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.

Shared and Unique Risk Factors Underlying Mathematical Disability and Reading and Spelling Disability.

High comorbidity rates have been reported between mathematical learning disabilities (MD) and reading and spelling disabilities (RSD). Research has identified skills related to math, such as number sense (NS) and visuospatial working memory (visuospatial WM), as well as to literacy, such as phonological awareness (PA), rapid automatized naming (RAN) and verbal short-term memory (Verbal STM). In order to explain the high comorbidity rates between MD and RSD, 7–11-year-old children were assessed on a range of cognitive abilities related to literacy (PA, RAN, Verbal STM) and mathematical ability (visuospatial WM, NS). The group of children consisted of typically developing (TD) children (n = 32), children with MD (n = 26), children with RSD (n = 29), and combined MD and RSD (n = 43). It was hypothesized that, in line with the multiple deficit view on learning disorders, at least one unique predictor for both MD and RSD and a possible shared cognitive risk factor would be found to account for the comorbidity between the symptom dimensions literacy and math. Secondly, our hypotheses were that (a) a probabilistic multi-factorial risk factor model would provide a better fit to the data than a deterministic single risk factor model and (b) that a shared risk factor model would provide a better fit than the specific multi-factorial model. All our hypotheses were confirmed. NS and visuospatial WM were identified as unique cognitive predictors for MD, whereas PA and RAN were both associated with RSD. Also, a shared risk factor model with PA as a cognitive predictor for both RSD and MD fitted the data best, indicating that MD and RSD might co-occur due to a shared underlying deficit in phonological processing. Possible explanations are discussed in the context of sample selection and composition. This study shows that different cognitive factors play a role in mathematics and literacy, and that a phonological processing deficit might play a role in the occurrence of MD and RSD.

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.

Asset correlations in single factor credit risk models: an empirical investigation

ABSTRACTThe internal ratings–based (IRB) approach (based on a single risk factor model) was designed by the Basel Committee on Banking Supervision (BCBS) to determine banks’ regulatory credit risk capital. Key inputs of the model – asset correlations – are prescribed by the regulator; relevant banks must use them for capital determination. To ascertain whether these correlations are too onerous or too lenient, empirical asset correlations embedded in loss data spanning different loss milieu were backed out of the regulatory model. Static and rolling correlations over a period of time were compared with the prescribed correlations for developed and developing economies and found to be significantly more conservative.

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 of commercial real estate credit risks

Modeling the probability of default for commercial real estate mortgages is more complicated than that for non-commercial real estate loans. This is because borrowers will default only if both the net operating income and the property value fall below the threshold levels. To make modeling more complicated, the property value at the time of default will determine the loss-given default. In this paper, I derive closed-form solutions for the probability of default and the expected loss of commercial real estate mortgages in a Merton framework. The model is in its essence still a single risk factor model, although there is a sector risk factor that influences both the net operating income and the property value. I obtain analytically the economic capital for the corporate-wide commercial real estate portfolio, with granularity adjustments for name concentration and sector concentration.

Credit Ratings in Structured Finance and the Role of Systemic Risk

This paper explores the implications of systemic risk in Credit Structured Finance (CSF). Risk measurement issues loomed large during the 2007-08 financial crisis, as the massive, unprecedented number of downgrades of AAA senior bond tranches inflicted severe losses on banks, calling into question the credibility of Rating Agencies. I discuss the limits of the standard risk frameworks in CSF (Gaussian, Single Risk Factor Model; GSRFM), popular among market participants. If implemented in a ‘static’ fashion, GSRFM can substantially underprice risk at times of stress. I introduce a simple ‘dynamic’ version of GSRFM that captures the impact of large systemic shocks (e.g. financial meltdown) for the value of CSF bonds (ABS, CDO, CLO, etc.). I argue that a proper 'dynamic' modeling of systemic risk is crucial for gauging the exposure to default contagion (‘correlation risk’). Two policy implications are drawn from a 'dynamic' GSRFM: (i) when rating CSF deals, Agencies should disclose additional risk information (e.g. the expected losses under stressed scenarios; asset correlation estimates); and (ii) a ‘point-in-time’ approach to rating CSF bonds is more appropriate than a ‘through-the-cycle’ approach.

