Loan portfolio loss distribution: Basel II unifactorial approach vs. Non parametric estimations

Transcription

1 MPRA Munich Personal RePEc Archive Loan ortfolio loss distribution: Basel II unifactorial aroach vs. on arametric estimations Analía Rodríguez Duuy Banco Central del Uruguay October 007 Online at htt://mra.ub.uni-muenchen.de/0697/ MPRA Paer o. 0697, osted 3. Setember :4 UTC

2 Loan ortfolio loss distribution: Basel II unifactorial aroach vs. on arametric estimations Analía Rodríguez Duuy 007

3 Abstract This aer analyzes the measurement of credit risk caital requirements under the new Basel Accord Basel II: the Internal Rating Based aroach IRB. It focuses in the analytical formula for its calculation, since its derivation to the main assumtions behind it. We also estimate the credit loss distribution for the Uruguayan ortfolio in the eriod , using a non arametric technique, the bootstra. Its main advantage is that we don t need to make any assumtions about the form of the distribution. Finally, we comare the requirements obtained using the IRB with the estimated ones, as an aroimation of the alication of the IRB in the Uruguayan financial system. P a g e

4 Inde I. Introduction...3 II. Basel II and the IRB aroach...6. IRB Fundamentals...7. Risk weight functions Princial assumtions behind the risk weight function The two main assumtions Asset correlations Loss given default Confidence level...6 III. on arametric estimation of credit loss distribution and comarison with IRB aroach.8. Data and methodology...8. on arametric estimation...8. Data...9. Results...0. Cororate ortfolio...0. Retail ortfolio...5 IV. Conclusions...7 Aendi A Proerties of the loss distribution...9 Aendi B Estimations for cororate ortfolio...37 Aendi C Estimations for retail ortfolio...40 References...43 P a g e

5 Loan ortfolio loss distribution: Basel II unifactorial aroach vs. on arametric estimations I. Introduction Financial institutions, and articularly banks, are eosed to different risks, which are inherent to the nature of their activities. Taking the simlest definition, we can define a bank as an institution whose habitual oerations consist in giving loans and taking deosits from the ublic. In this definition, one can observe that risk could derive from the counterarty as well as from the mismatch that emerges from the asset transformation that banks make. Princial risks can be resumed in: credit risk, market risk, liquidity risk and oerational risk. This aer will focus on credit risk analysis, which derives from the robability that the borrower defaults on his obligations. It is necessary to require banks to maintain a minimum of caital to cover otential losses due to this risk, which leads to the need for a roer system to measure it. In 988, Basel Committee roosed some recommendations to imrove banking regulation Basel I Accord, which were adoted by most art of world regulators and were considered as best ractices. This Accord reresented a first ste towards caital requirement based on credit risk, as it established fied weights according to the risk associated with every eosure. Different categories of eosures were determined in a simle way, and they did not allow for a roer measure of credit risk. As an eamle, all borrowers from non financial sector had the same weight. Financial system changed dramatically since that first Basel Accord. Minimum regulatory caital is calculated as: Regulatory Caital Risk Weight Eosure 8% Risk-weighted Assets 8% 3 P a g e

6 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations In 996, the Accord incororated an amendment to require caital to cover market risk, defined as the risk of losses in on and off-balance sheet ositions arising from movements in market rices. Basel Committee ermits to choose between two broad methodologies: a standardized manner which has been roosed in 993 and internal models VaR. Desite this breakthrough, the restrictions of the agreement of 988 require an adatation of the former, which is intended to be carried out within Basel II. The rincial urose is to make caital requirement more risk-sensitive, and also the romotion of the use of internal models to measure it. The Committee has built the suervisory rocess around three illars:. Minimum caital requirements. Suervisory review rocess 3. Market disciline In relation to the first illar, the Accord of 988 oted for a standardized aroach, in which different risk were weighted according to the borrower s category. Basel II incororates imortant changes in this illar, introducing caital requirements for oerational risk and significantly modifying the measurement of credit risk. Although it maintains the caital adequacy ratio at 8%, the way banks measure caital requirements is different. The rest of the illars are new; illar two refers to the suervision rocess, which must ensure that banks have adequate caital to suort all the risk in their business, and also encourage institutions to develo and use better risk management techniques in monitoring and managing risks. The Committee has identified four key rinciles of suervisory review. First, banks must be able to demonstrate that chosen internal caital targets are well founded and that these targets are consistent with their overall risk rofile. Secondly, suervisors must review and evaluate banks internal caital adequacy assessments and strategies, as well as their ability to monitor and ensure their comliance with regulatory caital ratios. Third and fourth rinciles refer to the ability of suervisor to require banks to hold caital in ecess of the minimum, and the early intervention to revent caital from falling below the minimum levels required to suort its risks. Lastly, illar three aims to romote a more cometitive and transarent market, which reinforces the two revious illars. These three illars work together towards ensuring the caital adequacy of institutions. They are more otent when working together within a common framework. This aer will focus on the analysis of the first illar, and articularly in caital requirements for credit risk. Related to this, the new accord has modified the riskweighted assets definition: the new aroaches for calculating risk-weighted assets are intended to rovide imroved bank assessments of risk and thus to make the resulting caital ratios more meaningful. 3 One of the main advantages of this new Basel Committee on Banking Suervision Overview of the new Basel caital accord, Consultative document; Basel Committee on Banking Suervision P a g e

