Eric on economic capital modeling

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Eric on economic capital modeling

  1. 1. Eric on Credit Risk EconomicCapital Modeling-Concept-Model Introduction-Applications2009/05/04Eric Kuo
  2. 2. Banks hold „Capital‟ to protect against “Unexpected Loss” and to maintain„Solvency‟, not for the regulatory compliance. AGENDA OF TODAY1. Role of bank Capital is not just for the regulatory requirement.capital Capital is a cushion to absorb unexpected loss. Basel 2 ‘s capital equation is a simplified ‘EC’ model, it comprises 4 factors :2. Interpreting  Vasicek modelBasel 2 capital  Correlationequation  EL  Tenor adjustment3.Model This model follows Basel’s approach and incorporate simulation skill toIntroduction & measure the EC.Applications Economic capital is wildly utilized in all aspects of bank’s management. Three key management applications :4. Role of EC in  Risk governancePillar 2  External communication  Internal management 2009 Eric Confidential 2
  3. 3. Bank capital serves as cushion to absorb losses and maintain solvency. Profit generation Credit loss is tiny Credit loss is huge When economy is healthy and When economy is stable or become In the extreme stressed situation, bank credit quality is solid, risk taking vulnerable, bank may encounter credit suffers huge credit loss that not only will generate profit and will loss and erode the profit. In this case, erode the profit but also consume all the strengthen bank capital. bank’s available capital is served as available capital that causes bank cushion and absorb the loss. Therefore, insolvent problem. Capital injection is bank capital decreases and this may immediate needed. affect bank rating or business. 40 Need to raise 60 40 30 80 capital immediately 30 40 30 40 -10 -50 30 10 20 -10Revenue Credit Profit Bank Bank Revenue Credit Profit Bank Bank Revenue Credit Profit Bank Bankafter Op loss available remaining after Op loss available remaining after Op loss available remainingcost capital capital cost capital capital cost capital capital Before add Before Before profit absorb absorb loss loss 2009 Eric Confidential 3
  4. 4. Basel 2 promote the risk measurement that based on bank‟s historical data. The EL is considered as cost of doing business, while as capital serves as reserve to protected against unexpected loss. Conceptual Generate risk parameters (PD,LGD,EAD) from historical loss data. Bank capital (risk or economic capital) is prepared as cushion to absorb the The expected loss estimation is the cost of doing loan business. unexpected credit losses. Capital is used to Target rating cover these Credit loss Credit Loss extraordinary loss Risk Appetite A Bank’s actual loss Unexpec- Risk experience ted loss Capital Average credit loss EL =PD * LGD *EAD EL Time ProbabilityNote : Expected Loss = PD*LGD *EADEL doesn’t necessary equal to the historical loss experience, due to the portfolio component may change. 2009 Eric Confidential 4
  5. 5. More capital a bank has, allows bank to absorb more unexpected loss… …but how much capital is sufficient ? Conceptual In theory the unexpected loss estimation is kind of prediction whether if obligors will default or even default together. Current Year EndObligors The potential outcome in the year endAssuming a 3 # of Default No LGD Loss Possi- 1. How canobligors, A B C , default default bility we 0 A,B,C 0 44% estimateeach has the joint 1 A B,C 50 10% default• 100 of exposure B A,C 50 10% event ?• 50% of LGD C A,B 50% 50 10% 2. What is the 31.5 possibility• 10% of PD 2 A,B C 100 5% of potential of each A,C B 100 5% event ? EL= 15 credit loss B,C A 100 5% 3 A,B,C 150 1%Note: EL = PD*LGD*EAD 2009 Eric Confidential 5
  6. 6. The amount of capital held by a bank reflects the risk appetite of a bank. Illustrative Based on the above example, this bank need to reserveProbability of • 15 for expected lossloss • 31.5 for unexpected loss Bank also can reserve more provision and Loss Distribution capital for the future uncertainty. A Credit Losses0 Tail Risk Expected Capital need to hold to loss protect Unexpected loss 15 31.5 253.5 2009 Eric Confidential 6
