Asymptotic Single Risk Factor Research Articles

Measurement and Calibration of Regulatory Credit Risk Asset Correlations

Vasicek’s asymptotic single risk factor (ASRF) model is employed by the Basel Committee on Banking Supervision (BCBS) in its internal ratings-based (IRB) approach for estimating credit losses and regulatory credit risk capital. This methodology requires estimates of asset correlations; these are prescribed by the BCBS. Practitioners are interested to know market-implied asset correlations since these influence economic capital and lending behavior. These may be backed out from ASRF loan loss distributions using ex post loan losses. Prescribed asset correlations have been neither updated nor recalibrated since their introduction in 2008 with the implementation of the Basel II accord. The market milieu has undergone significant alterations and adaptations since then; it is unlikely that these remain relevant. Loan loss data from a developed (US) and developing (South Africa) economy spanning at least two business cycles for each region were used to explore the relevance of the BCBS calibration. Results obtained from three alternative methodologies are compared with prescribed BCBS values, and the latter were found to be countercyclical to empirical loan loss experience, resulting in less punitive credit risk capital requirements than required in market crises and more punitive requirements than required in calm conditions.

Credit risk measures and the estimation error in the ASRF model under the Basel II IRB approach

In many standard derivation and presentations of risk measures like the Value-at-Risk or the Expected Shortfall, it is assumed that all the model’s parameters are known. In practice, however, the parameters must be estimated and this introduces an additional source of uncertainty that is usually not accounted for. The Prudential Regulators have formally raised the issue of errors stemming from the internal model estimation process in the context of credit risk, calling for margins of conservatism to cover possible underestimation in capital. Notwithstanding this requirement, to date, a solution shared by banks and regulators/supervisors has not yet been found. In our paper, we investigate the effect of the estimation error in the framework of the Asymptotic Single Risk Factor (ASRF) model that represents the baseline for the derivation of the credit risk measures under the Internal-Ratings Based (IRB) approach. We exploit Monte Carlo simulations to quantify the bias induced by the estimation error and we explore an approach to correct for this bias. Our approach involves only the estimation of the long run average probability of default and not the estimation of the asset correlation given that, in practice, banks are not allowed to modify this parameter. We study the stochastic characteristics of the probability of default estimator that can be derived from the Asymptotic Single Risk Factor framework and we show how to introduce a correction to control for the estimation error. Our approach does not require introducing in the Asymptotic Single Risk Factor model additional elements like the prior distributions or other parameters which, having to be estimated, would introduce another source of estimation error. This simple and easily implemented correction ensures that the probability of observing an exception (i.e. a default rate higher than the estimated quantile of the default rate distribution) is equal to the desired confidence level. We show a practical application of our approach relying on real data.

Relaxing the Assumption of Conditional Independence in a Asymptotic Single Risk Factor Model

Relaxing the Assumption of Conditional Independence in a Asymptotic Single Risk Factor Model

The Estimation Risk and the IRB Supervisory Formula

In many standard derivation and presentations of risk measures like the Value-at-Risk or the Expected Shortfall, it is assumed that all the model’s parameters are known. In practice, however, the parameters must be estimated and this introduces an additional source of uncertainty that is usually not accounted for. The Prudential Regulators have formally raised the issue of errors stemming from the internal model estimation process in the context of credit risk, calling for margins of conservatism to cover possible underestimation in capital. Notwithstanding this requirement, to date, a solution shared by banks and regulators/supervisors has not yet been found. In our paper, we investigate the effect of the estimation error in the framework of the Asymptotic Single Risk Factor model that represents the baseline for the derivation of the credit risk measures under the IRB approach. We exploit Monte Carlo simulations to quantify the bias induced by the estimation error and we explore an approach to correct for this bias. Our approach involves only the estimation of the long run average probability of default and not the estimation of the asset correlation given that, in practice, banks are not allowed to modify this parameter. We study the stochastic characteristics of the probability of default estimator that can be derived from the Asymptotic Single Risk Factor framework and we show how to introduce a correction to control for the estimation error. Our approach does not require introducing in the Asymptotic Single Risk Factor model additional elements like the prior distributions or other parameters which, having to be estimated, would introduce another source of estimation error. This simple and easily implemented correction ensures that the probability of observing an exception (i.e. a default rate higher than the estimated quantile of the default rate distribution) is equal to the desired confidence level. We show a practical application of our approach relying on real data.