7 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations accord is that it generates incentives for banks to develo more sohisticated risk management techniques. To assess credit risk, Basel II allows to choose between two methods: standardized aroach and internal rating based aroach IRB foundation and advanced. In the standardized aroach, which is similar to the current Accord, banks are required to slot their credit eosures into suervisory categories based on observable characteristics of the eosures e.g. whether the eosure is a cororate loan or a residential mortgage loan. Fied risk weights are established corresonding to each suervisory category and eternal credit assessments are used to enhance risk sensitivity. The IRB aroach differs substantially from standardized aroach, as banks internal assessments of key risk drivers are the rincial inuts to measure credit risk. Caital requirements are determined by combining quantitative inuts rovided by banks and formulas secified by the Committee. In Uruguay, regulation is based on the standardized aroach. Caital requirements are detailed in the Article 4. of the Comilation of Central Bank norms circulars for the regulation and control of the financial system, where is stated that: caital requirement for credit risk is equivalent to the 8% of risk-weighted assets. Risk weights for each category range from 0% to 5%. The aer aims to serve as a first aroimation to how the alication of the foundation IRB could be in Uruguayan financial system, given that regulation has develoed in line with Basel II sirit. During last years, and after financial crisis of 00, the Suerintendence of Financial Institutions SFI has substantially modified regulation with the urose of giving more information to markets, thus having a more transarent and cometitive banking sector as well as rotecting agents when takings their decisions. Related to caital requirements, market risk has been incororated in 006, and there have been advances in credit risk measurement. Although the standardized method is used, regulation about credits classification introduced the analysis of debtors cash flows, thus evaluating their ability to ay, and also requiring stress scenarios to assess it. This contributes to a better analysis of credit risk, and also rovides a useful data base in case banks decide to use IRB aroach. More recently, a regulation allows for the use of internal model to assess credit risk in small borrowers. All these changes make of the analysis an imortant tool to be aware of IRB imlications when using it to measure caital requirements. In the first art of the aer, we resent IRB rincial characteristics, emhasizing the analysis of the formula that Basel II roosed to calculate risk weights. Therefore, we focus on its main assumtions and its imlications for develoing financial systems The second art will make use of a non arametric technique to estimate credit loss distribution of banking ortfolio, to have a measure of eected and uneected loses VaR. Data covers rivate banking sector, during eriod , and we run different estimations distinguishing between cororate and retail ortfolio. Having obtained these estimations, we comare them with the ones that would emerge in the case of using IRB aroach formulas. 5 P a g e

8 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations II. Basel II and the IRB aroach Banks activity could be seen as taking risks. During certain eriod of time, for eamle a year, it is common to observe that some borrowers do not ay their obligations. The bank can not eactly calculate the amount of such losses, but it can estimate the loss it eects to have at the end of the year. That measure is called eected loss EL, and reresents the amount of caital that the bank could lose as a result of its eosure to credit risk, given a time horizon. Such losses can be seen as the cost of doing banking business, and should be covered by rovisions that banks make on each loan. However, losses could eceed that eected level, and more caital is thus needed to absorb them. These are known as uneected loss UL. Taking ortfolio loss distribution, we can reresent eected loss by its mean. Figure In Figure 4, uneected loss is defined as the difference between the Value-at-Risk VaR and eected losses. The likelihood that losses will eceed the sum of eected Loss EL and Uneected Loss UL - i.e. the likelihood that a bank will not be able to meet its own credit obligations by its rofits and caital - equals the hatched area under the right hand side of the curve. 00% minus this likelihood is called the confidence level and the corresonding threshold is called Value-at-Risk VaR at this confidence level. VaR analysis constitutes an imortant tool when measuring risks, and given that Basel II aims to have more risk-sensitive caital requirements, this kind of analysis has been incororated in the new accord. In this section we resent the main asects related to IRB aroach, and basic consideration one must take into account when alying it. We deeen in the analysis of the formula used to determine caital requirements for uneected loss. Last art of this section analyses rincial assumtions of the model, giving a better understanding of this aroach, esecially for emerging economies. 4 Etracted from An Elanatory ote on the Basel II IRB Risk Weight Functions, P a g e

