Portfolio Loss Distribution Research Articles

Simulating Credit Loss Distributions: Empirical Versus the Vasicek Model

Because credit losses can be substantial, managing credit risk is a focus area of risk measurement and management. It is important for financial institutions to select credit risk models that accurately forecast losses. The Basel Committee on Banking Supervision (BCBS) chose the closed-form single risk factor Vasicek model for regulatory capital calculations. In this article, its forecast accuracy is compared with empirical loss distributions using simulated probabilities of default and losses given default. The effect of altering probabilities of default on asset correlations was analysed and how this affects credit portfolio loss distributions. The robustness of the Vasicek model against five different portfolios with unique compositions was explored: results highlight two key findings. Firstly, the Vasicek model is a good approximation of credit losses for a portfolio that does not contain dominating loans (it is, after all, based on the assumption of large-scale homogeneity). Secondly, the Vasicek model is a good approximation for expected loss (ELs) but lacks accuracy when determining extreme unexpected losses (ULs). Finally, credit capital requirements as a function of two variables are presented which reveals novel ways of viewing these values.

A contagion process with self-exciting jumps in credit risk applications

The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the default probability of a single firm and probability of the total number of defaults by time $t$ in a homogeneous portfolio. We use a contagion process to model the arrival of credit events causing the default and develop a framework that allows firms to have resistance against default unlike the standard intensity-based models. We assume the point process driving the credit events is composed of a systematic and an idiosyncratic component, whose intensities are independently specified by a mean-reverting affine jump-diffusion process with self-exciting jumps. The proposed framework is competent of capturing the feedback effect. We further demonstrate how the proposed framework can be used to price synthetic collateralized debt obligation (CDO). Finally, we present the sensitivity analysis to demonstrate the effect of different parameters governing the contagion effect on the spread of tranches and the expected loss of the CDO.

Tail-GAN: Nonparametric Scenario Generation for Tail Risk Estimation

The estimation of loss distributions for dynamic portfolios requires the simulation of scenarios representing realistic joint dynamics of their components, with particular importance devoted to the simulation of tail risk scenarios. Commonly used parametric models have been successful in applications involving a small number of assets, but may not be scalable to large or heterogeneous portfolios involving multiple asset classes. We propose a novel data-driven approach for the simulation of realistic multi-asset scenarios with a particular focus on the accurate estimation of tail risk for a given class of static and dynamic portfolios selected by the user. By exploiting the joint elicitability property of Value-at-Risk (VaR) and Expected Shortfall (ES), we design a Generative Adversarial Network (GAN) architecture capable of learning to simulate price scenarios that preserve tail risk features for these benchmark trading strategies, leading to consistent estimators for their Value-at-Risk and Expected Shortfall. We demonstrate the accuracy and scalability of our method via extensive simulation experiments using synthetic and market data. Our results show that, in contrast to other data-driven scenario generators, our proposed scenario simulation method correctly captures tail risk for both static and dynamic portfolios.

Seismic risk assessments for real estate portfolios: Impact of engineering investigation on quality of seismic risk studies

Seismic risk evaluation studies for real estate portfolios conducted by technical professionals (often Civil and Structural Engineers) have become increasingly desirable and common in financial decisions. In this article, we develop a series of risk measures and ratings based on common outcomes from probabilistic portfolio seismic risk assessments. We first define two portfolio risk metrics: Portfolio Expected Loss (PELα) and Portfolio Upper Loss (PULα), where “α” is the annual exceedance probability, or the corresponding return period (“1/α”). PULα/PELα ratio characterizes the uncertainty in estimated portfolio risks which results from the uncertainty in seismic performance of the individual assets. Three uncertainty levels are defined, namely, low, medium, and high, based on the PULα/PELα. We then develop an asset risk metric, called Tail Contribution Index (TCIα), that characterizes the contribution of individual assets to the portfolio losses that fall within the high-consequence “tail” of the portfolio loss distribution. To describe the overall engineering efforts of a portfolio seismic risk study, we develop a portfolio risk metric, called Portfolio Level of Investigation (PLIα), that characterizes the effective level of engineering investigation. Three investigation levels are defined: low ( desktop), medium ( semi-engineered), and high ( engineered), based on the PLIα. Finally, based on the combination of uncertainty level and investigation level, we develop a rating scheme by which the quality (Qα) of a portfolio seismic risk study is characterized. Five quality levels are defined: very poor, poor, fair, good, and very good. These risk indices and ratings can help stakeholders and technical professionals better diagnose and communicate portfolio seismic risks, scope adequate studies, effectively utilize valuable resources, and base financial decisions on risk assessment results that have the desired reliability.