Credit Ratings in Structured Finance and the Role of Systemic Risk

This paper explores the implications of systemic risk in Credit Structured Finance (CSF). Risk measurement issues loomed large during the 2007-08 financial crisis, as the massive, unprecedented number of downgrades of AAA senior bond tranches inflicted severe losses on banks, calling into question the credibility of Rating Agencies. I discuss the limits of the standard risk frameworks in CSF (Gaussian, Single Risk Factor Model; GSRFM), popular among market participants. If implemented in a i?½statici?½ fashion, GSRFM can substantially underprice risk at times of stress. I introduce a simple i?½dynamici?½ version of GSRFM that captures the impact of large systemic shocks (e.g. financial meltdown) for the value of CSF bonds (ABS, CDO, CLO, etc.). I argue that a proper 'dynamic' modeling of systemic risk is crucial for gauging the exposure to default contagion (i?½correlation riski?½). Two policy implications are drawn from a 'dynamic' GSRFM: (i) when rating CSF deals, Agencies should disclose additional risk information (e.g. the expected losses under stressed scenarios; asset correlation estimates); and (ii) a i?½point-in-timei?½ approach to rating CSF bonds is more appropriate than a i?½through-the-cyclei?½ approach.

Systematic Risk in Recovery Rates - An Empirical Analysis of U.S. Corporate Credit Exposures

This paper presents an analytical and empirical analysis of a parsimonious model framework that accounts for a dependence of bond and bank loan recoveries on systematic risk. We extend the single risk factor model by assuming that the recovery rates also depend on this risk factor and follow a logit?normal distribution. The results are compared with those of two related models, suggested in Frye (2000) and Pykhtin (2003), which pose the assumption of a normal and a log-normal distribution of recovery rates. We provide estimators of the parameters of the asset value process and their standard errors in closed form. For the parameters of the recovery rate distribution we also provide closed-form solutions of a feasible maximum-likelihood estimator for the three models. The model parameters are estimated from default frequencies and recovery rates that were extracted from a bond and loan database of Standard&Poor's. We estimate the correlation between recovery rates and the systematic risk factor and determine the impact on economic capital. Furthermore, the impact of measuring recovery rates from market prices at default and from prices at emergence from default is analysed. As a robustness check for the empirical results of the maximum-likelihood estimation method we also employ a method-of-moments. Our empirical results indicate that systematic risk is a major factor influencing recovery rates. The calculation of a default?weighted recovery rate without further consideration of this factor may lead to downward-biased estimates of economic capital. Recovery rates measured from market prices at default are generally lower and more sensitive to changes of the systematic risk factor than are recovery rates determined at emergence from default. The choice between these two measurement methods has a stronger impact on the expected recovery rates and the economic capital than introducing a dependency of recovery rates on systematic risk in the single risk factor model.

Systematic Risk in Recovery Rates: An Empirical Analysis of Us Corporate Credit Exposures

This paper presents an analytical and empirical analysis of a parsimonious model framework that accounts for a dependence of bond and bank loan recoveries on systematic risk. We extend the single risk factor model by assuming that the recovery rates also depend on this risk factor and follow a logit?normal distribution. The results are compared with those of two related models, suggested in Frye (2000) and Pykhtin (2003), which pose the assumption of a normal and a log-normal distribution of recovery rates. We provide estimators of the parameters of the asset value process and their standard errors in closed form. For the parameters of the recovery rate distribution we also provide closed-form solutions of a feasible maximum-likelihood estimator for the three models. The model parameters are estimated from default frequencies and recovery rates that were extracted from a bond and loan database of Standard&Poor's. We estimate the correlation between recovery rates and the systematic risk factor and determine the impact on economic capital. Furthermore, the impact of measuring recovery rates from market prices at default and from prices at emergence from default is analysed. As a robustness check for the empirical results of the maximum-likelihood estimation method we also employ a method-of-moments. Our empirical results indicate that systematic risk is a major factor influencing recovery rates. The calculation of a default?weighted recovery rate without further consideration of this factor may lead to downward-biased estimates of economic capital. Recovery rates measured from market prices at default are generally lower and more sensitive to changes of the systematic risk factor than are recovery rates determined at emergence from default. The choice between these two measurement methods has a stronger impact on the expected recovery rates and the economic capital than introducing a dependency of recovery rates on systematic risk in the single risk factor model.