Institute for Development and Research in Banking Technology Hyderabad
Economic Capital Assessment for Consumer Credit Risk
Arushi Gupta
Integrated M.Tech, Mathematics and Computing IIT Delhi Project Trainee, 2013 IDRBT, Hyderabad
Project Guide Dr. Mahil Carr Associate Professor IDRBT, Hyderabad Certificate
This is to certify that Ms. Arushi Gupta, student from Indian Institute of Technology, Delhi , pursuing Integrated M.Tech in Mathematics and Computing, has undertaken a project as a summer project trainee at the Institute for Development and Research in Banking Technology (IDRBT), Hyderabad from May 13 to July 13, 2013.
It is thereby certified that this project report titled “Economic Capital Assessment for Consumer Credit Risk” is a bonafide record of the research work completed by her under my guidance.
Dr. Mahil Carr Associate Professor IDRBT, Hyderabad
Acknowledgement
I would like to express my sincere gratitude to the Institute for Development and Research in Banking Technology (IDRBT) and particularly Dr. Mahil Carr who was my guide in this project for providing me with the opportunity to learn all the nuances of a banking platform and carry out research on the topic of – “ Economic Capital Assessment of Consumer Credit Risk” which is a concern that the entire world is facing. I would like to add that this short stint in IDRBT has added a different facet to my life as this is a unique organization being a combination of academics, research, technology, communication services, crucial applications, etc., and at the same time performing roles as an arm of regulation, spread of technology, facilitator for implementing technology in banking and non-banking systems, playing a role of an NGO (without being one) and many more varied activities.
I am extremely grateful to Dr. Mahil Carr for his constant support and supervision in any manner whatsoever. I thank him for considering me capable enough to work under his esteemed guidance and on a project area that deeply interested me. I owe my sincere gratitude to the Chairman and the entire staff of Andhra Pradesh Grameen Vikas Bank, who willingly supported my research by providing me with all the necessary input data.
I am thankful for IDRBT for providing such an amazing platform for students to work in real application oriented research. Finally, I thank one and all who made this project successful either directly or indirectly.
Arushi Gupta
(IIT Delhi) Project Trainee
IDRBT, Hyderabad
Contents
Abstract …………………………………………………………………………………………………………………....6 1. Motivation……………………………………………………………………………………………………...……...7 2. Introduction…………………………………………………………………………………………………………. 8 2.1 Credit Life Cycle ...... 8 2.2 Credit Risk and its management ...... 8 2.2.1 New approaches to Credit Risk Management ...... 9 3. Background………………………………………………………………………………………………………… 10
3.1 Emergence of the Basel Accords ...... 10 3.1.1 The Purpose of Basel I ...... 10 3.1.2 Pitfalls of Basel I ...... 11 3.1.3 The Basel II Accord ...... 12 4. Basel II Model………………………………………………………………………………………………………14
4.1 Minimum capital Ratio ...... 14 4.2 Credit Risk Management Approaches...... 14 4.3 Economic Foundations ...... 16 4.4 Derivation of the Capital requirement ...... 19 4.5 Calibration of the Model ...... 21 4.5.1 Confidence Level ...... 21 4.5.2 Supervisory estimates of Asset Correlations ...... 21 4.5.3 Specification of the Retail Risk Weight Curves ...... 22 5. Theoretical Model using Monte Carlo Simulations………………………………………………….23
6. Empirical Model using Regression Analysis………………………………………………..………….24
6.1 Data Description...... 24 6.2 Model Design ...... 24 6.2.1 Risk factors Estimation ...... 24 6.2.1.1 Probability of Default Estimation ...... 24 6.2.1.2 Loss Given Default Estimation ...... 27 6.2.2 Asset Correlation Calculation ...... 30 6.2.3 Economic Capital Evaluation ...... 31 6.3 Results ...... 31
7. Analysis and Discussions……………………………………………………………………………………...32
7.1 Assessing Credit Portfolios ...... 32 7.2 Year-wise Analysis ...... 34 7.3 Trends Identified ...... 36 8. Conclusions and Future Work ………………………………………………………………………….…..39
9. References …………………………………………………………………………………….…………………….40
Abstract
Banks and financial institutions are faced with long-term future uncertainties that they intend to account for. It is in this context that the Basel Accords were created, aiming to enhance the risk management functions of banks and financial institutions. Basel II provides international directives on the regulatory minimum amount of capital that banks should hold against their risks, such as credit risk, market risk, operational risk and others. In this project, we first attempt a theoretical model, based on the Vasicek one-factor model, to compute the loss distribution of credit risk and henceforth, the probability of default of an obligor using vanilla Monte Carlo Simulations in MATLAB.
Next, we utilize statistical models of logistic regression and multiple linear regression respectively to estimate the probability of default of the obligor and the loss given default to the bank. To validate the model empirically, historical data of long term retail loans granted by Andhra Pradesh Grameen Vikas Bank (APGVB) from a period of 2000 to 2013 are collected and scrutinized.
Once the models are appropriately validated and their performance assessed, the trends of estimated default probabilities and the loss given default are examined according to loan-types, rating classes and year-wise. The results of the regression analysis are then employed to determine the economic capital requirement of Andhra Pradesh Grameen Vikas Bank in order to maintain its solvency.
1. Motivation
Compelled by the sporadic financial instabilities being faced by the world economy today, many banks are examining or implementing models now to help enhance their risk management efforts and align their capital requirements along the lines laid down by the new but much more complex Basel II Accord. This project employs empirical loan history data from a regional rural bank named Andhra Pradesh Grameen Vikas Bank (APGVB) to assess its adequate capital buffer.
These banks provide more credit to weaker sections of the society and are mainly focussed on the agro-sector. Since credit risk management forms the backbone for the survival of any banking institution in the world, we aim to develop an internal model for retail credit risk assessment of APGVB. In the words of Mahatma Gandhi, “Real India lies in villages”. The motivation behind the choice of a Grameen Bank stems from the fact that such regional rural banks cater to an important part of the Indian economy and without the upliftment of rural economy, objectives of economic planning and stability cannot be achieved.