EFFICIENT RISK MEASURES CALCULATIONS FOR GENERALIZED CREDITRISK+ MODELS

Numerical calculations of risk measures and risk contributions in credit risk models amount to the evaluation of various forms of quantiles, tail probabilities and tail expectations of the portfolio loss distribution. Though the moment generating function of the loss distribution in the [Formula: see text] model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (Value-at-Risk and Expected Shortfall) and risk contributions for the [Formula: see text] model poses technical challenges. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced [Formula: see text] model under the common background vector framework using the Johnson curve fitting method, saddlepoint approximation method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. Once we implement the systematic procedure of finding the saddlepoint within an approximate domain, the saddlepoint approximation schemes provide efficient calculation and accurate numerical results. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and efficiency with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an extra optimization algorithm is required.

Macroeconomic shocks and credit risk stress testing the Iranian banking sector

PurposeThe Iranian banking industry has been greatly affected by dramatic changes in macroeconomic conditions over the past several decades owing to volatile oil revenues, changing fiscal and monetary policies, and the imposition of US sanctions. The main objective of this paper is to estimate potential credit losses in the Iranian banking sector due to macroeconomic shocks and assess the minimum economic capital requirements under the baseline and distressed scenarios. The paper also contrasts the applications of linear and nonlinear models in estimating the impacts of macroeconomic shocks on financial institutions.Design/methodology/approachThe paper uses a multistage approach to derive the portfolio loss distribution for banks. In the first step, the dynamic relationship between the selected macroeconomic variables are estimated using a VAR model to generate the stress scenarios. In the second step, the default probabilities are estimated using a quantile regression model and the results are compared with those of the conventional linear models. Finally, the default probabilities are simulated for a one-year time horizon using Monte-Carlo method and the portfolio loss distribution is calculated for hypothetical portfolios. The expected loss includes the loss given default for loans drawn randomly and uniformly distributed and exposed at default values when loans are assigned a fixed value.FindingsThe results indicate that the loss distributions under all scenarios are skewed to the right, with the linear model results being very similar to those of quantile at the 50% quantile, but very unlike those at the 10% and 90% quantiles. Specifically, the quantile model for the 90% (10%) quantile generates estimates of minimum economic capital requirement that are considerably higher (lower) than those using the linear model.Research limitations/implicationsThe study has focused on credit risk because of lack of data on other types of risk at individual bank level. The future studies can estimate the aggregate economic capital using a risk aggregation approach and a panel data (not presently available), which could further improve the accuracy of the estimates.Practical implicationsThe fiscal and monetary authorities in developing countries, specially oil-exporting countries, can follow the risk assessment approach to assess the health of their banking system and adapt policies to mitigate the impacts of large macroeconomic shocks on their financial markets.Originality/valueThis is the first paper estimating the portfolio loss distribution for the Iranian banks under turbulent macroeconomic conditions using linear and nonlinear models. The case study can be applied to other developing and emerging countries, particularly those highly dependent on natural resources, prone to extreme macroeconomic shocks.

Efficient Risk Measures Calculations for Generalized CreditRisk+ Models

Numerical calculations of risk measures in credit risk models amount to evaluation of various forms of tail expectations of portfolio loss distribution. Though the moment generating function of the loss distribution in CreditRisk+ model is available in analytic closed form, efficient, accurate and reliable computation of risk measures (VaR and Expected Shortfall) and risk contributions for the CreditRisk+ model pose technical challenge. We propose various numerical algorithms for risk measures and risk contributions calculations of the enhanced CreditRisk+ model under the common background vector framework using the Johnson curve fitting method, saddle-point approximation method, curve fitting method, importance sampling in Monte Carlo simulation and check function formulation. Our numerical studies on stylized credit portfolios and benchmark industrial credit portfolios reveal that the Johnson curve fitting approach works very well for credit portfolios with a large number of obligors, demonstrating high level of numerical reliability and computational efficiency. The importance sampling in Monte Carlo simulation methods are easy to implement, but they compete less favorably in accuracy and reliability with other numerical algorithms. The less commonly used check function formulation is limited to risk measures calculations. It competes favorably in accuracy and reliability, but an external industrial optimization software is required.

