Toward Credit Portfolio Management

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  • 1. Toward Credit Portfolio Management Part I A Rudimentary Guide to Credit Portfolio Modeling: Theory Introduction Model Development Applications Draft Edition Eric Kuo
  • 2. Toward Credit Portfolio Management Part I A Rudimentary Guide to Credit Portfolio Modeling: Theory Introduction Model Development Applications Draft Edition Eric Kuo
  • 3. ii Copyright @ Eric Kuo 2008 All rights reserved No part of this document may be reproduced in any form, by Photostat, microfilm, xerography or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the copyright owner.
  • 4. iii To Liwen, for her patience and support To Tiffany, with joy and pride
  • 5. iv Preface Banks may fail for a lot of reasons. It may fail to compete with peers and gradually lose the market share. Or it may adopt wrong strategies and target at riskier segments. Or it may fail to manage the market risk and incur massive trading losses. Or it may underestimate the interest rate risk, liquidity risk and unable to manage the funding gap. Of all the possible reasons of failure, the most threatening risk, in my opinion, may be the credit risk. The massive credit losses arise from the credit risk that wipe out bank capital and cause bank failure. In my opinion, these credit losses sometimes are a result of concentration risk, such as single name concentration that one or few obligors account for significant portion of total portfolio. It also may be a result of industry concentration, product concentration or segment concentration, that several losses appear simultaneous during the economic malaise. Moreover, sometimes the credit losses may be the consequence of the problem of miss-pricing that the credit revenue cannot cover the credit losses. Although bankers conceited themselves as credit risk professionals, the banking industry in general doesn’t generate any profit to the shareholders, simply counting the credit losses occurred in the past decade. All theses lead us to conclude that the banking industry is a highly vulnerable, highly competitive and highly regulated business. The business that banks do is risk-taking; however, the amount of potential credit risk is almost unknown – both of the investors and bankers themselves. As banks have moved in the direction of new Basel Accord, the credit risk has become more important and will be more transparent to the investors. However, the measurement of credit risk parameters (such as probability of default, loss given default and exposure at default) is only just the first step toward the credit risk management. The estimation of the credit portfolio risk further allows bankers to understand the unexpected loss. Banks can apply the economic capital into many management applications, such as capital allocation, performance management and business strategic planning. This document is an overture of credit portfolio management. Part 1: Credit portfolio modeling – measuring the unexpected loss of portfolio. (Discussed in this document) Part 2: Credit correlation modeling- estimating the asset correlation among obligors.
  • 6. v Part 3: Credit portfolio analysis- diagnosing the performance of portfolio – assessing whether if risk and return is balanced. Part 4: Credit concentration risk management- managing the credit concentration risk through setting the limit boundary. This document provides a simple credit portfolio model for readers to estimate the credit portfolio unexpected loss. One important message that I’d like to deliver is that if you cannot measure risk, then you should not expect to manage the risk. This document is not aimed at technicians or quants. There are already many excellent books explore the technique of portfolio modeling. Instead, this document focuses on the application of the credit portfolio risk. How can bank leverage the simple model provided in this document to better understand the unexpected loss of bank’s credit portfolio. In my opinion, the efforts of risk measurement are vanity without promoting the measurement into management. In summary, sound credit risk management must be transparent and well perceived. Only when the board, CEO, rating agencies, equity analysts and the investors are well-informed and are confident at the bank’s risk management- they will not expect bankers perform miracles and they won’t be surprised by the losses in a economic recession. After all, banking business is a cyclical business with frequent expected loss and threatening unexpected losses. I could point out that the most opaque black box in many financial institutions is the quality of credit asset. The figures that bank provided in the annual report only depicts the past and have no value regarding the future asset quality. The disclosure of economic capital will give outsiders a clue. Also is an action of risk governance that will distinguish bank from her peers. Any error and unintentional deviation from the best practices remain my own responsibility. Eric Kuo, Sep, 2008
  • 7. Table of content Section 1: Foreword and Introduction .......................................................................... 1 Section 2: Why banks need capital .............................................................................. 3 The IRB capital equation...................................................................................... 4 Differences between AIRB capital and Economic Capital .................................... 9 Section 3: Theory & vendor models introduction ....................................................... 17 Portfolio Models Introduction.............................................................................. 18 KMV’s Portfolio manager ................................................................................... 19 Creditmetrics ...................................................................................................... 24 CreditRisk+ ........................................................................................................ 31 Creditportfolioview.............................................................................................. 32 Section 4: Methodology of simple portfolio model development ................................ 37 Unconditional and conditional PD ...................................................................... 37 Loss based or value based ................................................................................ 39 Correlation Model............................................................................................... 40 Model Design Methodology................................................................................ 42 Section 5: Model manual instruction .......................................................................... 47 Data requirement ............................................................................................... 47 Model screenshot............................................................................................... 47 Example ............................................................................................................. 48 Joint default observation .................................................................................... 51 Charts................................................................................................................. 52 Section 6: Applications of in-house credit portfolio model.......................................... 54 Effect of correlation ............................................................................................ 54 Effect of concentration ....................................................................................... 59 Effect of PD ........................................................................................................ 60 Effect of LGD...................................................................................................... 61 Section 7: Future improvements ................................................................................ 64 Correlation.......................................................................................................... 64 Risk contribution................................................................................................. 65 Section 8: Economic capital as management applications ........................................ 67 Risk governance ................................................................................................ 67 External communication..................................................................................... 69 Internal management ......................................................................................... 72
  • 8. Performance metrics .................................................................................. 72 Capital allocation ........................................................................................ 75 Limit setting ................................................................................................ 76 Section 9: Beyond economic capital .......................................................................... 77 Revisit the commercial banking business model ............................................... 77 Credit portfolio management as a business model ............................................ 79 Section 10: Helping the CEO’s sleep quality.............................................................. 82 Reference .................................................................................................................. 83 Appendix .................................................................................................................... 85 Interpreting the IRB capital equation.................................................................. 85 1. The Vasicek formula............................................................................... 86 2. Correlation estimation: R........................................................................ 88 3. The expected loss .................................................................................. 91 4. Maturity adjustment: ............................................................................... 91 VBA code ........................................................................................................... 93
  • 9. 1 Section 1: Foreword and Introduction Credit portfolio modeling is one of the most important topics in risk management and finance theory today. The last decade has seen the development of models to compute portfolio credit losses for bonds and loan portfolios. The important output from the credit portfolio model is so called economic capital which is used to gauge how many amount of potential unexpected loss a bank is exposed to given the current credit portfolio constitution. Banks hold ‘Economic Capital’ (or “Risk” Capital) to protect against “Unexpected Loss”. It is opposed to the Basel 1 and is different from the AIRB approach under Basel 2. Although, the Basle 2 has taken the first steps to amend the capital requirement and to promote banks to implement the internal credit risk models for better estimating the unexpected loss (regulatory capital). In the BIS regulatory model, the potential exposures are given by an add-on factor multiplying the notional of each transaction. It is simple to implement, but the model has been widely criticized because it does not accurately capture the diversification effect and concentration risk of portfolio. By contrast, credit portfolio models measure credit economic capital and are specifically designed to capture the portfolio effects, specifically obligor correlations. These models include the pioneers: KMV’s Portfolio Manager (1993), CreditMetrics (JP Morgan 1997), CreditRisk+ (Credit Suisse Financial Products 1997) and Credit Portfolio View (Mckinsey, Wilson 1997a and 1997b). Although superficially they appear quite different—the models differ in their distributional assumptions, restrictions, calibration and solution1. The major limitations, in my point of view, are: the vendor models are expensive and complexity. Expensive means the subscriber needs to pay for the software expense each year, unless bank has a big commitment on the use of economic capital. Most of the model comprises sophisticated mathematic modeling and are difficult to explain in a simple spreadsheet. For the user who is less trained in math will have an impression of ‘Black box’. Both of the above are the motivation of this document. In addition, it is vital for banks to estimate the unexpected loss of their credit portfolio to better understand the uncertainty. 1 Gordy (1998) and Koyluoglu and Hickman (1998) show an underlying mathematical equivalence among these models.
  • 10. 2 This document will begin at revisit the role of bank capital; why is it essential to the bank management- it is definitely not only to meet the regulatory compliance. Then, review the portfolio theory and introduce several vendors’ portfolio models. In the following, this document will explain the method that grounded on this simple credit portfolio model. There is one section elaborates how to leverage the model provided in this document to simulate the economic capital and gauge the sensitivity of portfolio loss based on bank’s internal risk parameters. The limitation of this model and directions of future improvement are also discussed. I also briefly introduce the management applications and the concept of active credit portfolio management in this document as well.
  • 11. 3 Section 2: Why banks need capital Banks as chartered financial intermediary institutions require taking many responsibilities and needs to meet many regulations. To prevent from the insolvency and result in financial crisis, banks need to reserve a certain amount of capital to protect from unexpected loss except for the provision reserve. Therefore, the estimation of bank capital is essential for the regulator to gauge the riskiness of bank’s asset and is important for bank itself to do business. In principle, the maximum amount of the credit loss a bank might face is the ‘Credit Exposure’ multiplying by the ‘Loss Given Default’. Maximum Loss = Exposure at Default * Loss Given Default Take a portfolio contains 500 billions of exposure and has an average 40 % of LGD as an example. The maximum loss is equal to 500 Bn*40% = 200 Bn. If the regulator takes a conservative stance in the regulatory capital policy, then bank needs to hold 200 billion for this 500 billion of loan portfolio. However, the occurrence of this maximum loss is close to zero. The event implies that all obligors are going to be insolvent in the same time. The essence of credit portfolio management is to establish portfolio balance with adequate diversification2. This mitigates the consequences of the portfolio's volatility of value (sometimes termed unexpected losses) to a level where an institution can survive such losses given its reserves and capital. The Basel committee, therefore, investigated the existed credit portfolio models and assistance from the best practices3. Finally, Basel committee decided to apply the Merton’s concept and come up with equation4 to estimate the risk weight for bank’s credit asset. 2 Usually comes from the correlation which is to estimate the default event relationship among the obligors. 3 Such as, J.P Morgan, CSFB, Bank of America and other pioneers in credit portfolio management. 4 Basel Committee on Banking Supervision,2004.An Explanatory Note on the Basel II IRB Risk Weight Functions. Page 5, page 6.
  • 12. 4 The IRB capital equation5 The calculation of AIRB capital requires a bank to utilize the past historical loss information to estimate the PD, LGD and EAD - the same concept as developing a credit scoring card or utilizing the data mining skills to find the customer behaviors. Restricted Bank capital is reserved as a cushion to absorb unexpected loss. Conceptual Generate risk parameters (PD,LGD,EAD) Bank capital (risk or economic capital) from historical loss data. is prepared as cushion to absorb the The expected loss estimation is the cost unexpected credit losses. of doing loan business. Target 1. Basel Capital is used to rating cover these Credit loss generates a Credit Loss extraordinary loss general form of Risk Appetite formula to A proxy the Bank’s actual loss ‘unexpected Risk Unexpec- experience Capital loss’ and as ted loss Average capital credit loss requirement. 2. It might over or EL under estimated the Time Probability risk. Note : Expected Loss = PD*LGD *EAD EL doesn’t necessary equal to the historical loss experience, due to the portfolio component may change. 2008 Eric Confidential 1 The EL is a concept of average credit losses based on the past experience and should be able to cover the credit losses in the most of the time6. The capital requirement is reserved for the losses that exceed the expected loss. Ideally, the capital requirement estimation should link to bank’s desire rating grade. The Basel committee generates a general form of formula to proxy the ‘unexpected loss’ and as capital requirement. It might over or under estimated the risk. For the corporate exposure, the Basel suggests the following formula: 5 Please refer to the appendix ‘Interpreting the IRB capital formula’ for more detail mathematics deduction. 6 The historical expected loss usually doesn’t equal to the current expected loss of the portfolio, for the following reasons : 1. The portfolio mix is different from the past portfolio : The PD and LGD of a portfolio may change over time. If the PD becomes better than the past, the expected loss ratio might be lower and visa’ versa. 2. The exposure may be different: for example the average credit loss is 100 out of 1,000 of exposure, while as the exposure may be expand to 2,000. To compare the absolute amount of expected loss may not be appropriate.
  • 13. 5 Restricted Basel committee generates a general form of unexpected loss formula for banks to calculate the capital. –”a simplified version of EC”. Basel estimates A General formula Inverse of the Factors in Basel2 For banks standard normal Standard normal distribution distribution (G) (N) applied to threshold and applied to PD to 1 Year PD is considered, conservative value of PD derive default systematic factor instead of cumulative PD threshold Subtract EL based Correlation K= provision Based on historical data LGD ⎡ ⎤ ⎡ ⎤ 0.5 ⎛R⎞ ⎢LGD× N ⎢(1 − R) × G(PD) + ⎜ ⎟ × G(0.999)⎥ − PD × LGD⎥ −0.5 ⎝1− R ⎠ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ Inverse of the ⎦ standard normal Current status of EAD EAD distribution (G) × (1 − 1.5 × b ) × [1 + (M − 2.5) × b ] −1 applied to confidence level to derive Tenor adjustment conservative value of systematic factor [0.11852 − 0.05478 × ln(PD)]2 Tenor RWA = K * 12.50 * EAD B= R= ⎡1 − e (−50× PD ) ⎤ ⎡ ⎛ 1 − e ( −50× PD ) ⎞ ⎤ Asset Capital = RWA * BIS Ratio ⎥ + 0.24 ⎢1 − ⎜ ⎟⎥ 0 .12 × ⎢ ⎜ ⎟ 1 − e (− 50 ) ⎦ − 50 ⎣ ⎝ 1− e Correlation ⎣ ⎠⎦ 2008 Eric Confidential 1 Restricted Basel 2 Capital estimation is a simplified version of EC (or Credit VaR) Basel set at 99.9% of confidence Subtract EL based Correlation provision ⎡ ⎤ ⎡ ⎤ 0.5 ⎛R⎞ ⎢ LGD× N ⎢(1 − R) × G(PD) + ⎜ ⎟ × G(0.999)⎥ − PD × LGD⎥ × (1 − 1.5 × b )−1 × [1 + (M − 2.5) × b ] −0.5 K= ⎝1− R ⎠ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ Tenor adjustment 2008 Eric Confidential Source : An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2 1 Higher the target BIS ratio, larger the capital required to reserve. The Basel 2 AIRB capital is a simplified version of VaR model 7 . From Basel committee’s perspectives, the confidence 7 An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2.
  • 14. 6 interval for protecting the risk of insolvency is set at 99.9%. This implies that there is only 0.1% of possibility that the loss will exceed ‘expected loss + unexpected loss’ within 1 year. The major difference between wholesale and retail bank’s capital computation can be further explained by the below chart: Restricted The major difference in Basel 2 capital estimation between Corporate and Retail Banking is ‘Correlation’. Retail example Corporate example Frequency Frequency Credit Loss Credit Loss 1. UL more important than EL (small number 1. EL more important than UL (large number of of relatively good quality loans) loans minimises Impact of fluctuations) 2. High correlation 2. Low correlation 3. Significant capital requirements 3. Capital required is relatively low: Constant correlation = 15% for Mortgage Correlation = = 4% for Revolving Correlation = 23.8% for obligor with 0.03% (AA-)of PD K= K= RWA= K * 12.5*EAD 2008 Eric Confidential 1 As illustrated in the chart, the Basel committee considered the retail products have a lower asset correlation than wholesales banking. Based on the equation; the asset correlation for the AA grade8 is around 23.82%. By contrary, the correlation for the B- grade is less than half of the 1st grade – 12%. We can find that the asset correlation function is built of two boundaries: correlations of 12% and 24% for very high and very low PDs. Correlations between these boundaries are modeled by an exponential weighting function that displays the dependency on PD. The exponential function decreases rather fast; its pace is determined by the risk weight equation; the so-called “k-factor”. The upper and lower bounds for the correlations and the functions are based on the empirical studies. On the other hand, the mortgage asset has a 15% of constant correlation; while as the revolving product has a 4% of correlation. Both of the above retail products’ correlations are lower than most of the corporate rating’s. The reason that the retail products have lower 8 PD of AA grade is 0.03%. PD of B- grade is 12.61%.
