CyRCE: A CreditRisk defaultm odelthat m easures concentration,single obligor lim its and bank capital adequacy.

Transcription

1 CyRCE: A CreditRisk defaultm odelthat m easures concentration,single obligor lim its and bank capital adequacy. JavierM árquez D iez-c anedo. M arch 2002

2 Index I. Introduction II. CyRCE model III. System ic CreditRisk analysis

3 Introduction One of the m ain concerns of regulators is having a proper assessm entofthe solvency ofthe FinancialSystem. Itis im portantto have an adequate picture ofthe levelofrisk ofthe Financial System as a w hole, w here risk is concentrated, and the individualbanks contribution to overallrisk. This represents a technically form idable problem for several reasons: Inform ation Creditrisk methodologies: No accepted paradigm Num ericaltechniques with heavy com putationalrequirem ents How to draw the picture?

4 Existing CreditRisk Methodologies Big inform ation req uirem ents. Substantialcom putationaleffortto obtain loss distribution. No explicitrelation between creditrisk and Capitaladequacy Concentration Single obligor lim its

5 Index I. Introduction II. CyRCE model III. System ic CreditRisk analysis

6 CyRCE:Properties CyRCE is a default Credit Risk M ethodology w hich avoids the use of com putationally dem anding num erical methods, by assum ing thatthe loan portfolio loss distribution can be characterized by its m ean and its variance: Closed form expression for Value atrisk (VaR) Explicitparam etrization ofallrelevantcreditrisk elem ents. C reditrisk related C apitala dequacy can be established in term s of: Defaultrates. A m easure ofconcentration and/or Single obligor lim its.

7 CyRCE:CapitalA dequacy and C reditrisk The m odelis builtup from a very sim ple case w here all loans have i.i.d default probabilities, and extended to a general situation where default probabilities of loans can differ, be correlated and the portfolio can be segm ented arbitrarily to detect risky concentration segm ents.

8 CyRCE:A sim ple Model Letf i be the i th loan am ountin the portfolio;i= 1,2,...,N Define N binary i.i.d random variables: X i = f i with probability p 0 with probability 1 - p The m ean and standard deviation ofthe totalportfolio loss are: µ = pv σ = p ( 1 p) i N = 1 f i 2 V = N i= 1 f i

9 CyRCE:Value atrisk A ssum e for the momentthat the loss distribution can be approxim ated by the Norm aldistribution,so that,the value atrisk with confidence levelα is: VaR α pv + z 1 α p ( p) = N i = 1 f i 2 EXPECTED LO SS µ UNEXPECTED LO SS z α σ

10 CyRCE:CapitalA dequacy CAPITALIZATION RATIO: ψ = Economic Capital Value of Loan Portfolio Capitaladequacy req uires: Economic Capital VaR α ψ p + z p( 1 p )H( F ) α w here a m easure ofconcentration em erges naturally: H = N ( ) i= 1 F 2 N i= 1 f f 2 i i Herfindahl-Hirschm an concentration index F = [ f 1 f 2 f N ] T loan portfolio vector

11 CyRCE:C oncentration Lim it The VaR expression establishes a lim iton concentration through H(F): H(F) = N i= 1 N i= 1 f f 2 i i 2 z 2 ( ψ p) 2 ( ) α p 1 p From which single obligor lim its can be obtained.

12 CyRCE:Single Obligor Lim its and Concentration K δ K ψ = K V is the capitalization ratio. f i δ K = i=1,,n K δ V = δ ψ V =θ V V So, single obligor lim its can be seton Capitalor on total portfolio value as long as: θ = δ ψ = δ (Capitalization Ratio)

13 CyRCE:Single Obligor Lim its and Concentration HHIProperty I: V θv f i θv i=1,,n H(F) θ Then,itfollow s that: f i θv i = 1,,N θ z ( ) 2 ψ p 2 ( ) α p 1 p ψ

14 CyRCE:The concentration index HHIProperty 2:H(F) θ θ Maximum loan θv The tradeo fffor lending the m axim um loan to a single debtor,is atthe expense ofcreditto allother debtors, which tends to zero as N increases. 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% This graph show s how fastdecreases the am ountofother loans under the hypothesis ofa m axim um loan when N increases.