A Study on Operational Risk and Credit Portfolio Risk Estimation Using Data Analytics*

ABSTRACTIn this article we consider operational risk and use data analytics to estimate the credit portfolio risk. Specifically, we consider situations in which managers need to make the optimal operational decision on total provision for risk to hedge against the potential risk in the entire supply chain. We build a new structural credit model integrated with data analytics to analyze the joint default risk of credit portfolio. Our model enables the decision maker to better assess the risk of a supply chain, so that they could determine the optimal operational decisions with total provision for risk, and react in a timely manner to economic and environmental changes. We propose an efficient simulation method to estimate the default probability of the credit portfolio with the risk factors having the multivariate t‐copula. Moreover, we develop a three‐step importance sampling (IS) method for the t‐copula credit portfolio risk measurement model to achieve an accurate estimation of the tail probability of the credit portfolio loss distribution. We apply the Levenberg–Marquardt algorithm to estimate the mean‐shift vector of the systematic risk factors after the probability measure change. Besides, we empirically examine the changes in the credit portfolio risks of 60 listed Chinese firms in different industries using our proposed method. The results show that our model can help the decision maker make the optimal operational decisions with total provision for risk, which hedges against the potential risk in the entire supply chain.

Efficient Routines for CDO Loss Calculations

We revive, discuss, and improve upon the excellent earlier work of Hull and White in efficiently building iterative loss distributions for debt portfolios of structured finance securitizations. After adjusting this Hull-White technique to conserve both probability and expected loss, we develop a straightforward mathematical formalism that produces this Modified Hull-White method from a different perspective. Specifically, we find that our formalism produces Modified Hull-White with the choice of jump interpolation. We create an alternative interpolation and show with examples that this exponential alternative provides greater accuracy in generating loss distributions. A final contribution is the adaptation of a recent correlated binomial methodology to this Hull-White concept of iterative construction of the debt portfolio analysis. TOPICS:Credit risk management, CLOs, CDOs, and other structured credit, portfolio theory, portfolio construction Key Findings • We revive, discuss, and improve upon the excellent earlier work of Hull and White in efficiently building iterative loss distributions for debt portfolios of structured finance securitizations. • After adjusting this Hull-White technique to conserve both probability and expected loss, we develop a straightforward mathematical formalism that produces this Modified Hull-White method from a different perspective. • A final contribution is the adaptation of a recent correlated binomial methodology to this Hull-White concept of iterative construction of the debt portfolio analysis.

Qualitative robustness of set-valued value-at-risk

Risk measures are defined as functionals of the portfolio loss distribution, thus implicitly assuming the knowledge of such a distribution. However, in practical applications, the need for estimation arises and with it the need to study the effects of mis-specification errors, as well as estimation errors on the final conclusion. In this paper we focus on the qualitative robustness of a sequence of estimators for set-valued risk measures. These properties are studied in detail for two well-known examples of set-valued risk measures: the value-at-risk and the maximum average value-at-risk. Our results illustrate, in particular, that estimation of set-valued value-at-risk can be given in terms of random sets. Moreover, we observe that historical set-valued value-at-risk, while failing to be sub-additive, leads to a more robust procedure than alternatives such as the maximum likelihood average value at-risk.

Simple Correlated Binomial Portfolio Loss Distribution

In the quantitative analysis of the credit risk of a portfolio of bonds and loans, one industry approach imposes pairwise correlation directly on the defaults of the debt instruments in order to approximate the portfolio loss distribution. A leading credit rating agency applies a correlated binomial model for this purpose. Motivated by this example, we create two alternative correlated binomial models. We demonstrate through comparative calculations that one of these alternatives, a mixture of two zero-correlation binomial models, is superior among these three models. This mixed binomial model is also far simpler in its logical foundation and mathematical expression. We also find that the rating agency model is conceptually flawed and produces unsuitable credit loss distribution results. <b>TOPICS:</b>Fixed income and structured finance, simulations, information providers/credit ratings, credit risk management

