mathematcal models for assessment and quantfcaton of the concentraton rsk under the Add-On approach. Ths study s based on the Granularty Adjustment (GA. Ths measure quantfes the dosyncratc rsk that s neglected by the Asymptotc Sngle Rsk Factor model (ASRF based on the nfntely granular assumpton of the portfolo. Ths work s about the approxmaton of ths measurement to smplfy the formula of GA usng the Ad-Hoc approach. We have mplemented emprcal tests to fnd the relaton between the GA and concentraton ndexes and we appled these results to the Boxx portfolo. Keywords Credt Rsk, Asymptotc Sngle Rsk Factor, Concentraton Rsk, Granularty Adjustment, Vascek Model, Credt Rsk+ Model. Introducton The Ad-Hoc approach does not take nto consderaton the specfc rsk factors lke the PD and LGD. On the other hand, t does not allow computng the provson charge of captal requrement to cover the concentraton rsk. Behnd ths, the GA represents all specfc rsks neglected by the ASRF model, so t s over than the concentraton rsk. However, we can use t as a metrc to measure ths knd rsk. Ths paper studes the modelng and the approxmaton of ths measure of concentraton rsk. We wll focus on the credt envronment that represents the bankng book and the source of rsks n the bank balance. We wll restrct on the name concentraton. Frst, we wll begn by modelng the name concentraton under the granularty ad- DOI: 0.436/jfrm December 6, 06

2 B. Slme justment. ext, we wll mplement ths approach n the Vascek and Credt Rsk+ models. Then, we wll suggest the approxmaton of the GA. Fnally, we wll mplement some tests to see the effcency of these approxmatons and we wll use these results on the Boxx portfolo to make t avalable.. The Formulaton of Granularty Adjustment (GA The GA was developed to underpn the Asymptotc Sngle Rsk Factor model (ASRF n order to cover the dosyncratc rsk under Internal Ratng Based model (IRB of Basel II. Indeed, the ASRF model supposes that the portfolo s nfntely granular and ths assumpton neglected the specfc rsk. The GA formula was computed by Wlde (00. Thereafter, Martn and Wlde (00 used the results of Goureroux et al. (000 to smplfy t. In ths secton, we wll compute the GA formulaton under the Vascek and Credt Rsk+ models. We deem as the one-dmensonal systematc factor and L as the portfolo loss wth loan, and gvng the followng notatons of the mean and the varance of the condtonal loss : µ = L et σ = L ( [ ] ( [ ] For ε =, the portfolo loss s equal to: L = µ + ε L µ ( ( ( Usng these notatons, the GA s defned as: ASFR GA L = VaR L VaR L ( ( ( ( ( ( q q q ( µ ε µ ( µ ( VaRq( µ ε ( L µ = VaR + L VaR q q By applyng the Taylor expanson on ( ( + wth second order accordng to the ε = 0 and by replacng the ε =, we get 3 : GA q ( L = VaR + L + VaR + L ε ( µ ( ε ( µ ( µ ( ε ( µ ( ( q q ε = 0 ε ε = 0 By computng the frst and the second dervatve terms, we fnd the followng results 4 : VaRq( = L = VaRq( ε Wth VaR ( = f x L = x ( ( ( [ ] q ε f x x f defnes the densty functon of. = µ, we get the followng results 5 : If we set ( x= VaR ( See Annex. See Annex. 3 See Wlde (00, Probng granularty, Rsk Magazne, Vol 4, o 8, pp See Goureroux, Laurent, & Scallet (000, Senstvty analyss of Values at Rsk, Journal of Emprcal Fnance. 5 See Martn, & Wlde (00: Unsystematc credt rsk, Rsk Magazne 5(, pp 3-8. q 47