9 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations. IRB Fundamentals The IRB aroach is based in both eected and uneected losses. Risk weights and caital requirements are determined by combining quantitative data from bank with formulas secified by the Committee. There are three key elements in IRB aroach. The first are the Risk Comonents, which can be divided in: Default Probability PD: quantifies the likelihood that the borrower will default in the coming twelve months. Loss Given Default LGD: is the loss that the bank will suffer if the counterarty defaults. It is eressed as a ercentage of the eosure. Eosure at Default EAD: is the eosure of the bank at the moment the obligor goes into default. Maturity M: the amount of time until the loan is fully due and ayable. The second element is Risk-weight Functions, in which risk comonents are transformed into risk-weighted assets and therefore caital requirements. The last comonent is Minimum Requirements; these are minimum standards that must be met in order for a bank to use IRB aroach. These standards are based fundamentally in rating and risk estimation systems and rocesses, which must rovide for a meaningful assessment of borrower and transaction characteristics, a meaningful differentiation of risk; and reasonably accurate and consistent quantitative estimates of risk. There are two tyes of IRB: foundation and advanced. In the first one, banks estimate PD and the rest of the arameters are established by the Committee. In the advanced IRB, all arameters are estimated by the financial institution. Under the IRB aroach, banks must categorize banking-book eosures into broad classes of assets with different underlying risk characteristics; these are: cororate, sovereign, bank, retail and equity. Within the cororate asset class, five sub-classes of secialized lending SL are searately identified, while for retail ortfolio three subclasses are determined. This is based in the fact that each eosure requires a different treatment because they have different risk-drivers. This work will focus on cororate and retail ortfolios. Basel II allows for the inclusion in retail ortfolios loans to small enterrises, if the eosure is below million. Once the eosures are categorized and PD s are calculated, one should aly the formula roosed by the Committee to determine caital requirements. That formula will be analyzed in net sections, and can be eressed as follows: K LGD * [ PD] [ 0.999] PD [ M.5] b PD.5b PD Where: 7 P a g e

10 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations K caital requirement LGD loss given default PD default robability assets correlation M maturity As we mentioned above, this formula generates caital requirements to cover uneected loss, while eected loss is treated searately. Banks alying IRB must comare the total amount of eligible rovisions with the total EL amount as calculated within the IRB aroach ELPD*LGD. Where the calculated EL amount is lower than the rovisions of the bank, its suervisors must consider whether the EL fully reflects the conditions in the market in which it oerates before allowing the difference to be included in Tier caital with a maimum of 0.6% of risk-weight assets. Where the oosite is true, the difference is subtracted from caital 50% from Tier and 50% from Tier.. Risk weight functions A first characteristic of the IRB aroach is that the model should be ortfolio invariant; that is, the caital required for any loan should only deend on the risk of that loan and must not deend on the ortfolio it is added to. Under this assumtion, secific characteristics PD, LGD and EAD of each borrower are enough to calculate caital requirement for each loan. It can be shown that 5 only Asymtotic Single Risk Factor ASRF models are ortfolio invariant; these models are derived from traditional credit models by the law of large numbers. When a ortfolio consists of a large number of small eosures, idiosyncratic risks associated with individual eosures tend to cancel out one-another and only systematic risks that affect many eosures have a significant effect on ortfolio losses. Vasicek 00 has demonstrated that under certain circumstances, Merton s model of 974 can be adated to an ASRF one. In this kind of models, all systematic risks that affect all borrowers, like industry or regional risks, are modeled with only one systematic risk factor. Consider a ortfolio consisting of n loans. The value of the assets of a borrower i can be described as a geometrical Brownian motion, as stated in equation 3. da µ A dt σa i i i dz i [3] where is a Wiener rocess,, Thus the value of assets at T can be eressed as: σ lnai T ln Ai 0 µ ; σ T so it can be established that: [4] 5 Gordy, P a g e

11 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations σ lnai T lnai 0 µ T σ Tz i [5] where z i is a standard normal random variable. Basel s II formula uses Merton s model interretation of default robability of the borrower i, so an obligor will default if the value of his assets falls below the value of his debt: lnbi lnai µ it σit [6] [ Ai T < Bi] [ zi< ci] c [ i] [ d] conc i σ T Remembering that [-d ] is default robability, and calling it by letter, then [c i ], and therefore c i - []. z i is assumed to have the following eression: zi by aεi i,,3,... n where y reresents systemic risk affecting the entire ortfolio and ε i is the secific risk for borrower i. It is assumed that both of them follow standard normal distributions, and the values for b and a are given by: b ;a measures the asset correlation between borrowers assets. It could be described as the deendence of the assets of a borrower on the general state of the economy; so all borrowers are linked to each other by the single risk factor. If we are talking about a dollarized economy, we can think in the echange rate as the single risk factor, because a strong movement in that variable substantially affects loan ortfolios. Under those conditions, default robability of any loan, conditional on the single risk factor y, can be written as: i [ y] Pby [ aε i < c ] P ε < i From equation [6], can be noticed that c i [ ] i c by a [8] [7], so relacing in [8] we have [ y] Pεi < c i by a Pεi < [ ] y [ ] y [9] Total ortfolio consists on n identical individuals, with the same articiation in total eosure. Being L i the gross ortfolio loss before recoveries for borrower i, so that L i if the obligor defaults and L i 0 if the oosite is true, total gross ortfolio loss can be eressed as: 9 P a g e