  7. 7. Basel committee generates a general form of unexpected loss formula for banks to gauge the unexpected loss and use as capital minimum requirement.Factors in Basel2 Inverse of the Standard normal distribution standard normal (N) applied to threshold and distribution (G) 1 Year PD is considered, conservative value of applied to PD to PD derive default instead of cumulative PD systematic factor threshold Subtract EL based K= Correlation Based on historical data provision LGD    R  0.5   LGD N 1  R  GPD     G0.999  PD  LGD 0.5    1 R        Inverse of the  Current status of EAD standard normal EAD  1  1.5  b  1  M  2.5  b 1 distribution (G) applied to confidence level to derive conservative Tenor adjustment value of systematic factor Tenor B= 0.11852  0.05478  ln( PD)2 RWA = K * 12.50 * EAD R= Asset 1  e 50PD     1  e 50PD   0.12    50   0.241   50   Capital = RWA * BIS Ratio Correlation  1 e    1 e  It might over or under estimated the risk. 2009 Eric Confidential 7
  8. 8. Basel 2 capital equation is a simplified method of EC and is originally fromVasicek‟s model. AGENDA OF TODAY1. Role of bank Capital is not just for the regulatory requirement.capital Capital is a cushion to absorb unexpected loss. Basel 2 „s capital equation is a simplified „EC‟ model, it comprises 4 factors :2. Interpreting  Vasicek modelBasel 2 capital  Correlationequation  EL  Tenor adjustment3.Model This model follows Basel’s approach and incorporate simulation skill toIntroduction & measure the EC.Applications Economic capital is wildly utilized in all aspects of bank’s management. Three key management applications :4. Role of EC in  Risk governancePillar 2  External communication  Internal management 2009 Eric Confidential 8
  9. 9. Vasicek generates a equation to estimate the probability distribution of the %loss on the bank loan portfolio in 1987.   G( PD )  R * G(99.9%)    PD  LGD * 1  1.5  b   1  M  2.5  b 1 k   LGD  N      1 R    1 2 3 4 Asset correlation Vasicek Model Expected loss Tenor adjustment subtraction where N = cumulative standard normal distribution G= inverse standard normal distribution R= asset correlation, used to gauge the relationship between obligors’ return and systematic risk (common factor) the G(99.9%) represents for the confidence level that bank capital is able to sustain the credit losses. 2009 Eric Confidential 9
  10. 10. Recall the CAPM theory that there is a one almighty factor that can explain the asset relationship across assets. Definition of default correlation If Fannie Mae files chapter 11.. ..will Lehman also has insolvency problem?Asset correlation Correlation between corporations Vasicek Model ..will this event affect to ..if Lehman did bankrupted and cause auto industry?... Lehman’s employer unemployed.. Expected loss Correlation between corporation and individuals CorrelationTenor adjustment between GM announces 500 individuals and temporary layoffs corporation 2009 Eric Confidential 10
  11. 11. Asset correlation is the dependence of asset value of obligors on the general state of economy. All obligors are linked to each other by this systematic factors. Let’s use the ‘SUN’ as an example…‘Sun’ has almighty power to all creatures on the earth. In different seasons or timing, ‘Sun’ has different influences on earth. The clothes Plant Sun becomes less burdensome in the winter or ..and of setting course to the human being Snowman Insect•The impact to the different creatures can be considered •In the different stage of economy, sameas asset correlation. obligors or asset classes also perform different•Different borrowers or asset classes show different dependency.degrees of dependency on the state of economy. 2009 Eric Confidential 11
  12. 12. Highly correlated with state of economy implies that the PD of a company is highly dependent on the economy situation. •100% of asset correlation means Implication the company’s asset return is perfect link to the economy. •Company defaults in the bad state of •No firm’s specific risk existed 100 % of correlation economy •Company shows strong financial High return 100% 100% performance in the booming economySystematic Company’s PDrisk Company’s Bad increasing asset economy Good along with the return economy event of bad economy Company State of economy Low return State of economy Bad economy Good economy 2009 Eric Confidential 12