3 B. Slme VaR ( VaR ε q ( µ ( = 0 σ = ( ( x f ( x µ ( q ε f x x x We fnd the general formula of GA basng on these results: ( GAq L x x x ( x= VaR q f ( x x = σ ( + σ σ µ x f ( x µ x ( ( µ ( ( ( ( x= VaR ( Therefore, f we want to explan ths formula we should use a rsk model. The most prevalent models for the bankng book to calculate the captal request for the credt rsk s: The Vascek and the Credt Rsk+ models. The frst one s deemed as a structural model, and the second one belongs to the ntensty model. In the followng paragraphs, we wll develop the GA formula under these models. The GA formula under the Vascek model: The Vascek 6 model supposes that the systematc factor s followng the Gaussan dstrbuton ~ ( 0,, and ths result leads to: f ( x f ( x = x Substtutng n the formula of GA, we get: Vascek µ x GAq ( L = ( ( ( ( xσ x σ x + σ ( x µ x µ ( x ( x ( x µ = + µ ( x µ ( σ ( x σ x x q ( ( x= Φ ( q x= Φ ( q Thus, we can compute the components that allow computng the GA 7 : ( x = [ L x] = s ( x = s [ LGD] PD( x µ µ σ = = = = = wth µ ( x [ LGD ] PD ( x [ LGD ] and ( x ( x [ L x] s σ ( x ( PD Φ ρx = = Φ ρ ( PD ρ Φ ρx µ ( x = [ LGD ] ϕ ρ ρ ( PD ρ Φ ρ x µ = µ ρ ρ ( x 6 See Vascek (987. Probablty of loss on loan portfolo, KMV Corporaton, San Francsco, USA. 7 See Annex. 48

4 B. Slme We have also: σ ( x = [ L x] = ( LGD D x ( [ LGD D x ] = ( LGD ( D x [ LGD] D x Wth C ( ( [ ] = ( [ ] + ( [ ] ( [ ] ( = Cµ ( x µ ( x [ LGD] + [ LGD] and D s the default varable [ LGD ] 8. ( LGD LGD LGD PD x PD( x = The dervatve functon regardng to x s equal to: ρ Φ ( PD ρx µ ( x = s µ ( x, µ ( x = s µ x = = ρ ρ ( x = s ( x ( C ( x, ( x = s ( x ( C ( x σ µ µ σ µ µ = = =Φ, we fnd the followng formula: Vascek GAq ( L = s ( ( ( ( ( µ q δ C µ q Φ Φ µ ( Φ ( q = µ ( Φ ( q ( C ( ( µ Φ q µ ( Φ ( q µ ( Φ ( q Wth δ = Φ ( q + µ ( Φ ( q The GA formula under the Credt Rsk+ model: As we have seen to compute the GA formula, we need to calculate the followng quanttes µ ( x, σ ( x, and f ( x that depend to the model. The assumpton of the Credt Rsk+ 9 model s that ~ Γ ( αβ, where α =. Then, we obtan the followng relaton: β f ( x = ( α x f x By developng the GA under x ( q ( We can explan the GA formula by computng the followng components: ( x = s ( x, ( x = s ( x µ µ σ σ = = The expresson of µ ( x s gven by 0 : µ ( x = [ LGD ] PD ( x = [ LGD ] PD ( w + w x ( x [ LGD ] PD ( w w x µ = +, = 8 See Annex. 9 See Credt Susse Fnancal Products (997. Credt Rsk+: A Credt Rsk Management Framework. London, See Annex. ( 49

5 B. Slme And µ ( x [ LGD ] PD w µ ( x = et = 0 = We have for the condtonal varance: σ ( x = ( LGD D x ( [ LGD D x ] = ( LGD ( D x [ LGD] PD x = ( LGD ( D x ( µ ( x [ LGD ] = Cµ ( x + ( µ ( x LGD Wth C [ LGD] + [ LGD] [ LGD ] = We conclude that: Therefore, we have: [ ] σ ( x = s µ ( x C + µ ( x = ( ( ( [ LGD ] [ LGD ] [ LGD ] [ LGD ] σ ( x = s µ ( x C + µ ( x = These results we lead us to the GA formulaton found t by Gordy and Lutkebohmert (007 : Wth, CR+ GAq ( L = s δc ( UL + EL + δ ( UL + EL UL = UL C + ( UL + EL [ LGD ] [ LGD ] ( [ ] [ ] ( [ LGD ] [ LGD ] EL = LGD PD, UL = LGD PD w VaR, UL = s UL q = ( q α = α + VaRq And δ VaR ( ( 3. The Granularty Adjustment Approxmaton The am of ths study s the mplementaton of algorthmc tests to test approxmatons of GA. These algorthmc tests wll be establshed on R and under the followng assumptons: See Gordy, & Lutkebohmert (007, Granularty adjustment for Basel II, Dscusson Paper Seres : Bankng and Fnancal Studes, Deutsche Bundesbank (. See Gordy, & Lutkebohmert (03, Granularty Adjustment for Regulatory Captal Assessment, Internatonal Journal of Central Bankng. See Lutkebohmert (009. Concentraton Rsk n Credt Portfolos. Sprnger. 50