Also, since these banks are primarily retail-focussed, the less differentiated classes of exposures as compared to large, corporate banks make the genesis of an internal model relatively simpler. An internal model for such a bank would be preferred since it allows choosing credit risk management approach according to bank’s aims and restrictions and sufficient for their own activities.
2. Introduction
2.1 Credit Life Cycle Bank operations involve sanctioning of loans and advances to customers for variety of purposes. These loans may be business loans for short or long term commitments and consumer finance for purchase of durables, property and vehicles. Other types of loans provided by banks are micro credit for small borrowers and contingent obligations that are off balance sheet transactions.
The loan sanctioning process commences when the bank receives a loan proposal from the customer. The loan requirement may be for equipment purchase or for working capital finance requirements. The loan proposal will be evaluated using an internal rating system or the credit rating offered by external rating agencies. After the loan sanction the bank needs to follow up and monitor the loan accounts. The loan sanctioned may become a default or a bad loan if the outstanding payments (either interest or installment of principle amount) are overdue by 90 days. Thus credit life cycle involves four stages, credit opportunity, credit assessment, credit management and credit implication.
2.2 Credit Risk and its Management A major topic in retail lending is the measurement of the inherent portfolio credit risk. Credit risk involves probability of loss of loan sanctioned. It is the risk of loss of principal or loss of a financial reward stemming from a borrower's failure to repay a loan or otherwise meet a contractual obligation. Credit risk arises whenever a borrower is expecting to use future cash flows to pay a current debt. Moreover, Consumer Credit Risk (or Retail Credit Risk) is the risk of loss due to a customer's non re-payment (default) on a consumer credit product, such as a mortgage, unsecured personal loan, credit card, overdraft etc.
Credit risk management is risk assessment that comes in an investment. Risk often comes in investing and in the allocation of capital. The goal of credit risk management is to maximize a bank’s risk-adjusted rate of return by maintaining credit risk exposure within acceptable parameters. Banks need to manage the credit risk inherent in the entire portfolio as well as the risk in individual credits or transactions. The risks must be assessed so as to derive a sound investment decision. And decisions should be made by balancing the risks and returns. For assessing the risk, banks should plan certain estimates, conduct monitoring, and perform reviews of the performance of the bank. So banks develop different models to assess and managing the losses due to credit risk and effective management of credit risk becomes a critical component of a comprehensive approach to risk management and essential to the long-term success of any banking organization.
2.2.1 New Approaches to Credit Risk Management
In recent years, a revolution has been brewing in risk as it is both measured and managed. Contradicting the relatively dull and routine history of credit risk, new technologies and ideas have emerged among a new generation of financial engineering professionals who are applying their model building skills and analysis to this area. The primary reasons for this growing need include:
Structural increase in bankruptcies Disintermediation More competitive margins Declining and volatile values of collateral Growth of off-balance sheet derivatives Technology The BIS Risk Based Capital Requirements
Thus, the risk of losses that result in the default of payment of the debtors is a kind of risk that must be expected. A bank must keep substantial amount of capital to protect its solvency and to maintain its economic stability. So, banks hold capital as a cushion against losses stemming from adverse credit, market, and operational circumstances. The greater the bank is exposed to risks, the greater the amount of capital must be when it comes to its reserves, so as to maintain its solvency and stability.
3. Background
3.1 Emergence of the Basel Accords Since exposure to credit risk continues to be the leading source of problems in banks world-wide, banks and their supervisors should be able to draw useful lessons from past experiences. Banks should now have a keen awareness of the need to identify, measure, monitor and control credit risk as well as to determine that they hold adequate capital against these risks and that they are adequately compensated for risks incurred.
From 1965 to 1981 there were about eight bank failures (or bankruptcies) in the United States. Bank failures were particularly prominent during the '80s, a time which is usually referred to as the "savings and loan crisis." Banks throughout the world were lending extensively, while countries' external indebtedness was growing at an unsustainable rate. As a result, the potential for the bankruptcy of the major international banks because grew as a result of low security. In order to prevent this risk, the Basel Committee on Banking Supervision, comprised of central banks and supervisory authorities of 10 countries, met in 1987 in Basel, Switzerland. The committee drafted a first document to set up an international 'minimum' amount of capital that banks should hold. This minimum is a percentage of the total capital of a bank, which is also called the minimum risk-based capital adequacy.
In 1988, the Basel I Capital Accord (agreement) was created. The Basel Committee issued this document in order to encourage banking supervisors globally to promote sound practices for managing credit risk. Basel encourages such sound practices set out in this document specifically address the following areas:
establishing an appropriate credit risk environment; operating under a sound credit granting process; maintaining an appropriate credit administration, measurement and monitoring process; and ensuring adequate controls over credit risk.
For smaller or less sophisticated banks, supervisors need to determine that the credit risk management approach used is sufficient for their activities and that they have instilled sufficient risk-return discipline in their credit risk management processes.
3.1.1 The Purpose of Basel I In 1988, the Basel I Capital Accord was created. The general purpose was to:
Strengthen the stability of international banking system. Set up a fair and a consistent international banking system in order to decrease competitive inequality among international banks.
Credit Risk is defined as the risk weighted asset (RWA) of the bank, which are banks assets weighted in relation to their relative credit risk levels. According to Basel I, the total capital should represent at least 8% of the bank's credit risk (RWA).
The basic achievement of Basel I has been to define bank capital and the so-called bank capital ratio. In order to set up a minimum risk-based capital adequacy applying to all banks and governments in the world, a general definition of capital was required.
3.1.2 Pitfalls of Basel I
Basel I Capital Accord has been criticized on several grounds. The main criticisms include the following:
Limited differentiation of credit risk. There are four broad risk weightings (0%, 20%, 50% and 100%), as shown in Figure1, based on an 8% minimum capital ratio. Static measure of default risk. The assumption that a minimum 8% capital ratio is sufficient to protect banks from failure does not take into account the changing nature of default risk. No recognition of term-structure of credit risk. The capital charges are set at the same level regardless of the maturity of a credit exposure. Simplified calculation of potential future counterparty risk. The current capital requirements ignore the different level of risks associated with different currencies and macroeconomic risk. In other words, it assumes a common market to all actors, which is not true in reality. Lack of recognition of portfolio diversification effects. In reality, the sum of individual risk exposures is not the same as the risk reduction through portfolio diversification. Therefore, summing all risks might provide incorrect judgment of risk. A remedy would be to create an internal credit risk model - for example, one similar to the model as developed by the bank to calculate market risk. This remark is also valid for all other weaknesses.