  • 15. 7 correlations is that the retail products are viewed as more diversified portfolio compare to corporate obligors. Several studies also confirmed the same result9. On the other hand, the mortgage has higher dependency with the real estate industry and is deeply influenced by the economy; therefore, the mortgage has higher asset correlation compare with other retail products10. Restricted Better grade has higher asset correlation under Basel committee’s assumption. ORR_Grade PD Correlation AA 0.03% 23.82% 25% 23.82% A+~A- 0.10% 23.41% BBB+ 0.16% 23.08% 20% BBB 0.26% 22.54% Mortgage BBB- 0.42% 21.73% 15% BBB- Negative 0.61% 20.85% Asset 12% Correlation perspective 10% BB+ 0.90% 19.65% Resolving Product BB 1.35% 18.11% 5% BB- 2.04% 16.33% 11 BB-Negative 3.15% 14.48% perspective 0% 1 2 3 4 5 6 7 8 9 10 11 12 13 B+ 4.93% 13.02% B 7.82% 12.24% CTCB Rating Grates B- 0.1261 12.02% 2008 Eric Confidential 1 The Basel deployed the correlation effect on the rating grade instead of on the country, industry. The difficulty that the Basel committee faces is that it’d be challenge to estimate average correlation for different country and different industry. Therefore, they turn to implement the correlation into the probability of default. The rational is that better rating obligor usually has larger asset size; larger asset usually has a higher dependency with the state of economy11. For example, the Honhai company has conducted many business across the world and is a major export contributor to Taiwan’s GNP. Therefore, if the economic declines, the Honhai company will be easier influenced by the economic downturn than SMEs or retail products may have. 9 Asset correlation of mortgage is range from 7% ~ 10% and 2%~5% for the retailing products, based on MKMV’s survey, Technical note. ‘Including non-corporate credit risk’, 2007. 10 Paul Calem and James Follain also found the 15% of correlation is reasonable and is supported by their empirical test. 11 Lopez,2002. The empirical relationship between average asset correlation,firm probability of default and asset size.
  • 16. 8 We can use the following example to illustrate the effect of correlation on the capital estimation. Restricted Same lending amount, different capital charge are result from correlation. Both have the same EL = PD * LGD * EAD Mortgage Corporate Clients = 1.35% * 45% * 100 Mn = 0.61 Million 1.35% 1.35% PD While as mortgage has Corporate has higher lower ‘K’ , due to lower ‘K’ correlation 45% K = 0.082 K = 0.0549 45% LGD RWA = K * 12.50 * EAD RWA = K * 12.50 * EAD NTD 100 Million NTD 100 Million EAD = 0.082 * 12.5* 100 Mn = 0.0549 * 12.5* 100 Mn = 102 Million = 68.7 Million 2.5 Years 2.5 Years Tenor Capital = RWA * BIS Ratio Capital = RWA * BIS Ratio =102 Mn * 10% =68.7 Mn * 10% 15 % 18.11% Correlation =10.2 Million =6.87 Million 2008 Eric Confidential 1 In this example, the PD, LGD, EAD and loan maturity are the same for both of the corporate loan and retail mortgage. As a result, the EL is the same for both exposures.12. Given the correlation formula, the corporate client has an 18.11% of asset correlation and result in a capital charge of 10.2 million higher than the 6.87 million of mortgage’s. Even though, the correlation difference between this corporate client and mortgage is merely 3.11% but this tiny variation result in a 3.15 million of capital divergence. 12 EL = PD * LGD* EAD
  • 17. 9 Restricted It also implies that Mortgage will faces longer tail risk of uncertainty. Probability of loss 98.65% Mortgage Corporate Case Case Best effort estimation if default 1.35% 1.35% Loss Most likely loss If default = Max loss =100million EAD * LGD =100mn*45% =45 Million Expected loss Unexpected loss Tail Risk Corporate 0.61 89.19 Million 10.2 Million Case Million Expected loss Tail Risk Unexpected loss Mortgage 0.61 92.52 Million Case 6.87 Million Million Total lending amount = 100 Million 2008 Eric Confidential 1 We can further depict the loss assumption under the Basel by using the above chart. The maximum amount of loss is the total principle, in our case it is 100 million. The most likely loss in the event of default is the EAD* LGD, in this case is 45 million. If the loan is still performing, the bank needs to reserve 0.61 million of EL as provision and requires to charge 10.2 million of capital13 for the unexpected loss in this corporate lending example. In oppose to the corporate loan, the mortgage also needs to reserve the same provision, but the capital is far lower. The amount that is not covered by the EL and UL is so called tail risk. The tail risk is a risk that rarely happens but once it does, it will cost you an arm and a leg. We can easy observe that the mortgage asset retains a longer tail than corporate client’s. The recent sub-prime credit lesson is a perfect example to demonstrate the importance of tail risk management. Differences between AIRB capital and Economic Capital There are five major differences between economic capital and regulatory capital, in my point of views: 1. The linkage to bank’s target rating: the AIRB capital is large depends on the BIS ratio, however, the BIS ratio doesn’t links to bank’s desire rating. The determination of economic capital requires bank to identify the confidence interval which direct links to bank’s target rating. 13 Assume BIS =10%
  • 18. 10 Restricted The amount of EC held by a bank reflects the risk appetite of a bank. Illustrative EC links to bank‘s target Probability of rating loss ‘A’ rating ‘AA’ rating Loss Distribution :Confidence of :Confidence of =99.9% =99.97% Better rating requires increased capital holding and Y demonostrates the appetite of a bank X Credit Losses 0 Loss Tail Risk Expected Unexpected loss loss = Economic Capital Regulator Capital is also used to cover unexpected loss.The Basel Comittee uses a general form of formula to proxy the UL 2008 Eric Confidential 1 In the above chart illustrates that determination of economic capital links to the bank’s target rating. The X axis represents for the amount of credit losses. On the other hand, the Y axis stands for the occurrences of the corresponding credit losses. If bank is aiming at ‘A’ rating grade bank, given the riskiness of the current credit portfolio, this bank requires X amount of economic capital. If this bank shooting for a better rating, say ‘AA’, then needs Y amount of economic capital. Better rating grade means bank need to hold more capital for protecting unexpected loss. The economic capital represents for risk governance of a bank. As shown in the chart below: the Winterthur illustrates that their risk exposure in line with risk taking capacity to a confidence of 99.97% over 1 year period. The 99.97% implies that the possibility of not be able to cover the losses is 0.03%. The 0.03% equal to the default probability of AA rating grade’s. This indicates the ‘Risk Appetite’ of Winterthur.
  • 19. 11 Restricted Risk appetite makes explicit how much risk the institution is willing to take. Winterthur The bank’s current available Link to its target rating capital is sufficient to cover 99.7% is equal to ‘AA’ 99.97% ‘s potential unexpected loss. The possibility of loss amount exceeds the current available capital is 0.03% Sources: Credit Suisse analyst day presentation 2006 2008 Eric Confidential 1 2. The consideration of diversification effect, which usually refers to the estimation of customized asset correlation, instead of using the constant correlation suggested by the Basel. Restricted Higher the correlation will have high relationship with the global economics, result in a higher impact to obligor’s business. 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. 2008 Eric Confidential Source : An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2 1
  • 20. 12 Higher the asset correlation of a portfolio, usually result in a higher unexpected loss; vice versa14. Use a constant correlation (or formula to estimate) doesn’t really capture Restricted the correlation nature. Different types of loan assets have different asset correlation. 2008 Eric Confidential 1 Conceptually, as shown in the above, that the large corporations have higher asset correlation than SME and retail banking products. Higher correlation usually results in a higher economic capital requirement under everything being equal. The reason behind this is that: large corporations generally have better rating and usually have larger amount of financing needs. Once default, the credit losses may result in a catastrophe. For example, a bank has a 400 billion on loan exposure- 200 billion lends to one corporate client and a 200 billion credit card portfolio; and with 100 billion capital reserve. Assuming both has 50% of LGD. Once the corporate client defaults the bank capital will be wiped out. Credit loss = 200 billion * 50% =100 billion =total bank capital Even under economic downturn, the credit card portfolio encounters all card holders claim insolvency event is rare. This may explain that the retailing products are considered a more diversified and enjoy a lower correlation. 14 An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2 working paper.
  • 21. 13 3. Estimation of concentration risk : concentration risk refer to a bank’s credit portfolio that concentrated on certain country, industry, rating grade, single name or collaterals. Once the above encounter systematic risk that will cause banks significant losses. In the worse situation, bank may not be able to raise capital and turn out to be insolvent. Credit concentration risk is the largest source of risk to the solvency of a bank. This can occur in the form of the default of a large customer, and causes the simultaneous default of a few sizeable customers, or a downturn in the industry the bank is exposed to. Credit expected loss risk is something that can be priced for in most circumstances, whereas concentration risk is simply too expensive to price and need to be covered by the bank capital. Paradoxically, banks tend to have concentrated exposure to their best customers, and hence underwriting standards alone would not be sufficient to control this form of credit risk. Below chart illustrates that obligor ‘Y’ has larger exposure with low PD. In the event of default, bank will suffer large credit loss from obligor ‘Y’ that may jeopardize bank’s operation. Obligor ‘X’ has higher risk but lower exposure, even bank faces the insolvency of obligor ‘X’, the loss will be covered with the bank’s provision. Restricted Paradoxically, banks tend to have concentrated exposure to their best customers or certain industries that might result in an extreme losses when Illustrative economic downturn. Our view Illustrative – Impact of concentration on portfolio loss distribution • Credit concentration risk is the largest source of risk to the solvency of a bank, and this can occur in the form of the Default probability default of a large customer, the Company X simultaneous default a few sizeable but Or industry weak customers, or a downturn in the High risk and high industry the bank is exposed to exposure • Credit expected loss risk is something that can be priced for in most Name level circumstances, whereas concentration concentration risk is simply too expensive to price for Or Industry in most cases Solvency concentration • Paradoxically, banks tend to have Company Y concentrated exposure to their best Low risk and very high customers, and hence underwriting exposure standards alone would not be sufficient to control this form of credit risk Provision Capital 2008 Eric Confidential 1 A good demonstration is Enron event occurred in 2002. J.P Morgan lost a total of perhaps $0.5 billion in the course of Enron meltdown. Yet, as events unfolded, shareholders lost confidence
  • 22. 14 and share price plummeted to all time low. JPMC’s share price fell from about $40-$15 dollars, destroying about $50 billion of market cap. This gives us a good example that the concentration risk will not only cause bank large credit loss but also result in: Stock price drops Market capitalization decline even more than the credit loss Downgrade by rating agency The recent sub-prime credit crunch is an example 15 . If banks’ revenue relies on certain products, industries or obligors in the recession, the concentration risk will penalize banks a lag and an arm sooner or later. In the worst case, bank may not be able to raise capital from the capital market, due to investors lose confidence on banks’ future. Restricted Usually the low possibility of ‘Large unexpected loses’ not only wiped out the profits but also results in significant market cap decline… – In addition to putting in danger bank’s target credit rating, large losses can erode shareholder confidence – Implications for • JPM Bank lost a total of market perhaps $0.5 billion in the capitalization course of Enron meltdown. Yet, can far exceed as events unfolded, actual losses shareholders lost confidence and share price plummeted to all time low • XX Bank’s share price fell from about $40-$15 dollars, destroying about $50 billion of market cap 2008 Eric Confidential 1 4. The estimation of economic capital requires bank to utilize the simulation skill to simulate the occurrences of loss and its corresponding credit loss amount based on the internal rating system (PD, LGD,EAD). The simulation will simulate different state of economy, under which the obligor may defaults or even results in a joint default event. The simulation result will form the loss distribution of a portfolio. 15 Several American and British banks have claim insolvent. This event also demonstrates these banks have over concentrated on the sub-prime segments.
  • 23. 15 5. Objective is different: The last but not least, the major objective of regulator capital is to assess the capital sufficiency. On the other hand, the regulatory capital is for the measure of capital sufficiency. The regulator needs to have easier way to estimate the capital requirement to supervise the banks and to measure whether if banks have sufficient cushion to sustain against unexpected loss. The economic capital is a better management tool for bank to demonstrate the capital efficiency to their shareholder. For example, under the AIRB capital estimation, bank may need to reserve 30 dollars of capital, after considering the diversification and concentration effect, the economic capital is only 27 dollars; as illustrate in the blow. Restricted The objective of economic capital is to measure the ‘Capital Efficiency’. While as the objective of regulatory capital is ‘Capital sufficiency’. Illustrative Total = 30 Total = 27 5 2 Credit Regulatory Capital Diversification Concentration - Economic Regulatory Bank’s current Effect- Benefit Punishment , due Capital Capital available Requirement from correlation to concentrated on capital certain name , industry… EC is the maximum amount of unexpected losses Regulatory capital is the minimum amount of capital to potentially arising from all sources that could be Definition meet regulator’s request. absorbed while remaining solvent, with a given level of confidence over a given time horizon. Capital efficiency : Shareholder’s interest, measures if Capital sufficiency : regulator’s Objective main concern. capital is utilized efficient. For example, given 100 dollar of exposure, A portfolio consumes 10 dollar of capital while as B only consumes 5 dollar. 2008 Eric Confidential 1 The regulator capital is what we need to comply with. By contrast, the economic capital exhibits the real risk capital given the current portfolio mix and we have enough capital to prevent against unexpected loss – we are a fine bank with high disciplined in risk management. We can further use one example to illustrate the capital efficiency: assuming there are 2 portfolios, each has 100 dollar of exposure but the portfolio components are different16 which result in ‘A’ portfolio has 10 dollar EC while as ‘B’ only consumes 5 dollar of EC. Therefore, we 16 The weights of exposures are different, such as industries, countries, ratings or collaterals.
  • 24. 16 17 find that portfolio ‘A’ has a 10% of capital ratio and portfolio ‘B’ has 5% of capital ratio. We, then, can conclude that portfolio ‘B’ is more efficient than portfolio ‘A’ in terms of use of bank’s capital.18 17 Estimated by divided EC with exposure. 18 We can also measure the capital efficiency by using the regulatory capital, such as AIRB approach. Using economic capital is more accurate in terms of the risk estimation.
  • 25. 17 Section 3: Theory & vendor models introduction In the past decade, four important credit portfolio models have been introduced to measure the credit portfolio risk: 1. KMV’s portfolio manager is the first model introduced in 1993 2. J.P Morgan’s Credit metrics in 1997 3. Credit Suisse First Boston introduced CreditRisk+ in 1997 4. Also in 1997, McKinsey brought out portfolioview in different approach There are other vendors and consulting firms provide solutions in this field. Some banks also developed their in-house models. According to the survey conducted by the IACPM 19 (International Association of Credit Portfolio Managers), 44% of the IACPM members have in-house models 20 . In terms of vendor model, the KMV’s portfolio manager is widely subscribed. Restricted Understand your MODEL and what is the ASSUMPTION before further implementation. Findings from credit portfolio management Comments study in 2004 1. Different models may have different results. Which models are used? • Each vendor model has his own In-house features. developed CSFB Credit Risk+ 6% 44 % • Understand what Credit Metric 6% are you doing when using vender’s Vendor model model. MKMV 42% 66% Banks need to understand Macro Model 2% what methods are you using in estimating EC and gain confidence internally before communicating with regulator and rating agency. 2008 Eric Confidential Source: IACPM -Survey of CPM practices 2004 1 This section provides brief introductions of the vendor models. 19 Source: IACPM -Survey of CPM practices 2004. Wedsite: 20 Some banks even maintain 2 models-vendor model and in-house model.