6 B. Slme The HKI (Hannah-Kay Index parameter s equal to 3. The HIS (Hammam-Slme Index parameter s equal to 0.5. The generaton of exposures follows the Log-normal dstrbuton. The parameter of the Gamma dstrbuton s equal 0.3. The quantle s equal to 99.9%. 3.. The Reduced Form of GA The authors of the GA formula below the Credt Rsk+ suggest a smplfcaton under the assumpton that quanttes of EL and UL are enough small. So, we can neglect UL + EL and UL ( UL + EL 0. The smplfed GA becomes: ( 0 CR+ q δ + UL = GA ( L s C ( ( UL EL UL By the same way, we can approxmate ths formula below the Vascek model gvng µ q 0 µ Φ q µ Φ q 0, by: ( the assumpton Φ ( and ( ( ( q ( ( ( ( δ µ ( ( µ ( ( Vascek ( GAq L s C Φ q Φ q µ Φ = Ths test allows verfyng the valdty of these approxmate formulas of GA. The Table summarzes the formulatons under the both models Vascek and Credt Rsk+. The test mplementaton s based on portfolo generatng of some = 000 exposures accordng to the Log-normal dstrbuton. Then, we compute the full and the approxmate GA under the both models Vascek and Credt Rsk+. We repeat ths operaton one thousand tmes to get 000 portfolos at the end. Test steps are descrbed on the followng algorthm: Generate 000 exposures accordng to the Log-normal (0, 3 dstrbuton. Generate 000 probabltes of default accordng to the unform dstrbuton. 3 Generate000 correlaton coeffcent accordng to the unform dstrbuton between 0. and Compute the full GA accordng to the two models. 5 Compute the approxmate GA accordng to the two models. Table. Summary of the GA formula dependng on model. GA q GA q GA Vascek q ( L Vascek = Φ Φ µ ( s µ ( ( q δ C µ ( ( q ( ( Φ q = ( Φ ( q ( C µ ( ( q ( Φ ( q µ Φ µ Vascek GAq ( L ( ( ( ( ( s C δ µ Φ q µ Φ q Φ = µ ( ( q GA CR+ q ( L Credt Rsk+ = s δc ( UL + EL + δ ( UL + EL UL = UL C + ( UL EL + UL [ LGD ] [ LGD ] [ LGD ] [ LGD ] CR+ GAq ( L s C ( δ ( UL + EL UL = 5

7 B. Slme 6 Iterate 000 tmes the steps from to 5. 7 Statstcal test of the average under the generated data of the full and the approxmate GA. 8 Statstcal test of the varance homogenety under the generated data of the full and the approxmate GA. Ths test allows us to determne the condtons of usng the approxmate GA n order to smplfy computng. Frst, we get n the Vascek model wth an nterval of default probabltes between 0 and %. We conclude that the two values are very close. Furthermore, the Studen test of the average and the Fsher test of the varance are conclusve and we fnd respectvely a p-value equal to 4% and 5.5%. Ths result underpns the approxmate formula of the GA. On the other hand, f we have the un-condtonal default probabltes go beyond of % then ths approxmaton doesn t more work. The Fgure reproduces the results of ths test: In regards to the Credt Rsk+ model, we can prove usng tests that the approxmaton formula of GA stll sutable when the probabltes of default are between 0 and 0%. We get n by the same way and we generate the PDs between 0 and 0%. The Student test on the average and the Fsher test on the varance gve respectvely a p-value of 49% and de 6%. On the other sde, ths result s no more sutable for the PDs beyond of 0%. As concluson, the condton that makes the approxmaton formula sutable for Vascek model s the PDs portfolo between 0% and %, and for the Credt Rsk+ model s the PDs portfolo between 0% and 0%. The Fgure shows the evoluton of the full and the approxmate GA. 3.. The Regresson of GA on the Concentraton Indexes The regresson of the GA on the Herfndahl-Hrschman Index (HHI: We fnd nto the GA formula the square of shares s, and these represent compo- Fgure. The evoluton of the GA under the Vascek Model accordng to number of smulatons. Fgure. The evoluton of the GA under the Credt Rsk+ Model accordng to number of smulatons. 5