In effect, over time there has been a growing concern that the effectiveness of the Accord has eroded as banks have devised ways to engage in regulatory capital arbitrage introducing mismatching of risks taken on and buffer capital held. In response to these concerns, the Basel Committee released its proposal for the future capital adequacy rules, i.e., the new Accord known as Basel II, which added operational risk and also defined new calculations of credit risk.
To summarise, the Basel I Capital Accord aimed to assess capital in relation to credit risk, or the risk that a loss will occur if a party does not fulfill its obligations. It launched the trend toward increasing risk modelling research; however, its over-simplified calculations, and classifications have simultaneously called for its disappearance, paving the way for the Basel II Capital Accord and further agreements as the symbol of the continuous refinement of risk and capital.
3.1.3 The Basel II Accord
To make this approach more differentiated, the new accord called Basel II was implemented in 2007. Its goal is to better align the required regulatory capital with actual bank risk. This makes it vastly more complex than the original accord. Basel II has multiple approaches for different types of risk. The purpose of Basel II is to introduce a more risk-sensitive capital framework with incentives for good risk management practices. Many banks are examining or implementing models now to help enhance their risk management efforts.
Basel II is Three Pillars
Basel II has three pillars: minimum capital, supervisor review and market discipline.
Minimum capital is the technical, quantitative heart of the accord. Banks must hold capital against 8% of their assets, after adjusting their assets for risk. Supervisor review is the process whereby national regulators ensure their home country banks are following the rules. If minimum capital is the rulebook, the second pillar is the referee system. Market discipline is based on enhanced disclosure of risk. This may be an important pillar due to the complexity of Basel. Under Basel II, banks may use their own internal models (and gain lower capital requirements) but the price of this is transparency.
Basel II Charges for Three Risks
The accord recognizes three big risk buckets: credit risk, market risk and operational risk. In other words, a bank must hold capital against all three types of risks. The details of the approaches offered for each of these are illustrated in Figure 1 below.
Figure 1
Not only is the implementation staggered globally, but the accord itself contains tiered approaches as visible in the figure above. For example, credit risk has three approaches: standardized, foundation internal ratings-based (IRB), and advanced IRB. Roughly, a more advanced approach relies more on a bank's internal assumptions. A more advanced approach will also generally require less capital, but most banks will need to transition to more advanced approaches over time.
The Basel II Accord attempts to fix the glaring problems with the original accord. It does this by more accurately defining risk, but at the cost of considerable rule complexity. The technical rules will be importantly supported by supervisor review (Pillar 2) and market discipline (Pillar 3). The goal remains: Maintain enough capital in the banking system to guard against the damage of financial shocks.
The Role of Adequate Capital
Bank regulators set minimum capital requirements so as to reduce the likelihood of bank insolvencies that are costly to the economy. Beside government supervision, deposit insurance and other regulatory conditions, capital requirements limit risks for depositors, and reduce insolvency and systemic risks. However, they also form important restrictions on the workings of banks. Unnecessary capital requirements restrain credit provision needlessly, whereas inadequate capital requirements may lead to undesirable levels of systemic risk. Thus, the determination of adequate capital required for a banking system form an important part in the process of effective risk management.
4. The Basel II Model
4.1 Minimum Capital Ratio
The main intension of the Basel II capital accord is to set the total minimum capital requirements for market, credit and operational risk. Three main elements are related to these requirements: the definition of regulatory capital, risk weighted assets and the minimum capital ratio.
The capital ratio is defined as the ratio between the regulatory capital and the total sum of risk weighted assets
Capital Ratio = Regulatory Capital/ ∑Risk Weighted Assets
The denominator of equation above is composed of three addends describing market, operational and Credit risk. The first two addends are multiplied with a factor of 12.5, whereas the third representing the the Credit risk contribution is described in the following sections. The capital ratio must be at least 8%. That is,
Total Capital Capital Ratio 8% RWA12.5* MRC 12.5* ORC Credit Market oprt
4.2 Credit Risk Management Approaches
For Credit risk, the Basel II capital accord proposes two methods: the Standardised and the Internal Ratings Based (IRB) approach. Hence, there are two ways to determine the risk weights serving as the input for Equation (1.1). Depending also on the category of the claims that form the basis of Credit risk, the calculation of risk weights is carried out differently.
Based on credit ratings of external agencies, the standardised approach uses a mapping between rating classes and risk weights. The mapping rules and the risk weights are defined by the authorities. As can be inferred from this, the standardised approach is very inflexible and this inflexibility encourages the use of the new approach as suggested by the Basel II accord. The advantages of choosing the Internal Ratings Based Approach are as follows:
Increased flexibility in banks Additional risk sensitivity Incentive compatibility
Applying this method, the risk weights are calculated using internal estimators for the risk factors. For the above mentioned categories of claims, three key elements form the basis of the IRB approach: risk components, risk weight functions and minimum requirements. The estimates of four risk factors used in the IRB approach are provided either by banks or by the supervising authorities, depending on whether the IRB foundation approach or the IRB advanced approach is applied. These risk components are the probability of default (PD), the loss given default (LGD), the exposure at default (EAD) and the effective maturity (M).
Probabilities of Default (PD) All banks whether using the foundation or the advanced methodology have to provide an internal estimate of the PD associated with the borrowers in each borrower grade. Probability of default per rating grade gives the average percentage of obligors that default in this rating grade under normal business conditions. Exposure At Default (EAD) It gives an estimate of the amount outstanding in cases the borrower default. In most cases, EAD is equal to the nominal amount of the exposure but for certain exposures - e.g. those with undrawn commitments - it includes an estimate of future lending prior to default. Loss Given Default (LGD) Loss given default rate, which is equal to one minus the recovery rate, gives the percentage of exposure the bank might lose in case the borrower defaults. Note that, average losses over long period of time understate LGD rates during an economic downturn. Therefore, it needs to be adjusted upward to appropriately reflect adverse entomic conditions.
