ASSESSING CREDIT LOSS DISTRIBUTIONS FOR INDIVIDUAL BORROWERS AND CREDIT PORTFOLIOS. BAYESIAN MULTI-PERIOD MODEL VS. BASEL II MODEL.

Transcription

1 ASSESSING CREIT LOSS ISTRIBUTIONS FOR INIVIUAL BORROWERS AN CREIT PORTFOLIOS. BAYESIAN ULTI-PERIO OEL VS. BASEL II OEL. Leonid V. Philosophov,. Sc., Professor, oscow Coittee of Bankruptcy Affairs Vilis Latsis Street, 25480, oscow, Russia. E-ails: August 9, 2004

2 Abstract. As developent of the New Basel Capital Accord (Basel II) approaches to its final, the proble of validation and calibration of the Basel II odels attracts increasing interest of scientists and practitioners. This is evidenced by the Basel Coittee press releases, recent past and forthcoing conference progras, scientific publication on the proble. The principal attention of the Basel II is concentrated on credit risk where validation probles see to be solvable, because credit risks of individual borrowers and portfolios are accessible for precise calculations. The current paper assesses validity of credit loss distributions, calculated by eans of the Basel II odel. The assessent ethod consists in parallel calculations the sae distributions by eans of exact probabilistic forulae. The exact calculation schee is realized within Bayesian ulti-period (BP) odel and Portfolio BP odel. We found that Basel II odel ensures correct assessents of credit loss distributions for ediu and large portfolios of thirty or ore one-period credits (credits without interediate interest etc. payents). For sall portfolios and individual one-period credits assessents of Basel II odel are rough or extreely rough. Basel II odel undervalues credit risk of portfolios of ulti-period credits (credits with active period covering several years and interediate payents). The study provides an alternative look at soe principal Basel II concepts, like Probability of efault (P) and Correlation. This can serve to ore correct understanding and flexible use of those concepts in assessing credit risks. Exact odels are ore coplex that Base ll odel, but perfectly available for quick coputer calculations. They can be used in bank practice ensuring flexible assessents of credit risk. Keywords: Credit risk, Credit loss distribution, Basel II, Bayesian odel. GEL codes: C, C3, C5, E58, G2, G33. 2

3 . Introduction. The Basel Coittee on Banking Supervision published docuent entitled International Convergence of Capital easureent and Capital Standards known also as the New Basel Capital Accord (Basel II). The docuent establishes the new fraework for assessent bank capital adequacy with strong ephasis on iproveent of bank capabilities to assess and anage risks. The new Accord will be put in action in OEC countries by the end of 2006 and its ost coplex parts y the end of A key eleent of the new fraework consists in calculation of bank s risk-weighted assets, which depend on probabilistic assessent of arginal losses that bank can bear due to influence of arket, credit, operational etc. risks. Principal attention of the New Basel Accord is focused on credit risk, which reflects bank s losses caused by clients defaults. Along with Standardized approach to credit losses based on external credit ratings of borrowers the New Accord proposes Internal Rating- Based (IRB) approach in its foundation and advanced versions. Both versions suppose preliinary deterination of four key inputs to the odel that estiates required capital and risk-weighted assets. Those inputs are: probability of clients default (P); loss given default (LG); exposure at default (EA); effective aturity (). Only one of the paraeters (P) directly characterizes a specific borrower, LG also ight characterize client but in the current odel this is the preset deterinistic (non-rando) paraeter depending on the seniority of debt. EA and are pure characteristics of a current credit. These inputs ust be substituted in forulae represented in the New Accord to calculate required capital and risk-weighted assets. The sae forulae are proposed for corporate sovereign and bank exposures as well as for client portfolios. The question arises if the single paraeter P is sufficient characteristic of borrowers and their risks. The current paper provides an attept of validation of Basel II credit risk odel, in application to individual borrowers and hoogenous portfolios of borrowers. To do this we develop an alternative - Bayesian ulti-period (BP) odel (based on exact probabilistic forulae) for assessing credit risk of individual borrowers and then aggregating the in hoogeneous portfolios within Portfolio Bayesian ulti-period (PBP) odel. Both odels use detailed description of tie schedules of true cash flows between bank and borrower and calculate conditional probabilities of a borrower s default at all stages of cash flow process. The approaches are based on studies L. Philosophov, V. Philosophov (2002) and L. Philosophov, J. Batten, V. Philosophov (2003), which assess tie horizons of a fir s bankruptcy basing on its current financial indices and schedule of repaying of longter debt. 3