15 CyRCE:Sim ple Modelcontinued The expression ψ p + z p( 1 p )H( F) α is attractive because: Loan portfolio value atrisk through: z α CapitalAdequacy: ψ Itrelates Single Obligor Lim its Single loan default probabilities: p Loan portfolio concentration: H(F )

16 CyRCE:Exam ple The average defaultprobability for the 25 loans is 10.89% , , ,480 () = L H F = ,164 Obligor Amount Obligor Amount A1 4,728 A14 5,042 A2 7,728 A15 15,411 A3 5,528 A16 1,933 A4 5,848 A17 2,317 A5 3,138 A18 2,411 A6 3,204 A19 2,598 A7 4,831 A A8 4,912 A21 1,090 A9 5,435 A22 2,652 A10 5,320 A23 4,929 A11 5,765 A24 6,467 A12 20,239 A25 6,480 A13 1,800 Total 130,164 A ssum ing Norm ality and a 5%,confidence level,capitaladequacy requires: ψ ( ) ( ) ( ) = % Or: K ( 26.6 % )( 130, 164 ) = $34, 603

17 Capitalization ratio: CyRCE:Exam ple (continued) Suppose K = 35,000,then: ψ = K V = 35, , 164 Is the portfolio too concentrated? Η(F) = z α W hat s the largestpossible loan? = 26.9% 2 2 ψ( p) ( ) 2 = 2 p( 1 p) ( 1.96) ( )( ) 26.6% required = f * = ( )( 130,164) = $34, 108 W ould there be single obligor lim itcom pliance? f i ,164 = $8,942

18 CyRCE:A GeneralModel Allloans have differentdefaultprobabilities:p 1,...,p N. The m ean and standard deviation ofthe portfolio loss per loan are: µ i = p i f σ i = 1,...,N i i = p i (1- p i ) f i 2. Allportfolio losses can be correlated to each other though a covariance m atrix. σ i,j = DefaultC ovariance betw een loan iand loan j = σ i σ j ρ i,j ρ i, j : defaultcorrelation betw een loan iand loan j

19 CyRCE:Value atrisk Approximating by the Norm aldistribution,the value atrisk with confidence levelα is: VaR α = i p i f i + z α i σ 2 i + i j σ σ i j ρ ij EXPECTED LO SS VARIANCE COVARIANCE Using m atrix notation: VaR α = π T F + z F T M F EXPECTED LO SS α UNEXPECTED LO SS π T = [p 1... p N ] T M: covariance matrix

20 CyRCE:CapitalA dequacy CapitalA dequacy relation is now : T π F ψ + V T F MF z α T H ( F) F F W eighted average default probability ofthe loan portfolio p _ T F M F R( F, M ) = T F F IHH Herfindahl-Hirschm an index Rayleigh s quotient Sum m arizes the variancecovariance effectfor portfolio losses

21 CyRCE:Single obligor lim its and Concentration Under the generalm odel, the concentration bound is: ψ p _ + z α R( F, M ) H( F ) H( F ) z 2 α p ( ψ ) 2 R( F, M ) Capitalization Ratio W eighted average defaultprobability of the loan portfolio 2 Confidence Level 2 Variance-C ovariance effect