Maturity (M) Where maturity is treated as an explicit risk component, like in the advanced approach, banks are expected to provide supervisors with the effective contractual maturity of their exposures.
For both, the foundation as well as the advanced IRB approach, PD is estimated by the bank. The other risk components are provided by the regulatory authorities for the foundation approach. If a bank uses the advanced IRB approach, it is required to estimate LGD, EAD and M in addition to PD. In this project, we propose a model based on the advanced IRB approach and provide internal estimates for all the risk components.
4.3 Economic Foundations
In the credit business, losses of interest and principal occur all the time - there are always some borrowers that default on their obligations. The losses that are actually experienced in a particular year vary from year to year, depending on the number and severity of default events, even if we assume that the quality of the portfolio is consistent over time. Figure 2 illustrates how variation in realised losses over time leads to a distribution of losses for a bank.
Figure 2
While it is never possible to know in advance the losses a bank will suffer in a particular year, a bank can forecast the average level of credit losses it can reasonably expect to experience. These losses are referred to as Expected Losses (EL) and are shown in Figure 3 by the dashed line. Financial institutions view Expected Losses as a cost component of doing business, and manage them by a number of means, including through the pricing of credit exposures and through provisioning.
One of the functions of bank capital is to provide a buffer to protect a bank’s debt holders against peak losses that exceed expected levels. Such peaks are illustrated by the spikes above the dashed line in Figure 2. Peak losses do not occur every year, but when they occur, they can potentially be very large. Losses above expected levels are usually referred to as Unexpected Losses (UL) - institutions know they will occur now and then, but they cannot know in advance their timing or severity. Interest rates, including risk premia, charged on credit exposures may absorb some components of unexpected losses, but the market will not support prices sufficient to cover all unexpected losses. Capital is needed to cover the risks of such peak losses, and therefore it has a loss-absorbing function.
Banks have an incentive to minimise the capital they hold, because reducing capital frees up economic resources that can be directed to profitable investments. On the other hand, the less capital a bank holds, the greater is the likelihood that it will not be able to meet its own debt obligations, i.e. that losses in a given year will not be covered by profit plus available capital, and that the bank will become insolvent. Thus, banks and their supervisors must carefully balance the risks and rewards of holding capital. There are a number of approaches to determining how much capital a bank should hold.
The IRB approach adopted for Basel II as discussed above, focuses on the frequency of bank insolvencies arising from credit losses that supervisors are willing to accept. By means of a stochastic credit portfolio model, it is possible to estimate the amount of loss which will be exceeded with a small, pre-defined probability. This probability can be considered the probability of bank insolvency. Capital is set to ensure that unexpected losses will exceed this level of capital with only this very low, fixed probability. This approach used to setting capital is illustrated in Figure 3.
Figure 3
The likelihood that losses will exceed the sum of Expected Loss (EL) and Unexpected Loss (UL) - i.e. the likelihood that a bank will not be able to meet its own credit obligations by its profits and capital - equals the hatched area under the right hand side of the curve. 100% minus this likelihood is called the confidence level and the corresponding threshold is called Value-at-Risk (VaR) at this confidence level. If capital is set according to the gap between EL and VaR, and if EL is covered by provisions or revenues, then the likelihood that the bank will remain solvent over a one- year horizon is equal to the confidence level. Under Basel II, capital is set to maintain a supervisory fixed confidence level.
So far the Expected Loss has been regarded from a top-down perspective, i.e. from a portfolio view. It can also be viewed bottom-up, namely from its components. The Expected Loss of a portfolio is assumed to equal the proportion of obligors that might default within a given time frame (1 year in the Basel context), multiplied by the outstanding exposure at default, and once more multiplied by the loss given default rate (i.e. the percentage of exposure that will not be recovered by sale of collateral etc.). Of course, banks will not know in advance the exact number of defaults in a given year, nor the exact amount outstanding nor the actual loss rate; these factors are random variables. But banks can estimate average or expected figures.
The Expected Loss (in currency amounts) can then be written as EL = PD * EAD * LGD or, if expressed as a percentage figure of the EAD, as EL = PD*LGD
4.4 Derivation of the Capital Requirement
The capital requirements of banking books are derived from risk weight formulas, which were developed from Asymptotic Risk Factor (ASRF) model. It is believed that the precursor to the formula is the working paper of. In the ASRF models, Zit is assumed the normalized asset return for firm i at time t. This variable can be decomposed in the following way:
ZX it t 1 it (1) where XNt ~ (0,1) and it ~N (0,1)
The component it represents the risk specific to institution i, and Xt represents a common factor to all firms in the portfolio. is the factor loading of the systematic risk and is often interpreted as the sensitivity to systematic. Under the assumption of distribution Normal(0, 1), the correlation between the normalized asset returns of any two borrowers is . If a borrower’s return falls short of some threshold c, i.e.
Zit < c Yit = 1 i = 1,.....,Nt, t = 1,…..,T (2)
Where Yit is an indicator variable and we assume that Y = 1 indicates that the firm defaults.
The unconditional probability of default as the form:
PDi = P(Yi = 1)
= P(Zit < c)
= P( Xt1 it < c) = ()c (3) Conditional on a realization x of the common random factor at time t then becomes the conditional probability of default: cx P(Y = 1|X = x) = P X x it t it t 1 cx = Xx t 1
c 1(0.001) P(Y = 1|X = x ) = (4) it t 99.9 1 where X denoted the systematic risk factor, x99.9 is the 99.9th percentile of the systematic risk factor, meaning that a worse outcome of the systematic risk factor only has a 0.01 percent change.
And the conditional expected loss is: c 1(0.001) E[Li|Xt = x99.9] = LGD · 1 11(PD ) (0.001) = LGD · i (5) 1 As mentioned in BCBS (2004a), the sum of expected loss(EL) and unexpected loss(UL) are derived from the exposure’s conditional expected loss (CEL) by the formula
UL + EL = E[Li|Xt = x99.9]
= LGD · (6)
Finally, in Basel II the calibration of the risk weight only to unexpected losses. Thus the formula of capital requirement given by BCBS(2004b) is of the form:
11(PD ) (0.001) 1b ( PD )( M 2.5) K = LGD · i PD 1 1 1.5b ( PD )
Where UL = (UL + EL) – EL are the unexpected losses given by,
UL = LGD · - LGD PD (7)
1b ( PD )( M 2.5) and is full maturity adjustment as function of PD and M, with 1 1.5b ( PD ) b(PD) = (0.11852 − 0.05478 × log(PD))2 means the smoothed (regression) maturity adjustment (smoothed over PDs).