4 We found that Basel II odel ensures correct assessents of credit loss distributions for ediu and large portfolios of thirty or ore one-period credits (credits without interediate interest etc. payents). For sall portfolios and individual one-period credits assessents of Basel II odel are rough or extreely rough. Basel II odel undervalues credit risk of portfolios of ulti-period credits - credits with active period covering several years and interediate payents. The study provides an alternative look at soe principal Basel II concepts, like Probability of efault (P) and Correlation. This can serve to ore correct understanding and flexible use of those concepts in assessing credit risks. In line with discussing necessary changes in Basel II, the current paper proposes BP and PBP odels as an alternative to Basel II odel. odels for assessent credit loss distributions known fro literature were developed ainly by ajor investent banks and rating agencies. Such odels of J.P. organ (997) Credit Suisse First Boston (997), oody s KV Corporation are categorized as coercial products and described ainly in hardly available working papers. As one can understand the odels are based on various abstract indexes of a client s risk. Gordy (998), who copared odels of J.P. organ and Credit Suisse First Boston has found that they have siilar underlying atheatical structure. According to his description both odels utilize rather siplified odel of bank s client and are interested in characteristics of portfolio of such clients. Bayesian ulti-period odel developed in this paper does not fall in any specific category of odels described in Basel Coittee docuent (999). This is structural ultiperiod odel; like erton-type odels it assesses probability of a fir s default at various tie horizons in future, but it does not use assuption of Brownian otion of arket prices of a fir s equity. In contrast it relies on fundaental characteristics of a borrower and involves in analysis its detailed financial data. This allows for calculation of ex-post default probabilities conditional on indices of fir s current financial position. This is why the odel was identified as Bayesian. The rest of the docuent is organized as follows. Sections 2,3 describe the BP odel and its iportant coponent parts. Section 4 copares principal features of BP and Basel II odel and credit loss distributions of the sae single borrower calculated by those odels. Section 5 calculates exact credit loss distributions for portfolio of one-period credits (PBP odel) and copares the with the sae distributions provided by Basel II odel. 4

5 One-period credits are understood as credits without interediate interest payents and other interediate cash flows. Section 6 calculates exact credit loss distributions for portfolio of ulti-period credits (by eans of PBP odel) and again copares the with Basel II distributions. Section 7 concludes. 2. Assessing credit loss distribution for an individual borrower. A Bayesian ulti-period odel. For each separate borrower we consider standard cash flow schedule where at tie oent t bank lends to a client U dollars. The loan ust be returned after years at 0 tie oent t and additionally client pays annual interest at rate (at tie oents t ). r If a client returns credit after years, balance of discounted cash flows between bank and client is equal to that represented by forula (): ru B = U + ( + d ) = U + ( + d ) We choose discount rate d be equal to interest rate on debt r so that after repaying the principal debt value U, balance of cash flows between bank and a borrower is equal to zero (B = 0 ). A client is subject to default, which occurs at unknown tie oent t. This event can take place within: tie interval T t t } with probability P T ) ; { 0 ( tie interval T t t } with probability P T ) ; 2 { 2 ( 2.. tie interval T t t } with probability P T ); { ( tie interval T t t } with probability P T ). + { ( + All intervals except the last one are supposed to be equal to one year, though other intervals can be also considered if necessary. The last tie interval corresponds to defaults, which occur after the debt is paid off. It includes situation when default does not occur; this situation is referred to as default at infinite tie (t ). Note that P (T + ) = P (T )... P (T ). () If a client defaults before the debt is paid off, cash flows between bank and borrower cease and balance becoes negative. The bank bears losses L = B. One can see fro () that loss can take one of the discrete values, naely: 5

6 loss L is equal to U ( L = U ) with probability P T ) ; (. loss is equal to = ru L = U with probability P j (T ) ; = ( + i ) j. loss is equal to L = 0 with probability P (T. (2) + + ) Cuulative loss distribution function F (L) is a stepwise function. Steps occur at values L = and have agnitudes P T ) : L F(L) ( = P (T + ) (L) + P (T ) (L L ), (3) = where function ( x ) is defined as: ( x ) = 0 if x < 0 and ( x ) = if x 0. Note that what we just obtained is not Loss Given efault but unconditional loss distribution function where default probabilities are already incorporated in loss distribution. Unconditional losses are of final interest to bank. One can see that unconditional loss distribution function adits possibilities of zero and non-zero losses. The first addendu in the right-hand side expression in (3) corresponds to finite probability of zero losses. The loss is zero if default does not occur or occurs after active credit period. Possibility of zero losses is absent in the current Basel II odel. The second addendu in (3) characterizes distribution of non-zero losses. Loss given default (LG) is also rando within this odel because defaults at different stages of credit process lead to different losses. Its cuulative distribution is (LG is easured as a percent of initial exposure): F(LG) = ( P (T + )) P (T ) (LG L / U ). (4) = One can also calculate that ean loss is L = P (T = ) L while ean Loss Given efault is LG = P P (T ) +, (5) = (T ) L / U. (6) 6