22 CyRCE:Exam ple Rating A B C D E F G Mean (%) Stand. Dev.(%) H() F = 6.61% p _ = 10.89% R( F, M ) = Obligor Amount Obligor Amount A1 4,728 A14 5,042 A2 7,728 A15 15,411 A3 5,528 A16 1,933 A4 5,848 A17 2,317 A5 3,138 A18 2,411 A6 3,204 A19 2,598 A7 4,831 A A8 4,912 A21 1,090 A9 5,435 A22 2,652 A10 5,320 A23 4,929 A11 5,765 A24 6,467 A12 20,239 A25 6,480 A13 1,800 Total 130,164 A ssum ing Norm ality and a 5% confidence level,capitaladequacy requires: ψ ( ) ( ) = % Or: K (42.78%)(130,164) = $55,685

23 CyRCE:Exam ple (continued) Suppose K = 60,000,then K 60, 000 Capitalization ratio: ψ = = = 46.10% 42.78% required V 130, 164 Is the portfolio too concentrated? Η (F) = z 2 α ( ψ p ) 2 R( F, M ) = ( ) 2 2 = ( 1.96) ( 0.401) W hat s the single obligor lim it? f * = (0.0805) ( 130,164) = $10,482

24 CyRCE:Portfolio Segm entation The loan portfolio can be partitioned arbitrarily in segm ents such that: Each group has its ow n defaultprobabilities. The covariance m atrix M includes two kinds ofcovariation: Idiosincratic:am ong defaults within the sam e segm ent. Extra-group:am ong defaults ofdifferentsegm ents. ρ 1,2 p ρ 1 ρ 1,3 p 2 p 3

25 CyRCE:Value atrisk The value atrisk with confidence levelα for each segm entjis: VaR α j = π T F + z F T R F α j j j EXPECTED LO SS UNEXPECTED LO SS π j is the vector ofdefaultprobabilities ofsegm entj F j is the vector ofthe am ounts ofthe loans in segm entj R j is the m atrix ofthe idiosincratic covariances in segm entjand the defaultcovariances betw een the loans ofsegm entjwith the loans ofother segm ents.

26 CyRCE:Value atrisk (continued) = 0 C 0 C M C 0 C 0 R j N, j,n j j,1 1,j j L L M M M M M L L M M M M M L L The m atrix R j has the follow ing stru cture: M j = Matrix ofthe idiosincratic covariances for the loans in segm entj. C j,i = Matrix ofdefaultcovariances betw een the loans of segm entjwith the loans ofsegm enti.

27 CyRCE:CapitalA dequacy After a bitofalgebra,the CapitalA dequacy relation per segm entis: π R(F j, M j ) T ψ jfj j + V j z α H ( F j ) +2 j i F j TC j,i F i V j 2 W eighted average defaultprobability of the loans in segm entj p _ j Rayleighs quotient for segm entj HHI j Adjustm entfor Correlation

28 CyRCE:Single obligor lim its and Concentration The concentration bound per segm entis: H( F ) j z 2 α p ( ψ ) 2 j j R( F, M ) j j -2 j i F j TC j,i F i V j2 R( F j, M j) Bound Adjustm entfor Correlation

29 Num ericalcom parison to CreditRisk+ For illustration purposes,an arbitrary sam ple of 1,320 loans w as picked from SENIC R EB and VaR w as calculated follow ing both CreditRisk+ m ethodology and CyRCE methodology,using a Norm aland a Gamma distribution Distribution VaR95 VaR97.5 VaR99 VaR99.5 CreditRisk Gamma Normal CR+ Gamma Normal

30 Index I. Introduction II. CyRCE model System ic CreditRisk analysis III.I Position III.II VaR analysis

31 Distribution ofthe loan portfolio by m ajor banks: SYSTEM November , % 28, % 20,054 6% SYSTEM = 364,150 MP 35,221 10% 98,357 27% 30,683 8% 32,570 9% 93,224 26% B1 B4 B2 B5 B3 B6 B7 Others