Focussed selection of the claim severity distribution

ABSTRACTRisk assessment is a core theme within non-life insurance and estimation of quantiles far out in the upper tail is therefore one of the main applications of the total loss distribution of a non-life insurance portfolio. The choice of claim severity distribution should therefore reflect this. Therefore, we have explored how the focussed information criterion, FIC, aimed at finding the best model for estimating a given parameter of interest, the focus parameter, works as a tool for selecting the claim size distribution. As a quantile cannot be used directly as a focus parameter, we have tried different proxy focus parameters. To see how the FIC performs in this setting, compared to the other commonly used model selection methods AIC and BIC, we have performed a simulation study. In particular, we wanted to investigate the effect of the heaviness of the tail of the claim size distribution and the amount of available data. The performance of the different model selection methods was then evaluated based on the quality of the resulting estimates of the quantiles. Our study shows the best of the focussed criteria is the FIC, based on one single quantile from the claim severity distribution. Further, the performance of the FIC is mostly either comparable to or considerably better than that of the BIC, which is the best performing of the state of the art approaches. In particular, the FIC works well when the data are heavy-tailed, when the sample size is rather low and when the parameter of interest is the quantile far out in the tail of the total loss distribution.

How Sensitive are Tail-Related Risk Measures in a Contamination Neighbourhood?

Estimation or mis-specification errors in the portfolio loss distribution can have a considerable impact on risk measures. This paper investigates the sensitivity of tail-related risk measures including the Value-at-Risk, expected shortfall and the expectile-quantile transformation level in an epsilon-contamination neighbourhood. The findings give the different approximations via the tail heaviness of the contamination models and its contamination levels. Illustrating examples and an empirical study on the dynamic CRIX capturing and displaying the market movements are given.

On a Quest for Robustness: About Model Risk, Randomness and Discretion in Credit Risk Stress Tests

In this paper we study the impact of model uncertainty, which occurs when linking a scenario to default probabilities, on reduced-form credit risk testing. This type of uncertainty is omnipresent in most macroeconomic testing applications due to short time series for banks' portfolio risk parameters and highly collinear macroeconomic covariates. We quantify the effect of model uncertainty on supervisory and bank tests in terms of predicted portfolio loss distributions and implied capital shortfalls by conducting a full-edged top-down credit risk test for over 1,500 German banks. Our results suggest that the impact of model uncertainty on predicted capital shortfalls can be huge, even among models with similar predictive power. This leaves both banks and supervisors with uncertainty when calculating impacts and implied capital requirements. To mitigate the impact of uncertainty, we suggest a modeling approach which filters the model space by combining the standard Bayesian model averaging (BMA) paradigm with a structural filter derived from the Merton/Vasicek credit risk model. Applying our testing framework, the dispersion decreases and the median effect is reduced from -5.0pp of CET1 ratio under the BMA model to -2.5pp under the structurally augmented BMA model, while the predicted capital shortfall is reduced by 70 %. The structural filter eliminates extreme outcomes on both sides of the forecast distribution, leading in our application to the German banking sector to a reduction in impact compared to the model without the stress testing plausibility filter.

Importance Sampling for Credit Portfolio Risk with Risk Factors Having t-Copula

This paper proposes an efficient simulation method for calculating credit portfolio risk when risk factors have a heavy-tailed distributions. In modeling heavy tails, its features of return on underlying asset are captured by multivariate [Formula: see text]-Copula. Moreover, we develop a three-step importance sampling (IS) procedure in the [Formula: see text]-copula credit portfolio risk measure model for further variance reduction. Simultaneously, we apply the Levenberg–Marquardt algorithm associated with nonlinear optimization technique to solve the problem that estimates the mean-shift vector of the systematic risk factors after the probability measure change. Numerical results show that those methods developed in the [Formula: see text]-copula model can produce large variance reduction relative to the plain Monte Carlo method, to estimate more accurately tail probability of credit portfolio loss distribution.

Default Correlations and Large-Portfolio Credit Analysis

A factor model with sparsely correlated residuals is used to model short-term probabilities of default and other corporate exits while permitting missing data, and serves as the basis for generating default correlations. This novel factor model can then be used to produce portfolio credit risk profiles (default-rate and portfolio-loss distributions) by complementing an existing credit portfolio aggregation method with a novel simulation–convolution algorithm. We apply the model and the portfolio aggregation method on a global sample of 40,560 exchange-listed firms and focus on three large portfolios (the U.S., Eurozone-12, and ASEAN-5). Our results reaffirm the critical importance of default correlations. With default correlations, both default-rate and portfolio-loss distributions become far more right-skewed, reflecting a much higher likelihood of defaulting together. Our results also reveal that portfolio credit risk profiles evaluated at two different time points can change drastically with moving economic conditions, suggesting the importance of modeling credit risks with a dynamic system. Our factor model coupled with the aggregation algorithm provides a useful tool for active credit portfolio management.