7 Suppose now that after a defaulting client s recovery procedures are over, bank receives back part β of lost aount. If β is fixed, bank s losses L (if they are nonzero) are reduced to L ( β ). One can obtain now cuulative loss distribution function given recovery rate β as: F(L / β) = P (T + ) (L) + P (T ) (L L ( β )). (7) = Cuulative distribution of LG is now F(LG / β) = ( P (T + )) P (T ) (LG L ( β ) / U ). (8) = Though it is not explicitly stated, Basel II considers nonzero but nonrando recovery rates. They are introduced by constant values depending on a seniority of debt. Coparing (7, 8) with (3, 4), one can see that such rates copress graphs of cuulative loss distributions β ties along horizontal axis; this holds for Basel II credit risk odel also. Taking this in account we can further consider zero recovery rates ( β = 0 ) only. 3. Assessing probabilities P T ). ( ethodology of assessing probabilities ) of a fir s default within any given tie intervals was developed in the studies L. Philosophov, V. Philosophov (2002) and L. Philosophov, J. Batten, V. Philosophov (2003). Nuerical data in the studies relate to a fir s bankruptcy (identified as event of filing a bankruptcy petition). efault is slightly different concept but the ethodology is fully applicable. Quantitative data see to be applicable also, though soe further specification is desirable. The difficulty consist in fact that data on firs defaults is less publicly available, through this difficulty does not relate to banks and their clients. While assessing probabilities P (T P (T ) one ust distinguish between ex-ante (unconditional) and ex-post (conditional) probabilities. An interediate case can be also identified. The ost suitable ex-ante probabilistic characteristic of default is the default rate π, which is defined as the conditional probability of a borrower s default prior to the end of the tie interval of interest (usually one year), given that it was acting at the beginning of the tie interval. In practice this probability ay be deterined approxiately as the percentage of clients operating at the beginning of the tie interval, which defaulted during that tie interval 7

8 Given default rates π, for a successive nuber of tie intervals (years) one π can calculate the probabilities of client s default during each of these years. The probability of a client s default during the P (T -th year can be calculated as ) = ( π )... ( π ) π, (9) and the probability of default within P (T +-th year or later is + ) = ( π )... ( π ). (0) If default rate is constant, P P (T ) = ( π) π, (T ) = ( π ) () +. (2) odels of this type are very popular in Probability Theory. Their underlying assuption of the independence of default events eans that the distribution of tie reaining until default does not (ex-ante) depend on client s history, i.e. on how long it was operating as an active borrower before, Feller (966). If default rates are deterined on econoy-wide (country-wide) level and averaged over the long tie periods one can refer to probabilities rate π P (T ) as unconditional. Being based on default statistics collected fro restricted subsets of borrowers, default can reflect current characteristic of the acroeconoic environent depending on its fundaental (priary) paraeters. Altan (982) identified four paraeters influencing the bankruptcy (business failure) rate - econoic growth activity, oney supply, capital arket activity, new business foration rate. Line of corporate business is also of principal iportance for default rate. Increased default rates for specific subsets of bank clients reflect interdependence (correlation) between their defaults. Within terinology of Basel (999) docuent probabilities deterined by eans of (9, 0,, 2) for restricted subsets of firs ust be qualified as conditional, though we would like to refer the as sei-conditional. Truly conditional probabilities can be obtained with accounting for current financial position of each (corporate) client. They are uch ore preferable than unconditional ones because enable assessing individual risk of each borrower. etailed description of ethodology of calculation of probabilities P conditional on the set (vector) f of current financial indices of a borrower is represented in L. Philosophov, V. Philosophov (2002) and L. Philosophov, J. Batten, V. Philosophov (2003). P (T ) ( T / f ) 8