  • 26. 18 Portfolio Models Introduction In the last decade, a whole range of modeling techniques has been developed to analyze portfolio credit risk. Broadly viewed, there are three groups of portfolio credit risk models. The first group is ’structural’ and based on Merton’s21 model of firm capital structure: individual firms default when their assets’ value fall below the value of their liabilities. Examples of such a microeconomic causal model are CreditMetrics and KMV’s PortfolioManager. The second group consists of econometric factor risk models, like McKinsey’s CreditPortfolioView 22 . McKinsey’s model is basically a logistic model where default risk in ’homogeneous’ subgroups that determined by a macroeconomic index and a number of idiosyncratic factors. These two model types apply similar Monte Carlo simulations to calculate portfolio risk, as both are ’bottom-up’ models that compute default rates at either the individual firm level or at sub-portfolio level. Both thus require a similar kind of aggregation. The third group contains actuarial models, like Credit Suisse’s CreditRisk+23, that make no assumptions with regard to causality. The credit portfolio models usually construct the portfolio loss density in two stages. First, one has to derive the credit risk on the level of individual asset. Second, these risks have to be aggregated to the portfolio level. The first stage needs to have obligor’s PD and rating transition metrics to capture default and migration risk. In the event of default, model estimates the credit loss by applying the LGD. If obligor upgrades or downgrades due to the rating migration instead of insolvent, then estimate the value of loan correspondingly.24 The second stage requires take the correlation into consideration. Usually leverage the factor model to capture the default dependency and estimate the joint probability for all obligors. Most models employ the Monte Carlo simulation technique to derive the portfolio loss. 21 Merton, Robert, (1974), On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance, Vol 29, pp. 449-470. 22 Wilson, Thomas, (1997), Portfolio credit risk (I), Risk,Vol. 10, No. 9. 23 CreditRisk+ - a credit risk management framework,1997. 24 Similar to the bond valuation. When bond being upgraded, the price goes up and visa’ versa. Models apply the Net Present Value and risk neutral methodology to evaluate the value of loan. These models also called ‘Value’ based model.
  • 27. 19 KMV’s Portfolio manager Portfoliomanager is the most comprehensive tool to accomplish a measurement of three objectives 25 : diversification, optimization and valuation, though it is complex in terms of methodology. The estimation of economic capital require user to input the PD, LGD, EAD, tenor, credit spread26 and obligor’s information: weighting of country, industry as minimum requirements.27 In the first step, KMV’s model applies Merton’s concept that debt behaves like a short put option on the value of firm’s asset. In Merton’s world, default occurs when the value of the firm’s asset falls below default point – determined by the structure of the individual firm and its asset volatility. KMV modifies the Motern’s model and assumes that asset values follow a log-normal process with a specific growth rate to calculate the distance to default of obligor. Combing the simulation, KMV then simulate a credit quality transition table (KMV’s used the term : Distance to Default Dynamic) of each obligor instead of leveraging the rating transition metrics that provided by the rating agency. The loan’s value depends on the financial condition of each facility, such as credit spread, upfront fee, tenure and payment type. The individual loan or facility can have a range of possible values at future dates depends on the obligor’s change of default probability. The following chart provides the logic behind the generation of such value distribution in portfolio manager.28 KMV assumes that, at some time in the future or the horizon, the value of firm’s asset will follow a lognormal distribution. Furthermore, individual value for the firm’s assets at the horizon will correspond to values the facilities (loans or bonds). In other words, if the firm’s asset value increased, there is a high chance that the firm’s rating will be upgraded and result in a higher value of the facilities. If the value of the firm’s assets falls below the default point, then the firm will default and the value of the facilities will be the recovery value. 25 Brian Ranson, Credit risk management,2005. page 10-22. Printed by Sheshunoff. 26 The user can choose to use KMV’s EDF as a measure of PD, or use internal rating system. Credit spread is used for loan valuation. 27 The more detail can be found from the PM’s preprocess documentation. 28 More detail can be found in the ‘Modeling Portfolio Risk’ and ‘Credit portfolio management’, Charles Smithson.
  • 28. 20 In the step 1 of the below chart demonstrates how the KMV simulates the firm’s future value and proxies the firm’s credit quality. Based on the credit quality and the credit related spread, the KMV estimates the value of the loan (the step 2). Final step is to map the loan value with probability – the value distribution of a firm’s facilities29. Restricted Log( Asset Value) Log( Asset Value) Step 1 Step 2 Step 3 Probability Value distribution 2008 Eric Confidential 1 To estimate the portfolio value distribution requires to take asset correlation into consideration. KMV decomposes asset returns into systematic and idiosyncratic factor. The asset return is derived from equity prices30 and incorporated into the firm’s liability. The correlation model (so called GCorr ) incorporates 120 factors to capture the macro factor, country factor and industry factor. This model estimates asset correlation among obligors and can translate into default correlation and joint probability of default. Please refer to the following chart. As we can see in the next chart that KMV’s GCorr decomposes the firm’s asset return into country’s, industry’s, sector’s, globe’s and firm’s specific risk. Moreover, user can easy to see the asset and default correlation. It is well known that if higher portion of systematic risk, the easier to be influenced by the macroeconomic fluctuation. The source of the diversification comes from the firm’s specific risk. The GCorr also contains a feature that can estimate 29 An obligor may have multiple credit lines (facilities) in a bank in the same time for different purpose and bank charges different rate for the risk taking activities. 30 KMV collects the stock index around the world to calculate the market value of firm, by adding the liability of firms, KMV can estimate the asset value of firms. Therefore, KMV can utilize these information and to re-contracture a benchmark index and to estimate the asset return of firms.
  • 29. 21 correlation in matrix. Restricted Bank can extend the one factor model into multiple factors model to estimate the correlation. Systematic Risk Country Risk Industry Risk US Electronic UK Manufacturing Taiwan Service Korea Real estate . . • Bank can extend the on factor into multiple factors model Common Firm Specific Countries Industries Risk 14 45 61 rk = ∑ βkf rf + ∑βkcεc +∑βkiεi + εk f =1 c=1 i=1 2008 Eric Confidential 1 Restricted 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 2008 Eric Confidential 1
  • 30. 22 Restricted 2008 Prepared by Eric — CTCB Confidential 1 The portfolio value distribution is calculated simply by summing up all the facility values31. Then, the loss distribution is obtained by using the risk-free rate to calculate the future value of the portfolio at the horizon and subtracting the simulated value of the portfolio at the horizon: Portfolio Loss at Horizon = Expected Future Value of Current Portfolio– Simulated Value of the Portfolio at Horizon The portfolio value distribution and the loss distribution are mirror images of each other. The below chart32 in the next page shows how the two distributions that are related graphically. The value-distribution highlights several reference points and depict the boundary of the portfolio: VMax—Maximum possible value of the portfolio at the horizon (assuming there are no defaults and every obligor upgrades to AAA). VTS—Realized value of the portfolio if there are no defaults and all borrowers migrate to their forward EDF (maintain at the same credit rating grade). VES—Realized value when the portfolio has losses equal to the expected loss (or earns the 31 Considering the correlation effect in this step. For example, all obligors maintain solvency, then A obligor defaults and causes B obligor defaults and so on. Then evaluate the recovery value of the loan. 32 Source, ‘Modeling Portfolio Risk’ and ‘Credit portfolio management’, Charles Smithson. Page 117.
  • 31. 23 expected spread over the risk-free rate)—expected value of the portfolio. VRF—Realized value when credit losses wipe out all the spread income—zero spread value of the portfolio. VBBB—Realized value when the losses would consume the entire portfolio’s capital if it were capitalized to achieve a BBB rating (equivalent to approximately a 15 bps EDF). Restricted Portfolio value distribution and the loss distribution are mirror images of each other. Expected value Value Distribution VES VBBB VRF VTS VMax Expected loss Loss Distribution 2008 Eric Confidential 1 KMV defines two loss distributions. One is based on the expected spread for the portfolio, so the loss is that in excess of the expected loss. The other is based on the total spread to the portfolio, so it is the loss in excess of total spread. Expected loss is expressed as a fraction of the current portfolio value. The economic capital estimation requires user to pre-define the target rating before running the simulation. EC is the difference between unexpected loss and expected loss. This capital value is calculated at each iteration, and is binned and portrayed graphically as the tail of the loss distribution. It answers the question: “Given the risk of the portfolio, what losses should we be prepared to endure?”33 33 KMV also provide portfolio optimization and trade optimization and can provide information which obligor should reduce or increase exposure. We won’t spend time to introduce these functions here.
  • 32. 24 Creditmetrics In April 1997, J.P. Morgan introduced CreditMetrics, a model that first developed by the banking practitioner and sponsored by the KMV and other financial institutions. Like KMV’s model is based on Merton’s concept - debt behaves like a short put on the value of the firm’s assets, the stochastic variable in CreditManager is the value of the firm’s assets. 2 4 1 3 5 Inputs The approach can be explained as following34: First, calculate the different exposure profiles and dynamics for each exposure type with the consideration of the volatility of value due to credit quality migration for each individual exposure.35 The below chart explains the rating change within 1 year. That the BBB grade has an 86.93% will remain at the same grade. 6% of possibility will be upgraded and 0.18% of possibility may be insolvent within 1 year. 34 We summarized the steps into 2 steps to keep it simple and consistent in this paper, although Credit manager has many, many steps. Please refer to the ‘Introduction of CreditMetrics,1997’ for more details. 35 Source, Creditmertics’s technical note.
  • 33. 25 1 Creditmetrics then estimates the value of the loan corresponding to credit quality of the obligor – as shown in the below charts. 1
  • 34. 26 2 3 Second, calculate the volatility of value due to credit quality migration across the entire portfolio and different approaches. The estimation of correlations of credit quality migrations plays the core. Consequently, portfolio effects – the benefits of diversification and costs of concentrations – can be properly quantified36. 36 Refer to the ‘4’, ‘5’.
  • 35. 27 4 Similar to KMV’s correlation approach, the major differences are illustrated below: Moody’s–KMV Model CreditMetrics Group’s Model 37 Asset value driven Equity index proxy Default correlation derives Systematic risk based on Default correlation derives Default correlation derives from asset correlation from correlation in the proxy (equity returns) Assuming obligor ‘s return is related to 2 systematic factors described as below : rA= uA+wA1f1+wA2f2+εA where : f1 ~ N(0,σ21) , f2 ~ N(0,σ22), εA ~ N(0,σ2A) σ1 ,σ2 is the standard deviation of factor 1 ,2 and σA is the firm-specific risk standard deviation. The correlation between two obligors A and B’s returns are given by: 37 Equity correlation would be a perfect proxy if the value of the debt remained fixed. The RiskMetrics Group argues that the approximation is good as long as the firm’s volatility is primarily driven by equity fluctuations, and the volatility of the debt level is relatively small in comparison to the equity fluctuations.
  • 36. 28 w A1 wB1σ 12 + w A2 wB 2σ 2 + ( w A1 wB 2 + w A 2 wB1 )σ 1σ 2 ρ ( f1 , f 2 ) 2 ρ (rA , rB ) = σr σr A B Where σrA=(wA12σ21+ wA22σ22+σ2A)1/2 and similarly for obligor B. What is left is to determine the relationship between the correlation of the returns and the default event correlation for two obligors. We see that correlation depends both on the weights on the factors and on the correlation between the factors38. 4 Utilized the correlation model, Creditmetrics measures the joint migration probability based on the asset correlation, as exhibited above. Next step is to apply the Merton’s model to estimate the firm’s credit rating based on the joint migration. 4 38 General assumption is the correlation between the factors is zero.
  • 37. 29 The process reiterates many times and then can generate the value distribution of the portfolio. 5
  • 38. 30 The differences between the Creditmetrics and KMV are as follows: 1. Rating migration : User of KMV can choose to use the Distant to Default Dynamic or applies the rating agency’s rating transition matrix, whiles as the credit metrics only provide rating agency’s rating transition matrix to measure the rating migration likelihood. 2. Correlation : KMV constructs index by their own and estimate the asset return. The creditmetrics utilize the equity index39 to estimate the asset correlation. A core assumption of both of the KMV and CreditMetrics models is the multivariate normality of the latent variables. The asset return correlations are calibrated by assuming that asset returns follow a factor model, where the underlying factors are interpreted as a set of macro-economic variables40, such as country, industry etc. At each simulation, the return will determine if the obligor’s credit quality (credit migration) and then estimate the loan value of the obligor. The result is a value distribution of the portfolio. Restricted KMV and Creditmatrics run numerous simulations to generate a ‘Value Distribution’ and then translate it into loss distribution. For Each Simulation Draw Calculate Recover Correlat individual Sum Calculate y Rate in ed asset Obligor’s asset across Save the Obligor’s Simulati the case value value return, r, obligors to result new rating on of returns given index get (loss given asset #3436 default for each returns and portfolio #3436) return and index ∆I idiosyncratic losses calculate component loss Portfolio Value Distribution Tabulated Simulation Results ∆Portfolio Simulation # Company A Company B Value 1 Default; loss = 100 No Default 100 2 No default No Default 0 3 Default, loss = 60 Default, loss = 50 110 . . . . . . . . . . . . . . . . 2008 Eric Confidential 1 39 MSCI global index or local equity index. User of creditmetrics needs to estimate the correlation in the implementation stage, while as KMV embedded the GCorr into the Portfoliomanager already. 40 For further reading see papers by Koyluoglu and Hickman (1998), Gordy (2000) and Crouhy, Galai, and Mark (2000).
  • 39. 31 CreditRisk+ Creditrisk+ is an actuarial model also called a reduced form model. Economic causality is ignored—there is no “story” to explain default. Consequently, specific asset values and leverage details for specific firms are irrelevant. Actuarial models specify a distribution for the default rate and apply statistics to obtain a closed form expression for the joint distribution of loss events. By extension the expected severity can be incorporated to arrive at a distribution of losses. An attractive feature of Credit Risk+ is that it requires only limited data. There are three required inputs: 1. PD of the obligor. 2. Volatility of default rate for the obligor—the model is very sensitive to this parameter; and this parameter is difficult to accurately measure and seldom can be provided by the user. 3. Facility exposure (amount at risk net of recovery)—Credit Risk+ takes the loss given default as fixed. The user inputs a net figure taking into account usage at default (for committed lines) and the amount of recovery. Unlike the Moody’s–KMV model and the RiskMetrics Group’s model, there is no simulation of how much is actually lost when default occurs for the recovery rate. Given the data input, it doesn’t require the calculation of individual default risk, and neither does it look at changes in market value in the even of upgrade or downgrade- as oppose to the KMV and creditmetrics. In this model, correlation is handled through the aggregation of like assets sorted by industry and country or the termed sector. Sectors allow users to influence the degree of default correlation between obligors: Specific risk sector—Placing some of an obligor’s risk in the specific risk sector means that that risk can be fully diversified away. Systematic sectors (maximum nine)—Within each sector, the default rates of each obligor are correlated. Across the sectors, the default rates are independent. For the loss aggregation, Credit Risk+ assumes that the default rate may vary, thus introducing the concept of “default rate volatility” for each obligor. Because this implies an underlying distribution for the average default rate, the developers made the assumption that the mean
  • 40. 32 default rate (for each sector) is governed by a gamma distribution. Though there is no upper bound for the gamma distribution, this is permissible because the default rate for a sector can be greater than one, as opposed to a default probability. The actuarial approach has the appearance of precision because results are calculated via mathematical model rather than a simulation; however, just the opposite is true. Actuarial models are closed form approximations to the true distribution of defaults. However, Credit Risk+ is subject to at least two criticizes: 1. It is possible for a credit to default more than once. 2. The approximation used to calculate the portfolio distribution from the individual loss distributions relies on default rates being small. This means, for example, that noninvestment grade credits of longer. Creditportfolioview The first widely discussed macrofactor model was introduced by McKinsey & Company and was called CreditPortfolioView.41 The model starts not with the individual obligors’ information but rather with the view about the situation of the economy. The major difference between is the structured model ( KMV, CreditMetrics) estimates risk parameters (for example, PD and migration probabilities) as un-conditional, the Creditportfolioview’s approach focuses on conditional modeling conditional on the state of economy. The model requires the country risk of the economy, the industry risk within the economy and the rating of the obligor in order to predict default. To gauge the diversification, this model segments the portfolio based on the number of firms in the segment index. The systematic (non-diversifiable) risk – risk of economy – has a large predictable impact on credit migration and on default probabilities. That implies that we should use conditional rather than unconditional models! As an example Wilson42 presents the following table, in which he has found a linear regression model forecasting defaults and having the highest explaining power and only one explanatory variable. 41 Designed by the Thomas Wilson 42 Wilson, T. (1997a) Measuring and Managing Credit Portfolio Risk: Part I: Modelling Systematic Default Risk. The Journal of Lending and Credit Risk Management, July, 61 – 72. Wilson, T. (1997b) Measuring and Managing Credit Portfolio Risk: Part II: Portfolio Loss Distributions. The Journal of Lending and Credit Risk Management, August , 67 – 78.