A credit portfolio framework under dependent risk parameters: probability of default, loss given default and exposure at default

This paper introduces a credit portfolio framework that allows for dependencies between default probabilities, secured and unsecured recovery rates and exposures at default (EADs). The overall approach is an extension of the factor models of Pykhtin (2003) and Miu and Ozdemir (2006), with respect to differentiated recovery rates and the inclusion of dependent exposures. As there is empirical evidence for dependence between these risk parameters and observations for the EAD, and since the secured and unsecured recovery rates are available only in the case of a default, we propose a multivariate extension of the selection model of Heckman in order to estimate the unknown parameters within a maximum likelihood framework. Finally, we empirically demonstrate the effects of the dependence structure on the portfolio loss distribution and its risk measure for a hypothetical loan portfolio.

The Two Defaults Scenario for Stressing Credit Portfolio Loss Distributions

The impact of a stress scenario of default events on the loss distribution of a credit portfolio can be assessed by determining the loss distribution conditional on these events. While it is conceptually easy to estimate loss distributions conditional on default events by means of Monte Carlo simulation, it becomes impractical for two or more simultaneous defaults as then the conditioning event is extremely rare. We provide an analytical approach to the calculation of the conditional loss distribution for the CreditRisk + portfolio model with independent random loss given default distributions. The analytical solution for this case can be used to check the accuracy of an approximation to the conditional loss distribution whereby the unconditional model is run with stressed input probabilities of default (PDs). It turns out that this approximation is unbiased. Numerical examples, however, suggest that the approximation may be seriously inaccurate but that the inaccuracy leads to overestimation of tail losses and, hence, the approach errs on the conservative side.

Loss distributions: computational efficiency in an extended framework

Credit risk models are difficult to implement in an actionable form. While near realtime results are required for pricing credits, making origination decisions and optimizing portfolio allocation, the complexity of credit models (which are often employed on portfolios spanning millions of exposures) typically requires either expensive Monte Carlo simulations or the imposition of inflexible assumptions to compute the portfolio loss distribution. The contribution of this paper to the credit risk literature is twofold: Credit Suisse's CreditRisk+ framework is significantly generalized, and an algorithm from the option pricing literature is introduced to retain precision and speed for the computation of the loss distribution even for very large portfolios. This generalization allows for time-dependent portfolios, fully accounts for granularity and concentration within the credit portfolio and does not rely on assumptions of asymptotically large credit portfolios. The algorithm allows even a standard laptop to precisely compute an entire bank's loss distribution. The core results of this paper are as follows: the model uses a stochastic process as the state variable yet retains tractability; the model integrates liquidity risk into the credit loss distribution; the algorithm from the option pricing literature is leveraged for efficient computation.

Computation of credit portfolio loss distribution by a cross entropy method

Quantification and management of credit risk is always crucial for the financial industry. Computing credit risk is generally a challenging task while correlated defaults exist. Traditional approaches such as exponential twisting are model specific and often involve difficult analysis, therefore computational methods are sought to estimate the credit risk when analysis is unavailable. The accurate measurement of credit risk is often a rare-event simulation problem, i.e., calculating probabilities (which are usually small) of extreme losses. It is well-known that the Monte Carlo (MC) method may become slow and expensive for such problems. Importance sampling (IS), a variance-reduction technique, can then be utilized for rare-event simulation for credit risk management. In this work, we propose the implementation of a special IS procedure, the cross-entropy (CE) method, to simulate credit risk models. More specifically, we obtain iteratively biasing probability density functions (PDF’s) for credit portfolio losses by the CE method, and then combine the results from each stage by the technique of multiple importance sampling to obtain a complete PDF. The main advantage of this method is that it can avoid the nontrivial analysis required by a general IS method, and therefore simplifies the estimation of loss distributions. Moreover, this approach is generic and can be applied to a wide variety of models with little modifications. In particular, we apply this approach to a normal copula model and a t-copula model to estimate the probabilities of extreme portfolio losses under the models. Numerical examples are provided to demonstrate the performance of our method.