9 The ethodology is based on enhanced Bayesian ulti-alternative odels that account for non-norality and interdependence of predictive (explanatory) variables. The proper choice of financial indices f is of great iportance because inforative prognostic variables enhance default prediction and increase accuracy of assessing loss distribution function, capital requireent and risk-weighted assets. L. Philosophov, J. Batten, V. Philosophov (2003) proposes two groups of predictive variables. Group of indices of a fir s current financial position includes four financial ratios: f - Working Capital / Total Assets; f 2 - Retained Earnings / Total Assets; f 3 - Earnings Before Interest and Tax / Total Assets; f 4 - Interest Payents / Total Assets. Good predictive power of above ratios was confired by parts in any studies starting fro Altan (968). Another group of indices is first proposed in L. Philosophov, J. Batten, V. Philosophov (2003). It is derived fro a schedule of paying off a fir s long-ter debt and includes: g - portion of long-ter debt due within the first year starting fro the date of the fir s last financial stateents; g 2 - portion of long-ter debt due within the second year; g - portion of long-ter debt due within the -th year. The set of factors g,... ust coprise all available data including those g n corresponding to zero payents; variable g includes all fir s long-ter payents due in k -th year, not only those fro current credit. k Relevant literature studies also variables of arket anticipation of bankruptcy of a publicly traded fir. They are not studied in above cited papers but can be easily incorporated in the proposed odels. 4. Assessing credit loss distribution for a single borrower. Coparing Bayesian ulti-period and Basel II odels. The coparison of the Basel II and BP odels in easuring credit risk of a single borrower is executed along two lines: these are coparison of principal features of the two odels and coparative nuerical estiations that they provide. Principal features of the odels are represented in the table. 9

10 Table. AIN FEATURES BASEL II OEL BP OEL Input data Probability of default in one year (P). aturity tie (). Loss given default (LG). Exposure at default (EA). Last accounting stateents of a fir. Schedule of repaying of long-ter debt Schedule of planned bank-client cash flows Probabilities of One-year unconditional Calculated conditional default default probability (P) set in accordance with a fir s rating probabilities for successive years until debt aturity depending on fir s initial financial indices. ebt aturity Effective (ean weighted) True tie schedule of cash flows aturity Loss given default Preset, non-rando Rando, calculated in dependence on non-executed part of cash flow schedule and recovery rate. Outputs arginal loss, risk-weighted arginal loss, risk-weighted assets assets One can see that BP odel uch ore closely traces econoic and financial essence of lending process and account for individual characteristics of a borrower. Coparison of quantitative outputs of the two odels in typical situations is represented in the figure. Curves in the figure represent cuulative credit loss distributions of corporate borrower calculated by eans of Basel II odel (curve 4) and by BP odel (,2,3). A fir borrows U dollars for five years and ust pay annual interest at rate 0%. Basel II credit loss distribution includes expected and unexpected losses and is calculated as dependence between Capital Requireent K (paragraph 24 of the New Basel Capital Accord, version of 2003) and varying confidence probability P, which in original Basel II odel is set constant 0,999. Nuerical data used to calculate this distribution are given in the caption to the figure. Credit loss distribution () is ex-ante distribution calculated by eans of BP odel. For better visibility annual default rate is chosen relatively high 4,5% as it really was in USA in 200 (see oody s Investor Service ). The sae value was assigned to P. Curves 2,3 represent ex-post distributions calculated for the sae default rate and individual c 0

11 financial indices of firs with good (Lechters Inc., 993) and bad (Iperial Sugar Co., 2000) financial positions. Figure. Cuulative loss distribution functions for single corporate borrowers. BP odel stepwise curves,2,3: ex-ante distribution with annual default rate 4,5% 2 ex-post distribution for Lechters Inc., ex-post distribution for Iperial Sugar Co., Basel II odel continuous curve (4). = 5; P = 4,5% - true USA default rate in 200; LG =,0; R = 0,33 - as calculated by Basel II odel. One can see that losses assessed by the two odels greatly differ fro each other; the only reseblance is that all the vary in range 0. All BP distributions adit zero losses that correspond situations when a default does not occur. Basel II distribution does not adit zero losses. Horizontal dotted line in the figure corresponds to confidence level P c = 0,95, which is chosen relatively low for better visibility. Abscissa of its intersect with the graph of cuulative loss distribution deterines arginal loss. One can see that arginal loss is case of ex-ante distribution (curve ), L = 2 0,0 in case of Lechters Inc. (curve 2, in this case probability of zero losses is ore than confidence probability),. L,0 in case of = 3 L = 0,9 in