8 B. Slme nents of the HHI ndex. Furthermore, n the case of a homogeneous portfolo regardng to specfc rsk factors, we get a lnear relaton between the GA and the HHI: µ ( Φ ( q Vascek GAq ( L = δ ( C µ ( Φ ( q µ ( Φ ( q µ ( Φ ( q ( C µ ( Φ ( q HHI µ ( Φ ( q CR + [ LGD] GAq ( L = δc ( UL + EL + δ ( UL + EL UL [ LGD] [ LGD] UL C + ( UL + EL HHI [ LGD] Vascek GAq ( L = Coeff ( PD, LGD, q HHI CR+ GAq ( L = Coeff ( PD, LGD, w, q HHI where HHI = s. = The Fgure 3 shows the evoluton of the GA accordng to the HHI ndex n the case of homogenous portfolos ( PD = 5%, LGD = 45%, w = % : The goal of ths test s to verfy the valdty of ths relaton on the non-homogeneous portfolo. For ths, we establsh the followng test: Generate 000 exposures accordng to the Log-normal (0, 3 dstrbuton. Generate 000 probabltes of default accordng to the unform dstrbuton (5%, 0%. 3 Generate 000 correlaton coeffcent accordng to the unform dstrbuton between 0. and Compute the full GA accordng to the two models (Vascek and Credt Rsk+. 5 Compute the HHI ndex. 6 Iterate 000 tmes the steps from to 5. 7 Apply the lnear regresson under the smulated GA accordng to the smulated HHI. If we take an nterval of PDs between 0% and 0%, we obtan the followng results n the Fgure 4. The Table summarzes the characterstcs of the lnear regresson. From these results, we can deduce that the relatonshp of lnearty between the GA and the HHI remans vald for mnmum concentratons. Otherwse, you can have qute substantal dspersons around the regresson for farly major ndexes. The regresson of the GA on the Hannah-Kay Index (HKI: We couldn t fnd drectly the relaton between the GA and the HKI even though n case of a homogeneous portfolo. Therefore, we wll use an emprcal approach to get ths relaton. The HKI 3 s defned by: See Herfndahl (950. Concentraton n the U.S. Steel Industry, Dssertaton, Columba Unversty. See Hrschmann (964. The paternty of an ndex. Amercan Economc Revew, 54, 5, pp See Hannah, & Kay (977. Concentraton n modern ndustry. Mac Mllan Press, London. 53

9 B. Slme Fgure 3. The evoluton of GA regardng to HHI n case of a homogeneous portfolo. Fgure 4. The evoluton of GA regardng to HHI wth PD [ 0,0% ]. Table. Summary of lnear regresson of GA on HHI. Coeffcent Standard Resdue R-Squared GA Vascek GA Credt Rsk ( α α HKI = s avec α > 0 et α = Basng on the emprcal experence, we get a non-lnear regresson relaton: α Vascek V α V GAq ( L = a HKI + a HKI α CR+ CR α CR GAq ( L = a HKI + a HKI ( α α ( α α Wth α s the HKI parameter We process n the same way to the last mplementaton. Indeed, we generate = 000 exposures wth the Log-normal and we compute the GA and the HKI ndex. The descrpton of the algorthm steps s: Generate 000 exposures accordng to the Log-normal (0, 3 dstrbuton. Generate 000 probabltes of default accordng to the unform dstrbuton (5%, 0%. 3 Generate 000 correlaton coeffcent accordng to the unform dstrbuton between 0. and Compute the full GA accordng to the two models (Vascek and Credt Rsk+. 5 Compute the HKI ndex. 6 Iterate 000 tmes the steps from to 5. 7 Apply the nonlnear regresson under the smulated GA accordng to the smulated HKI. In the case of homogenous portfolos, the Fgure 5 shows the evoluton of the GA accordng to the HKI ndex, and coeffcents of the non-lnear regresson are respectvely a = 0.99, a =.36 and a = 0.8, a =.748 ( PD = 5%, LGD = 45% V V CR CR, w = % : 54