32 Loan Portfolio Segm entation S Y S T E M B A N K S E C O N O M I C S E C T O R AGR/LIVESTOCK INDUSTRY CONSTRUCTION COMMERCE COMMUNICATION & TRANSPORT SERVICES OTHERS RATED LOANS Agriculture Livestock,Forestry,Fishing & H unting Extractive Food Textile W ood/paper Chemical Products Non-Meta Row Materials lic Minerals & Plastics Metals Mortgage/C & Meta onsum lic Prod. ption/c reditcards Machinery & Equipm ent OtherIndustries Financial Professional/Technical/Personal Exam Recreational/Hotels/Restaurants ple:foreign Loans Social& Communal Group by five differentrates

33 Distribution ofthe loan portfolio by Econom ic Activity:SYSTEM November 2001 Total Loan Portfolio: 364,150 MP 30% 25% 20% Non-rated loans: 304,911 MP (84%) Rated loans*: 59,238 MP (16%) 15% 10% 5% 0% Social & Commun. Services Mortgage & Consum ption FinancialServices ProfessionalServices Commerce of Products Construction Comm. & Transp. I. Metals & Metallic Prod. I. Food Industry Commerce of Raw Materials I. Non-Metallic Min. & Plastics I. Machinery & Equip. Other Industries Hotels, Rest. & Recr. Services I. Wood & Paper Ind. I. Textile Ind. I. Chemical Ind. Others I. Extractive Ind. Agriculture Livestock * Rated by Standard & Poor s,m oody s y Fitch 100,000 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0

34 Index I. Introduction II. CyRCE model System ic CreditRisk analysis III.I Position III.II VaR analysis

35 DefaultProbability,C oncentration Index and R ayleigh s Q uotient:system 2.0% 1.8% 1.6% 1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% R(F,M) 2.0 Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Jul-00 Ago-00 Sep-00 Oct-00 Nov-00 Dec-00 Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Ago-01 Sep-01 Oct-01 Nov-01 p, IHH Default Probability (p) Concentration index (IHH) Rayleigh s quotient (R(F, M))

36 W eighted average defaultprobability by Econom ic Activity 4.0% 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% Nov-01 Jan-00 Feb-00 Mar-00 Apr-00 May-00 Jun-00 Jul-00 Ago-00 Sep-00 Oct-00 Nov-00 Dec-00 Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Ago-01 Sep-01 Oct-01 SYSTEM Agr/Livestock Industry Construction Commerce Com. and Trans. Services Others

37 Risk,DefaultProbability,C oncentration Index and R ayliegh s quotient:system R ayleighs quotient VaR/Econom ic capital VaR MDP 50,000 40,000 30,000 27% % % 29% % 32% 33% 34% 30% 31% 31% % 25% 26% 27% % 1.6% 1.4% 1.2% 1.0% p,ihh 20, % 0.6% 10, % 0.2% 0 Mar-00 Jun-00 Sep-00 Dec-00 Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Ago-01 Sep-01 Oct-01 Nov % VaR 31,744 38,938 32,283 35,828 41,344 40,230 38,183 40,032 41,029 44,305 45,519 33,279 34,477 36,965 41,584 p 1.6% 1.6% 1.5% 1.2% 1.3% 1.3% 1.2% 1.2% 1.2% 1.2% 1.5% 1.3% 1.2% 1.3% 1.4% IHH 1.2% 0.8% 0.8% 0.8% 0.8% 0.8% 0.8% 0.7% 0.8% 0.7% 0.7% 0.7% 0.7% 0.7% 0.8%