12 Iperial Sugar Co. (curve 3). Basel II odel does not capture these situations. arginal loss corresponding to Basel II distribution is L 0, 6 in all above cases. = 4 The data of the figure evidence also that accounting for the individual financial position of a (corporate) borrower is very iportant. In attept to reveal sources of discrepancy between the two odels I checked the Wehrspohn s inference of Basel II forulae. Original Basel II inference is unavailable, but the doubtful ter is identical in Basel II odel and in Wehrspohn (2003). Given very specific for of this ter one can be sure that inferences are highly correlated. It was found that forula for capital requireent K in Basel II ite 24 (The New Basel Capital Accord, ver. 2003) is erroneous (detailed analysis one can find in Appendix). Erroneous forula in Basel II notation is K = LG N[( R ) 0.5 G( P ) + (R /( R ) 0.5 G(0.999 )] ; in ore usual notation of Wehrspohn (2003) it is K Φ = LG Φ( ( p) + ρ Φ ρ (0,999 ) ), where Φ( ) is cuulative standard noral distribution function and Φ ( ) inverse function to Φ ( ) ; p = P, ρ is the sae as R. In both forulae 0,999 is specific value of confidence probability P. Soe Basel II forulae for capital requireent contain additional aturity adjustent ter. Correct expression for K in Basel II odel (for an individual borrower) is uch ore siple: K = 0 if P < and P c K = LG if P >. P c Note that after corrections correlation factor R disappeared fro Basel II forulae for required capital and risk-weighted assets. This is quite natural for a single borrower. c In soe forulae for Capital requireent K Basel II odel uses additional ultiplier T = (.5 b(p)) ( + ( 2.5 ) b(p)), whose destination is aturity adjustent. In the current exaples its influence is insignificant. The above one and other exaples evidence that in assessing credit risk of a single borrower Basel II odel is very iprecise and highly undervalues credit risks. 2

13 odel. To iprove the situation radically one can use BP odel instead of the Basel II 5. Portfolio of one-period credits. Consider now portfolio of one-period credits. A bank lends to each borrower U dollars at tie t 0 and at the end of credit period receives the loan back with extra interest payent. Tie oent t can vary for different clients in portfolio. 0 This case corresponds to P (T ) = π and P T ) = π. ( + = Cuulative loss distribution (as follows fro (7, 8) is F(L) = ( π) (L) + π (L U ( β )), and cuulative distribution of LG is F(LG ) = (LG ( β )). in above forulae, and fro (, 2) one can find The last expression eans that LG = β with probability. Within active credit period each of borrowers in portfolio can either default with probability π or stay solvent with probability π. Each default brings to a bank loss U ( β ). If default events for different clients are independent, portfolio status at the end of credit period is equivalent to one of possible issues of Probability of exactly k defaults in portfolio is equal to P(k ) n! k! (n k )! n n Bernoulli tests. k n k = π ( π ), (3) and bank s loss in this case is equal to: L = k ( β ) U. (4) Taking in account (3, 4) cuulative loss distribution of the portfolio can be deterined as: n F(L) = P(k ) (L k ( β ) U ), (5) k = 0 where again function ( x ) = 0 if x < 0 and ( x ) = if x 0. To bring expressions (3, 5) to conditions of Basel II, one ust also take in account correlation. Basel (999) docuent explains effect of correlation by (coon for all borrowers) influence of acroeconoic environent that changes their propensity to default. For portfolio BP (PBP) odel this eans that default rate π ust be considered rando. Basel II inference proposes for π the following odel (see Wehrspohn (2003) and Appendix to this paper): 3

14 Φ (P) R Y π = π(y ) = Φ( ), (6) R where P is interpreted as probability of default within one year, R correlation, Y - rando variable with standard noral distribution N(Y ), Φ - function of cuulative standard noral distribution, Φ - inverse function of Φ. As the result cuulative loss distribution of portfolio of one-period credits is deterined by the following forula: n n! k n k F(L) = (Y ) ( (Y )) N(Y ) dy (L k ( ) U ) k! (n k )! π π β. (7) k = 0 Taking soe care with factorials (which are very large nubers), one can use (7) to calculate exact loss distribution of portfolio of n one-period credits. This distribution is represented in figure 2 (stepwise curve) for portfolio of 00 borrowers and LG = ) - no recoveries. β = 0 (and hence Continuous curve represents cuulative loss distribution deterined by Basel II odel. This again is dependence between Capital Requireent K and varying confidence probability P c, which in original Basel II odel is set as constant 0,999. In this odel we also have set LG =. 4