  13. 13. Rating agencies do try to utilize the actual the joint default probability out of their data base to estimate the asset correlation…. S & P‟s historical default study Asset correlation estimation Year # of # obligors Joint Statistics suggests the asset correlation is a defaults default function of joint default events and obligor rate defaults, depict as below 1981 0 1070 0.0000% Jo int_ Default _ Pr obability ij 1982 2 1099 0.0002%  ( Default _ threshold i , Defaulr _ threshold j 1983 1 1122 0.0000% , Asset _ Correlatio nij ) 1984 2 1181 0.0001% 1985 0 1216 0.0000%  = Bivariate standard normal distribution ….. …. … … 2003 3 2998 0.0001% In this case, the asset correlation from 2004 0 3117 0.0000% S & P ‘s study is 3.88% 2005 1 3264 0.0000% …but past experience tends to be less useful for the current and future, in terms of correlation estimation.Source: Diane Vazza and Devi Aurora: Annual 2005 Global Corporate Default Study AndRating Transitions, 2006, Standard & Poors 2009 Eric Confidential 13
  14. 14. Basel committee adopts the single factor model to modeling the asset correlation. The asset correlation is a matter of systematic risk. One Factor Model Rationale of joint default event •Joint default event is driven by the correlation This is the ‘factor’ •Borrower i’s asset value Ai depends on the common factor Z and an idiosyncratic factor εi Ai  Wi Z  1  Wi  i 2 •The parameter Wi represents the obligor’s dependence to the market index. asset return of .. explained by •The Wi can be estimated from an OLS regression by .. the remainder a single state of the utilizing the Equity return of obligor i borrower over a economy (√1-W 2 ) is idiosyncratic Correlation one year (systematic factor) risk (firm’s specific risk) ‘Z’ at ‘W’ % Systematic factor horizon can be (market factor) .. cov( i ,  j )  0 where _ i  j; idiosyncratic cov( Z ,  i )  0,  i risk Obligor 1 Obligor 2Note : A detailed explanation of the capital requirement formula can be found in Basel Committee onBanking Supervision, 2005, An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel. 2009 Eric Confidential 14
  15. 15. Market practice leverages equity correlation for the estimation of asset correlation.•Credit Metrics suggests to utilize the MSCI to estimate the equity correlation.•Suggest equity correlation is a good proxy of asset correlation. It is also possible to utilize the Taiwan stock index to proxy the asset correlation for CBG.Source: J.P. Morgan,1997.Introduction to CreditMetrics technical document. 2009 Eric Confidential 15
  16. 16. Moodys KMV estimate the asset value by adding the market cap with firm‟s liability. Moreover, KMV extends the one factor model into multiple factors model to estimate the asset correlation.KMV begins at collecting equity indices cross ..KMV finally identified 120 factors in gauging theglobe and estimate the market cap for each firm systematic risks.before adding firm‟s liability to come up with The systematic factors are the sources of „correlation‟ .firm‟s asset value.. Systematic Risk Firm Specific Country Risk Industry Risk Risk 14 45 61 US Electronic rk   kf rf   kc  c   ki  i   k UK Manufacturing f 1 c 1 i 1 Taiwan Service Korea Real estate 14 Common 45 Countries 61 Industries … macro … factors 2009 Eric Confidential 16
  17. 17. Higher the asset correlation easier to be influenced by the state of the economy. Asset correlation between obligor and the state of the economy Asset correlation between 2 obligors Joint default probability of 2 obligors Default correlation between 2 obligors Systematic Risk Can be Further Diversified Through Firm Specific Add more risk obligors 2009 Eric Confidential 17
  18. 18. Asset correlation estimation can be translate into default correlation and joint probability of default. Market Value of Illustrative AssetsTaiwan Business bank Default Point Insurance Autohas a high assetcorrelation (51.57%) Taiwan Business Bank High Asset Correlation iswith the state of bad, but low PD creates aeconomy very low probability of both defaulting --- resulting in lower default correlation PD of Taiwan Business Bank =0.87% Default Point Taiwan Business bank Market Value of Insurance Auto Assets Joint probability of default Insurance Auto has a =0.00445% low asset correlation Measures the probability of both companies defaulting (11.29%) with the state PD of Insurance Auto at the same time =0.24% of economy 2009 Eric Confidential 18