10 B. Slme Fgure 5. The evoluton of GA regardng to HKI n case of a homogeneous portfolo. Fgure 6. The evoluton of GA regardng to HKI wth PD [ 0,0% ]. If we take an nterval of PDs between 0% and 0%, we obtan the followng results n the Fgure 6. We can conclude that ths relatonshp between the GA and the HKI remans vald for mnmum concentratons. Otherwse, you can have qute substantal dspersons around the regresson for farly major ndexes. The regresson of the GA on The Hammam-Slme Index (HSI: We can t drectly fnd the relaton between GA and HSI even though n case of a homogeneous portfolo. Therefore, we wll use an emprcal approach to get ths relaton. The HSI 4 s defned by: + = HSI = s α ;0< α Usng the emprcal study, we get a non-lnear regresson relaton: ( α + Vascek V α V α GAq ( L = a HSI + a HSI Wth α s the HSI parameter ( α + CR+ CR α CR α GAq ( L = a HSI + a HSI We process n the same way to the last mplementaton. Indeed, we generate = 000 exposures wth the Log-normal and we compute the GA and the HSI ndex. The descrpton of the algorthm steps s: 4 See Slme, & Hammam (06. Concentraton Rsk: The Comparson of the Ad-Hoc Approach Indexes. Journal of Fnancal Rsk Management, 5,

11 B. Slme Generate 000 exposures accordng to the Log-normal (0, 3 dstrbuton. Generate 000 probabltes of default accordng to the unform dstrbuton (5%, 0%. 3 Generate 000 correlaton coeffcent accordng to the unform dstrbuton between 0. and Compute the full GA accordng to the two models (Vascek and Credt Rsk+. 5 Compute the HSI ndex. 6 Iterate 000 tmes the steps from to 5. 7 Apply the nonlnear regresson under the smulated GA accordng to the smulated HSI. In the case of homogenous portfolos, the Fgure 7 shows the evoluton of the GA accordng to the HSI ndex, and the coeffcents of the non-lnear regresson are respectvely a = 4.7, a = 3.7 and a = 6.65, a = 4.49 ( PD = 5%, LGD = 45%, V V CR CR w = %. If we take an nterval of PDs between 0% and 0%, we obtan the followng results n the Fgure 8. We can conclude that ths relatonshp between the GA and the HSI remans vald for mnmum concentratons. Otherwse, you can have qute substantal dspersons around the regresson for farly major ndexes. Fgure 7. The evoluton of GA regardng to HSI n case of a homogeneous portfolo. Fgure 8. The evoluton of GA regardng to HSI wth PD [ 0,0% ]. 56

12 B. Slme 4. Applcaton: Boxx Portfolo In ths secton, we wll apply the obtaned results under an Boox portfolo. We wll buld some portfolos gven the composton of ths ndex. We wll deem that the portfolo buldng ths ndex s the market portfolo. The Boox contans 663 exposures over 0 sectors and 36 countres. The total amount of debt s trllon Euros. The Fgure 9 and Fgure 0 show reparttons by sector and by countres (the dsplayed data are dated 30/06/05. We can also have the repartton by ratng n the Fgure. Fgure 9. Graph of exposures by sector. Fgure 0. Graph of exposures by countres. Fgure. Graph of exposures by ratng. 57