  • 41. 33 Country R Square Explanatory variable Belgium 96.9 % Unemployment rate France 89.1% GDP growth Germany 95.7% Long-term interest rate Spain 95.3% Foreign exchange rate Swiss 89.3% Public disbursement U.K 88.1% Aggregate savings rate The above table tells us that the default rates, as also migration probabilities are not constant in time, and they are largely explained by changes in general economic conditions – by the systematic risk or the state of the economy. The second point is that systematic risk is not explained by one explanatory variable, but instead by several factors. Wilson shows the following as an example of need of a multi-factor model: Total systematic risk explained by risk factors (% of total) Explained by the United States United Kingdom Germany 1’st risk factor 23.9 % 56.2 % 66.8 % + the 2’nd factor 60.1 % 62.1 % 74.0 % + the 3’rd factor 92.6 % 82.1 % 90.7 % The third important points is, that different economic sectors react differently to cyclical variations of the economy. The table below offers another example. Default prediction models, Germany, only the best explanatory variable included Industry R-Square Explanatory variable Agriculture 85.4 % Unemployment rates Banking/insurance 91.1 % Credit to the economy Construction 91.8 % GDP growth Energy 28.2 % Long term interest rates Process 95.9 % Foreign exchange rate Services 94.1 % Aggregate savings rate The historical default rates for industry/country combinations are described as a function of
  • 42. 34 macroeconomic variables specified by the user. For example, the default rate for German construction firms could be modeled as a function of different macroeconomic “factors.” (Probability of Default)German Construction =f(GDP growth, Unemployment rate…) CreditPortfolioView captures the fundamental intuition that economy-wide defaults rise and fall with macroeconomic conditions. It also captures the concept of serial correlation in default rates over time. Given the data and the specification of the relation between macrovariables and default/transition probabilities, the McKinsey model can calculate timevarying default and transition matrices that are unique to individual industries and/or countries. The following chart illustrates the Mckinsey’s idea on construct loss distribution43. Restricted McKinsey’s portfolio view estimation structure 1.Determine the state 2. Determine segment probability of default 3. Determine loss distributions 2008 Eric Confidential 1 1. Determine the state: For a given period, the first step is to determine the health of the macroeconomy. Assuming there are three possible states of the economy can occur under portfolio manager’s educated guess: 1. Expansion : with GDP growth of +1 % 2. Average year: with GDP growth of 0 % 3. Recession: with GDP growth of -1 % Each of these states can occur with equal probability (33.33 %). 43 Source,Thomas C. Wilson ,FRBNY Economic Policy Review / October 1998.
  • 43. 35 2. Determine segment probability of default: Translate the economic assumptions into conditional probabilities of default 44 for each customer segment based on the estimated relationships described earlier. Assume there are two counterparty segments: 1. “Low-beta” segment: with probability of default reacts less strongly to macroeconomic fluctuations (with a range of 2.50 % to 4.71 %) 2. “High-beta” segment: has a strong relationship with the macroeconomic fluctuations (with a range of 0.75 % to 5.25 %). 3. Determine loss distributions: Then Mckinsey tabulates the loss distribution for portfolios that are constant over their life, cannot be liquidated with known recovery rates, including both diversified and non-diversified positions. Next step is to relax each of the assumptions within the framework of this model in order to estimate more accurately the expected losses and risk capital from credit events. Numerical example from Thomas C. Wilson: 1.Determine state State GDP Probability of default Expansion +1 33.33% Average 0 33.33% Recession -1 33.33% 2.Determine State Low-Beta probability High-Beta segment probability of default A probability of of default default B Expansion 2.5% 0.75% Average 2.97% 3.45% Recession 4.71% 5.25% 3.Determine loss distribution The conditional loss distribution is based on the independent draws result from states of the economy: 1. A defaults 2. B defaults 3. A+B defaults 44 The explanation of conditional and unconditional PD will be introduced in the next section.
  • 44. 36 4. no one defaults Unfortunately, CreditPortfolioView specifies only the functional form of the model. It does not provide guidance on the correct macrovariables or estimated weights for the industry/country segment. Furthermore, given a functional form, it is unlikely that the available data would be sufficient to estimate the needed model parameters except in the most liquid market segments of developed countries.
  • 45. 37 Section 4: Methodology of simple portfolio model development In this section, we will first discuss some observations on the methodology of vendor portfolio models. Then develop a simplified method to estimate the economic capital and explain on how to use spreadsheet to develop this portfolio model. Unconditional and conditional PD Before comparing the vendor models, we need to clarify a concept that is widely used by the above models and is seldom understood by the most of the credit risk practitioners – the unconditional and conditional PD An essential concept in any risk-factor model is the distinction between unconditional and conditional event probabilities45. An obligor’s: Unconditional default probability: also known as its PD or expected default frequency which is the probability of default before some horizon ( for example,1 year) given all information currently observable. In Basel’s term the through the cycle PD. Conditional default probability: is the PD we would assign the obligor if we also knew what the realized value of the systematic risk factors at the horizon would be. The point in time PD. Let’s take the following example, consider a simple credit cycle in which the systematic risk factor takes only three values. The “bad state” corresponds to a recession at the risk horizon, the “good state” to an expansion, and the “neutral state” to ordinary times. Say that the three states occur with probabilities of 1/4, 1/4 and 1/2 (respectively) at the risk horizon. Consider an obligor which defaults with probability 2% in the event of a bad state, probability 1% in the neutral state, and probability 0.4% in the event of a good state. The “conditional default probability” is then 0.4%, 1%, or 2%, depending on which horizon state 45 More detail please refer to: Michael B. Gordy, Board of Governors of the Federal Reserve System ,A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules,Oct 22,2002,
  • 46. 38 we condition upon. The PD is the probability-weighted average default rate, or 1.1%. Restricted Conditional PD reflects the state of economy. State of Description Occurrence Probability economy of default 2% Bad a recession 1/4 2% at the risk horizon Good expansion 1/4 0.4% Weighted average= 1.1% 1% Neutral ordinary 1/2 1% times 0.40% Weighted 1.1% average Bad Good Neutral 2008 Eric Confidential 1 The Basel committee made a conclusion that46 : “In a sense, all models are conditional: they seek to incorporate some current information about the credit quality of each borrower and credit facility. That being said, it is nevertheless possible to distinguish between unconditional models that reflect relatively limited borrower or facility-specific information, and conditional models that also attempt to incorporate information on the state of the economy, such as levels and trends in domestic and international employment, inflation, stock prices and interest rates, and even indicators of the financial health of particular sectors.” In other words, the portfolio models introduced in the previous section utilize the through the cycle PD, which is the average value of the conditional default probability across all possible realizations of the systematic risk factors as default threshold. Take the credit migration of obligors into consideration to proxy the conditional PDs under different economic situation. The portfolio models usually follow the following steps: 46 April , 1999, Basel Committee : Credit risk modeling: Current practices and applications. Page 28.
  • 47. 39 Calculate obligor’s default threshold by using the obligor’s through the cycle PD. Estimate the conditional credit quality of obligors by considering the credit migration (upgrade, down grade, default or maintain at the same rating) 1. In default stage: apply LGD to estimate the loss. 2. In non-default stage: calculating the loan value. Model of firm value and migration Credit migration Value of BBB firm at horizon date Joint default event: apply the asset correlation model to gauge the joint default event47- the conditional PD considering the default effect result from other obligors. The asset correlation used by the vendors’ are considered as a proxy of the state of the economy. KMV and Creditmetrics use the multi-factor model, the Mckinsey uses the econometrics method and the Creditrisk+ applies the actuarial method. All roads lead to the Rome and guide to a principle that the default events are not independent, there is a lot of portions are driven by the systematic factors (the macroeconomic condition and industrial cycle). Loss based or value based Another observation from the vendor models are they can divided by two groups: loss based and value based. Loss based: do not consider the loan value, such as McKinsey’s portfolioview and CSFB’s CreditRisk+. In this method, the view on the loss is taken a simplified view because they disregard the coupon or interest income may receive from the obligor. Loss based method only consider the default event of obligors. Value based: Value based methods are generally mark to model , due to lack of market 47 The historical observation of joint default co-movement is either difficult or less accurate.
  • 48. 40 price for loans. In this method, the loan value will be appreciated or depreciated according to the obligor’s credit migration. In addition, the credit spread and credit related fee incomes are also taken into consideration. For example: Consider a credit line with 100 dollar of exposure and 1 dollar of interest income per month with 1 year maturity. The maximum loss a bank can face is the obligor claims default in the first day of the loan agreement. The bank will lose 100 dollar. It is also possible that the bank receives the 11 months of interest income and the obligor claims insolvent in the 12th month. The bank’s net lost is 89 (100-1*11). In this way, the value based needs to modeling the timing of obligor’s default instead of simply considering the default event. This adds more challenge other than modeling the obligor’s credit migration48. In this document, I apply loss-based based method for the portfolio model as the rudimentary attempt, because of loss based is easy to implement and more intuitive. Correlation Model It is difficult to observe joint-default events when obligors remain financial healthy. Observing the historical default data is the most direct way49, however, the historical experiences rarely can represent for the future state. I utilize the one-factor model method for the estimation of asset correlation50. The one factor method is widely quoted and adopted by the Basel committee51 52 . Conceptually, this method borrows a single systematic risk factor to estimate the asset correlation across obligors. The asset values are assumed to be normally distributed with correlations that go back to a single common factor53. Formally, borrower i’s asset value Ai depends on the common factor Z and an idiosyncratic factor εi: 48 As previous introduces, an upgrade or downgrade may have impacts on the bond’s value. 49 Several papers utilize the Rating’s observed historical default rate by rating grade and estimate the default correlation by rating. For example, Akira Ieda, Kohei Marumo, and Toshinao Yoshiba,2000.A Simplified Method for Calculating the Credit Risk of Lending Portfolios,Monetary and economic studies. 50 Gordy, M., 2000, A comparative anatomy of credit risk models, Journal of Banking and Finance 24, 119–149.Frey, R. and McNeil, A.J., 2003, Dependent defaults in models of portfolio credit risk,Journal of Risk 6, 59–92 51 Basel committee on banking supervision, 1999b. A new capital adequacy framework. No.50 June 52 More detail factor model for the estimation can be found in the, Alfred,Thilo, Daniel ‘Credit risk factor modeling and the Basel 2 IRB approach’, Deutsche Bundebank.2003. 53 More detail description, please refer to the appendix: interpreting the IRB capital formula.
  • 49. 41 Ai = Wi Z + 1 − Wi 2 ε i , cov(ε i , ε j ) = 0, i ≠ j; cov(Z , ε i ) = 0, ∀ i where the market index Z and the idiosyncratic terms εi are independent standard normal random variables. The parameter Wi represents the obligor’s dependence to the market index. The Wi can be generated from an OLS regression by utilizing the Equity return of obligor i, for example TSMC, with Taiwan’s Stock index. RTSMC = α + β * StockIndex + ε The correlation between TSMC and stock index: ρTSMC,Stock requires the β from the Index regression, market return volatility σStock index and obligor’s volatilityσTSMC. Recall that: σ TSMC ,StockIndex σ TSMC ,StockIndex β= ρ TSMC ,StockIndex = , σ TSMCσ StockIndex σ StockIndex 2 Therefore, we estimate the correlation by solving: σ StockIndex σ TSMC ,StockIndex σ StockIndex σ TSMC ,StockIndex ρ TSMC ,StockIndex = β = = σ TSMC σ TSMC σ StockIndexσ TSMC σ StockIndex 2 The notation of ρTSMC,StockIndex here, is corresponding to the W2i 54 . We can further apply the same concept into measuring the industry correlation and apply it for the non-public listed firms. For example: RSemi−Conductor = α + β * StockIndex + ε W2i Industry Semi-Conductor 50% Real-estate 50% ……. ..% The correlation between 2 obligors is completely determined by the factor sensitivities W: cov(Wi Z + 1 − Wi 2 ε i ,W j Z + 1 − W j2 ε j ) cov( Ai , A j ) ρ = = asset σ ( Ai )σ ( A j ) ij 1 *1 = cov(Wi Z ,W j Z ) = WiW j var(Z ) = WiW j 54 Please refer to appendix (correlation estimation) for more detail explanation.
  • 50. 42 That is to say the default co-movement is modeled by the weights of systematic factor among obligors and the correlation between asset moves for obligors i and j is equal to Wi*Wj . To obtain the joint probability of default, we need to first determine the default threshold: PDi = P( Ai < α i ) = N (α i ) where N denotes the normal cumulative density function. α i = N −1 ( PDi ) where N() denotes the cumulative standard normal distribution function. The joint default probability is PDi , j = Pr ob( Ai < α i , A j < α j ) = N 2 ( PDi , PD j , ρ ij ) asset where N2 denotes the bi-variate standard normal distribution function with correlation. I don’t estimate the correlation in this document, because of the purpose is to develop a portfolio model. The correlation estimation is not an easy task and requires time and resource to develop. However, I suggest several ways when estimating the economic capital in this portfolio model: 1. Utilize the Basel 2 correlation equation or suggested correlation figures. Apply a 20% of constant asset correlation55 for the corporate obligors that proposed in 2. the first draft of Basel II. Recent study done by RMA56 suggests a correlation of 2%~ 5% for non-mortgage 3. products and a range from 5% ~10% for the mortgage product. Model Design Methodology Both of KMV and CreditMetrics are based on the Merton’s theory and are structured method, we considered to apply the structured method because it is widely accepted and easier to apply and link to our internal rating model- the 1-year PD of obligor. 55 Here means the Wi2 . 56 Retail credit economic capital estimation-best practices, RMA,2003.
  • 51. 43 Log of market Probability value distribution of asset Expected asset values over time Default point The structured portfolio model doesn’t mean that we need to model the default probability by using the concept of Merton. Instead we apply the concept into our portfolio model development. The Merton suggests that the default probability of an obligor can be described as: ⎡ ln L − ln At − ( µ − σ 2 / 2)(T − t ) ⎤ Pr ob( Default ) = Φ ⎢ ⎥ σ T −t ⎣ ⎦ ⎡ ln( L / At ) − ( µ − σ 2 / 2)(T − t ) ⎤ = Φ⎢ ⎥ σ T −t ⎣ ⎦ ln At + ( µ − σ 2 / 2)(T − t ) − ln L DD = σ T −t Prob(Default)=N[-DD] =N (PD Internal Rating) A borrower defaults if and only if its asset return falls below a threshold value. Having the default threshold, we can apply the one factor correlation model that described in the previous to estimate the conditional PD and then simulate the portfolio loss.