12 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations [0] We can establish that the ercentage of default on total ortfolio equals the number of individuals that default on their obligations. If n is large enough, by the law of the large numbers it can be stated that the fraction of clients L that default on their debts is equal to the default robability conditional on y, [ y] P[ L / y] i [ ] y [] Thus the cumulative distribution function of loan losses on a very large ortfolio is in the limit 6 : [] The VaR at 99.9% confidence level is thus: 99.9% 99.9% [3] And default robability can be eressed like follows: 99.9% [ ] [ 0.999] [4] Until now we have derived most art of Basel formula. The comlete risk weight function for caital requirements to cover uneected loss is: K LGD* [ PD] [ 0.999] PD [ M.5] b PD.5b PD [5] 6 Convergence of the ortfolio loss distribution to the limiting form actually holds even for ortfolios with unequal weights. Let the ortfolio weights are w i, with w. The ortfolio loss n L wl i i conditional on Y converges to its eectation Y whenever w i 0 this i is a necessary and sufficient condition; inn other words, if the ortfolio is not concentrated. n i i n i 0 P a g e

13 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Calling PD the default robability reviously defined as, we can observe that the formula generates caital requirement for uneected loss, given that the LGD is multilied by the difference between the PD s VaR at 99.9% confidence level and the eected PD. That arameter is estimated by banks, in the foundation IRB as well as in the advanced. It is worthy to comment some roerties of this distribution function 7. The cumulative distribution is given by the eression: F ; ; [6] Thus, if we want to obtain the density function we must calculate the derivative of eression [6], The measures of osition for this density function are: if </ Variance is given by: being the bivariate normal distribution function. Grah function rho 0., PD ,04 0, 0,9 0,7 0,34 0,4 0,49 0,57 0,64 0,7 0,79 0,87 0,94 7 Demonstrations are resented in Aendi A. P a g e

14 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations When >/, density is U-shaed, which means that when correlation is high, bank result could be very good, in case all firms erform well, or really bad, in case all of them incur in default. Grah illustrates this case. Grah function rho 0.8, PD ,07 0,4 0, 0,8 0,35 0,4 0,49 0,56 0,63 0,7 0,77 0,84 0,9 0,98 One can think about a dollarized economy, where banks do not have roblems in case they face a stable echange rate, but in case of a negative realization of this risk-driver, the financial system immediately would show bad results, thus being in the other etreme of the distribution. In the articular case for /, function is monotone. Grah 3 function rho 0.5, PD ,005 0,05 0,045 0,065 0,085 0,05 0,5 0,45 0,65 0,85 0,05 0,5 0,45 Lastly, when correlation is erfect, so that, density tends to the binomial distribution,. In Grah we can see that loss distribution is asymmetrical. This attern has imlications when requiring caital, as it generates higher requirements that in the normal distribution s case. Correlation coefficients were determined based on G0 data. Two main assumtions are made. The first one is that correlations have an inverse relationshi with PD. The higher the PD is for a firm, the lower its correlation to the single risk factor. This is an emirical observation, as it is observed that when PD is high, it means that idiosyncratic risk revails, so the risk is driven by its own characteristics and does not deend on the general state of the economy. The second assumtion is that size of the firm is directly related to PD. The bigger the comany, the higher the PD. As comanies P a g e

15 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations become bigger, they are more deendent on the state of the economy. Correlation is then calculated as follows: Where 0. corresonds to maimum PD 00% correlation, and 0.4 is the correlation for lowest PD 0%. Each of them is multilied by eonential weights, which dislay the deendency on PD. Last term corresonds to a size adjustment, which affects borrowers with annual sales between 5 million and 50 million. For borrowers with 50 million annual sales and above, the size adjustment becomes zero, and for borrowers with 5 million or less annual sales, the size adjustment takes the value of 0.04, thus lowering the asset correlation from 4% to 0% best credit quality and from % to 8% worst credit quality. For most retail loans, 8 correlation is determined as stated below: Last term of equation [5] is a maturity adjustment 9, as it was assumed that all loans had a one-year maturity. Emirical evidence shows that long-term loans are riskier that short-term ones, so caital requirements should be higher for longer maturities. This could be seen as the additional requirements that would emerge for ossible credit downgrades, which are more likely to haen in long-term loans. The adjustment has the following form: Where is: [ ln ] b PD PD Adjustments are linear and increasing in M. The reason is that the lower de PD, the loan is more likely to be downgraded, so it is riskier. The sloe of the adjustment function with resect to M decreases as the PD increases. et Figure resents a matri which contains the values of the adjustment for different PD and M. 8 Ecet for mortgage and revolving loans, where correlations are fied, taking values of 0.5 and 0.04 resectively. 9 This adjustment does not aly for retail eosures 3 P a g e

