Asymptotic Single Risk Factor Model Research Articles

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.

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.

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.

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.

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.

Concentration Risk under Pillar 2: When are Credit Portfolios Infinitely Fine Grained?

Summary Concentration Risk under Pillar 2: When are Credit Portfolios Infinitely Fine Grained? The ongoing debate concerning credit concentration risk is mainly driven by the requirements on credit risk management due to Pillar 2 of Basel II since risks (e. g. concentration risk) that are not fully captured by Pillar 1 should be adequately considered in the banks' risk management. This instruction is indeed relevant since quantifying credit portfolio risk in Pillar 1 is based on an Asymptotic Single Risk Factor (ASRF) framework in which concentration risk is not covered. Against the background of the ASRF model, we determine the number of credits up to which concentration risk leads to a significant estimation error so that the assumption of an infinitely fine grained portfolio is inadequate. We conclude that the critical portfolio size varies from 22 up to 35,986 debtors, dependent on assets correlation and probability of default. Using a modified valuation function (granularity adjustment) it is possibl...