  • 52. 44 Model of firm values and its default threshold Default Probability threshold of density Default Scenario Asset return within one year : Estimated by using estimated by applying the one the firm’s PD factor model The portfolio model is breakdown as the below steps: Step 1 : Estimating the unconditional default threshold The first step is to produce default threshold. The αi defines the unconditional default threshold which is derived from the obligor’s 1 year PD57. Note that all obligors in the same PD band have the same unconditional probability of default58.The unconditional default probability of obligor i is given by PDi = P( Ai < α i ) = N (α i ) and α i = N −1 ( PDi ) where N denotes the normal cumulative density function. Step 2 : Applying one factor correlation model to simulate the Conditional default Consider a 1 year horizon. In this horizon model a scenario corresponds to a state of the economy. At the end of the horizon, the scenario is defined by the systemic factors, the credit drivers, which influence the credit worthiness of the obligors in the credit portfolio. The conditional probability of default is then the probability that the credit worthiness index falls below the threshold in a given scenario. In the asset value approach, the standard way is to run a Monte Carlo simulation. Recalled the single factor model59: 57 The same definition under Basel 2 (the through the cycle). 58 For example, all the AAA obligors have the same probability of default within 1 year horizon. The information is from an internal model or from an external agency. 59 this is the same model underlying the regulatory proposal of the Basel committee (Basel Committee on
  • 53. 45 Ai = Wi Z + 1 − Wi 2 ε i , cov(ε i , ε j ) = 0, i ≠ j; cov(Z , ε i ) = 0, ∀ i Where the Z represents for the systematic factor and εi stands for the idiosyncratic (firm’s specific risk factor), both are independent. The idiosyncratic factor deserves their name because they are independent across firms; the distribution of the index is standard normal. Moreover, the Wi2 Is the asset correlation and the Wi stands for the weight of the asset correlation. The conditional default threshold is the default probability conditional on the systematic factor Z: a function of the idiosyncratic component of obligor i and the credit worthiness index, then we have Pi ( Z ) = P( Ai < α i | Z ) = P(Wi Z + 1 − Wi 2 ε i < α i | Z ) = P(Wi Z + 1 − Wi 2 ε i < N −1 ( PDi )) (ε i ) −1 The conditional PD of obligor i can be described as = Wi Z + 1 − Wi * N 2 Therefore, we called obligor default if Wi Z + 1 − Wi 2 * N −1 (ε i ) < N −1 ( PDi ) The conditional probability is obtained by simulating the single factor factors. In the Merton model, default occurs when the assets of the firm fall below a given boundary or threshold, generally given by its liabilities. We consider that an obligor defaults when its credit worthiness index, Ai, falls below the obligor’s unconditional probability of default (the threshold) . Step 3 : Obligor exposures and recoveries in a scenario With the asset values and the default threshold in hand, we can decide whether a loan defaulted in the scenario or not. If it defaulted, the associated loss is LGD × EAD. Step 4 Aggregation of losses in all scenarios By summing up the individual loan losses we then can obtain the portfolio loss. The aggregation of losses is generally obtained by performing a Monte Carlo simulation on Banking Supervision 2001,Wilde 2001).
  • 54. 46 the risk factor returns (the Wi). Further considering the number of occurrences and its associated loss amount, we can derive the portfolio loss distribution.
  • 55. 47 Section 5: Model manual instruction In this section, we will describe how to apply the portfolio model into Excel spreadsheet. Data requirement The data required for this model are: 1. PD of an obligor : bank’s internal rating model 2. Average LGD of an obligor: estimate the average LGD of an obligor. If one obligor has more than one facility, user needs to calculate the average LGD before running the model. 3. EAD of an obligor : the total EAD of an obligor. Model screenshot Restricted Confidence interval # of Simulation Simulation Systematic results Credit VaR factor Z Data input Weighting of Default This area allow user area correlation threshold to observe the joint default event. Conditional PD threshold Limited to 255 columns though. Estimate Loss if obligor default : LGD * EAD Call default if Conditional PD < Default threshold 2008 Prepared by Eric — CTCB Confidential 1
  • 56. 48 Example Step 1 : Unconditional default threshold estimation In this example all obligors have the same PD, LGD, EAD and correlation. 1. PD: 10% 2. LGD: 50% 3. EAD: 100 Correlation weight(Wi)60: 50% 4. Weights of Default Loan PD LGD EAD Correlation EL Threshold 1 10% 50% 100 50% 5 -1.28 2 10% 50% 100 50% 5 -1.28 3 10% 50% 100 50% 5 -1.28 4 10% 50% 100 50% 5 -1.28 5 10% 50% 100 50% 5 -1.28 6 10% 50% 100 50% 5 -1.28 7 10% 50% 100 50% 5 -1.28 8 10% 50% 100 50% 5 -1.28 9 10% 50% 100 50% 5 -1.28 10 10% 50% 100 50% 5 -1.28 Recall the unconditional default probability of obligor i is given by PDi = P( Ai < α i ) = Φ (α i ) Where N denotes the normal cumulative density function. α i = N −1 ( PDi ) In Excel, we can utilize the statistical function. Using the example here: =NORMSINV(PD) =NORMSINV(10%)=-1.28 ; the default threshold for the 10% of probability of default is -1.28. Step 2 : Apply single factor correlation model to simulate the Conditional default: Apply the one-factor model to estimate the conditional PD of the obligor. 60 In this example it means wi =50%. Moreover, the I apply the KMV’s suggestion that 25% of R square for corporate clients. The 50% is the square root of R square.
  • 57. 49 Recall the one factor model: Ai = Wi Z + 1 − Wi 2 ε i , cov(ε i , ε j ) = 0, i ≠ j; cov(Z , ε i ) = 0, ∀ i Conditional PD = Wi * Z + √(1- W2i) * εi Utilizing the EXCEL function, we can transform the above formula into the following: 1. Systematic factor Z: is a random number that we utilize to proxy the state of the economic. NORMSINV(RAND()) 2. Idiosyncratic factor (firm’s specific factor) : εi is also a random number and can be described as NORMSINV(RAND()). Weight of correlation: utilize the constant 50% of correlation for all6162. 3. Conditional PD = Wi * NORMSINV(RAND()) + √(1- W2i) * NORMSINV(RAND()) 4. =50% * NORMSINV(RAND())+√(1- 50%2i)* NORMSINV(RAND()) 5. If conditional PD < default threshold, call obligor default Default Conditional Default Threshold PD Loss Factor Z (1:default) -1.28 -2.07 50 -1.0458 1 -1.28 -1.27 0 Portfolio loss 0 -1.28 -0.01 0 100 0 -1.28 -0.15 0 Count default 0 -1.28 -0.59 0 2 0 -1.28 0.67 0 0 -1.28 -0.32 0 0 -1.28 -1.74 50 1 -1.28 -0.20 0 0 -1.28 -0.43 0 0 In this scenario, we find the conditional PD below the default threshold, therefore we call obligor default. Recall the below chart that describes the Merton’s concept that firm faces the risk of insolvency once a firm’s asset falls below a certain threshold (Distance to default). The 61 50% of weight is equal to the 25% of asset correlation. Recall the weight of correlation = wi, while as 2 2 the asset correlation:wi , therefore,50% =25%. 62 Let me emphasize again, this paper is not aiming at estimating the correlation for corporate obligors nor does for the retailing products. I assume the constant correlation in this example. User can either apply the Basel’s asset correlation, vendor’s or other empirical study results from academic paper. Conducting customized correlation will require time, resource and knowledge and suggest treating it separately.
  • 58. 50 driver that determines that firm’s asset value is firm’s asset return which is a function of systematic factor and firm’s specific factor. Restricted Applying the single factor correlation model to simulate the Conditional default. Model of firm value and its default threshold Probability of density Default threshold Default area -2.07 -1.74 -1.28 -0.01 0 Estimated by using the firm’s Asset return within one year: PD Estimated by applying the one factor model α i = N −1 ( PDi ) = N −1 (10%) = −1.28 through simulation 2008 Eric Confidential 1 Step 3: Estimate the loss of an Obligor in a scenario Loss of an obligor in the event of default = 100 * 50% =50 Step 4 Aggregation of losses in all scenarios All the simulation results will be pasted into the simulation results area.
  • 59. 51 Restricted 2008 Eric Confidential 1 Step 5. Determine the Economic capital The economic capital is linked to a bank’s target rating. In this example, we target at ‘A’ grade- VaR loss at 99.9% of confidence interval and after the EL of the portfolio. 1. apply the EXCEL function: Percentile (portfolio loss, 99.9%) to estimate the VaR. Confidence Percentile 90% 150.0 95% 200.0 Simulated VaR 99% 250.0 loss 99.9% 300.1 distribution 99.97% 370.0 99.99% 390.0 2. Estimate the EL : PD* LGD*EAD =50 3. EC = VaR- EL = 325-50= 275 Joint default observation This simple model allows user to observe the joint default event in the simulation result area, although with a limitation.
  • 60. 52 Restricted Obligor 1,2,3,4,5 defaulted in the same time User can observe the joint default event However, it is constrained by the Excel’s limitation 2008 Eric Confidential 1 Charts The spreadsheet allows user to draw chart and to diagnose the joint default event and loss distribution. Restricted 2008 Prepared by Eric — CTCB Confidential 1
  • 61. 53 Restricted Under a 50% of correlation, there are a max ‘55’ segments/ obligors will have a joint default event (0.1% of probability). The EC=2,200. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 10% 9% 8.3% 9% Target Rating =A 8% Cumulative 8% 7% probability =99.9% 7% 61% 6% 6% 2.4% of 5% There is a 0.1% of possibility result 5% possibility that 55 in a ‘900’ 4% segments or potential loss 4% obligors will default within 1 year 0.1% of possibility together within 1 3% 3% the loss will exceed year 2,200 2% 2% 1% 1% 0.1% 0% 0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 1950 2100 2400 2650 0% 55 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 48 53 $ of Loss Amount 500 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 2,700-500 =2,200 = Economic Capital 2008 Eric Confidential 1 There are 2 charts can be used to diagnose the portfolio risk from this model. The first one is to observe the joint default event (please to the left hand side of the above chart). In this chart, the Y axis represents for the frequency of joint default event; while as the X axis stands for the number of obligor that might default together. We can observe that the probability of no obligor default within the 1 year horizon is 8.3%. There is chance, though very low possibility: less than 0.1%, that there 55 obligors will default together within one year. Moreover, users can draw a loss distribution from the outcome. As we can see from the right hand side of the above chart that the Y axis describes the frequency of loss; on the other hand, the X axis is the amount of credit loss. From the chart, reader can find that the EL based provision can cover 61% of credit loss within one year. In other words, there is a 39% of possibility that the credit loss amount might exceed the EL based provision. If the bank targets at single A rating grade, the bank requires to reserve 2,200 of economic capital to protect against the unexpected loss. The chance of the loss exceeds the 2,200 is merely 0.1%.
  • 62. 54 Section 6: Applications of in-house credit portfolio model Although this is a simplified portfolio model, user still can utilize this model to better understand the effect of internal rating system on the portfolio unexpected loss. Moreover, users can leverage this model into portfolio stress testing by changing the risk parameters or correlation. In this section, I’ll brief illustrate how the change of the parameters may have profound impact on the shape of loss distribution. Effect of correlation First of all, we extend the size of the portfolio into 100 obligors; each has the same PD, LGD and EAD. Risk parameters Portfolio statistics PD=10% Portfolio EAD= 10,000 LGD=50% EL =500 EAD=100 Target rating =A Assuming the correlation weight(Wi) for all obligors is 25% correlated to the systematic factor. We’ll find that: 1. Worst joint default event will have a 33 obligors default together within one year; however, the probability is lower than 0.1%. 2. There is a high possibility, 8.2%, that there are 8 obligors might claim insolvent in the next 12 months. 3. The possibility of no default within 1 year is 0.8%, in this example. 4. If bank reserve the provision based on the EL, there is a 61% of possibility that the credit losses will within the EL based provision. 5. The VaR at the 99.9% of confidence interval is 1,400. Translate into the EC = VaR- EL=1,400-500=900 6. The target rating ‘A’ or the 99.9% of confidence interval implies that the chance of the credit losses exceed 1,400 is only 0.1%; the 0.1% also is the probability of default of A rating.
  • 63. 55 Please refer to the chart below. Restricted Under a 25% of correlation weight, there are a max ‘33’ segments/ obligors will have a joint default event (0.1% of probability). The EC=900. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 10% 9% 8.2% 9% Target Rating =A 8% Cumulative 8% 7% probability =99.9% 7% 3.5% of 6% possibility result 6% in a ‘800’ There is a 0.9% of potential loss 5% possibility that 24 5% within 1 year segments or 4% obligors will default 4% together within 1 0.1% of possibility 3% year 3% the loss will exceed 2,000 2% 2% 0.9% 1% 0.8% 1% 0% 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1700 0% 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 $ of Loss Amount 500 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 1,400-500 =900 = Economic Capital 2008 Eric Confidential 1
  • 64. 56 Now, assume the correlation weight (Wi) for all obligors is 10%. We’ll find that: 1. Worst joint default event will have a 26 obligors default together within one year; however, the probability is lower than 0.1%. 2. The joint default distribution exhibits that there is a highest chance, 13.1% of possibility, that there are 9 obligors will claim insolvent within one year. 3. There is a 58% of possibility that the credit losses will within the EL based provision. 4. The VaR at the 99.9% of confidence interval is 1,100. Translate into the EC = VaR- EL=1,100-500=600 Refer to the below chart. Restricted Under a 10% of correlation weight, there are a max ‘26’ segments/ obligors will have a joint default event (0.1% of probability). The EC=600. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 14% 14% 13.1% Target Rating =A 12% 12% Cumulative probability 10% =99.9% 10% 8% 8% 6% There is a 0.1% of 6% possibility that 26 0.1% of possibility segments or 4% the loss will exceed obligors will default 4% 1,100 together within 1 year 2% 2% 0.1% 0% 50 150 250 350 450 550 650 750 850 950 1050 1300 1500 0% 1 3 5 7 9 11 13 15 17 19 21 26 $ of Loss Amount 500 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 1,100-500 =600 = Economic Capital 2008 Eric Confidential 1
  • 65. 57 If assuming the correlation weight (Wi) for all obligors is 0%. We’ll find that: 1. Worst joint default event will have a 21 obligors default together within one year; however, the probability is lower than 0.1%. 2. There is a 58% of possibility that the credit losses will within the EL based provision. 3. The VaR at the 99.9% of confidence interval is 1,000. Translate into the EC = VaR- EL=1,000-500=500 Refer to the below chart. Restricted Under a 0% of correlation, there are a max ‘21’ segments/ obligors will have a joint default event (0.1% of probability). The EC=500. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 16% 16% Target Rating =A 13.5% 14% 14% Cumulative probability 12% 12% =99.9% 10% 10% 8% 8% There is a 0.1% of possibility that 21 6% segments or 6% 0.1% of possibility obligors will default the loss will exceed together within 1 4% 950 year 4% 2% 2% 0.1% 0% 100 200 300 400 500 600 700 800 900 1000 1000 0% 2 4 6 8 10 12 14 16 18 20 $ of Loss Amount 500 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 1,000- 500 =500 = Economic Capital 2008 Eric Confidential 1
  • 66. 58 So far, we have test the correlation weight (Wi) from 25% to 0%, one interesting question is what the loss distribution will be if the correlation for all obligors is 100%: 1. In the 100% of correlation among the obligor, there is only 2 outcomes: 0 default within 1 year or one insolvent event causes chain effect and result in all obligors default. 2. The chances of all obligors go default will close to the probability of default of the portfolio, in our case demonstrated here: close to 10%. 3. There is no VaR can be estimated, since one default event will cause all portfolio failure, which means either there is no loss or the credit losses will be the EAD * LGD. The EC = EAD * LGD- EL=5,000-500=4,500 Refer to the below chart. Restricted Under a 100% of correlation, it is all or nothing, and require EC=4,500. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 100% 100% 90% 90% 80% 80% 70% 70% There is a 12.5% of possibility that 60% 60% all segments or obligors will default 50% 50% together within 1 year 40% 40% 30% 30% 20% 10.23% 20% 10.23% 10% 10% 0% 0% 0 5000 0 100 $ of Loss Amount # of Joint Defaulted Segment or Obligors Unexpected Loss = 5,000- 500 =4,500 = Economic Note: Assume EAD=100 for all obligors Capital 2008 Eric Confidential Based on portfolio simulation model 1
  • 67. 59 In conclusion, we list the results of the above scenarios to demonstrate the importance feature of correlation – lower the correlation, lower the economic capital. Corr Corr Corr Corr weight weight weight weight =25% =10% =0% =100% # of joint default obligors in the worst case 33 26 21 100 Economic capital 900 600 500 4,500 $ of max credit loss 1,650 1,300 1,050 5,000 63 Capitalization rate = EC / EAD 9% 6% 5% 45% Effect of concentration Concentration risk means that a small number of segments, industries or names have a significant amount of exposure in a portfolio. We utilize the same example and assume that one of the obligor accounts for significant share of the total exposure. Risk parameters Portfolio statistics PD=10% Portfolio EAD= 10,000 LGD=50% EL =500 EAD= 90 for most of the obligors Target rating =A Only 1 client has significant exposure: 1,090 Correlation weight is set at 25% for all and account for 10.9% of total. obligors 1. Worst joint default event will have a 34 obligors default together within one year; however, the probability is lower than 0.1%. 2. There is a highest possibility, 9.2%, that there are 9 obligors will claim insolvent within one year. 3. The possibility of no default within 1 year is 0.12%, in this example. 4. The VaR at the 99.9% of confidence interval is 1,850. Translate into the EC = VaR- EL=1,850-500=1,350 higher than the previous diversified scenario, 900. 5. The reason of higher EC is resulting from the single name concentration. One obligor 63 Exposure of this portfolio is 10,000
  • 68. 60 accounts for 10.9% of total exposure. Therefore, if this obligor defaults, his credit loss is 12 times64 of other obligor. Not to mention the joint default event may result in other obligors’ insolvent or vice versa. Restricted The concentration effect shows that, there are a max ‘34’ segments/ obligors will have a joint default event (0.1% of probability). The EC=1,350. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 10% 10% 9.19% 9% 9% Target Rating =A Cumulative 8% 8% probability =99.9% 7% 7% 6% 6% 5% 5% 4% 4% 3% 3% 2% 2% 1% 1% 0% 0 180 360 540 675 765 855 945 1035 1125 1220 1350 1490 1670 0% 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 34 $ of Loss Amount 500 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 1,850-500 =1,350 = Economic Capital 2008 Eric Confidential 1 Effect of PD In this scenario, we will illustrate the effect of PD- the lower the PD of a portfolio, the lower the unexpected loss. The obligors of the portfolio are the same, 100 obligors; each has the same PD, LGD, EAD. But the PD is lower. Risk parameters Portfolio statistics PD= 5% Portfolio EAD= 10,000 LGD=50% EL =250 EAD=100 Target rating =A 1. Worst joint default event decreases from 33 to 23 obligors. 2. The possibility that the EL can cover most of the credit loss within one year is 60%. 3. The possibility of no default within 1 year is 3.2%, in this example. 64 Concentrated obligor has 1,090 of exposure while as the rest has a 90 of exposure.