16 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Figure : Maturity adjustment Probabilidad de default M %.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 0.00% BPD It should be noticed that for M,. From Figure, it can also be seen that there eists a negative relationshi between the adjustment and PD for any given M. 3. Princial assumtions behind the risk weight function 3. The two main assumtions There are two strict assumtions behind Basel s II formula: bank s credit ortfolio is infinitely fine-grained, and there is a single systematic risk factor which drives all deendence across credit losses in the ortfolio. First assumtion imlies that any single obligor reresents a very small share of the ortfolio s total eosure. For the traditional Merton s model to be adated to an ASRF model, Vasicek 00 has demonstrated that it is necessary that the loan ortfolio consists of a large number of small eosures. If this assumtion is not met, then there will be an undiversified idiosyncratic risk, which results in an underestimated caital requirement. In view of this kind of situation, Vasicek 00 rooned a granularity adjustment which can be alied when the ortfolio is not sufficiently large for the law of large numbers to take hold. In equation [4], granularity adjustment could take lace by taking δ - instead of. Being δ n w i i The second main assumtion is the eistence of a single systematic risk factor. Diversification effects, like regional or industry branch, are not considered. This failure to recognize the diversification effects could result in an overestimation of ortfolio risk, esecially for banks that have roerly diversified its risks. Industries are subject to different kind of risks and cycles, and they should be searate modeled. Some recent aers suggest the use of multi-factor frameworks to reflect diversification effects. Césedes et al 005 estimate a model based on what the called diversification factor, which is a function of two arameters that broadly cature size concentration and the average cross-sector correlation. Tasche 005 incororates diversification effects by including a diversification inde, where VaR contributions of 4 P a g e

17 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations each factor are calculated. Caital requirements can be substantially reduced when the ortfolio is well-diversified. 3. Asset correlations Another crucial factor that affects caital requirements is the calculation of asset correlation. That formula assumes that there eists a ositive relationshi between asset correlation and size of the firm. This is based on the idea that smaller firms have a higher comonent of idiosyncratic risk, thus having less correlation with the general state of the economy. Most research in this area confirms this assumtion 0. However, an argument against this evidence could be found in Bernanke et al 996. They claim that bigger firms have access to financial markets in case of negative shocks, while small and medium comanies do not have it, thus being more eosed to state of the economy. Debtors with higher agency costs in credit markets small firms will burden the costs of economic recessions; the so called flight to quality. To calculate correlations, it is also assumed that the relationshi between PD and asset correlation is negative. In contrast with the revious assumtion, there is no consensus about this. Düllmann y Scheule 003 analyzed asset correlation as a measure of systematic credit risk from a database of balance sheet information of German comanies, finding that it increases with firm size but the relationshi with PD is not unambiguous. Dietsch y Petey 003 estimated asset correlations in two large oulations of French and German SMEs. They found that, on average, the relationshi between PDs and correlations is not negative as assumed by Basel II. It is U shaed in France, and ositive in Germany. They conclude that caital requirements could be too high for SMEs, where correlation is lower. Critics also come from PD s modellization. Rosch 00 argues that PD is not constant over time, and that it deends on macro economical conditions, so when calculating PD s one must incororate roy variables for the business cycles. That would reduce uncertainty about PD, and also correlations and caital requirements. Hamerle et al 003 also introduce macro economical factors in their estimation, and roose a time-deendent PD, thus obtaining lower correlations. In conclusion, caital requirements are said to be highly sensitive to asset correlations; if they are lower than the ones that result from Basel II, caital requirements are much lower and vice versa. Parameter s calibration is therefore crucial; Basel II has used data from the grou of ten major suervisors G-0, which may not be suitable for emerging economies. In articular, eonential weights could be too high. The ace of the eonential function is determined by the k-factor, which is set at 50, generating a fast decline. From Grah 4, it is noticeable how changes in k-factor can smooth the function, thus enlarging the range between minimum and maimum PD. 0 Dietsch and Petey 003, Düllmann and Scheule P a g e

18 Correlacion Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Gah Correlation For Different K-Factors K50 K40 K30 K0 K0 K PD 3.3 Loss given default To determine the LGD, the Committee roosed a determinist value, when one can think that LGD is a random variable which take values between 0 and, and also that there may eists a deendence relationshi between LGD and PD. Altman et al 00 found a ositive relationshi, and they argue that it must be considered in the analysis, as it increases caital requirements. The same factors that affect PD also affect LGD. Hillebrand 006 estimated a model for the ortfolio loss including deendence of PD and LGD on the economic cycle. He states that LGD deends on the general state of the economy, and it has to be incororated when measuring credit risk. In a recession, for eamle, the value of collateral decreases considerably, and therefore the LGD is much higher. Criticism also relates to the LGD estimation, as banks may have different measures according to the model they use. This alies also for PD s estimation. 3.4 Confidence level Confidence is set at 99.9%, recognizing that it may result too high. Basel s argument is that this eigency covers ossible measurement errors in estimation institutions with well-secified and calibrated models will be unished with higher caital requirements that they may not need. 6 P a g e

19 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Grah 5 caital requirement 0% 8% 6% 4% % 0% 8% 6% 4% % 0% Caital requirements and significance level 0 0, 0,4 0,6 0,8, K 99% K 99.9% PD As it can be observed in Grah 5, if the confidence level is reduced by 0.9%, caital requirement is significantly lower. This conservative criterion is also observed when calculating market risk caital requirement, where the VaR is multilied by a minimal factor of 3, which also retends to reflect measurement error in estimations. Previous grah also shows how from certain levels of PD, caital requirement starts decreasing rather fast, as a consequence of the increase of eected loss comonent, which becomes zero when PD equals unity that is, when the borrower is in default, as there is not uneected loss. 7 P a g e