Calibration of rating grades to point-in-time and through-the-cycle levels of probability of default

The paper argues for the need for and importance of the dual calibration of a probability of default (PD) model (ie, calibration to both point-in-time (PIT) and through-the-cycle (TTC) PD levels). It explains why such a calibration should happen at the rating grade level, and how PD estimates can be assigned to a bank’s existing rating grades, ie, without introducing any changes to either the underlying (hybrid) rating function or the rating scale. This is done by first translating the traditional asymptotic single risk factor model underlying the Basel risk-weight formulas from its original obligor level to the rating grade level, and then determining the model’s parameters by using a maximum likelihood estimation. Standard deviations of the estimates obtained then provide a link to the regulatory margin of conservatism. Rating migration adds a new dimension to the model and leads to procyclical deviations of TTC PDs per grade from their long-term average counterparts. A link between rating migration and calibration levels gives a new definition to the rating function’s degree of PIT-ness, which becomes a key parameter responsible for the TTC stability at the portfolio level. The closed-form solutions obtained are flexible with respect to different patterns of rating migration and explicitly account for a prediction error in the business cycle model. The proposed technique is illustrated on a sample of corporate customers.

Estimating the Probability of Default for No-Default and Low-Default Portfolios

This article proposes a sequential Bayesian updating approach to estimate default probabilities on rating grade level for no- and low-default portfolios. Bayesian sequential updating allows to obtain default probabilities also for those rating grades for which no defaults have been observed. The advantage of the proposed approach is that it preserves the rank order of rating grades in case of no defaults. Rank-preservation is not ensured when using an identical prior distribution across all rating grades. We discuss Bayesian sequential updating for the beta-binomial model and a model incorporating the asymptotic single risk factor model of the Basel Accord. Practical aspects such as incorporating information from external sources and the margin of conservatism are addressed.

Convergence of sums of dependent Bernoulli random variables: an application from portfolio theory

Generalizations of the Central Limit Theorem to N dependent random variables often assume that the dependence falls off as . In this paper we present an example from mathematical finance where convergence is independent of N. Specifically, we consider dependent Bernoulli variates that represent N loans and probabilistic dependence among loans is based on an underlying economic model for default. A simple model for correlated default is the Vasicek Asymptotic Single Risk Factor (ASRF) framework. Our results showcase an example of “fast” convergence of dependent variates to this limiting non normal distribution with rate .

Stochastic loss given default and exposure at default in a structural model of portfolio credit risk

In this paper, we develop a factor-type latent variable model for portfolio credit risk that accounts for stochastically dependent probability of default (PD), loss given default (LGD) and exposure at default (EAD) at both the systematic and borrower specific levels. By employing a comprehensive simulation study, we set our results in contrast to those obtained using the asymptotic single risk factor (ASRF) model that underlies Basel II and III. Several sensitivity and robustness analyses for different parameter assumptions are conducted to break down our results. As required by the regulator, we show how to map our portfolio credit loss quantiles with correlated PD, LGD and EAD into values for downturn LGD and EAD. Our analyses reveal that stochastically dependent defaults, LGD and EAD increase a credit portfolio’s tail risk significantly. Estimating risk markups separately can therefore lead to substantially underestimating the inherent risk of a portfolio with nondeterministic exposures. Further, we show that relative downturn markups on LGD and EAD depend strongly on asset correlations and, in particular, that credit portfolios with low asset correlations might be prone to an underestimation of additional capital charges for stochastically dependent LGD and EAD. Thus, our results are of economic significance for banks and regulators when setting up minimum capital requirements.