  19. 19. With asset correlation, bank can then compare the retail segments with corporate customer. Prob of default together is 0.0079% Asset correlation The asset between TSMC return of & Mail Loan is TSMC is 16%highly related to global asset return Default correlation between TSMC & Mail Loan is 0.4% Country risk Systematic Industry risk Risk Sector risk Region risk Global risk Firm Specific risk 2009 Eric Confidential 19
  20. 20. Even the mortgage segment has the same correlation with mail loan segment, thehigher PD of mail loan result in a high joint PD. Prob of default together is 0.0079% Asset correlation between Mortgage & Mail Loan is 5% Default correlation between Mortgage & Mail Loan is 1.3% 2009 Eric Confidential 20
  21. 21. Low the PD accompany with low asset correlation, generate a lower the joint EDF and default correlation. Lower the PD lower the joint EDF and default correlationMortgage Mortgage 2009 Eric Confidential 21
  22. 22. Correlation plays important role in identifying the „Diversification Effect‟ and serves as a driver to estimate the economic capital. - Illustrative - Case1 :Different industry but in same country 26 % This portion of the Saving risk has been diversified away Diversified within the portfolio. Basel 2 5.2 MM 19.7 MM Obligor‟s Diversifiable 11.5 MM Standalone AIRB Capital + Firm Specific Economic Capital = 14.5 MM 8.2 MM Risk Contribution of Systematic China Airline =3 MM Bank Airline Case2 :Different industry and different country 29.5 % This portion of the Saving risk has been Diversified diversified away 4.1 MM within the portfolio. 13.9 Obligor‟s total + Diversifiable Economic Capital 8.2 MM Economic Capital = 9.8 MM 5.7 MM Firm Specific (stand-alone) Systematic Risk Contribution of Insurance Airline Insurance (US) Auto = 1.6 MMAssume : Lend NT 100 MM to each borrowers at LGD=45% of collateral, given their different PD. 2009 Eric Confidential 22
  23. 23. For retail portfolios, leverage the historical time series of default or loss information (by product, segments etc.) is the major practice on developing asset correlations for retailing .Default Rate Default Rate by Product – Client Example0.35% Installment Line of Credit Real Estate Credit Card0.30%0.25%0.20%0.15%0.10%0.05%0.00% Jul-01 Jul-02 Jul-00 Jan-03 Mar-03 Jan-02 Jan-01 Mar-01 Mar-02 Sep-01 May-01 May-03 Nov-02 Sep-02 Sep-00 Jan-00 Nov-00 Nov-01 Mar-00 Sep-00 May-02 Nov-00 May-00 2009 Eric Confidential 23
  24. 24. Empirical studies show that : Higher the correlation will have high relationship with the global economics, result in a higher impact to obligor‟s business. Therefore, requires more capital to protect against „unexpected loss‟. Higher the correlation higher the UL, therefore, bank needs to reserve higher capital requirement Low correlation High correlation Basel Committee generates the asset correlation through „Reverse Engineering‟ – Empirical experimental through several banks‟ EC. 2009 Eric ConfidentialSource : An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2 24
  25. 25. Different types of loan assets have different asset correlation. 2009 Eric Confidential 25
  26. 26. Basel has difficulty to estimate correlation for different countries and industries. Therefore, Basel come up with a equation for corporate obligor based on rating and suggests a constant for retail products. •Asset correlation decrease with increasing PDs. This is based on empirical evidence. Better grade has higher asset correlation •Intuitively, higher the PDs, higher the firm specific risk. Default risk depends less on the overall state of the under Basel committee‟s assumption. economy. 25% 23.82% ORR_Grade PD Correlation 1.AA- or better 0.03% 23.82% 2.A+~A- 0.10% 23.41% 20% Mortgage 3.BBB+ 0.16% 23.08% 4.BBB 0.26% 22.54% 15% 15% 5.BBB- 0.42% 21.73%AssetCorrelation 6.BBB- negative 0.61% 20.85% perspective 10% 7.BB+ 0.90% 19.65% Revolving Product 8.BB 1.35% 18.11% 5% 9.BB- 2.04% 16.33% 4% 10.BB- negative 3.15% 14.48% perspective 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 11.B+ 4.93% 13.02% 12.B 7.82% 12.24% CTCB Rating Grates 13.B- 0.1261 12.02% 2009 Eric Confidential 26