The risk-weighted assets and capital requirements are related in a straightforward manner and the resulting formula is the following:
RWAi = K(PD, ) · 12.5 · EAD (8)
4.5 Calibration of the Model
Within the above model, two key parameters have to be determined by supervisory authorities: the confidence level supervisors feel comfortable to live with, and the asset correlation that determines the degree of dependence of the borrowers on the overall economy.
4.5.1 Confidence Level
Since it is neither practical nor desirable to completely indemnify the banking system against all insolvencies; instead, an “acceptable” level of risk is necessary to prevent moral hazard considerations that would encourage banks to take on excessive risk exposures. The proposed BIS II Internal Ratings-Based model sets this risk threshold at the 99.9 percentile; that is, the capital charge is sufficient to cover losses in all but the worst 0.1 percent of adverse credit risk events. Whence, the confidence level is fixed at 99.9%, i.e. 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.
4.5.2 Supervisory Estimates of Asset Correlations
Asset correlation is an important component of the Basel II Accord for regulatory capital requirements of credit risk portfolios. It means the correlation of a given firm’s assets with the risk factor that summarizes general economic conditions and is a key parameter to determine the shape of the risk weight formulas that is based on the Asymptotic Single Risk Factor (ASRF) model. As mentioned in eq (1) above, gives the average asset correlation between the normalized asset returns of any two borrowers.
According to the Advanced Internal Ratings Based Approach (A-IRB) of the accord, the asset correlation parameter R is a function of PD:
1 exp( c PD ) 1 exp( c PD ) ab 1 (9) 1 exp( cc ) 1 exp( ) where the parameters a, b, and c depend on borrower type. For example, the Basel average asset correlation for retail exposures ranges from 3% to 16%, as a decreasing function of PD. This function suggests that a retail borrower with a lower credit score has lower asset correlation than a borrower with a higher credit score.
The asset correlation function implies two systematic dependencies: Asset correlation decrease with increasing PDs. Asset correlation increase with firm size.
4.5.3 Specification of the Retail Risk Weight Curves
The retail risk weights differ from the corporate risk weights in two respects: First, the asset correlation assumptions are different. Second, the retail risk weight functions do not include maturity adjustments. The asset correlations that determine the shape of the retail curves have been “reverse engineered” from (i) economic capital figures from large internationally active banks, and (ii)historical loss data from supervisory databases of the G10 countries. Both data sets contained matching PD and LGD values per economic capital or loss data point.
The banks' economic capital data have been regarded as if they were the results of the Basel risk weight formulas with their matching PD and LGD figures being inserted into the Basel risk weight formulas. Then, asset correlations that would approximately result in these capital figures within the Basel model framework, have been determined. Obviously, the asset correlation would not exactly match for each and every bank, nor for each and every PD-LGD-Economic Capital triple of a given bank, but on average the figures work. With the second data set (supervisory time series of loss data), Expected Loss (as the mean of the time series) and standard deviations of the annual losses were computed.
Both analyses showed significantly different asset correlations for different retail asset classes. They have led to the three retail risk weight curves for residential mortgage exposures, qualifying revolving retail exposures and other retail exposures, respectively. The three curves differ with respect to the applied asset correlations: relatively high and constant in the residential mortgage case, relatively low and constant in the revolving retail case, and, similarly to corporate borrowers, PD- dependent in the other retail case:
For Residential Mortgages: 15% For qualifying Revolving Retail Exposures: 4% For other Retail Exposures: 1 exp( 35 PD ) 1 exp( 35 PD ) 0.03 0.16 1 (10) 1 exp( 35) 1 exp( 35)
In the above analysis, both the economic capital data from banks and the supervisory loss data time series implicitly contained maturity effects. Consequently, the reverse engineered asset correlations implicitly contain maturity effects as well, as the latter were not separately controlled for. Thus, the maturity effects have been left as an implicit driver in the asset correlations, and no separate maturity adjustment is necessary for the retail risk weight formulas.
5. Theoretical Model using Monte Carlo Simulations
Monte Carlo Simulation is a very flexible and powerful tool for estimating integrals and expected values. Monte Carlo Simulation is a random algorithm. A different run of simulation will yield a different estimate. Different Monte Carlo schemes can be generically described as follows.
Let H be a random variable such that µ = E[H]. Then the corresponding Monte Carlo 1 n estimate is given by, Hi where H1, H2,….,Hn are iid copies of H. n i1
Consider the one-factor portfolio credit risk model. The loss distribution and hence the default probability can be estimated from using a plain Monte Carlo Simulation as described below.
Pseudo code for i=1,2,….n Generate independent samples Z (systematic risk factor) and
12 , ,..., midiosyncratic risk factors from N(0,1). Compute 2 for k=1,2,…,m where is the XZk k 1 k k k exposure to the common factor. m Compute Lc 1 where ck is the default from the kth obligor. k Xxkk k1
Set Hi = 1 if L > h; set Hi = 0 otherwise HHH ..... Compute the estimate v 12 n n n 1 22 Compute the standard error S.. E Hi nv nn( 1) i1
An interesting observation is that as h gets larger, the probability gets smaller and the quantity of estimates deteriorates. The reason is that only for a very small fraction of samples will the total loss exceed the large threshold h. With so few hits, the estimate cannot be accurate. Hence, the plain Monte Carlo is inefficient for estimating small probabilities. The above algorithm is coded in MATLAB but due to unavailability of the specific inputs required and the inaccuracy of the estimate due to small number of defaults, this model could not be utilised to estimate the default probability efficiently.