Assessing the Basel II internal ratings-based approach

PurposeThis study aims to render a fundamental assessment of the Basel II internal ratings-based (IRB) approach by taking readings of the Australian banking sector since the implementation of Basel II and comparing them with signals from macroeconomic indicators, financial statistics and external credit ratings. The IRB approach to capital adequacy for credit risk, which implements an asymptotic single risk factor (ASRF) model, plays an important role in protecting the Australian banking sector against insolvency.Design/methodology/approachRealisations of the single systematic risk factor, interpreted as describing the prevailing state of the Australian economy, are recovered from the ASRF model and compared with macroeconomic indicators. Similarly, estimates of distance-to-default, reflecting the capacity of the Australian banking sector to absorb credit losses, are recovered from the ASRF model and compared with financial statistics and external credit ratings. With the implementation of Basel II preceding the time when the effect of the financial crisis of 2007-2009 was most acutely felt, the authors measure the impact of the crisis on the Australian banking sector.FindingsMeasurements from the ASRF model find general agreement with signals from macroeconomic indicators, financial statistics and external credit ratings. This leads to a favourable assessment of the ASRF model for the purposes of capital allocation, performance attribution and risk monitoring. The empirical analysis used in this paper reveals that the recent crisis imparted a mild stress on the Australian banking sector.Research limitations/implicationsGiven the range of economic conditions, from mild contraction to moderate expansion, experienced in Australia since the implementation of Basel II, the authors cannot attest to the validity of the model specification of the IRB approach for its intended purpose of solvency assessment.Originality/valueAccess to internal bank data collected by the prudential regulator distinguishes this paper from other empirical studies on the IRB approach and financial crisis of 2007-2009. The authors are not the first to attempt to measure the effects of the recent crisis, but they believe that they are the first to do so using regulatory data.

Credit Name Concentration Risk: Granularity Adjustment Approximation

During the last subprime mortgage crisis, the concentration risk issue has become increasingly important in the world of finance. This risk is defined as the loss that we can incur from a large exposition of a single name counterparty, a sector or a product. This paper represents some mathematical models for assessment and quantification of the concentration risk under the Add-On approach. This study is based on the Granularity Adjustment (GA). This measure quantifies the idiosyncratic risk that is neglected by the Asymptotic Single Risk Factor model (ASRF) based on the infinitely granular assumption of the portfolio. This work is about the approximation of this measurement to simplify the formula of GA using the Ad-Hoc approach. We have implemented empirical tests to find the relation between the GA and concentration indexes and we applied these results to the iBoxx portfolio.

Probabilities of Default for Impairment Under IFRS 9

Probabilities of default built for regulatory purposes cannot be applied directly to expected credit losses impairment calculations under the IFRS 9 new standard. This is because the regulatory framework requires stressed through-the-cycle (TTC) probabilities, so as to avoid a procyclical capital charge calculation, while IFRS 9 requires point-in-time (PIT) probabilities that include forward looking information. However the model (Asymptotic Single Risk Factor model or similar) that is generally used for the construction of stressed TTC probabilities of default can still be used to build forward looking PIT probabilities of default, provided it is made explanatory so as to obtain probabilities that are conditional on current and forecast economic conditions.This note explores how this can be achieved, thus leveraging on developments that institutions have already made in order to comply with regulatory requirements.