  27. 27. How does the Basel Committee come up with the correlation equation??? Technically, Basel utilize the CAPM theory…1 Recall the CAPM model that : 3 Defined A, Z, ε are normalized random variables ri=αi+βirm+εi ri   i   i E (rm ) where A  ri = return of firm i. i rm= return of the market rm  E (rm ) i εi= firm’s specific risk- a random error Z  m  ( i ) Manipulating CPM by Re write the CAPM tells us that .. Define the variables normalizing equation 4 m  ( i ) 2 A  ( i )Z   Normalizing the above equation, we have: i i ri   i   i E (rm )  m rm  E (rm )  ( i )  i Where  ( i )  i i m  i  ( i )  m 2  ( i ) 2 1  ( i ) ( ) Where  i ,  m ,  ( i ) i i are the standard variance of firm i asset, the return of market and the random error, respectively. 2009 Eric Confidential 27
  28. 28. … to derive the one factor correlation model. 5 7 A ordinary regression Correlation One factor model is between I and derived from CAPM Yi=α+βiXm+ε m is  im A  WZ  1  W 2   i   i  im  m i m2  W 2 = asset correlation W  ( i m ) =correlation weight i Recall the regression in Re-arranging the ..finally, we have One statistics parameters.. factor model 6 m  ( i ) A  ( i )Z   i i  m 2  ( i ) 2 Because 1  (  i ) ( ) i i     We’ll have (  i m ) 2  ( im m ) 2  ( im ) 2   i 2 i m i 2  m i  ( i ) 2  ( )  1  (  i m ) 2  1  i2  1  W 2 m i i Set W  ( i ) i  ( i ) the  1W 2 i 2009 Eric Confidential 28
  29. 29. The empirical asset correlation estimation done by Thomas and Zhiqiang support Basel‟s correlation equation.Hugh Thomas and Zhiqiang Wang examinethe average asset correlation of United In the first draft of Basel Accord, Basel suggested a constantStates by using the one factor model and asset correlation 20% , for all corporate obligors. Later the banks in US and Europe provide their survey and empiricalfound the current asset correlation is close to study to Basel and urged to refine the asset correlation.their finding.. 2    v  i 2   i m      i  20% 2  20%   1 *   46%   18.9% Asset where the market return Correlation volatility m = 20 % per annum; the average stock volatility is i = 46 % per annum i =1 PDSource: Hugh Thomas and Zhiqiang Wang "Interpreting the Internal Ratings-Based Capital Requirements in Basel II" Journal of Banking Regulation, 2005. 2009 Eric Confidential 29
  30. 30. Nobel prize winner Merton suggests that default can be viewed as a function of the underlying asset value of company…and Vasicek modified Merton‟s model and became the cornerstone of Basel‟s capital equation. Definition of default correlation Under Merton’s model assumption, underlying firm value is random with Asset correlation normal distribution. If the value of assets decreases below the default threshold (amount of liabilities outstanding), it will be impossible for the firm to satisfy its obligations and it will thus default. Vasicek Model The PD of an obligor can be translated into the default threshold. For example: the threshold of PD= 2% is -2.053 Default threshold Expected loss 2% of probability will fall below 98% of confidence interval Tenor adjustment NORMSINV(2%)= -2.053Note: Vasicek is one of the founders of KMV. 2009 Eric Confidential 30
  31. 31. Vasicek combines Merton‟s model with one factor correlation model to measure the credit loss. 1 2 3 The company defaults on its But we’d like to know the PD of an Basel set the Z (state of economy) at the loan if the value of its assets obligor fluctuate with the economy; poorest condition (99.9%), to test how drops below the contractual therefore, we rewrite the equation by many capital reserve is required by bank value of its obligations α incorporating the state of the to sustain in the worst economic situation. economy: Therefore we have (default threshold) payable at PDi=P(Ai<αi) one year horizon. We thus Z =N-1(99.9%)=G(99.9%) have: Wi Z  1 Wi  i <αi) 2 = P( Recall the Wi   PDi=P(Ai<αi)   Wi Z i = P( i  ) And Basel use the symbol ‘R’ to represent 1  Wi 2 for asset correlation The default threshold can be Therefore, we have derived : Substituting the α and expressed Wi    R αi=N-1(PDi) (1  Wi )  1  R the above with cumulative standard 2 normal distribution: PDi= P(Ai<αi) And finally we have the Basel’s K factor =Note:  G( PD )  R * G(99.9%) Vasicek, O, 1987. Probability of loss on loan =N ( N 1 ( PD )  Wi Z ) N portfolio. KMV Corporation.  1 R Vasicek, O, 1991. Limiting loan loss probability 1  Wi 2distribution. KMV Corporation. 2009 Eric Confidential 31