20 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations III. on arametric estimation of credit loss distribution and comarison with IRB aroach When estimating caital requirements to cover uneected losses due to credit risk it is necessary to obtain the arameters of the distribution function of loss ortfolio. There eists different methodologies to calculate them; in this aer we ot for the rocedure roosed by Carey 998, 00, where he estimated ortfolio loss distribution using a non arametric technique, known as bootstra. The main idea is that the samle itself is the best guide to infer the distribution. The rincial advantage of this method is that we do not need to make assumtions about the functional shae and the arameters of that distribution; the only assumtion is that the samle is reresentative enough. The bootstra consists on simulating a large number of ortfolios, etracting, with reosition, the loss rate for each of them. The frequency distribution of that loss rates is the estimation of the relevant distribution function. Majnoni, Miller y Powell 004 conducted a boostraing eercise for three countries Argentina, Meico and Brazil to relicate the distribution of credit losses revailing at a secific eriod of time. Their aer concludes that caital requirements that emerge from IRB are lower that the estimated ones for the cases of Argentina and Meico, while the oosite is true for Brazil. It should be noticed that results are based in date from only one year, while this technique requires having a larger eriod, to cover economical cycles. Jacobson, Lindé y Roszbach 005 also alied Carey s nonarametric method to two banks comlete loan ortfolios, to comare the risk associated with small and medium enterrises SME with big cororate ortfolios. Lastly, the aer of Gutiérrez Girault 007 estimates conditional and unconditional loss distributions for loan ortfolios of argentine banks, to comare with Basel s II requirements. His eercise controlled by tye of borrower and tye of bank, and covered the eriod Data and methodology. on arametric estimation The aroimation to estimate credit loss distribution follows the work of Carey 998, 00. In his aer of 998 he estimates the credit risk associated with rivate debt ortfolios, reorting non arametric estimates of the size of losses in the bad tail, using Monte Carlo resamling methods. Portfolios are simulated etracting different assets from total samle, reeating this rocedure times. Emirical losses are comuted for each drawn ortfolio, and the frequency distribution of such losses is the relevant loss distribution. The term bootstra was introduced in 979 by Efron, although the tye of methodology was being used since time before. It is a comuter-intensive method, which allows to make statistical inference without making hyothesis about oulation distribution F. This method starts from the concet of bootstra samle. Taking as the emirical distribution, a bootstra samle is defined as a random samle of size n, obtained from, 8 P a g e

21 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations * *, * n * *, * n * The sura inde * means that we are not dealing with the original data set, as it is a random version or resamling of. This values are a samle of size n etracted with relacement from the original samle,,.., n, so each bootstra samle consists of n values of the original samle, where some of them could aear more than once or even not be included. For every bootstra samle, the same function alied to the original data, s., is then calculated. For eamle, if s is the samle mean, then s* is the bootstra samle mean,. Both the construction of the bootstra samle and the calculus of the relevant statistic are reeated B times, constructing a frequency distribution which will constitute the relevant robability function.. Data Data is obtained from the Credit Registry of the Suerintendence of Financial Institutions SIIF - Central Bank of Uruguay. Banks and other financial intermediaries send to the Credit Registry information about transactions, which include data about borrowers name, business activity, document and about their debts amount, tye of facility, collateral. From the total database, we only consider those credits that were erforming at the beginning of every eriod, ecluding those that were in non erforming categories. Then we observe the erformance of each loan, articularly if it has been included in non erforming categories. To determine if the credit is nonerforming, we follow the same criterion of the SIIF s Accounting Scheme; that is, if the loan ayments are ast due by 60 days or more. If the credit has been written off, it is also considered as default, while if the credit has been cancelled it is considered as a recovery. The analysis will cover the eriod , thus covering the dee crisis that the Uruguayan economy suffered in 00, as well as its following recovery. By this way, unconditional distribution will be reresentative. Data come from rivate banks only, thus ecluding the activity of ublic banks. Portfolio will be classified according to the tye of borrower, following SIIF regulations. It allows for the distinction between cororate and retail sectors. This segmentation aims to match Basel s II categorization. Eosure at default is determined by total risks of a borrower, minus the coverage of liquid collaterals as they are totally recovered after a default event. It is also assumed As starting oint of each eriod we took December of every year. 9 P a g e