Stress testing and modeling of rating migration under the Vasicek model framework: empirical approaches and technical implementation

Under the Vasicek asymptotic single risk factor model, stress testing based on rating transition probability involves three components: the unconditional rating transition matrix, asset correlations, and stress testing factor models for systematic downgrade (including default) risk. Conditional transition probability for stress testing given systematic risk factors can be derived accordingly. In this paper, we extend Miu and Ozdemir’s work ([14]) on stress testing under this transition probability framework by assuming different asset correlation and different stress testing factor model for each non-default rating. We propose two Vasicek models for each non-default rating, one with a single latent factor for rating level asset correlation, and another multifactor Vasicek model with a latent effect for systematic downgrade risk. Both models can be fitted effectively by using, for example, the SAS non-linear mixed procedure. Analytical formulas for conditional transition probabilities are derived. Modeling downgrade risk rather than default risk addresses the issue of low default counts for high quality ratings. As an illustration, we model the transition probabilities for a corporate portfolio. Portfolio default risk and credit loss under stress scenarios are derived accordingly. Results show, stress-testing models developed in this way demonstrate desired sensitivity to risk factors, which is generally expected.

Modeling systematic risk and point-in-time probability of default under the Vasicek asymptotic single-risk-factor model framework

Systematic risk has been a focus for stress testing and risk capital assessment. Under the Vasicek asymptotic single risk factor model framework, entity default risk for a risk homogeneous portfolio divides into two parts: systematic and entity specific. While entity specific risk can be modelled by a probit or logistic model using a relatively short period of portfolio historical data, modeling of systematic risk is more challenging. In practice, most default risk models do not fully or dynamically capture systematic risk. In this paper, we propose an approach to modeling systematic and entity specific risks by parts and then aggregating together analytically. Systematic risk is quantified and modelled by a multifactor Vasicek model with a latent residual, a factor accounting for default contagion and feedback effects. The asymptotic maximum likelihood approach for parameter estimation for this model is equivalent to least squares linear regression. Conditional entity PDs for scenario tests and through-the-cycle entity PD all have analytical solutions. For validation, we model the point-in-time entity PD for a commercial portfolio, and stress the portfolio default risk by shocking the systematic risk factors. Rating migration and portfolio loss are assessed.

Modeling Intra and Inter Correlations in Credit Default Losses

Comparing alternatives for a simultaneous incorporation of intra and inter correlations into the credit portfolio loss distribution within the asymptotic single risk factor (ASRF) model and showing that the resulting distribution depends on the type of a dominant correlation: whether it is of intra d-d/lgd-lgd or inter d-lgd type. Showing that the classic Vasicek distribution (derived originally for intra d-d correlations only), modified to embrace both intra and inter correlation types by properly constructing composite mean and correlation parameters, offers an analytic solution which covers an entire range of correlations and is easy to use as opposed to a more formal approach of a bivariate joint default probability density which requires numerical averaging over the underlying latent factors and applies only when intra correlations dominate.

Book-to-market equity, operating risk, and asset correlations: Implications for Basel capital requirement

We examine the effect of book-to-market equity (BE/ME) on asset correlations in the asymptotic single risk factor (ASRF) framework under the Basel II Accord on regulatory capital requirement. Over a sample period from 1988 to 2007, we find that BE/ME is negatively related to asset correlations after controlling for firm size, default probability, and industry effects. Decomposing BE/ME into risks of operating leverage and financial leverage according to Penman et al. (2007) reveals that higher (lower) operating risk are related to lower (higher) asset correlations. Our findings suggest that incorporating risk of operating leverage into the estimation of asset correlations may improve the measurement of a bank's regulatory capital requirement and potentially reduce regulatory capital arbitrage.