  32. 32. Basel defined that the adequate capital must be sufficient at 99.9% of the time… to make up the Vasicek‟s assumption. Major assumption of Vasicek‟s model for calculate the State of economy loan loss distribution •Consider a portfolio consisting of n loans in equal dollar amounts : Diversified and no concentration risk existed. •Let the probability of default on any one loan be ‘P’ : each Poor state Good state of obligor has the same PD of economy economy •Assume that the values of the borrowing companies’ assets are correlated with a coefficient ρ for any two 0.1% of probability companies : constant asset correlation will fall below 99.9% •Assuming that the loan generates no income: Net interest of confidence interval income and fee income is not considered. • When a loan goes into default, there is no recovery: LGD=100% •Firm’s asset returns are normally distributed : Normal 0 distribution -NORMSINV(99.9%)= •The company defaults on its loan if the value of its assetsNORMSINV(0.1%) = -3.09 Mean of state of economy drops below the contractual value : Merton’s model Vasicek assumes if a bank has diversified portfolio : sameAsset value follows Normal distribution is a strong PD, LGD, EAD and same correlation. The only thingassumption. To prevent from this flaw, Basel set the needs to worry is the ECONOMY, borrowers have highcapital cushion at a high level that can protect bank at probability in the bad state of economy : source of99.9% of probability. unexpected loss. 2009 Eric Confidential 32
  33. 33. EL based provision comes with a proper reason. If banks are not comply with EL basedprovision then banks may need to prepare more capital to protect credit losses. Banks should price the expected loss based o the risk and as provision reserve. Otherwise, banks need to hold more capital to prevent from insolvency.Asset correlation Basel set at 99.9% of confidence Vasicek Model =100% - 99.9% = 0.1% Expected loss Subtract EL based Correlation provision    R  0.5   LGD N 1  R  GPD    G0.999  PD  LGD 0.5 Tenor adjustment  K=    1 R         2009 Eric Confidential 33
  34. 34. Longer term credit facilities are riskier than short-term credits. As a consequence, the capital requirement should increase with maturity. The final adjustment in the IRB formula is the adjustment for the average maturity.Asset correlation   G( PD )  R * G(99.9%)    PD  LGD * 1  1.5  b   1  M  2.5  b 1 k   LGD  N      1 R    Vasicek Model b= 0.11852  0.05478  ln( PD)2 As we all know that the longer term credit facilities are riskier Expected loss than short-term credits. As a consequence, the capital requirement should increase with maturity. The maturity adjustments can be interpreted as anticipations of additional capital requirements due to downgrades. Downgrades are more likely in case of long-term credits and hence theTenor adjustment anticipated capital requirements will be higher than for short- term credits. 2009 Eric Confidential 34
  35. 35. 4 objectives in this topic : Rational of model development, Demo the model,understanding the importance of risk parameters and estimation of CTCB‟s EC. AGENDA OF TODAY1. Role of bank Capital is not just for the regulatory requirement.capital Capital is a cushion to absorb unexpected loss. Basel 2 ‘s capital equation is a simplified ‘EC’ model, it comprises 4 factors :2. Interpreting  Vasicek modelBasel 2 capital  Correlationequation  EL  Tenor adjustment3.Model This model follows Basel‟s approach and incorporateIntroduction & simulation skill to measure the EC.Applications Economic capital is wildly utilized in all aspects of bank’s management. Three key management applications :4. Role of EC in  Risk governancePillar 2  External communication  Internal management 2009 Eric Confidential 35
  36. 36. Model introduction & application Ai  Wi Z  1  Wi 2  iTheory introduction Pi (Z )  P( Ai   i | Z )  P(Wi Z  1  Wi 2  i   i | Z )Model demonstration  P(Wi Z  1  Wi 2  i  N 1 ( PDi )) Conditional PD  Wi Z  1  Wi * N ( i ) 2 1Model applications Wi Z  1  Wi 2 * N 1 ( i )  N 1 ( PDi )Limitation 2009 Eric Confidential 36