15 Figure 2. Cuulative loss distribution functions for portfolio of 00 borrowers. Loss is calculated as the share of total portfolio exposure: Exact (Portfolio BP) odel stepwise curve, Basel II odel continuous curve. = ; n = 00; P = 4,5% - true USA default rate in 200; LG =,0; R = 0,33 - as calculated by Basel II odel. For paraeter values given in the caption to the figure, arginal loss, corresponding to the confidence probability P c = 0,999, is assessed by the exact odel as 0,28 of total portfolio exposure (28 borrowers of 00 in portfolio can default siultaneously with probability ore than P = 0,00 c ), while Basel II odel assesses this loss as 0,27 of total exposure; accuracy of Basel II is %. One can see that (in contrast with the previous section) for portfolio of 00 clients confority between exact (Portfolio BP) and Basel II odels is rather good. Using exact forulae (6, 7) one can assess how portfolio size influences accuracy of Basel II odel. In figure 3 exact cuulative loss distribution and those deterined by Basel II odel are calculated for portfolio of 30 borrowers. Exact odel assesses that loss will be zero with probability 0,37. This is probability that no one of 30 borrowers will default while Basel II odel does not adit zero losses. Figure 3. Cuulative loss distribution functions for portfolio of 30 borrowers. Loss is calculated as the share of total portfolio exposure: 5

16 PBP odel stepwise curve, Basel II odel continuous curve. = ; n = 30; P = 4,5%; LG =,0; R = 0,33 (as calculated by Basel II). For paraeter values given in the caption to the figure arginal loss, corresponding to confidence probability P = 0,999 c, is assessed by the exact odel as 0,3 of total portfolio exposure (9 borrowers of 30 in portfolio can default siultaneously with probability ore than P c ), while Basel II odel again assesses this loss as 0,27 of total exposure; accuracy of Basel II is 0%. In figure 4 the sae probability distributions are calculated for portfolio of 0 borrowers. In this case distinctions between exact and Basel II distributions are still ore significant. Figure 4. Cuulative loss distribution for portfolio of 0 borrowers. Loss is calculated as the share of total portfolio exposure: Exact odel stepwise curve, Basel II odel continuous curve. = ; n = 0; P = 4,5%; LG =,0; R = 0,33. arginal loss, corresponding to confidence probability P c = 0,999, is assessed by the exact odel as 0,4 of total portfolio exposure (4 borrowers of 0 can default siultaneously with probability ore than P c ) while assessent of Basel II does not change of total exposure (accuracy of Basel II is 32,5%). 6

17 Finally the sae distributions can be calculated for a single borrower. Forula (7) for cuulative loss distribution is this case reduces to F(L) = ( π ) (L) + π (L U ), (8) where π = π(y ) N(Y ) dy - ean value of π. One can calculate that π = P. The resulting cuulative distribution (8) is represented in figure 5 together with Basel II distribution which is the sae curve as before.. Exact distribution has only one step that corresponds to cuulative probability F(L) = P. arginal loss in exact odel is assessed as U - credit can be fully lost with probability ore than P, while L 0, 999 = c Basel II again assesses arginal loss as 0,27 which is obviously incorrect. Figure 5. Cuulative loss distribution for a single borrower. Exact odel stepwise curve, Basel II odel continuous curve. = ; n = ; P = 4,5%; LG =,0; R = 0,33. The odel, described by exact expressions (6,7), (taking in account its good coincidence with Basel II when portfolio is large) presents alternative look on concepts of Probability of efault (P) and Correlation (R) considered within Basel II approach. 7

18 In exact odel P is unobservable paraeter, while observable is current default rate π. As we noted earlier P is ean value of π. Hence tie base used for deterination of P ust be very large, given relatively slow alteration of default rates. One ust also note that distribution of default rate π, used in Basel II odel, is rather specific. Cuulative distribution as deterined by (6) is: Φ (P) R Φ ( π ) F( π ) = Φ( ). R It is based on very indirect considerations and ignores rich statistical data concerning default rates. On the other hand using current default rates π instead of average rate P is preferable because it increases accuracy of assessing current credit risk. Hence accounting for correlation in Basel II odel is unnecessary. In this case forulae (3, 4, 5) can be used to exactly assess credit loss distributions. 6. Portfolio of ulti-period credits. In the ulti-period case credit process for each borrower has + possible issues, each of the with its specific probability P T ) and lost aount L, as described in the section 2. For portfolio of n ulti-period credits one ust consider polynoial (instead of binoial) tests (Feller 966). In coon case one can find in portfolio k defaults within the first year, k defaults within the -th year, k borrowers avoid default within active credit period ( k k + k + = ( + n ). Probability of such issue is polynoial probability n! P(k,...k,k ) = k k k + + P (T )... P (T ) P (T + ) k!... k! k!, (9) and total portfolio loss is + L p(k,...k,k + ) k L k L + k + L + =. (20) Note that in accordance with forulae (9,0) probabilities P T ) depend on annual default rates P (T ) = P (T / π,... π ). To take the axially near to Basel II we consider π rando but the sae during all credit period π π. Probabilistic properties of default rate π are again deterined by forula (6). As the result we obtain following expression for cuulative loss distribution: ( F(L), (2) n! k k + =... P (T / (Y ))... P (T + k!...k! π k k + / π(y )) P(Y ) dy (L L p ) 8