6. Empirical Model using Regression Analysis
6.1 Data Description
Empirical data of 2500 long term retail credit portfolios granted from January, 2000 till June 17, 2013 by the Ashok Nagar Branch of Andhra Pradesh Grameen Vikas Bank was collected and employed for the analysis and testing of the model proposed. The regression models for estimating all the parameters are carried out using IBM SPSS.
6.2 Model Design
The model suggests the following steps in order the access the economic capital buffer that needs to be kept aside by bank: Risk factors Estimation Asset Correlation Calculation Economic Capital Evaluation
6.2.1 Risk factors Estimation
The two risk components are estimated using regression analysis as detailed below. The Exposure at Default is taken as the Current Outstanding amount of the different loans.
6.2.1.1 Probability of Default Estimation
Binary logistic regression is an appropriate technique to use on these data because the “dependent” or criterion variable (the thing we want to predict) is dichotomous (loan default vs. no default). Logistic regression has replaced discriminant analysis as the preferred tool for dichotomous outcomes, largely because logistic regression has assumptions that are less restrictive, and so less frequently violated. This model allows for predictors that need not be normally distributed and that are nonlinear in their effects on the probability of an event, and for dependent variables with different sized groups.
Regression Form The logistic regression takes the following form:
p K log kk x (11) 1 p k1 where p is the probability of the even occurring, and K independent variables, x, are each weighted by a coefficient, β.
This logit function preferable to the linear regression function because it limits p, the probability of default, to be between 0 and 1 (or zero and 100%), where applying the conventional linear regression to dichotomous outcomes would instead allow nonsensical results like probabilities greater than 1 or less than zero. A transformation of (11), obtain the logistic model which gives the probability of default as:
K kkx e k1 p K kkx 1 e k1
An estimating algorithm is used to find the coefficients, β's that best satisfy the relationship expressed in the regression equation for the estimation data sample. The technique used to find those coefficients for logistic regression, was using maximum likelihood estimation. Basically, method tries coefficients until it finds the set that maximizes the value of a mathematical function that gives the joint probability of observing the given data.
In this model, the dependent variable is the probability of default whereas the independent variables or the predictors include loan type, limit amounts sanctioned, Age of the loan, irregularities observed, and the current outstanding amounts. A decision was made to include only parameters with insignificant correlation in the model.
This is the Wald chi-square test that tests the null hypothesis that the constant equals 0. This hypothesis is rejected because the p-value (listed in the column called "Sig.") is smaller than the critical p-value of .05.
Using this statistic, the variables that are found to be significant for the given portfolio among the various risk indicators are the limit amount sanctioned, age of loan, irregularities amount, and 5 significant loan types (Loan against Deposits, Over Draft loans, Housing loans, Personal loans and Staff housing loans)
Exp(B) column gives out the odds ratios for the predictors. They are the exponentiation of the coefficients. We can interpret EXP(B) in terms of the change in odds. If the value exceeds 1 then the odds of an outcome occurring increase; if the figure is less than 1, any increase in the predictor leads to a drop in the odds of the outcome occurring.
Prediction Equation
Like linear regression, logistic regression provides a coefficient ‘b’, which measures each predictor’s partial contribution to variations in the dependent variable.
p log 1.62 0.001*Age _ of _ loan 4.46* IDL 1.28* I HL 1 p
3.1*IIIOD 1.35* PL 4.49* STF where IPL is the indicator for personal loan type and similarly for all other loan types. The negative coefficients for the 5 significant loan types indicate that these loan types are less risky for the bank since an obligor from any of these loan types decreases the odds against default and hence are less likely to default.
ROC Curve
Sensitivity and specificity, which are defined as the number of true positive decisions/the number of actually positive cases and the number of true negative decisions/the number of actually negative cases, respectively, constitute the basic measures of performance of diagnostic tests. To deal with these multiple pairs of sensitivity and specificity values, one can draw a graph using the sensitivities as the y coordinates and the 1-specificities or FPRs as the x coordinates. Each discrete point on the graph, called an operating point, is generated by using different cutoff levels for a positive test result. An ROC curve can be estimated from these discrete points, by making the assumption that the test results, or some unknown monotonic transformation thereof, follow a certain distribution. For this purpose, the assumption of a binormal distribution is most commonly made. The resulting curve is called the fitted or smooth ROC curve.
Discrimination refers to the “ability of the model to distinguish correctly the two classes of outcomes”. In other words, it is a measure of the ability of the model to “separate subjects with different responses. The area under the ROC curve, which ranges from zero to one, provides a measure of the model’s ability to discriminate - the larger the area under the ROC curve, the more the model discriminates. The ROC curve of this logistic regression model is illustrated below and constitutes an area under ROC curve of 0.755.
Summary of Model Performance Analysis
A logistic regression analysis was conducted to predict the default probability for 250 credit portfolio holders using current outstanding amounts, loan types, limits sanctioned, irregularities, and age of loans as predictors. A test of the full model against a constant only model was statistically significant, indicating that the predictors as a set reliably distinguished between acceptors and decliners of the offer (chi square = 24.096, p < .000 with df = 2). Overall prediction success increased from 79.3 to 89.6% in comparison to the null model. Nagelkerke’s R2 of .737 indicated a moderate relationship between prediction and grouping. The Wald criterion demonstrated resulted in 5 significant loan types and 3 other significant predictors. Current outstanding was not a significant predictor. EXP(B) value indicates that if a housing loan is granted i.e. the indicator for housing loans takes the value 1, the odds ratio is 0.277 times as large and therefore obligors are less times likely to default. The area under ROC curve of 0.755 (close to 1) indicates the model offers acceptable discrimination.
6.2.1.2 Loss Given Default Estimation
Only loans that had previously defaulted i.e. loans with payments due more than 90 days are considered in this prediction. The loss given default for these loans is estimated using multiple linear regression. The two major questions to justify the use of this method are answered below. Is the regression model meaningful? And if yes, then which of the variables contribute meaningfully?
The following screening procedures were applied to validate the assumptions behind a multiple linear regression model and thereby ensure the meaningfulness of the regression model.
(a) Linear relationship between the predictor variable and the response variable.* Test: The observed and predicted values of the dependent variables must be symmetrically distributed along the diagonal line.