  37. 37. Integrate the bank „internal rating system‟ into one factor model with simulation skill to depict the loan loss distribution. Applying one-factor Estimating the Simulating and Data Estimating Default model to estimate the credit loss in the aggregating allRequirement threshold default event event of default scenariosObligor based or segment based For example :• PD ‘AAA’ Corporation has• LGD - 1% PD based on bank’s• EAD This model requires to treat AAA as Internal rating • PD=1% system • EAD=200 - 2 credit lines • LGD = 75% 1. EAD=100 with LGD =50% 2. EAD =100 with LGD =100% 2009 Eric Confidential 37
  38. 38. Estimating default threshold means to translate the unconditional PD into threshold value. Applying one-factor Estimating the Simulating and Data Estimating Default model to estimate the credit loss in the aggregating allRequirement threshold default event event of default scenarios Log of market Probability value distribution Default threshold of asset values Expected 2% of probability asset values will fall below 98% over time Today’s of confidence firm interval valueDefaultpoint Refers to this area Default threshold = NORMSINV(2%)= -2.053 This area translate into PD,say 2% The company defaults on its loan if the value of its assets dropsThe structured portfolio model doesn’t mean that below the contractual value of its obligations α (default threshold)we need to model the default probability by using payable at one year horizon. We thus have: PDi=P(Ai<αi)the concept of Merton. Instead we apply the The default threshold can be derived :concept into our portfolio model development. αi=N-1(PDi) 2009 Eric Confidential 38
  39. 39. The obligor‟s PD is significantly influenced by the state of economy. A PD that considered the state of economy is called „Conditional PD‟ Applying one-factor Estimating the Simulating and Data Estimating Default model to estimate the credit loss in the aggregating all Requirement threshold default event event of default scenarios The conditional default threshold is the default probability Recall the One Factor Model describe conditional on the systematic factor Z: a function of the as below : idiosyncratic component of obligor i and the credit worthiness index, then we have : Ai  Wi Z  1  Wi 2  i The PD of obligor i considering the state of economy 1 Pi (Z )  P( Ai   i | Z )  P(Wi Z  1  Wi 2  i   i | Z ) Asset State of the W 2 : asset Idiosyncraticreturn of economy correlation risk (firm’s  P(Wi Z  1  Wi 2  i  N 1 ( PDi ))obligor i (systematic between specific risk) 2 The conditional PD of obligor i can be described as : factor) obligor i and Conditional PD  Wi Z  1  Wi * N ( i ) 2 1 state of economyWhere : 3 We call obligor default if : cov( i ,  j )  0 Wi Z  1  Wi 2 * N 1 ( i )  N 1 ( PDi ) where _ i  j; 4 To model and to simulate the conditional PD of cov( Z ,  i )  0,  i obligor i, we simulate ‘Z’ & ‘ε’ 2009 Eric Confidential 39
  40. 40. Conditional PD reflects the state of economy. Unconditional PD refers to the „Through The Cycle‟ PD estimation. Conceptual State of Description Occurrence Probability economy of default 2% Bad a recession at 2% the risk ¼ horizon Good expansion 0.4% ¼ Weighted average= 1.1% 1% Neutral ordinary 1% times ½ 0.40% Weighted 1.1% average Bad Good NeutralNote: More detail can be found in Basel Committee : Credit riskmodeling: Current practices and applications. Page 28.April , 1999. 2009 Eric Confidential 40
  41. 41. Applying the single factor correlation model to simulate the Conditional default. Model of firm value and its default threshold Conceptual Probability of density Simulation will try all valid Default threshold combinations of the current given obligor’s PD portfolio, given the PD, LGD, EAD and asset correlation Default area -2.07 -1.74 -1.28 -0.01 0 ………………………… Asset return within one year: Estimated by using the firm‟s PD Estimated by applying the one factor model through simulation : i  N 1 ( PDi )  N 1 (10%)  1.28 Conditional PD = W i * NORMSINV(RAND()) + √(1- W 2i) * NORMSINV(RAND()) 2009 Eric Confidential 41