19 where k take all possible values that satisfy the restriction (... + k + k n ) and i k + + = L p ust be taken fro (20). Expression (2) is rather coplex and bulky, but rational organization of coputations allows for its quick calculation. In figure 6 we represent cuulative loss distributions calculated by eans of exact (PBP) odel () and Basel II odel (2) for portfolio of 00 three-year credits with annual interest payents. Reeber that for one-year credits both odels reveal good confority of assessed credit loss distributions in such a large portfolio. Figure 6. Cuulative distributions of credit losses for portfolio of three-year credits according to exact () and Basel II (2) odels. = 3; n = 00; P = 4,5%; β = 0 - no recoveries; ean LG = 0,938 (as calculated by PBP odel); R=0,33 (as calculated by Basel II odel). In case of three-year credits Basel II odel undervalues credit losses; aturity adjustent of the odel is insufficient. Note that in PBP odel LG is calculable value; the sae value was used in Basel II odel. In the figure 7 the sae curves are represented for portfolio of five-year credits. One can see that in this case Basel II odel undervalues credit losses still ore significantly. 9

20 Figure 7. Cuulative distributions of credit losses for portfolio of five-year credits according to exact () and Basel II (2) odels. = 5; n = 00; P = 4,5%; β = 0 - no recoveries; ean LG = 0,883 (as calculated by PBP odel); R=0,33 (calculated). 7. Conclusion and propositions. The above calculations and graphs evidence that area of correct assessents of the Basel II credit risk odel is restricted by one-period credits and ediu or large portfolios of about 30 or ore borrowers. For portfolios of lesser size down to individual borrowers stepwise character of true credit loss distribution is poorly approxiated by continuous Basel II curves. For single corporate borrowers their individual financial characteristics drastically influence their credit risks. For ulti-period credits Basel II odel undervalues credit losses the ore the ore periods (interediate payents) covers active credit interval. aturity adjustent in the Basel II odel appears to be insufficient (if at all possible). The study proves that accounting for correlation in assessing risk of individual borrowers and credit portfolios is unnecessary. Correlation is caused by changes in default rate (coon for all borrowers) in dependence on acroeconoic situation. Current default rate π can be easured and used instead of ean default rate P in credit risk estiations. 20

21 Exact BP and PBP odels are ore coplex than Basel II odel, but nevertheless they are easily available for PC calculations that take no ore than one inute for one distribution. Such odels can be used in bank practice and flexibly account for peculiarities of specific portfolios. The odels can be ipleented as a standardized software progras, developed under the auspices of Basel Coittee and distributed aong relevant banks. 2

22 Literature. Altan E., 982. Corporate Financial istress, a Coplete Guide to Predicting, Avoiding and ealing with Bankruptcy, (John Wiley, New York.). Basel Coittee on Banking Supervision, 999, Credit Risk odelling: Current Practices and Applications, Basel Coittee on Banking Supervision, 2003, The New Basel Capital Accord, Basel Coittee on Banking Supervision, 2004, International Convergence of Capital easureent and Capital Standards JP organ 997, Crediteri cs T, Technical docuent,. Credit Suisse First Boston 997, CreditRisk Fraework, T + 997, A Credit Risk anageent Finger C.C., 2000, A Coparison of Stochastic efault Rate odels, SSRN. Feller W., 966, An Introduction to Probability Theory and its Applications, (John Wiley & Sons, Inc., New York, London, Sydney). Gordy.B., A Coparative Anatoy of Credit Risk odels, Journal of Banking and Finance 24, 2000,9-49. oody s Investor Service, 2002, efault & Recovery Rates of Corporate Bond Issuers. Philosophov L., Philosophov V., 2002, Corporate bankruptcy prognosis: An attept at a cobined prediction of the bankruptcy event and tie interval of its occurrence, International Review of Financial Analysis,, Philosophov L., Batten J., Philosophov V., Philosophov, "Assessing the Tie Horizon of Bankruptcy Using Financial Ratios and the aturity Schedule of Long-Ter ebt", (Noveber 2003), Philosophov L.. Bayesian ulti-period odel for Assessing Credit Loss istributions vs. Basel II odel, (eceber 2003), Saidenberg., Schuerann T., The New Basel Capital Accord and Questions for Research, (ay 2003), Wharton Financial Institutions Center Working Paper No Wehrspohn, U., "Analytic Loss istributions of Heterogeneous Portfolios in the Asset Value Credit Risk odel" (April 2003). T Wilson 997, CreditPort folioview, Portfolio Credit Risk, 9,0. 22