(b) Normally distributed and homoscedastic residuals. Test: The error distribution as can be seen in the histogram of residuals must be approximately normal. Be alert if in a plot of residuals vs. the predicted values, the residuals are getting larger (more spread out) either as a function of time or as a function of predicted values.
(c) The residuals are independent of one another. Test: If the value of the Durbin-Watson statistic is approximately 2, then the residuals are uncorrelated.
(d) No significant outliers are present in the data.* Test: The Cook’s Distance must be <1.00 to ensure no outliers.
(e) Lack of multi-collinearity between the predictors.* Test: Tolerance values for every predictor must be > 0.20 to ensure no multi-collinearity in the data.
(f) The expected value of the errors of prediction in the population regression function is equal to 0. Test: The average residuals value from residual statistics is equal to 0.
* Some of the results could not be displayed here as the SPSS trial version had expired on the system so the outputs could not be retrieved.
(g) The predictors consist of continuous or nearly continuous data. Test: Since loan type is a categorical variable, dummy coding was applied for entering it into a standard regression model.
Regression From The multiple linear regression model takes the form:
yx ii where y is the losses occurring due to default, and K independent variables, x, are each weighted by a coefficient, β. Apart from considering the age of loan in default in place of age of loan from the time it is sanctioned, two additional risk indicators were also considered in this model: the IRAC status and the interest rate.
Coefficients
Multiple linear regression employs ordinary least squares (OLS) estimates to compute the weights of the prediction equation (by minimising the squared differences between the actual and predicted values). The unstandardised coefficients (B) are the regression coefficients. The standardized coefficients (Beta) are what the regression coefficients would be if the model were fitted to standardized data.
The t statistic tests the hypothesis that a population regression coefficient β is 0, that is, H0: β = 0. It is the ratio of the sample regression coefficient B to its standard error. The statistic has the form (estimate - hypothesized value) / SE. Since the hypothesized value is 0, the statistic reduces to Estimate/SE. Sig. labels the two-sided P values or observed significance levels for the t statistics. The null hypothesis is rejected if the 2 tailed p- values are less than the critical p-value i.e. 0.05 and the associated coefficient is thus declared to contribute significantly in the model.
In our model, the significant variables selected on similar lines as detailed above are the current outstanding amounts, and 2 loan types (Personal Gold Loan and Term Loans).
Prediction Equation
LGD0.062*OT U 91725.45*I 164495.9*I PGL TL
The positive coefficients associated with the outstanding amount indicate that the losses given default are in a positive relationship with the outstanding amounts. It is also evident from the prediction equation obtained above that if the bank grants a loan in either of these two significant loan types, there is a rapid rise in the losses given default.
Model Performance Analysis
R is the square root of R-Squared and is the correlation between the observed and predicted values of dependent variable. R-squared is an overall measure of the strength of association, and does not reflect the extent to which any particular independent variable is associated with the dependent variable. R-Square is also called the coefficient of determination. The adjusted R-square attempts to yield a more honest value to estimate the R-squared for the population. Adjusted R-squared is computed
(1RNsq )( 1) using the formula 1. These values for our linear regression model are Nk1 as presented below.
6.2.2 Asset Correlation Calculation
The default probabilities estimated from logistic regression model are then utilised to obtain the asset correlation factors for different retail credit portfolio types depending on the different exposure types as specified in eq. (10) above. The asset correlations obtained are as follows.
Table 1
Loan Types Default Probabilities Asset Correlation Factors Over draft loans 0.034483 0.068885812 Loans against deposits 0.041667 0.060240723
Staff housing loans 0.08 0.037905308
Personal loans 0.089597 0.035649909
Housing loans 0.091286 0.035325595
6.2.3 Economic Capital Evaluation
The minimum capital requirement is then determined for the retail portfolios using the asset correlation factors calculated above. Being retail portfolios, the economic capital requirement contains no explicit maturity adjustment term. So, the capital buffer required to cover the unexpected losses equals the unexpected losses itself and is given by eq. (7) in Section 4.4. The buffer capital requirements for the different loan types obtained are as follows.
Table 2
Loan Types Capital buffer required Limits sanctioned Over draft loans 2131391.756 75030000 Loans against deposits 258385.9928 23179000
Staff housing loans 588333.537 18212000
Personal loans 1246758.09 244036000
Housing loans 7555838.782 306009000
6.3 Results
The project delivers a minimum capital requirement assessment to enable credit risk management in Andhra Pradesh Grameen Vikas Bank (Ashok Nagar Branch) based on Basel II Internal Ratings Based Approach using historical loan portfolio data for the period 2000 to 2013. Some useful results obtained as a result of carrying out the above experiments are enumerated in the table below.
Table 3
Parameters Values
Average probability of default 6.74 % Total limit sanctioned 59,89,39,000 Total current outstanding 45,00,27,868 Capital requirement 1,17,80,708.16 Capital (as a % of EAD) 2.6178
7. Analysis and Discussions
A detailed analysis of the bank’s financial position is carried out by examining various patterns and trends obtained in different credit portfolios, rating classes and over the years.
7.1 Assessing Credit Portfolios
Five different loan types are selected and assessed on the basis of the results obtained above. The loan types are as enumerated below:
OD – Over draft Loans, DL – Loans against Deposits STF – Staff Housing Loans PL – Personal Loans HL – Housing Loans
(a) Figure 4 shows that default rate varies significantly across loan types. Among the different loan types considered, the housing loan type has the highest average default probability of 9.13% whereas the over draft loans pose the lowest value of 3.45%.
0.1 8.96% 9.13% 8% 0.08
0.06 4.17% 3.45% 0.04 Estimated Default Probability 0.02 0 OD DL STF PL HL Loan Types
Figure 5
(b) Figure 2 below show that the loss rates also vary significantly across loan types. It is interesting to notice that even though the default probability is the lowest for Over draft loans, the loss rates are observed to be maximum for this credit portfolio. Loans granted against deposits attain the lowest value in this case.
Figure 6
(c) Figure 7 below portrays a comparison between the limit amounts sanctioned versus the losses given default observed for the different loan types. With the maximum amount sanctioned for housing purposes by the bank, the losses observed too are also maximum for Housing loans and quite large in amounts as compared to other loan types.