38 Contribution to System Risk by Institution: Novem ber 2001 VaR 99/EC VaR 99 acum ulate VaR99/EC average:30.6% 100% 100% 100% 90% 90% 87% 100% 90% 81% 99% % 100% 80% 90% 80% 81% 87% 80% 100% 70% 90.00% 87% 80% 81% 90% 70% 70% 70% 87% 99% 81% 70% 58% 99% 100% 80.00% 80% 70% 70% 60% 60% 60% 58% 60% 56% 57% 70.00% 60% 70% 57% 54% 70% 58% 54% 60% 50% 60.00% 50% 57% 57% 54% 50% 60% 56% 60% 58% 40% 54% 41% 56% 42% 40% 40% 40% 50.00% 40% 50% 29% 40% 41% 40% 42% 30% 40% 41% 29% 42% 40.00% 33% 30% 40% 40% 30% 40% 17% 30% 33% 25% 29% 30.00% 20% 33% 20% 20% 14% 10% 11% 25% 30% 20% 17% 17% 20% 33% 29% 25% 20% 10% 11% 17% 12% 20.00% 17% 10% 9% 11% 6% 12% 20% 17% 20% 10% 10.00% 11% 10% 3% 12% 14% 4% 10% 6% 9% 11% 2% 6% 2% 0% 0% 0% 10% 0% 0% 0.00% 0% 0% B1 B1 B16 B2 B1 B2 B2 B3 B4 B3 B3 B6 B17 B4 B4 B5 B5 B5 B18 B6 B7 B6 B7 B7 B8 B19 B8 B9 B8 B1 B9 B10 B20 B11 B9 B2 B12 B21 B10 B11 B12 B13 B13 B14 B10 B22 B3 B14 B15 B15 B16 B11 B23 B4 B17 B18 B24 B12 B19 B5 B20 3% 4% 6% 2% 2% 0% 0% 0% B25 B13 B21 B22 B26 B23 B14 B24 B25 B27 B26 B15 B28 B27 B28

39 DefaultProbability,C oncentration Index and R ayleigh s Q uotient:n ovem ber 2001 VaR 99/EC Defaultprobability A verage p:1.38% C oncentration index A verage IHH :0.81% R ayleighs quotient A verage R:2.6% 100% 100% 90% 90% 2.0% 80% 70% 60% 50% 40% 30% 20% 10% 0% % % 1.4% 0.6% 0.3%0.4% 0.6%0.5% 1.3% 1.0% 14.4% 1.0% 1.0% 0.6% 0.7% 0.8% 0.9% 0.8% 5.2% 5.5% 5.7% % % 4.3% 0.4% 3.8% 0.4% 0.4% 0.6% 2.0%2.0% 0.3% 0.3% % 1.4% 0.2% 0.2% 1.5% 1.0% 1.0% % 0.7% 0.03% % 0.03% B1 B1 B2 B2 B3 B3 B4 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 B15 B16 B17 B18 B19 B19 B20 B20 B21 B21 B22 B22 B23 B23 B24 B24 B25 B25 B26 B26 B27 B27 B28 B % 17.5% % 0.2% % % 0.00

40 Contribution to System Risk by Econom ic A ctivity:n ovem ber 2001 VaR 99/SYSTEM VaR VaR 99 acum ulate 25% 20% 15% 10% 5% 0% 22% 11% 9% 7% 6.3% 5.7% 4.5% 4.4% 81.5% 98.5% 4.1% 4% 3.7% 3.5% 3.2% 3.1% 2.5% 2.3% 1.2% 1.1%1.1% 0.4% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Mortgage & Consum ption FinancialServices Construction ProfessionalServices I. Metals & Metallic Prod. Comm. & Transp. Commerce of Raw Materials I. Wood & Paper Ind. Other Industries Commerce of Products I. Maquinery & Equip. I. Food Industry Hotels, Rest. & Recr. Services I. Textile Ind. I. Non-Metallic Min. & Plastics I. Extractive Ind. Livestock I. Chemical Ind. Agriculture Social & Commun. Services Others

41 Capitaladequacy :(EC VaR)/Loan portfolio 0 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Ene-00 Feb-00 Mar-00 Abr-00 May-00 Jun-00 Jul-00 Ago-00 Sep-00 Oct-00 Nov-00 Dic-00 Ene-01 Feb-01 Mar-01 Abr-01 May-01 Jun-01 Jul-01 Ago-01 Sep-01 Oct-01 Nov-01 SYSTEM B1 B2 B3 B4 B6

42 END