23 Appendix. Basel Coittee on Banking Supervision (2004) docuent International Convergence of Capital easureent and Capital Standards in its ite 272 presents forula for calculation of required capital K, which in this docuent is equal to unexpected losses. To receive full (arginal) losses one can add expected (ean) losses. The forula for full losses was presented in the preceding version of Basel II (The New Basel Capital Accord (2003), ite 24). This forula is: K = LG N[( R ) 0.5 G(P ) + (R /( R ) 0.5 G(0.999 )], and in ore usual notation and suitable for perception for can be rewritten as K Φ = LG Φ( ( p) + ρ Φ ρ (0,999 ) ), (A.) where Φ( ) is cuulative standard noral distribution function and Φ ( ) inverse function to Φ ( ) ; p = P, ρ is the sae as R. Forula of this type was inferred in Wehrspohn (2003) for arginal credit loss (percentile) in portfolio of defaulting clients. Given specific for of this expression quite probable supposition is that Basel II has inferred it in the sae underlying assuptions. The following coent proves that the forula (A.) for K is invalid for a single client and presents the iproved forulae. Wehrspohn s inference relates to portfolio of clients; in the following text his forulae are reduced to a single client as it is done in the Basel II. In his paper Wehrspohn assigns to a client (fir) a rando risk index X, which has standard noral distribution N (0, ) with zero ean and variance equal to. A fir defaults if X falls below threshold α ; probability of this event is known value p (or P in Basel II notation) and hence α = Φ ( p), where again Φ ( ) denotes inverse of cuulative noral distribution. Further he decoposes X into two additive parts Y, Z : X = ρ Y + ρ Z, where Y represents systeatic risk and Z - idiosyncratic risk. Both Y and Z have the sae standard noral distribution N(0, ). If a borrower defaults a bank bears loss (LG) denoted in Wehrspohn s paper as λ, which is non-rando predeterined value. 23

24 Y is Next Wehrspohn freezes Y and founds that probability of client default conditional on Φ ( p) ρ Y P(client defaults Y) = Φ ( ). ρ Then in course of proving Theore he states that loss given Y is deterined by the expression: Φ ( p) ρ Y Loss Y = λ E Φ( ), ( E = EA ). (A.2) ρ In this place we ust note this is ean loss. In this point of inference loss giveny is the rando variable, which in application to a single client takes only two possible values: loss is L =λ E (client defaults) with probability π π Φ = Φ( ( p) ρ ρ Y ) (A.3) and L = 0 (client does not default) with probability π. ean loss is ean Loss Y = 0 ( π) + λ E π This is just the sae as forula (A.2). = λ E π Further Wehrspohn considers ean loss given Y fro (A.2) to be rando because of randoness of Y and calculates percentiles of resulting distribution of ean Loss, which can be correct only for large portfolios, not for a single client. In fact, to deterine arginal loss one ust calculate percentiles of unconditional loss distribution density, which is deterined as: P(L) = P(L Y ) P(Y ) dy. (A.4) In this forula P (L Y ) is probability density (not ean value) of Loss giveny - above described rando variable with two possible values. One can write following for: where P(L Y ) δ ( ) = ( π ) δ (L) + π δ (L λe ) P(L Y ) in the is well known in atheatics sybolic delta-function, which ay be thought of as a noral probability density with very sall (alost zero) variance. δ (L) is concentrated in very narrow area around L = 0, while δ( L λe ) is concentrated around L = λe. As the result of averaging in (A.4) we obtain P(L) = ( π ) δ (L) + π δ (L λe ) (A.5) where π is ean value of π (π is dependent on Y ). 24

25 π = π(y ) P(Y ) dy. (A.6) The later integral can be calculated analytically. The result is very interesting and quite predictable: π = P - probability of default. This result one could receive directly considering distribution of X without its decoposition into Y,Z ; the decoposition has little sense for a single client. For practical needs of the New Capital Accord is iportant to note that unconditional distribution P(L) deterined by (A.5) stays to be discrete with only two possible values L = 0 and L = λe. The 99,9%-quantile (99,9-percentile) of L is L 0 if 99,9% = P < 0, 999 = 0,00 and L = E if P > 0, ,9% λ Capital requireent K in Basel II forulae (ites 24, 299, 30) becoes trivial: K = 0 if P < 0.00 and K = LG if P > These expressions could be ore correct if P corresponds to probability of a client s default within years - period until debt aturity (not within one year as it is established at present tie). 25