22 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations that the rate of recovery for mortgage collaterals is 30% of the balance sheet value. Lastly, LGD is assumed to be 50%. We defined a variable L i which take the value L i in case of default and L i 0 if the loan remains erforming or has been cancelled. loss. rate n i i n L Eosure i Eosure i 50% Defining the eosure as follows: i Eosure Total Risks Liquid Collaterals 30% Mortgage Collaterals on arametric estimations were done for each year during the eriod , and for each tye of borrower. Simulations included reeats for conditional as well as unconditional distribution. The size of the ortfolios was determined according to the observed data for every year, taking the average size for rivate financial system. For the unconditional distribution, the average of was taken. The rocess can be summarized in net icture. original samle: erforming loans at the beginning of each year selection of B samles: B0.000 comute observed loss rate for each of the orfolios and construct a frequency calculate main statistics: mean, standard deviation an 99.9th ercentile. Results. Cororate ortfolio et we resent the results obtained from the bootstraing for cororate ortfolio. Table reorts the main indicators of loss distribution: eected loss, standard deviation and the 99.9th ercentile VaR measure. The first three columns show estimations in million dollars, but we are interested in losses relative to the total eosure, so the net three columns calculate that. Last column measures the The rocess was also calculated taking the size of the ortfolio as the total number of observations for every year and results were not significantly different. 0 P a g e

23 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations uneected loss, which is the difference between the 99.9th ercentile and the eected loss. Table : Estimation for cororate ortfolio Losses in million USD Loss rate Uneected Eected St. Dev. 99.9th ercentile Eected St. Dev. 99.9th ercentile loss ,89,47 0,90,% 0,67% 4,93%,8% ,84,44 0,50,9% 0,60% 4,35%,44% ,40 6,57 6,0,8%,40% 0,8% 8,99% ,49 0,8 5,65,47% 0,5% 3,68%,% ,60 9,65 59,30 4,40% 3,9% 7,64% 3,4% ,84 6,8 40,0,5%,49% 3,45% 0,94% ,95,69 0,70,8%,00% 6,3% 5,05% incondicional,0 4,40 34,80 4,7%,73% 3,% 8,50% Looking at the eriod , eected loss of ortfolio is USD 4,89 million. The credit VaR at 99,9% confidence level is USD 0,9 million. The frequency distribution is resented below, where it can be noticed the asymmetrical shae that characterizes credit risk distributions 3. As we mentioned before, this asymmetry imlies that the likelihood of high losses is higher than in the case of a normal distribution. Grah 6- Credit loss distribution for eriod e-07.0e e e07.50e07 erdidas From Table one can also etract statistics for the loss rate, which has an eected value of,%, the VaR at 99.9% is 4,93% and the uneected loss is thus,8%. It is noticeable from Table that financial crisis of 00 had an imortant imact in our estimations. Eected loss increased significantly, as well as the volatility of the distribution. For the net eriods we observe a higher standard deviation too, which could be indicating that financial system became more fragile. It should be mentioned that the low values obtained for eriod could be attributed to the fact that after financial crisis the loans that remain in the samle were only the good ones. To illustrate this argument, in there were.000 observations while for the years they reduced by almost a half. Once obtained the non arametric estimations, we comared estimated requirements with IRB aroach ones. To calculate IRB formula, it is assumed that LGD is 45% defined in Basel Committee final document and the values for PD are the ones that emerge from bootstra estimation. In other words, taking into account that eected loss can be defined as: 3 Aendi B resents estimations for each year. P a g e

24 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations EL PD*LGD Using EL estimations from Table, one can obtain the value of PD consistent with a LGD of 45%. Basel II also roosed a size adjustment to be included in the formula. When analyzing the ertinence of doing so for Uruguayan comanies, we observed that for May and June of 007, more than a half of firms were small and medium enterrises SMEs, whose annual sales are below USD 5 million. So it is necessary to reflect this situation when calculating caital requirements. To do that, we consider that before the crisis of 00, the ercentage of SMEs was similar to actual rates, while in the two years after crisis, only big comanies survived. A maturity adjustment also needs to be done, to reflect the fact that during all the eriod of analysis most art of loans had a maturity of less of one year. Results are reorted in Table. Table : Comarison of IRB and on arametric estimations for cororate ortfolio eriod K IRB IRB/Estim ,57% 3, ,3% 3, ,54%, ,49% 3, ,93%, ,0%, ,95%,38 unconditional,7%,43 First column in Table reorts the caital requirement generated from IRB formulas, while in the second column that requirement is divided into the estimated one by the bootstraing rocess. As it can be observed, results are not homogeneous. Before 003, Basel s II requirements are much higher than estimated ones, while after that year IRB requirements seems to be closer to the non arametric estimation. A factor that may be elaining those differences could be the measurement of asset correlations. The formula roosed by Basel deends on PD, and as mentioned before, it is assumed that correlation with systematic risk has a negative relationshi with PD. et grah shows that relationshi, and also how correlation arrives at its minimum for low values of PD thus the so called k-factor results too high. For low values of PD, correlation is too high and the range of intermediate correlations is too short. Therefore, it is indicating that arameters calibration was thought for develoed economies, where low PDs revail. P a g e