13 B. Slme The Table 3 dsplays the mappng between the probabltes of default and the ratng 5. Frstly, we can study the concentraton of the Boxx portfolo to get a global vew of the concentraton. The Lorenz curve, n the Fgure, allows us to have the dsperson of exposures by counterparty. Basng on the graph, we have an almost equal dstrbuton between exposures. We can make a frst feelng that the name concentraton s small. Therefore, we use also the other metrcs to confrm ths concluson. Indeed, we compute the tree concentraton ndexes and the GA. The Table 4 summarzes the result compute of these metrcs. Gvng these results, we can conclude that the name concentraton s neglected. After ths study, we wll take a small portfolo wth 00 exposures to see the mpact of the number of exposures on the name concentraton under these metrcs. For ths, we wll do a random selecton from the Boxx composton. We can use regressons of the GA on concentraton ndexes to compute the name concentraton rsk. We use the same algorthms n the thrd secton. The Fgure 3 below shows the smulaton result. The Table 5 summarzes the obtaned results: Table 3. The mappng table between the ratng and the PDs. Ratng PD AAA 0.09% AA 0.0% A 0.75% BBB.0% Fgure. Lorenz curve of the Boxx portfolo. Table 4. The computatonal result of the Boxx portfolo. HHI 0.07% HKI 0.08% VaR 6.4% EC 5.87% GA 0.07% Approxmate GA 0.08% 5 Moody s Investor Servce,

14 B. Slme Fgure 3. The regresson of GA on ndexes. Table 5. The concentraton measure recaptulatve of the credt portfolo. HHIP.7% HKIP.5% HSIP 3.4% VaRP 6.% ECP 5.7% GA (HHI.83% GA (HKI.9% GA (HSI.95% There s a concentraton rsk rather mportant consderaton at the GA, as t ncreases the costs n terms of provson approxmately %. Ths result s consstent wth the HSI ndex, unlke the HHI and HKI ndexes. 5. Concluson Ths paper s dedcated, frstly, to model the name concentraton under the Add-On approach; secondly, to approxmate the GA usng the concentraton ndexes. We establshed tests to fnd the relaton between the GA and the ndexes. These approxmatons 59

15 B. Slme allow us some smplfcaton of the GA formula. As applcaton, we chose the Boxx composton as the credt portfolo. These tests on the GA approxmaton enabled us to make the relaton between the Ad-Hoc and the Add-On. We retaned the regresson between the GA and concentraton ndexes. Furthermore, the HSI ndex gave a more consstent measurement of portfolos wth a small number of exposures. However, these approxmatons can be used to smplfy the GA calculaton under the sector concentraton. Indeed, the formulaton of GA s more complex n the sector concentraton than the name concentraton. References Goureroux, C., Laurent, J. P., & Scallet, O. (000. Senstvty Analyss of Values at Rsk. Journal of Emprcal Fnance, 7, Gordy, M., & Lutkebohmert, E. (007. Granularty Adjustment for Basel II. Dscusson Paper Seres : Bankng and Fnancal Studes, Deutsche Bundesbank. Gordy, B., & Lutkebohmert, E. (03. Granularty Adjustment for Regulatory Captal Assessment. Internatonal Journal of Central Bankng, 9, Hannah, L., & Kay, J. A. (977. Concentraton n Modern Industry. London: Mac Mllan Press. Herfndahl, O. (950. Concentraton n the U.S. Steel Industry. Dssertaton, Columba Unversty. Hrschmann, A. (964. The Paternty of an Index. Amercan Economc Revew, 54, 76. Lutkebohmert, E. (009. Concentraton Rsk n Credt Portfolos. Berln: Sprnger. Martn, R., & Wlde, T. (00. Unsystematc Credt Rsk. Rsk Magazne, 5, 3-8. Merton, R. (974. On the Prcng of Corporate Debt: The Rsk Structure of Interest Rates. The Journal of Fnance, 9, Slme, B., & Hammam, M. (06. Concentraton Rsk: The Comparson of the Ad-Hoc Approach Indexes. Journal of Fnancal Rsk Management, 5, Vascek, O. A. (987. Probablty of Loss on Loan Portfolo. San Francsco: KMV Corporaton. Wlde, T. (00. Probng Granularty. Rsk Magazne, 4, Credt Susse Fnancal Products (997. Credt Rsk+: A Credt Rsk Management Framework. London. 60