  42. 42. Apply LGD * EAD in the event of default in each scenario. Illustrative Applying one-factor Estimating the Simulating and Data Estimating Default model to estimate the credit loss in the aggregating all Requirement threshold default event event of default scenarios Default Simulation Loss AggregationAssume a constant 10% of probability of default for 100 segments ..We can also see there are events of joint default, that explainor obligors. After a 10 thousands of simulation, the following table when some obligors /segments insolvent , will influence otherillustrates the default event under the given correlation…. obligors / segments. These joint default event will occurs ‘UNEXPECTED LOSS’ Loan 1 2 3 … 100 Loan 1 2 3 … 100 PD 10% 10% 10% 10% LGD 50% 50% 50% 50% Scenario Total Scenario Total 1 1 0 0 .. 0 1 1 50 0 0 .. 0 50 2 0 1 0 .. 0 1 2 0 50 0 .. 0 50 3 0 1 1 .. 0 2 3 0 50 50 .. 0 100 4 1 1 1 .. 0 3 4 50 50 50 .. 0 100 5 0 0 0 .. 0 1 5 0 0 0 .. 0 .. ... ... 10,000 0 0 1 .. 0 1 10,000 0 0 50 .. 0 50 # of Default 1,000 1,000 1,000 … 1,000 xxx $ of Loss 50,000 50,000 50,000 … 50,000 xxx # of Survive 9,000 9,000 9,000 … 9,000 xxxx Avg Default ~10% ~10% ~10% .. ~10% xx Frequency Avg LGD ~50% ~50% ~50% .. ~50% xxNote: Assume EAD=100 for all obligorsBased on portfolio simulation model 2009 Eric Confidential 42
  43. 43. The outcome of simulation can allow bank to draw „Loss distribution‟ and „Joint default event distribution‟ as well. Illustrative Applying one-factor Estimating the Simulating and Data Estimating Default model to estimate the credit loss in the aggregating all Requirement threshold default event event of default scenarios Default Simulation Loss AggregationDiagnosing the joint default event allows bank to understand By summing up the individual loan losses, we then can obtainthe potential default contagion driven by the state of economy. the portfolio loss. Further considering the number of occurrences and its associated loss amount, we can derive theFurthermore, bank can draw joint default distribution. portfolio loss distribution. Loan 1 2 3 … 100 Loan 1 2 3 … 100 PD 10% 10% 10% 10% LGD 50% 50% 50% 50% Scenario Total Scenario Total 1 1 0 0 .. 0 1 1 50 0 0 .. 0 50 2 0 1 0 .. 0 1 2 0 50 0 .. 0 50 3 0 1 1 .. 0 2 3 0 50 50 .. 0 100 4 1 1 1 .. 0 3 4 50 50 50 .. 0 100 5 0 0 0 .. 0 1 5 0 0 0 .. 0 .. ... ... 10,000 0 0 1 .. 0 1 10,000 0 0 50 .. 0 50 # of Default 1,000 1,000 1,000 … 1,000 xxx $ of Loss 50,000 50,000 50,000 … 50,000 xxx # of Survive 9,000 9,000 9,000 … 9,000 xxxx Avg Default ~10% ~10% ~10% .. ~10% xx Frequency Avg LGD ~50% ~50% ~50% .. ~50% xxNote: Assume EAD=100 for all obligorsBased on portfolio simulation model 2009 Eric Confidential 43
  44. 44. Excel based and allows user to play around.Theory introductionModel demonstrationModel applicationsLimitation 2009 Eric Confidential 44
  45. 45. Confidence interval Credit VaR # of Simulation Simulation EC Systematic results factor Z This area allow user to observe the joint default event. Default Limited to 255Data input Weighting of columns though. area correlation threshold Conditional PD threshold Estimate Loss if obligor default : Call default if LGD * EAD Conditional PD < Default threshold 2009 Eric Confidential 45
  46. 46. Output Enter ‘Bucket’ Joint Default Distribution Loss Distribution 2009 Eric Confidential 46
  47. 47. Correlation plays an important role in determining the joint default event. Effect of correlationTheory introduction Effect of concentrationModel demonstration Effect of PDModel applications Effect of LGDLimitation 2009 Eric Confidential 47
  48. 48. This is a simple but still useful for bank to depict how the risk or EC will deviate by changing the risk parameters. Goal of this exercise Assuming bank has a 100 obligors, each has identical risk parameters describe as below: • Observing how the EC change if change the assumption of correlation. I’ll set the asset Risk parameters Portfolio statistics correlation at following and observe how the PD=10% Portfolio EAD= 10,000 EC changes LGD=50% EL =500 Asset Correlation Correlation Weight EAD=100 Bank target rating = A 1. 6.25% 25% 2. 25% 50% 3. 1% 10% 4. 100% 100% 5. 0% 0%Note : Defined ‘W 2 ‘ as asset correlation, ‘W‘ as correlation weight 2009 Eric Confidential 48

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