25 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Grah 7 - PD and correlation in IRB aroach PD and C orrelation asset correlation 0,0 0,7 0,4 0, 0,08 0,05 0,00 0,03 0,06 0,09 0, 0,5 0,8 0, 0,4 0,7 0,30 0,33 0,36 default robability The differences we obtain thus indicate that for years revious to the financial crisis, correlations calculated from IRB are too high, therefore generating a higher caital requirement than the estimated one. However, during last years of analysis results show that firms are more deendent on the general state of the economy, as the caital requirements is closer to estimations. Given that is hard to obtain data to estimate asset correlations for Uruguayan firms, we used bootstraing estimations to infer the asset correlations imlicit in them. That is, taking the estimated caital requirement, we use Basel s formula to determine which correlation is required to yield that result. Table 3: Asset correlations estimated Basel's ,5% 3,% ,0% 3,4% ,0%,0% ,3% 4,3% ,5%,% ,7%,7% ,5% 5,% From Table 3 it is clear that correlation for years before 003 are too high, while they tend to converge at the end of the eriod, coinciding with a better adjustment of IRB requirements in relation to estimated ones. Asset correlation thus results time deendent; revious to the crisis, firms erformance deended on themselves, so correlation is low. After the crisis of 00, as the economy was recovering, comanies deend more on the realization of the systematic factor; therefore, a higher correlation is eected 4. In view of that, it is necessary to adjust the coefficient according to the business cycle hase. Again, IRB s formulas are thought for develoed economies where macro economical fluctuations are not as usual as in develoing ones, and where PD is much lower. 4 During the eriod 00-00, correlation imlicit in caital requirement is low, which can be attributed to the high value of PD for that eriod giving Basel s formula, correlation is much lower. It could also be attributed to the fact that increases in correlation have certain lags, thus aearing some years after the crisis took lace. 3 P a g e

26 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations Another imortant attern observed in IRB s caital requirements is their rocyclicality. In recession times, it increases significantly, while in eansions is lower. Grah 8 Caital Requirements IRB 8% 6% 4% % 0% 8% 6% 4% % 0% GDP variation and caital requirements % 0% 5% 0% -5% -0% GDP variation -5% IRB GDP var. In Grah 8 one can see how the requirement increases as the GDP variation is lower, reaching its maimum in the eriod when GDP fell by %. It should be remembered that eriod is not reresentative, as it has fewer observations. As the economy recovers showing ositive variations of GDP, caital requirements fall. Having reached this oint, it is ertinent to distinguish between estimation oint in time PIT and estimation through the cycle TTC. The former emerges from using the PD in each eriod conditional estimations, while the TTC estimation attemts to measure credit quality in a long time horizon, thus incororating cyclical asects of the economy. Therefore, the rocyclicality could be reduced when using TTC estimations of PD. Grah 9 Requerimientos Point in time oint vs in Through time y through the cycle the cycle 8% 6% 4% % 0% 8% 6% 4% % 0% Req. conditional condicional estimacion unconditional incodicional Grah 9 resents the oint-in-time requirements, which are the ones that have been calculated in conditional distributions. Horizontal line reresents the unconditional estimation, which could be interreted as the through-the-cycle requirement. If we want caital requirements to reflect the risk rofile of institutions, then rocyclicality should not be a roblem. It is reasonable that in recession, when credit quality deteriorates, the risk associated is higher and so the caital requirement is. There is a trade-off between rocyclicality and risk measurement. If we calculate 4 P a g e

27 Loan ortfolio loss distribution: Basel s II unifactorial aroach vs on arametric estimations requirements with the standardized aroach with fied risk weights, we observe that they do not vary with economic cycle, thus being useless as an indicator of risk. Grah 0 Standardized aroach and GDP variation 5,0 5% USD millions 4,0 3,0,0, Risk-weighted assets GDP variation 0% 5% 0% -5% -0% -5% GDP variation Lastly, as mentioned in Part I, a risk-insensitive requirement does not allow for inferring about the economic caital of a bank, and does not reflect changes in ortfolio s risk rofile, thus making that market agents unable to monitor institutions and imeding the imlementation of risk management olicies.. Retail ortfolio The retail ortfolio was not segmented by tye of facility. This could lead to differences as some arameters are fied for mortgage and revolving loans. The former have an asset correlation of 5% while for the last ones it is set at 4%. Table 7 resents estimations for this kind of borrower, and grahs for each distribution function could be found in Aendi C. Table 7: Estimation for retail ortfolio Losses in million USD Loss rate Uneected Eected St. Dev. 99.9th ercentile Eected St. Dev. 99.9th ercentile loss ,3 0,79 7,86 3,9% 0,56% 5,6%,43% ,8 0,58 8,6 3,84% 0,37% 5,%,7% ,90 3, 69,00 7,86% 0,9% 0,7%,85% ,54 0,8 4,47,0% 0,9%,8% 0,6% ,5 0,3 3,66,69% 0,34% 3,88%,9% ,49 0, 3,9,70% 0,3% 3,47% 0,77% ,85 0,,3,66% 0,%,43% 0,77% incondicional 3,0,35 8,40 6,78% 0,64% 9,9%,4% In contrast with cororate ortfolio, distribution for retail has a major comonent of eected loss, which results in substantially lower caital requirements. By looking at grahs corresonding to each kind of debtor 5 it is clear that cororate ortfolio has a more asymmetrical distribution, thus indicating more volatility and higher requirements due to uneected loss. 5 Presented in Aendi B and C. 5 P a g e