16 B. Slme Annexes The Vascek model: In 987, Vascek used the Merton model (974 to modelng relatons between the default events to get the assessment of the credt rsk. We denote λ as the lablty of the borrower. The asset value of ths borrower wth a gvng tme t follows a geometrc Brownan moton and verfes the followng stochastc dfferental equaton (SDE: m dv = V µ dt σ dw ηdb + + t, t, k, kt, t, k= Wth µ, σ,, σm, η are constant and W, t,, Wmt,, B s an Independent Brownan moton. Wkt, t,, k =,, m represent the macroeconomc component (systematc rsk and B s the specfc factor (dosyncratc rsk. The Black & Scholes theory t, wth a one year horzon gves us the soluton of the SDE: V V m, =,0 exp µ + σk, k σk, + η η k= where,,, m are..d (ndependent and dentcally dstrbuted and follow a Gaussan dstrbuton. The model supposes that default varables D are Bernoull: s V, < λ D = 0 s V, λ Indeed, the default probablty s equal to: m PD = ( V, < λ = V,0 exp µ + σk, k σk, + η η k < λ = m m λ = σk, k + η < ln + σk, + η µ k= V,0 k= λ m m ln + σ k=, σ k, k η V k + η µ +,0 k= = < m m σ k k, + η = σ k k, + η = λ m ln + σ k k, η µ = + V,0 = Φ m σ k k, + η = Therefore, the borrower s n default when: m m ln ln k= k, σ k, k + η V,0 k= < m m σ k k, + η σ k, k + η = = λ + σ + η µ 6

17 B. Slme If we set: m σ k, σ k = k, et α m k, m σ k k, + η = σ k = k, ρ = = m σ k, Zk+ η k = We get: = ρα + ρ m σ k, k + η = Wth α = ( α,,, αm, and = (,, m The default condton becomes: Wth ( PD s ρ α + ρ <Φ D = 0 s ρ α + ρ Φ ( PD ( PD λ m ln + σ k k, η µ = + V,0 Φ = Then, we conclude that: m m σ k k, + η = [ 0, ],, α ~ ( 0, ρ α = ρ + ρ k, k = The Vascek model use one systematc factor some borrower condtonally to ths factor s equal to: =. The default probablty of ( ( ( ρ α ρ ( PD x = D = = x = + <Φ PD = x Φ = ( ρx + ρ<φ ( PD = < We can deduce that PD ( x ( PD Φ ρx = Φ ρ ( PD ρ ρx Gvng these results and under the assumpton that borrowers loss are ndependent. The loss rate of the whole portfolo s: L = s LGD { ρα + ρ <Φ ( PD } = We can obtan the expected loss condtonally to the systematc factor under the assumpton that the loss gvng default LGD and the default event D = ρ α + ρ <Φ ( PD are ndependent: { } ( PD Φ ρα [ L ] = slgdφ = ρ We can use the Monte Carlo smulaton on the systematc factor to compute ths value. The Credt Rsk+ model: The Credt Rsk+ model was had developed by Credt Susse Fnancal Products (CSFP. Ths model s the one of most used n the IRB Approach and he s one of reduced form models. The default rate s a stochastc varable and the default varable 6

18 B. Slme follows the Bernoull dstrbuton: s l emprunteur fat défaut à T D = 0 autrement Credt Rsk+ supposes that default probabltes are hazardous and systematc factors follow the Gamma dstrbuton wth the followng functon densty:, ( x e x x α Γ αβ =, pour x 0 et β = α β Γ α α ( Wth [ ] = αβ = and [ ] = α In the case that the default frequency as ntensty, we get: ( ( PD x D follows the Posson dstrbuton wth k ( D = k = x = exp PD x, k = 0,,, k! The default varable and the default frequency meet wth the followng relaton D = { D }. Therefore, the condtonal probablty s defned as: PD( = ( D = = D ( ( D = = 0 ( PD = exp PD Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc. A wde selecton of journals (nclusve of 9 subjects, more than 00 journals Provdng 4-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: Or contact jfrm@scrp.org 63