  • 69. 61 4. The VaR at the 99.9% of confidence interval is 1,000. Translate into the EC = VaR- EL=1,000-250=750 lower than the previous scenario (PD=10%), 900. 5. The effect of PD is significant, as the EC declines 16.7%, from 900 to 750. Restricted Under a 25% of correlation and lower the PD result in a lower EC=750, there are a max ‘23’ segments/ obligors will occur a joint default event. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 16% 16% 11.5% Target Rating =A 14% 14% Cumulative probability 12% 12% =99.9% 10% 10% 3.2% of possibility result 8% There is a 0.1% of 8% in a ‘500’ possibility that 23 potential loss segments or 6% within 1 year 6% obligors will default together within 1 4% year 4% 2% 2% 0.1% 0% 0 100 200 300 400 500 600 700 800 900 1150 0% 0 2 4 6 8 10 12 14 16 18 23 $ of Loss Amount 250 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 1,000-250 = 750 = Economic Capital 2008 Eric Confidential 1 Effect of LGD In this scenario, we will exhibit the effect of LGD- the better recovery of a portfolio, the lower the unexpected loss. The obligors of the portfolio are the same, 100 obligors; each has the same PD, LGD and EAD. But the LGD is lower. Risk parameters Portfolio statistics PD= 10% Portfolio EAD= 10,000 LGD=25% EL =250 EAD=100 Target rating =A 1. The EL can cover 60% of the credit loss occurrence within one year. 2. The lower LGD doesn’t have significant effect on the joint default event but reflect on the unexpected loss. 3. The VaR at the 99.9% of confidence interval is 800. Translate into the EC = VaR-
  • 70. 62 EL=800-250=550 lower than the previous scenario (LGD=50%), 900. 4. The effect of PD is significant, as the EC declines 38%, from 900 to 550. Restricted Under a 25% of correlation and lower the LGD(25%) result in a lower EC=550, there are a max ‘32’ segments/ obligors will occur a joint default event. Illustrative Joint Default Distribution Loss Distribution Frequency of Loss Frequency of Joint Default 12% 10% 9% Target Rating =A 10% Cumulative 8% probability =99.9% 7% 8% 6% There is a 0.1% of 6% 5% 2% of possibility that 32 possibility result segments or 4% in a ‘500’ obligors will default 4% potential loss together within 1 within 1 year 3% year 2% 2% 1% 0.1% 0% 0% 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 $ of Loss Amount 250 # of Joint Defaulted EL Segment or Obligors Unexpected Loss = 800-250 = 550 = Economic Capital 2008 Eric Confidential 1 In sum, the change of risk parameters will have vital influence on the unexpected loss. The below table summarize the effects65. Based case Concentration PD =5% and LGD (PD=10%, scenario rest =25% LGD=50%, assumptions EAD=100, Corr are same as weight =25%) based case # of joint default 33 34 23 32 obligors in the worst case Economic capital 900 1,350 750 550 $ of max credit loss 1,650 2,120 1,150 1,000 Capitalization rate = 9% 13.5% 7.5% 5.5% EC / EAD 65 Note here, the simulation results even under the same assumption may have different outcomes.
  • 71. 63 As we can observed from the results of the above table that: 1. In the Lower PD scenario, the EC decreased by 16%, from 900 to 750. 2. In the lower LGD scenario, the EC decreased by 38%, from 900 to 550. 3. In the concentration scenario, we found that the EC will increase significantly, from 900 to 1,350, increased by 50%. There is no impact on the joint default event. 4. In the above scenarios, the portfolios have same exposure; however, the risk parameters are different. One has a single name concentration risk, one portfolio has a lower PD than the based case and one has lower LGD. These differences result in a wide range of EC, from 550 to 1,350. In conclusion, the size of a portfolio doesn’t matter, what matter is the risk parameters of the portfolio which will decide the economic capital.
  • 72. 64 Section 7: Future improvements As the old saying said that ‘No one is perfect, only God is perfect’. This simple model has its limitations and drawbacks. The more sophisticated portfolio models consider: 1. Important sampling to facilitate the simulation time 2. Random LGDs that takes the variation of LGD into account 3. Rating migration that allows one to incorporate the changes in credit quality along with the effects on the market value of the facilities or instruments in the portfolio (also called mark to market approach). Here, I‘d like to discuss two aspects for the future improvement. One is the use of the constant correlation; the other one is the estimation of risk contribution of each obligor. Correlation As I have mentioned earlier that this document is aim at modeling the portfolio loss as the first step of estimating economic capital. The estimation methods of the correlation won’t be discussed here. However, I’d like to touch the following issues regarding the correlation: 1. Constant correlation 2. Multi-factors model 1. Constant correlation To use of the constant correlation doesn’t stand for use the same correlation figure for all obligors and all retailing products. Instead, it represents for the use of the correlation from the empirical study. I proposed to use the suggested correlations either from the Basel committee or study conducted by vendor or RMA66. 66 Retail Credit Economic Capital Estimation -Best Practices, Page 19; RMA − the Risk Management Association. Feb,2003.
  • 73. 65 Basel Vendor RMA Corporate clients Asset Correlation : General asset 3~30% correlation for the corporations : 25%67 The range is from 12% to 24%. 7%~10%68 Mortgage 15% 5%~10% Revolving 4% Credit card 2%~5% 2%~3.2% Auto loan 2%~5% 2%~5% Consumer loan 3%~5% 3%~30% 2 Multi-factors The one-factor model that we used is widely used in practice and also can be extended to multi-factors, as long as one can find enough data set and meaningful relationship between obligor’s asset return with independent variables69. Ai = W1 Z1 + W2 Z 2 + ...Wn Z n + 1 − (W12 + W22 + ...W 2 n ) * ε i where Z1, . . . , Zn are independent systematic risk factors, each having an N(0, 1) (standard normal) distribution εi is an idiosyncratic risk associated with the Ai obligor, also N(0, 1) distributed; W1, . . . , Wn are the factor loadings for the ith obligor. The underlying factors Zi are sometimes derived from economic variables, such as inflation risk, GDP change or interest rates. Risk contribution This model provided here allows estimating the VaR of a credit portfolio. It is aim at estimate the credit loss at the portfolio level. The risk contribution of each obligor is not able to be estimated by this model. The risk contribution means to estimate the ‘capital usage’ of a obligor, it is used to gauge the share of the risk contributed to the total economic capital. There are 67 KMV utilizes the GCorr, the 25% of asset correlation (R2) for non-public listed firm. 68 Include prime and sub prime mortgage. 69 For further detail, recommend to read the ‘Credit Risk Factor Modeling and the Basel II IRB Approach’, Deutsche Bundesbank, 2003.
  • 74. 66 70 many articles discussed this topic . Below is brief introduction of the estimation risk contribution. Risk contribution can be defined in more than one way, the most common definition is the estimation of marginal standard deviation—the amount of variation that a particular asset (call it “A”) adds to the portfolio. If the asset has a weight of wA, then its risk contribution is defined by: ∂σ p RC A = w A ∂w A The fraction on the right-hand side after “wA” contains the “partial derivative” symbol (“∂ ”). All this means is that (in the fraction) we are calculating the change in the portfolio standard deviation with respect to a very small change in the weight (i.e., dollar amount or number of shares) of the particular asset. Another way of saying this is that this derivative tells us the sensitivity of the portfolio standard deviation with respect to asset A. Calculating RCA in this way is equivalent to adding up all the elements of the row (or column, since it is symmetric) corresponding to asset “A” of the covariance matrix we introduced earlier and dividing this sum by the portfolio standard deviation. This confirms that the sum of all the risk contributions (RCi) is equal to the portfolio standard deviation—that is, N σ p = ∑ RCi i =1 In short, one may need to redevelop a model for estimating the risk contribution71. 70 One of the most cited is Glasserman, Paul, ‘Measuring Marginal Risk Contributions in Credit Portfolios’. Journal of Computational Finance, Vol. 9, No. 2, Spring 2006 71 Refer the interested reader to the literature : 22. Paul Glasserman, Measuring marginal risk contributions in credit portfolios. Journal of computational finance, Winter 2005/06.
  • 75. 67 Section 8: Economic capital as management applications The “Economic Capital” has been evolved into many banking management applications. Three key management applications from my opinions: Risk governance External communication Internal management Restricted Although there are many applications of Economic Capital developed in the past decade, I consider the key management applications are as follows: Key applications The most common applications of economic Risk Governance : Determine risk appetite of 1 capital are as following : a bank – Demonstrate the desire rating of a bank which 1 shows the risk appetite… Risk Governance : – …and therefore, determine the amount of Economic Determine risk appetite of Capital that required to hold for protecting a bank ‘Unexpected loss’ . Internal Capital Adequacy Assessment 2 2 3 Process : Pillar 2 compliance Internal – Utilize the EC to compare with the regulatory Capital Adequacy Communi- capital and to illustrate the sufficiency of solvency. cate with Access- menet Communicate with Rating agency 3 Rating Process agency – Capital ratio doesn’t have a direct link to bank’s – Pillar 2 rating. complian – Rating agency welcome more information to ce demonstrate the transparency of the risk. Performance & investors communication 4 4 – RAPM :To gauge the return over the risk of line of business Internal Performance Management & – Use a basis of capital allocation and strategic decision process. investor communication – Limit Setting / Pricing 2008 Eric Confidential 1 Risk governance The determination of risk appetite of a bank is an action of risk governance. To estimate the economic capital requires banks to identify their target rating which unveils banks’ risk appetite. To assess the amount of economic capital to be held by bank need to know the desire rating of a bank. The below chart exhibits the loss distribution of a credit portfolio. If bank is aiming at being a single ‘A’ rating bank, then bank require covering a potential loss at 99.9% of confidence interval. In this case, bank needs to hold ‘X’ amount of economic capital. Better rating comes with cost. If bank is targeting at double A rating, then bank requires to hold more
  • 76. 68 economic capital to protect against unexpected loss. A typical loss distribution of a loan portfolio can be divided by three parts: 1. Expected loss: expected loss is the cost of doing business and usually has a close to 50% of loss occurrence. 2. Economic capital: economic capital link to bank’s target rating and serves as a cushion to protect from unexpected loss. 3. Tail risk: the final part is the tail risk or so called extreme loss that has insignificant possibility of occurrence. This probability of this occurrence is difficult to estimate. However, if bank adopts economic capital as capital reserve, then one can view the possibility might be 1 minus confidence interval that will equal to bank’s target rating.72 Restricted 1 Risk Governance of a bank : Determine risk appetite of a bank The amount of EC held by a bank reflects the risk appetite of a bank. Illustrative EC links to bank‘s target Probability of rating loss ‘A’ rating ‘AA’ rating Loss Distribution :Confidence of :Confidence of =99.9% =99.97% Better rating requires increased capital holding and Y demonostrates the appetite of a bank X Credit Losses 0 Loss Tail Risk Expected Unexpected loss loss = Economic Capital Regulator Capital is also used to cover unexpected loss.The Basel Comittee uses a general form of formula to proxy the UL 2008 Eric Confidential 1 It has been a common practice in many banks to unveil the risk appetite and as part of risk governance. Let’s elaborate it by leverage the following information. The Winterthur, a subsidiary of Credit Suisse, claims that her risk exposures in line with risk taking capacity to a confidence of 99.97% over one year period. The 99.97% means the Winterthur will hold an amount of capital that can cover losses at 99.97%. The possibility of loss amount exceeds this amount of capital is only 0.03%; which links to the target rating: AA. 72 If bank is aiming at single A rating, the PD is 0.1% over one year. This probability equal to the cumulative probability of the area that exceeds the expected loss and economic capital in a loss distribution.
  • 77. 69 The Winterthur illustrated her available capital (The orange color in chart) on hands and amount of economic capital based on 99.97% of confidence level (shown in pink color). Risk appetite makes explicit on how much risk a financial institution is willing to take. The process of risk appetite identification will enforce bankers to contemplate upon their business strategy and revisit the resource allocation – is considered a risk governance. Restricted Risk appetite makes explicit how much risk the institution is willing to take. Winterthur The bank’s current available Link to its target rating capital is sufficient to cover 99.7% is equal to ‘AA’ 99.97% ‘s potential unexpected loss. The possibility of loss amount exceeds the current available capital is 0.03% Sources: Credit Suisse analyst day presentation 2006 2008 Eric Confidential 1 External communication Many banks have been utilizing the economic capital to communicate with equity investors, rating agencies and regulators for many years. Internal capital adequacy assessment process As we all knew that the regulatory capital’s primary purpose is for regulatory minimum, not reflect on bank’s current loan portfolio, business combition, strategies and capital planning. That’s why in the pillar II73 of the new Basel Accord suggests bank to develop a process for assessing their overall capital adequacy in relation to their risk profile and a strategy for 73 Principle 1 and 2 under Pillar II .
  • 78. 70 maintaining their capital levels. It also recommends supervisors should review and evaluate banks’ internal capital adequacy assessments and strategies, as well as their ability to monitor and ensure their compliance with regulatory capital ratios. The capital adequacy can be proved by comparing economic capital with bank available capital. Again, we use Credit Suisse as an example. In the annual report, this financial institution demonstrates the different types of risk, by organizations and most recent three years. The important message delivered here is that the economic capital is not a simple mathematic addition. It shows the diversification benefit across the different risks and compare with bank available capital. Restricted 2 Internal Capital Adequacy Assessment Process Capital adequacy is typically shown from a regulatory perspective and by comparing EC with available book equity. Illustrative Example of Credit Suisse Breakdown of economic capital by risk type, showing diversification benefit Historical comparison shows like-for-like evolution Sources: Credit Suisse annual report 2005 2008 Eric Confidential 1 Communicate with rating agency The common fallacy is that higher capital ratio may lead to better rating grade. In a study conducted by the Oliver Wyman that proves that historically, capital ratios have been inversely correlated to the rating. The below chart investigates the rating information and tier 1 capital ratios of banks’. We can find that some banks with A+ rating have higher tier 1 ratio than some AAA rated banks. This reveals that the rating agencies have more considerations in assign rating except for the capital ratio.