신용등급 전이행렬을 활용한 위기상황분석에 관한 실증분석

본 논문은 우리나라 기업의 신용등급 전이행렬을 활용하여 부도율과 신용 등급 전이에 내재된 체계적 요인을 추출하는 방법을 소개하고, 이률 활용한 위기상황분석에 관한 연구를 수행하였다. 본 논문의 주요 발견은 등급전이행렬에 내재된 체계적 요인의 변동은 경기 동행성이 뚜렷하고, 실제 경기 변동을 설명하는 것을 확인할 수 있었다. 특히, 투자적격등급과 투기등급별로 경기에 반응하는 속도가 상당히 다르다는 것을 확인할 수 있었다. 신용등급 전이행렬에 내재된 체계적 위험을 고려한 위기상황분석은 부도확률에만 초점을 맞추는 방법에 비해 위기상황에 대한 포트폴리오의 변화를 파악할 수 있기 때문에 개념적으로 우월하고, 분석 결과 등급 전이를 고려한 위기상황분석이 부도확률만을 고려하는 방법에 비해 예상손실에 상당한 차이가 있음을 발견하였다. In this paper, we estimate systematic risk from credit migration (or transition) matrices under "Asymptotic Single Risk Factor" model. We analyzed transition matrices issued by KR(Korea Ratings) and concluded that systematic risk implied on credit migration somewhat coincide with the real economic cycle. Especially, we found that systematic risk implied on credit migration is better than that implied on the default rate. We also emphasize how to conduct a stress test using systematic risk extracted from transition migration. We argue that the proposed method in this paper is better than the usual method that is only considered for the conditional probability of default(PD). We found that the expected loss critically increased when we explicitly consider the change of credit quality in a given portfolio, compared to the method considering only PD.

The asymmetric behavior and procyclical impact of asset correlations

This paper examines the behavior of asset correlations with the market returns in the asymptotic single risk factor (ASRF) approach of the Basel II accord on regulatory capital requirement. Over a sample period from 1988 to 2007, we find that asset correlations are positively related to firm size, but negatively related to firm default probability. Asset correlations are also industry specific, as firms in media, transportation, pharmaceutical, and semiconductor industries exhibit higher asset correlations than those in retail and consumer staples. Most importantly, asset correlations are asymmetric and have a procyclical impact on the real economy after controlling for these effects. They tend to rise during economic downturns, but decline during economic upturns. The average magnitude of the rise is larger than that of the decline. These findings suggest that asset correlations may be underestimated during economic downturns, and may provide policy implications for the capital requirement framework.

Sector concentration risk: A model for estimating capital requirements

The 2004 Basel Committee on Banking Supervision Accord (known as Basel II) provides a common framework for banks to determine their minimum capital requirements for solvency purposes. For credit risk (the most important one for banking) Basel II uses an asymptotic single risk factor (ASRF) model and, as we demonstrate in the paper, assumes two fundamental hypotheses: Firstly, that there is only one risk factor common to all banks; and secondly, that the number of debtors in bank portfolios is high enough to ensure that no single debtor’s behaviour can have a significant impact on the portfolio value as a whole. This allows capital requirements to be estimated by using a model based on the percentage of defaulting borrowers ( x ) . The model only requires values for two variables: the probability of default and loss if default occurs. Using a 99% likelihood and assuming that all sectors are equally correlated, the model estimates x through the cumulative distribution function for the Gaussian distribution. But many bank portfolios do not fit these hypotheses, and therefore the ASRF model underestimates actual capital requirements. Thus, a surcharge for concentration risk is required. There are two kinds of concentration risk (sector and name concentration risk), each one corresponding to the violation of one of the above mentioned hypotheses. Supervisory authorities are currently developing models to incorporate this surcharge into banking solvency rules. In Spain, the Spanish Central Bank bases its surcharge proposal for sectorial concentration on the Herfindahl–Hirschman Index (HHI). In this paper we show that HHI treats all sectors as equally risky and propose an alternative index (CI) in which sectors are weighted according to risk. Moreover, our index also incorporates the relations between each pair of sectors (in the HHI framework no sectorial relationship is considered). Our proposal is based on an adjusted variance–covariance matrix, in which negative covariances have been equalled to 0. We demonstrate the HHI is a particular case of our proposed index, by means of simplifying hypotheses. As we will show, the proposed index has two fundamental properties: it is lower and upper bounded; and it decreases as concentration and/or risk decreases. These properties allow the index to be incorporated into bank risk management models. In this way bank estimations can improve upon those based on the supervisory model and, according to banking rules, can also be used for determining the capital surcharge for sectorial concentration.