Estimated losses 400000000
300000000
200000000 Limit Alloacated 100000000 Loss Given Default 0 OD DL STF PL HL Loan types
Figure 7
(d) Figure 8 assesses the different portfolios from two perspectives simultaneously, the default chances as well as the capital lost in case of default. As can be seen, the housing loans emerge to be the riskiest type of credit for the bank with the highest probability of default and a considerably high loss rate on default. From the figure 2 above, we saw that the loan against deposits point to the lowest loss rates. With a quite low default probability of 4.2%, these come forth as the safest loan type for the bank.
Comparison of Loan Types
0.1 PL HL Riskiest STF 0.08 0.06 DL 0.04 OD 0.02 Safest
0 Probability of Probability Default 0.692 1.917 3.643 5.143 7.24 Loss Rates (%)
Figure 8
7.2 Year-wise Analysis
(a) Figure 9 below depicts that there is quite some movement over time in the average default rate of the bank’s portfolio. The maximum rates of default within the sample period are reached in the period of 2003 to 2005. This points to an economic crisis in the region as a result of which the obligors have been unable to make the repayments.
Default Probability 0.4
0.3
0.2
0.1
0
2011 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2012 2013
Figure 9
(b) Figure 10 demonstrates that high loss rates have been observed by the bank over the period ranging from 2005 to 2007. This can be explained as a result of the economic crisis being faced from 2003 to 2005 as the result of which the payments have been long overdue.
Loss Rates 60 50 40 30 20 10
0
2010 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2011 2012 2013
Figure 10
(c) Figure 11 below displays the correlation between the estimated and the actual average default rates of the bank over the years. A correlation coefficient of 0.67 indicates that the logistic regression model satisfactorily depicts the actual probabilities of default.
Year-wise Default Probability 0.5 R2 = 0.6731 0.4
0.3 Estimated PD 0.2 Actual PD 0.1
0
2006 2000 2001 2002 2003 2004 2005 2007 2008 2009 2010 2011 2012
Figure 11
(d) Similarly, Figure 12 below displays the correlation between the estimated and the actual loss rates of the bank over the years. A correlation coefficient of 0.73 again indicates that the linear regression model satisfactorily depicts the actual loss rates.
Year-wise Loss Given Default 2500000 R2 = 0.7339 2000000
1500000 Estimated LGD 1000000 Actual LGD
500000
0
2009 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2010 2011 2012 2013
Figure 12
7.3 Trends Identified
(a) Figures 13 and 14 below illustrate a counter-intuitive trend both in case of housing and personal loans. Intuitively, as the amount of loan sanctioned increases the chances of the obligor to default should also increase, but that is not what we observe in the two retail portfolio distributions that we consider. This can be attributed in the case of APGVB to comparatively much lesser number of loans with higher limit values. The lesser the number of large loans sanctioned, the lesser the chances of default.
Housing Loans Loss Distribution
0.4 0.3 0.2 0.1
0
Default Probability Default
3to4 8to9 1to2 2to3 4to5 5to6 6to7 7to8
9to10
13to14 30to40 10to11 11to12 12to13 14to15 15to20 20to25 25to30 Loan Amount Sanctioned (lacs)
Figure 13
Personal Loans Loss Distribution
0.3 0.25 0.2 0.15 0.1 0.05
0
Default Probability Default
50 to80 50
below 50 below
80 to100 80
150 to 170 to 150 100 to 120 to 100 150 to 120 200 to 170 230 to 200 250 to 230 270 to 250 300 to 270 330 to 300 350 to 330 370 to 350 400 to 370 450 to 400 500 to 450 Loan Amount Sanctioned ('000 Rs.)
Figure 14
(c) Figure 15 below establishes a trend that the default risk is not constant within rating classes over time. The rating classes are specified according to the IRAC status of the obligors.
Default Risk varies within rating class over time
0.5
0.4
0.3 Rating Class 2
0.2 Rating Class 1 Rating class 0
0.1 Default Probability Default 0 2008 2009 2010 2011 2012
Figure 15
(d) Figure 16 compares the capital to limit ratio of the various loan types. As can be seen, the highest ratio of capital to limit ratio is required for housing loans. This falls in line with the result that the housing loans are indeed the riskiest type of loans for the banks and thus, maximum capital needs to be buffered for them.
Loan-wise Capital Requirement
0.12 0.1 0.08 0.06 Capital/Limit Ratio 0.04 0.02 0 OD DL STF PL HL
Figure 16
8. Conclusions and Future Work
The project successfully complies by the objectives that we laid down in the beginning. The primary aims of the project that were indeed fulfilled are as stated below.
Development of econometric models in order to estimate the default probabilities of obligors associated with retail loans and the Loss Given Default for the facility (bank) employing statistical techniques such as logistic regression and multiple linear regression respectively.
Determination of the adequate buffer capital basis for credit risk on the basis of the internal model developed on the lines of the Advanced Internal Ratings Based Approach (A-IRB) of Basel II.
In this research, it was found that not all data fields stored in the loan files of banks are significant in regression modelling to estimate the risk components. Different fields were found significant for different regression estimates. For example, Current outstanding amounts were found to be significant only for predicting the losses given default whereas total limit sanctioned in the beginning was an important factor for the default probability. Another interesting observation were the adjacent periods of high default and loss rates from a period ranging from 2003 to 2007, thereby indicating an economic crisis for the obligors during that time. Assessing the relative riskiness of the different credit portfolios and the corresponding capital reserves that must be kept aside by the banks were other crucial outcomes of the project. The objective of this whole process was to incorporate the loan history data stored by Grameen banks and for the first time, attempt to analyse their financial stability and solvency.
The models provided in this paper help to point a credit analyst in the right direction, however, more validation techniques need to be provided on the past predictions of the model. For the time being, the model does have high correlations with true values of the risk components estimated, but still lacks the testing and calibration techniques of a true quantitative analyst.
In future, these models can be thoroughly tested and trained with a unique and richer data set including different bank branches to arrive at more accurate predictions. Moreover, various other statistical and non statistical models can be applied to a bank’s data depending on the input data fields available to propagate a continuous improvement in the credit risk management facilities at all levels.
9. References
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