  • 79. 71 Agency do reward economic capital disclosure, in a special report published by the Moodys’ 74 encourage banks to disclosure the economic capital, stress testing and earning volatility information as risk transparency. Restricted 3 Communicate with Rating agency Rating agencies welcome EC disclosure. Higher capital ratio doesn’t necessary lead to a better rating. Tier 1 capital ratio by S&P rating Observation Capital ratio is one of the rating criteria. Higher ratio • Capital ratios are only one of multiple doesn’t necessary rated a better grades. factors taken into consideration by rating agencies Tier 1 ratio Some A+ rating • Agencies reward EC disclosure : 13% banks have higher Moody’s* has revised their rating tier 1 ratio than 12% methodology and welcome the disclosure AAA’s of economic capital, stress testing and 11% earning volatility. 10% • Historically, capital ratios have been 9% inversely correlated to ratings – with 8% highly rated banks being more leveraged than lower rated banks. Upper quartile 7% • Factors that rating agency considers Average 6% important : Wider range of 5% Lower quartile Tier 1 Ratio 1. Extensive franchise 4% 2. Diversified business mix AAA AA+ AA AA- A+ A A- BBB+ BBB 3. Strong risk management Source :Mercer Oliver Wyman analysis based on a sample of 160 banks in 2003 obtained from Bankscope 4. Stable, predictable earnings * Moody’s Special Comment: Risk Disclosures of Banks and 2008 Eric Confidential Securities Firms, May 2006 1 Investor communication The incorporation of both of expected and unexpected loss information into the performance measurement extends the conventional accounting metrics. Many banks have adopted the so called RAPM (Risk adjusted performance management) into their daily business management and disclosure it to investors. 74 Moody’s Special Comment: Risk Disclosures of Banks and Securities Firms, May 2006.
  • 80. 72 Restricted The disclosure of RAROC to investors and to demonstrate the return of taking risk has become common. Moreover, it is an important part of future business planning . Return on economic capital and return on Demonstrate target income growth rate and invested capital shown by business segment RAROC.. Allows investors to distinguish between different balance growth and return and as business risk-return profiles planning Target Income growth rate Hurdle rate or cost of capital Note: Citigroup ‘s investor’s presentation. 2008 Prepared by Eric 1 In this example, Citibgroup disclosed the RAROC information (in Citigroup‘s term: net income over risk capital) by each business group, to allow investors to explore the performance of risk taking. Moreover, Citigroup exhibited the cost of equity is around 10% to 12%. This figure is an internal benchmark to measure the performance of each group and also serves as a minimum return for Citigroiup’s investors. Internal management Performance metrics There are many management applications of economic capital. The most common one would be utilized the economic capital into performance measurement (as have shown in the previous chart) and embedded related applications firmly within their businesses, helping provide clarity to decision making. Furthermore, some have linked the economic capital into the reward system.
  • 81. 73 Restricted 4 Performance & investors communication RAROC and Economic Profit extend the traditional ROE measure by incorporating risk. RAROC Economic Profit Interest Income Interest Income – Funds transfer price – Funds transfer price + Non-Interest Income + Non-Interest Income – Operating Expenses – Operating Expenses – Expected Loss* – Expected Loss Economic Capital Attributed in Risk-adjusted profit Risk-adjusted profit relation to risk – EC Economic Capital Hurdle Rate Hurdle Rate Bank’s minimum = RAROC (%) = EP ($) Return on Capital EC = Economic Capital *Risk-Adjusted Return on Capital * : Some banks have implemented credit transfer pricing and utilize the CDS to hedge the credit risk, in this case, the EL is equal to the CDS price. If the hedge activity is fully hedged through CDS or similar instruments, then the EC will be zero. 2008 Eric Confidential 1 Under the risk adjusted performance measurement framework, the practitioners use the expected loss to replace the actual loss or provision to measure the risk adjusted profit. The estimation of RAROC (Risk Adjusted Return on Capital) leverages the economic capital as denominator to calculate the return on risk investment. RAROC is an important performance metrics for assessing whether if the risk return is justified and compare the RAROC with the benchmark- hurdle rate or some use the cost of capital. Economic capital further subtracts the capital cost which is a cost by multiplying the economic capital usage and the hurdle rate. Economic profit (some uses the term Shareholder Value Added) reflects the size of the value contribution.
  • 82. 74 Restricted RAROC is an important performance measurement for assessing if the risk / return is justified. On the other hand, economic profit reflects the size of the value contribution. Illustrative Risk Adjusted Return On Capital Economic Profit •RAROC estimates the return over the invested •One advantage of economic profit or SVA over RAROC capital and gives picture if the return exceed bank’s is that EP reflects the size of the value contribution. hurdle rate (Chinatrust bank uses cost of capital). •Some transactions or segments may have large EP •Some transactions or segments may have higher transactions and businesses even though they may not RAROC but limited investment opportunities. have the highest RAROC values. Not the highest RAROC RAROC but contributed the most of the EP A A Cost of Cap 10% 100% of Capital 80% 80 % of Cap 20 % of Cap Total EP of Cap Note : some uses the phrase SVA(Shareholder Value Added) instead of EP. EP = (RAROC – Cost of capital) * Invested capital 2008 Eric Confidential 1 By comparing these two performance metrics, banks will have a clearer picture regarding the value contribution of each business unit, products or customer segmentations. On the left hand side of the above chart demonstrates relationship of the RAROC and the capital usage. We can see that the ‘A’ industry generates a return above the cost of capital (or hurdle rate) and consumed 10% of total credit economic capital. There are some industries which consumed 20% of capital but not be able to create return exceed the cost of capital. For these industries, managers may like to dig into the reasons before taking actions. The right hand side of the chart illustrates the economic profit of each industry. The positive economic profit comes from the industries that have a RAROC higher than the cost of capital. Moreover, we can find that though the ‘A’ industry is not the highest RAROC generator, this industry does contribute the most of the economic capital. The advantage of economic profit over the RAROC is that the economic profit reflects the size of value added. We can conclude that if two transactions have the same RAROC (assume return exceeds the hurdle rate), then we should look for the one that can generate the higher economic profit if resource is limited. On the other hand, if two transactions have the same economic profit, then we should look for the transaction that has the higher RAROC.
  • 83. 75 Capital allocation The utilization of risk adjusted performance metrics also can be applied to capital allocation decision. The allocation process requires both of the return and capital information. One may draw a scatter chart that incorporates the RAROC and economic capital information by the dimension. The below chart exhibits that the capital usage has been reverse correlated with the return- higher the RAROC lower the capital usage, even though all the segments have generate a return exceeds the hurdle rate. With this situation, the management team may begin to consider whether if to invest more resource into the high RAROC segments and constraint the capital consumption for some capital heavy users. The financial controller or credit risk manager may conduct scenario analysis by utilizing the economic capital model to run through different situation to see if the risk return can reach optimization75. Restricted The management team will have a better picture on how to re-allocate the limited capital through analyzing the relationship between RAROC and capital consumption. Illustrative = Industry or Line of Business Low Observation High Capital usage •Usually the capital heavy users High y = -9E-11x + 0.5702 don’t outperform the peers. R2 = 0.6437 •Higher the capital usage lower the RAROC, though the RAROC still exceed the cost of capital (hurdle Contributed majority of Risk- rate) adjusted profit Action can be taken : RAROC •Invest in high RAROC segments Average RAROC of and increase return on the capital portfolio heavy users. Challenge may face •Limited opportunities in high RAROC segments Cost of Capital •Strong bargaining power in the Low capital heavy users. Average allocated capital 2008 Eric Confidential 1 75 For example, managers can estimate the economic capital by assuming different scenarios in which the exposure breakdown by rating grades or industry are different. Some vendors’ models are able to estimate the economic capital by incorporating the credit related revenue and have optimization function,.
  • 84. 76 Limit setting The development of limit setting will force bank to revisit his ‘risk appetite’ and can prevent major adverse surprises. Limits are typically applied where default correlations are high or stressed events may cause a ‘catastrophic loss’. Therefore, to leverage the economic capital is a better choice than the other risk metrics, such as exposure or regulatory capital. Because the economic capital limits can not only take into account portfolio correlation effects, but also can utilize to derive management team’s appetite of loss.
  • 85. 77 Section 9: Beyond economic capital Estimating the economic capital by applying sophisticated portfolio model enables bank to explore the potential unexpected loss of current portfolio. However, knowing the risk doesn’t prevent bank from the risk. To prevent from the catastrophe, bank needs to react to the risk. Revisit the commercial banking business model Commercial banking business model has been remaining the same since many centuries. Bank as an intermediary and make tiny profit margin by collecting deposit and providing loans that were held to maturity. The loan business usually focuses on bi-lateral relationships and banker cross-sells other products to deepen relationship with clients. However, lending consumes bank ‘Capital’ and the regulatory legislation watches BIS ratio and NPL. This model has been under pressures especially in the economic recession. Restricted Traditional commercial banking model relies on lending as ‘Entry product’ and aims at cross selling to deeper relationship with clients. Conceptual Facing the same down-side risk but the up-side is totally Traditional model of commercial banking different that makes the stock investment – an attractive investment. Obligor Return Stock Deposit Loan With Bank possibility of ‘high’ Different compensation return Cross-sell profit Banker as the intermediary – collecting deposit and providing Credit income loans that were held to maturity – With high Loan consumed ‘Capital’ 0 possibility of – Regulatory legislation ‘constant’ return watch BIS ratio and NPL – Bank focuses on bi-lateral But faces the relationships same down side loss with low - 100 % possibility This model has been under pressure for the past 40 years Facing the same loss 2008 Eric Confidential 1 If we think in other way, considering that loan origination as an investment and compare the loan with the equity investment. We will find that the loan business is a “loser’s game” and the equity investment is a “winner’s game”. The reason is that both of the loans and equities face the same downside risk - both may lose the principle value in the worst scenario. However, the
  • 86. 78 upside return is totally different. Banks only can receive a pre-determined coupon or interest revenue. On the other hand, equity may generate double or triple return. The argument here doesn’t imply or suggest commercial banking should follow the investment banking’s model, but it tells the truth of the unfairness. Lending embeds a lot of uncertainties and the credit quality is highly related to the economic cycle. Restricted The credit crisis tends to be a cyclic event… …even though, we are still in the 1st decade of the 21th century… Too many Examples of losses by financial institutions No downgrades $ Millions banks triggered by these Loss size* losses 1,025 826 Too many losses 531 451 260 247 U.S. AmEx – Fleet JP BONY – Capital Sub-Prime Crisis Bancorp – high Boston – Morgan telecom One – airline yield Argentina Chase – loans and subprime exposure bonds bonds Enron bonds lending after 9/11 2001 2002 2007 Loss/ book 7% 7% 3% 1% 4% 7% equity** * Pretax ** Assuming book equity in 2003 Source: Literature searches; 2008 Eric Confidential 1 If we count how many credit crises do we have since year 2000, you will find there are at least 3 crises and losses are tremendous. Moreover, there is no downgrade triggered by these losses and the early warming mechanism that banks developed are malfunctioned. All these facts and history explain why the shape or the outcome of the credit portfolio is asymmetric: limited upside and unlimited downside. As depict in the below chart, the maximum gains or return a loan portfolio may generates is a scenario that there is no loss and bank can successful receive the net interest revenue, and this may not be the general case. The ordinary situation a bank may face is: loan portfolio has expected loss need to be taken into account. In addition, a small number of bad loans may wipe bank’s capital out. As shown in the chart76, the long tail of the credit portfolio is several times of the portfolio gains77. No matter from the history or through the sophisticated portfolio 76 Source: Creditmetrics technical document, 1997. 77 The upside gains of a credit portfolio is under a situation that there is no default and bank well-received the net interst income from the obligors.
  • 87. 79 model, it tells us that the credit crunch and crisis are not ’Once in a blue moon’. Credit portfolio management as a business model The only remedy, from my viewpoint, for banks is to active manage credit portfolio management given the severe challenges banks face. Restricted Active credit portfolio management is a new business model – it suggests to well utilize the risk mitigation tools instead of abusing it. Illustrative Active credit portfolio management approach Traditional credit management •Loan Credit Origination selling Management Origination Relationship •CDS Relationship Monitor Active Management •CDS Primary Management Workout Primary Secon- Credit Index dary Portfolio Market Market •CLO Manage- Market ment •Insuranc Credit e Credit Approval / Approval / •Portfolio Rating Rating Trading Source : ERisk, 2008 Eric Confidential 1 The traditional credit risk management focuses on the loan underwriting and risk measurement. Credit officer investigates the quality of the obligor and gauge the riskiness based on the rating
  • 88. 80 tool. The post-lending management a bank can have is to monitoring and works out the loan in the event of default. In the active credit portfolio management approach, bank analyzes the portfolio loss based on the economic capital to identify the concentration risk and the miss-pricing obligors or segments. The action taking to mitigate the concentration is to either conducting loan-sell, buying protection or securitizing the loan asset. To manage the miss-pricing obligors or segments, bank may consider the following actions: 1. Rising the price based on the risk 2. Cross-selling other fee-based product to increase return 3. Asking for higher recovery rate collaterals to reduce the risk 4. Shortening the loan maturity 5. Reducing the credit line The active portfolio management doesn’t symbolize cash out-flow activities. To achieve the portfolio optimization, sophisticated portfolio managers utilizing the credit market to sell protection or invest in credit to enhance the portfolio return and to diversify the portfolio. Restricted Portfolio Improvement is achieved by both reweighing existing exposure holdings and by hedging unwanted risk. Risk-Return Optimization Conceptual Efficient Frontier Same Risk Portfolio The efficient frontier for the portfolio is calculated Current Portfolio by optimizing within the portfolio. The frontier can be moved by expanding the portfolio assets to Same Return Portfolio include diversifying exposures. Optimal Sharpe Ratio By introducing new exposures that diversify the portfolio risk-return tradeoff will improve, allowing for the construction of optimal portfolios . 2008 Eric Confidential 1 The efficient frontier for the portfolio is calculated by optimizing within the portfolio. The frontier can be moved by expanding the portfolio assets to include diversifying exposures or by reducing the miss-pricing assets. In another word, bank can improve credit portfolio by both reweighing existing exposure holdings through hedging unwanted risk or investing the credit
  • 89. 81 assets in other countries via credit instruments. Restricted Portfolio theory suggests us that we can enhance portfolio performance by re-weighting, and banks finally can adopt the theory into practice. Conceptual Portfolio Performance Enhancement 12% Hedge : Reduce risk / uncertainty 1 -Tool : Buy CDS, Loan Sell, Insurance, Securitization Efficient Frontier 10% Enhance Yield : Invest in High Yield 2 Return given the same ‘risk’ 4 8% -Tool : Secondary loan, Sell 2 CDS, CDS Index, securitization. 3 Current Portfolio 1 6% 3 Swap Asset Synthetics : Reduce risk and utilize the 4 4% free up capital to invest in credit. 5% 15% 25% 35% Risk 2008 Eric Confidential 1
  • 90. 82 Section 10: Helping the CEO’s sleep quality “There are many methods for predicting the future. For example, you can read horoscopes, tea leaves, tarot cards, or crystal balls. Collectively these are known as quot;nutty methods.quot; Or you can put well-researched facts into sophisticated computer models, more commonly referred to as a complete waste of time.” --Scott Adams, The Dilbert Future. Unlike the manufacturing or beverage companies whom sell goods to clients, the goods that commercial banks have are loans. Loans, in nature, have predetermined coupon rate and with uncertainty of credit risk. Although bank’s loan / credit portfolio has very low incidence of problems. The credit losses may be significant once the ‘blue moon’ becomes reality; and this may cause the CEOs of banks insomnious. The key to good credit risk management lies in the construction of the portfolio, not in the individual credit decisions that made up the portfolio. The CEO must be properly informed on that risk (both of the expected and unexpected loss) and return of portfolio and serve as a guide to its construction. The better way, in my opinion, is to estimate the economic capital of portfolio that takes both of the diversification and concentration effects into consideration. The regulatory capital is there need to be complied with, not contained any future information. There is no doubt, although there is denial, which banks are very risky businesses, but after all, risk management is at the very heart of banking business. The objective of risk management is not to minimize the credit loss, but to achieve risk and return balanced. The objective of bankers should not be outperformed the peers in one time, but to be outperformed over time or over the economic cycle. Look at the earnings multiples that the equity markets give to commercial banks. They are far below those of most other industries most of the time. In large part, investors suspect the current high earnings of the banking industry may be a prelude to unpleasant years with high credit losses. Achieving excellence in risk management will take time and some significantly investments, but the prize may be very significant: a market multiple will make all the shareholders very happy and make the CEOs good sleep at night.