An Empirical Study of the Returns on Defaulted Debt and the Discount Rate for Loss-Given-Default

Prudential management of credit risk and supervisory requirements call for the accurate measurement of loss conditional upon default (LGD). In the case of banks, in order to achieve Advanced Internal Ratings Based (AIRB) compliance under the Basel II minimum regulatory capital framework, loss arising fro m counterparty default must be estimated. However, the discount rate to be applied to post-default cash flows is a largely unsettled issue, amongst both practitioners and bank supervisors. In this study we survey various methodologies extant in the literature for determining an appropriate discount rate. We propose an approach in which the discount rate is conditional upon the level or undiversifiable risk inherent in the recovery cash flows associated with defaulted facilities. We present a stylized theoretical framework for understanding such an approach. This is followed by an empirical exercise that utilizes a comprehensive and commercially available database of workout recoveries (Moody’s Ultimate Recovery Database - MURD), in which we analyze the returns on marketable bonds and loans, having market prices at default and at the resolution of the default event. We propose alternative empirical measures of the recovery risk inherent in post-default cash-flows: the annualized simple return on defaulted debt (RDD) and discount rates implied from a structural model of credit risk incorporating systematic recovery risk, a generalization of the asymptotic single risk factor (ASRF) framework (Gordy, 2003). We find our empirically derived estimates to be significantly higher than what has been found in the previous literature, as well as what is used in industry, mean RDD of 29.2%; this compares to benchmarks such as the 15% reported by Araten et al (2003), the 200bps over the risk-free rate suggested by Machlachlan (2004), or rates in the range of 10-15% derived from model-based estimations (one or two factor structural credit models in conjunction with an assumption on the systematic risk factor). Principal findings are that returns on defaulted debt, which can be interpreted as an appropriate discount rate for workout recoveries, vary significantly according to various factors. There is some evidence that discount rate metrics are elevated for loans having better collateral quality rank or better protected tranches within the capital structure; and for obligors rated higher at origination, more financially levered or having higher Cumulative Abnormal Returns (CARs) on equity just prior to default. However, the discount rate is increasing in market implied loss severity at default. We also find evidence that LGD discount rates vary procyclically, as they increase with industry default rates, but there tends to be some asynchronousity in this relationship; further, they are inversely related to short-term interest rates. However, for other demographics the results are inconclusive, such as the industry group of the obligor. Finally, we conduct an analysis of the impact of the discounting method upon the distribution of estimated LGD and regulatory capital. We find that a regression model based discounting, for a sub-sample of the MURD database, results in a capital charge 73 bps greater than discounting at a constant punitive rate of 25%, and 113 bps larger than discounting at the contractual coupon rate (where the capital charge ranges in 7-8%). This conservativeness of the risk-sensitive RDD model, as well as the evidence that the risk in recovery cash flows contain a significant non-diversifiable component, supports the appropriateness of this framework for regulatory capital calculations.

Extending the Basel II Approach to Estimate Capital Requirements for Equity Investments

Under the Basel II banking regulatory capital regime the capital requirements for credit exposures are calculated using the Asymptotic Single Risk Factor (ASRF) approach. The capital requirement is taken to be the contribution of an exposure to the unexpected loss on the bank's diversified portfolio. Here we extend this approach to calculate capital requirements for equity investments. We show that in the case when asset values have a normal distribution an analytical formula for the unexpected loss contribution may be developed. We show that the capital requirements for equity investments are quite different to those of credit exposures, since equity investments can suffer substantial loss of value even when the underlying company has not defaulted. Unexpected loss is commonly used as a measure of capital requirements, but it ignores the ability of earnings to absorb loss. We propose a definition of capital requirement that recognises the expected earnings on assets, and show how to combine the ASRF model and the Capital Asset Pricing Model to compute this quantity for credit and equity exposures.