  • 91. 83 Reference 1. Akira Ieda, Kohei Marumo, and Toshinao Yoshiba,2000. A Simplified Method for Calculating the Credit Risk of Lending Portfolios,Monetary and economic studies. 2. Alfred Hamerle,Thilo LiebigDaniel R sch, 2003. Credit Risk Factor Modeling and the Basel II IRB Approach,Discussion Paper ,Series 2: Banking and Financial Supervision. 3. Basel Committee on Banking Supervision,2004.An Explanatory Note on the Basel II IRB Risk Weight Functions. 4. Basle Committee on Banking Supervision,1999.CREDIT Risk modeling: Current practices and applications. 5. Brian Ranson, Credit risk management,2005. 6. Charles Smithson,2003 Credit portfolio management. 7. Christopher C. Finger,The One-Factor CreditMetrics Model In The New Basel Capital Accord.RiskMetrics Journal. 8. Gunter Löffler,Peter N. Posch. 2007.Credit risk modeling using Excel and VBA 9. Hugh Thomas and Zhiqiang Wang,2005 quot;Interpreting the Internal Ratings-Based Capital Requirements in Basel IIquot; Journal of Banking Regulation. 10. H. Ugur Koyluoglu and Andrew Hickman , 1998. “Reconcilable Differences”, Risk, October, p 56-62. 11. Ian Iscoe, Alex Kreinin and Dan Rosen,1999 An Integrated Market and Credit Risk Portfolio Model,Algo research quarterly 12. J.P. Morgan,1997.Introduction to CreditMetrics technical document. 13. Jongwoo Kim,Hypothesis Test of Default Correlation and Application to Specific Risk. RiskMetrics Journal 14. Jochen Felsenheimer, Philip Gisdakis, Michael Zaiser, Active credit portfolio management: A practice guide to credit risk management strategies. 15. Jose A. Lopez,2002.The Empirical Relationship between Average Asset Correlation, Firm Probability of Default and Asset Size.Economic Research Department Federal Reserve Bank of San Francisco. 16. Kenneth Carling Tor Jacobson Jesper Lindé Kasper Roszbach,2002.Capital Charges under Basel II: Corporate Credit Risk Modelling and the Macro Economy.Sveriges Riksbank Working Paper Series. 17. Merton, Robert, 1974. On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance, Vol 29, pp. 449-470. 18. Michael B. Gordy, 2000. A comparative anatomy of credit risk models. Journal of Banking & Finance.
  • 92. 84 19. Michael B. Gordy, 2002. A risk-factor model foundation for ratings-based bank capital rules. 20. MKMV,2006. Accommodating user-input correlation models in portfolio manager. 21. MKMV, 2001. An empirical assessment of asset correlation models. 22. Moody’s,2006. Special comment: Risk disclosures of banks and securities firms. 23. Paul S. Calem and James R. Follain, 2003. The asset-correlation parameter in Basel II for mortgages on single-family residences. 24. Paul Glasserman, 2005/2006 Measuring marginal risk contributions in credit portfolios. Journal of computational finance, 25. Philippe Jorion,2004. Credit risk management. 26. RMA,2003.Retail Credit Economic Capital Estimation --Best Practices. 27. Thomas Wilson, 1997a. Measuring and managing credit portfolio risk: Part I: Modeling systematic default risk. The Journal of lending and credit risk management, p 61 – 72. 28. Thomas Wilson, 1997b. Measuring and managing credit portfolio risk: Part II: Portfolio loss distributions. The Journal of lending and credit risk management, p 67 – 78. 29. Thomas Wilson. (1998) Portfolio Credit Risk. Federal Reserve Bank of New York Policy Review, p 71 – 82. 30. Vasicek, O, 1987. Probability of loss on loan portfolio. KMV Corporation. 31. Vasicek, O, 1991. Limiting loan loss probability distribution. KMV Corporation. 32. Wilde, T. 2001. IRB approach explained. Risk. p 87-90.
  • 93. 85 Appendix Interpreting the IRB capital equation78 In order to establish regulatory capital requirements applicable across institutions and across credit risk models. The regulatory AIRB capital proposed by the Basel Committee followed a general modeling framework, known as the asymptotic single risk factor (ASRF) approach which is originally from Vasicek’s work79. In this appendix, I will briefly explain the origin of IRB approach. The IRB ‘s K factor is defined as the following: ⎡ ⎤ ⎡ G( PD) + R * G(99.9%)⎤ − PD × LGD⎥ *×(1 − 1.5 × b) × [1 + (M − 2.5) × b] −1 k = ⎢LGD× N ⎢ ⎥ 1− R ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ where N = cumulative standard normal distribution G80= 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. We can divide the formula into four parts81: IRB K factor = ( Vasicek formula82 and Correlation – expected loss)* Maturity adjustment 78 Interested reader can refer to the ‘Hugh Thomas and Zhiqiang Wang quot;Interpreting the Internal Ratings-Based Capital Requirements in Basel IIquot; Journal of Banking Regulation, 2005.’ 79 Vasicek is a founder of KMV. Probability of loss on loan portfolio. KMV Corporation. 1987. Limiting loan loss probability distribution. KMV Corporation 1991. 80 Some uses the symbol: N-1 or Φ-1 81 Interested reader can refer to : Wilde, T. IRB approach explained. Risk. May 2001: 87-90 Hugh Thomas and Zhiqiang Wang quot;Interpreting the Internal Ratings-Based Capital Requirements in Basel IIquot; Journal of Banking Regulation(2005). 82 Vasicek set LGD=1
  • 94. 86 1. The Vasicek formula: estimation of capital required as % of exposure ⎡ G ( PD) + R * G (99.9%) ⎤ N⎢ ⎥ 1− R ⎣ ⎦ 2. Correlation estimation: R 3. The expected loss [ ] Maturity adjustment: (1 − 1.5 × b ) × 1 + (M − 2.5)× b −1 4. 1. The Vasicek formula The Vasicek formula forms the heart of the IRB equation and also called the asymptotic single risk factor approach83, recall the one factor model that we described in the previous: Ai = Wi Z + 1− Wi 2 ε i where Ai is the asset return of a single borrower over a one year horizon. Z denotes state of the economy systematic factor and εi denotes a company specific risk; both of the Z and ε are standard normally distributed random variables. The Wi is the weight of the borrower on the systematic factor, while as the Wi2 is the asset correlation. The company defaults on its loan if the value of its assets drops below the contractual value of its obligations α (default threshold) payable at one year horizon. We thus have: PDi=P(Ai<αi) Since we know the value of unconditional PD, we can determine the value of α by taking the inverse normal of the probability of default: αi=N-1(PDi)84 But we’d like to know the PD of an obligor fluctuate with the economy; therefore, we rewrite the equation by incorporating the state of the economy: 83 Lopez, JA. The empirical relationship between average asset correlation, firm probability of default and asset size. Economic Research Department, Federal Reserve Bank of San Francisco 2002. Vasicek. Probability of loss on loan portfolio. KMV Corporation. 1987 84 For example, if the borrower had a probability of default of 2%, then critical value of y would be α=N-1(2%)=-2.053. One can use the function NORMSINV (2%) in Excel.
  • 95. 87 α − Wi Z i ε i <αi) = P( ε i < PDi=P(Ai<αi) = P( Wi Z + 1− Wi 2 ) 1 − Wi 2 Substituting the α and expressed the above with cumulative standard normal distribution: N −1 ( PD) − Wi Z PDi= P(Ai<αi) =N ( ); 1 − Wi 2 the Vasicek formula estimates the loss distribution of portfolio depending on the state of the economy situation: е, while as the Basel has mandated the state of economy should be set at the poorest situation: Z =N-1(99.9%)=G(99.9%) . Substituting the Z, then we have the Basel’s k factor: ⎡ G ( PD) + R * G (99.9%) ⎤ N⎢ ⎥ 1− R ⎣ ⎦ The above formula also implies that the Basel committee expects bank to hold sufficient capital to maintain the solvency at the 99.9% of confidence level85. 85 As in the ‘An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel Committee’ describes that ‘The confidence level is fixed at 99.9%, an institution is expected to suffer losses that exceed its level of tier 1 and tier 2 capital on average once in a thousand years. This confidence level might seem rather high. However, Tier 2 does not have the loss absorbing capacity of Tier 1. The high confidence level was also chosen to protect against estimation errors, that might inevitably occur from banks’ internal PD, LGD and EAD estimation, as well as other model uncertainties. The confidence level is included into the Basel risk weight formulas’
  • 96. 88 Restricted Basel defined that the adequate capital as sufficient 99.9% of the time. Good state of economy Poor state of economy 0.1% of probability will fall below 99.9% of confidence interval 0 - NORMSINV(99.9%)= -3.09 Mean of state of economy 2008 Eric Confidential 1 2. Correlation estimation: R Default dependency or default correlation plays the central in the portfolio modeling. The estimation of default correlation aims at quantifying the joint probability among obligors. The market standard for the default correlation is so called factors model that one or more factors influence the default dependency in a state of economy. The rational behinds this approach is based on the structural model. In Merton’s world, default is triggered if firm’s asset value falls below the default barrier- firm’s debt. Moreover, firm’s asset value change is affected by two types of risk: firm’s specific risk and the state of the economy (could be one factor or multi-factors). The firm’s specific risk can be diversified away through adds more firms into the portfolio, while as the state of the economy (or so called systematic risk) is not be able to prevent. Therefore, assuming there are common factors that govern all firms’ asset value, will allow us to estimate the default dependency. Though, there is a board debate among market participants on how to parameterize the correlation. The common solution is to calculate it through equity return and assume that the firm’s equity and asset correlation are identical. For example, CreditMetrics approach. More sophisticated approach incorporate the equity value into firm’s asset value before deriving the
  • 97. 89 asset correlation, such as KMV. Recall the CAPM theory that uses a single factor model to explain the expected return of equity as a function of market risk premium. The single factor borrows the same concept to model the firm’s asset return as a function of market return. This idea also embedded in the Basel capital function: if bank’s credit (or loan) portfolio is diversified, then only the economy-wide systematic risk requires bank capital to protect the bank against borrows’ credit risk. Recall the CAPM model that : ri=αi+βirm+εi where ri = return of firm i. rm= return of the market εi= firm’s specific risk- a random error Normalizing the above equation, we have: ri − α i − β i E (rm ) σ m rm − E (rm ) σ (ε i ) ε i = (β i + ) σi σi σm σ i σ (ε i ) σ i , σ m , σ (ε i ) Where are the standard variance of firm i asset, the return of market and the random error, respectively. Defined A, Z, ε are normalized random variables and ri − α i − β i E (rm ) εi rm − E (rm ) ε= A= ,Z = , σi σm σ (ε i ) σm σ (ε i ) The equation will become: A = ( β i ε )Z + σi σi σ m 2 σ (ε i ) 2 where 1 = ( β i ) +( ), σi σi σ im σ im βi = ρi = As we know that in a liner regression that and . σm σ mσ i 2 σ m 2 σ im σ m 2 σ The ( β i ) = ( im ) 2 = ρ i ) =( 2 2 σi σm σi σ mσ i σm σ σ (ε i ) 2 set the W = ( β i ) , reconstruct 1 = ( β i m ) 2 + ( ); σi σi σi
  • 98. 90 σ (ε i ) 2 σ ) = 1 − ( β i m ) 2 = 1 − ρ i2 = 1 − W 2 we conclude ( σi σi σ (ε i ) = 1−W 2 the σi ε As we state the one factor model as A = WZ + 1 − W 2 ρi 2 or W2 in the above equation. The R in the k factor is refer to the Hugh Thomas and Zhiqiang Wang86 examine the average asset correlation of United States by using the above equation. 2 ⎛ σ ⎞ ⎛ 20% ⎞ 2 = ⎜ β i m ⎟ = ⎜1 * ρ v = ρi ⎜ σ ⎟ ⎝ 46% ⎟ = 18.9% 2 ⎠ ⎝ i⎠ σm σi = where = 20 % per annum; the average stock volatility is 46 % per annum and βi =1 the 18.9% of asset correlation is close to the 20 % of asset corelation that proposed in the first draft of Basel II. Restricted The asset correlation estimation done by Thomas and Zhiqiang is close to the upper boundary in the Basel equation. Asset Correlation PD 2008 Eric Confidential 1 86 9. Hugh Thomas and Zhiqiang Wang quot;Interpreting the Internal Ratings-Based Capital Requirements in Basel IIquot; Journal of Banking Regulation, 2005.
  • 99. 91 3. The expected loss The curve below illustrates the likelihood of credit losses of a portfolio. The area under the entire curve is equal to 100%. 100% minus the likelihood is called the confidence level and the corresponding threshold is called Value-at-Risk (VaR) at this confidence level. The Basel committee sets the confidence level at 99.9% and this implies the possibility of bank insolvency according to Basel’s k’s factor is merely 0.1%. Restricted EL based provision comes with a proper reason. If banks are not comply with EL based provision then banks may need to prepare more capital to protect credit losses. Basel set at 99.9% of confidence =100% - 99.9% = 0.1% Subtract EL based Correlation provision [ ]2 b = 0.11852 − 0.05478 × ln( PD) ⎡ ⎤ ⎡ ⎤ 0.5 ⎛R⎞ ⎢LGD× N ⎢(1 − R) × G(PD) + ⎜ ⎟ × G(0.999)⎥ − PD × LGD⎥ × (1 − 1.5 × b ) × [1 + (M − 2.5) × b ] −0.5 −1 K= ⎢ ⎝1− R ⎠ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ Tenor adjustment 2008 Eric Confidential Source : An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2 1 The expected loss is subtracted from the k factor, because 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. 4. Maturity adjustment: The final adjustment in the IRB formula is the adjustment for the average maturity. As we all know that the longer term credit facilities are riskier 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 the anticipated capital requirements will be higher than for short-term credits.
  • 100. 92 [ ] Maturity adjustment: (1 − 1.5 × b ) × 1 + (M − 2.5) × b −1 where b = (0.11852 − 0.05478 * ln( PD)) 2
  • 101. 93 VBA code Sub simsheet() Dim M As Long, i As Long Dim J As Long, K As Long Application.ScreenUpdating = False Application.Calculation = xlCalculationManual /**Clear output and formulae from previous runs**/ Range(quot;N10:N3000quot;).Clear Range(quot;L10:L3000quot;).Clear Range(quot;M10:M3000quot;).Clear Range(quot;L10:DJ30000quot;).Clear /**C3 = number of Simulation**/ M = Range(quot;c3quot;) /**Conduct trials till the indicated number of simulation in C3**/ For i = 1 To M Calculate Number of Simulation Current simulation time /** **/ Application.StatusBar = i /** Write loss of each trial into the result area **/
  • 102. 94 /** J12 sum up the credit loss of each simulation **/ Range(quot;Lquot; & i + 9) = Range(quot;J12quot;) /** J14 sum up the number of default obligors of each simulation **/ Range(quot;Mquot; & i + 9) = Range(quot;J14quot;) /** ‘i’ indicates which simulation**/ Range(quot;Nquot; & i + 9) = i /** Copy each simulation result into result area, this allows user to observe which obligors default together**/
  • 103. 95 Range(quot;O9:DJ9quot;).Select Selection.Copy Range(quot;Oquot; & i + 9).Select Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:= _ False, Transpose:=False Next i /** estimate the confidence interval from the loss result **/ Range(quot;H3quot;) = quot;=PERCENTILE(L$10:L$65536,G3)quot; 'Range(quot;H3:H6quot;).FillDown Application.Calculation = xlCalculationAutomatic End Sub