Conditional Default Probabilities Research Articles

Estimating actual probability of default from structural models

In this paper, we study the probability of default, the credit default swap (CDS) implied probability of default, and the estimated actual probability of default from the structural models. We first show that the conditional risk-neutral probability of default is totally different from the CDS implied probability of default, and the CDS implied probability of default provides biased estimation of a corporation’s actual probability of default. Hence, the common belief that the CDS implied probability of default gives unbiased (or reliable) estimations of the firm’s probability of default is false. Furthermore, we use the credit spread risk premium as the difference of the physical measure and the risk-neutral measure of the credit spread to estimate the actual bond spread, and relate the credit spread risk premium in terms of the market price of risk and the credit spread volatility. Using the credit spread volatility and the risk-neutral credit spread and the market price of risk in hybrid HPD model, we obtain the estimated actual probability of default in a structural model. The estimated actual probability of default would be useful for credit rating agencies, investors, credit risk managers and regulators.

Credit Risk: Simple Closed-Form Approximate Maximum Likelihood Estimator

Interpretable, Computationally Tractable Approximate Parameter Estimation for Corporate Defaults

A two‐stage Bayesian network model for corporate bankruptcy prediction

AbstractWe develop a Bayesian network (LASSO‐BN) model for firm bankruptcy prediction. We select financial ratios via the Least Absolute Shrinkage Selection Operator (LASSO), establish the BN topology, and estimate model parameters. Our empirical results, based on 32,344 US firms from 1961–2018, show that the LASSO‐BN model outperforms most alternative methods except the deep neural network. Crucially, the model provides a clear interpretation of its internal functionality by describing the logic of how conditional default probabilities are obtained from selected variables. Thus our model represents a major step towards interpretable machine learning models with strong performance and is relevant to investors and policymakers.

네트워크 모형을 이용한 우리나라 은행부문의 시스템리스크 측정 (Measurement of the Korean Banking Sector Systemic Risk using Network Model)

Korean Abstract: 글로벌 금융위기 이후 시스템리스크의 중요성이 부각되고 있다. 본 논문은 은행부문에서 개별 은행의 신용위험을 측정하고 은행 사이의 네트워크 구조에 따른 위험 전이를 반영한 시스템리스크 측정방법을 사용하였다. 먼저 개별 금융기관 재무 상태를 반영한 신용위험의 시계열적 변화를 살펴보았다. 또한 신용위험 측정 과정에서 추산된 개별 은행 자산가치 사이의 횡단면적 연계구조를 상관계수, 조건부 부도발생확률, 결합부도확률, Granger 인과관계 등 다양한 방법으로 추정하여 은행 부문 시스템리스크를 측정하고, 도출된 시스템리스크의 시계열적 움직임을 살펴보았다. 다음으로 시스템리스크 내에서 개별 은행들이 가지고 있는 파생적 특성을 찾고, 각 은행이 시스템 연계구조 네트워크에서 어떤 역할을 하고 있는지, 어떤 은행이 중심적 위치에 해당하는 지에 대해 분석하였다. 이는 기존 시스템리스크 연구들이 시도하지 못한 부분이다. 본 논문이 사용한 분석방법은 은행부문 시스템 안정성을 관리 감독해야 하는 감독당국에게 다양한 시스템리스크 척도를 제공할 뿐만 아니라 상시로 시장을 감독할 수 있는 유용한 수단을 제시한다. 결론적으로 시장 전체 특성을 파악하는 척도와 함께 개별은행의 미시적 특성도 같이 제공한다는 측면에서 본 연구 의의가 있다. English Abstract: The importance of systemic risk has been highlighted since the 2008 global financial crisis. In this paper, we propose a model that considers the cross-sectional and time series aspects of systemic risk simultaneously. Using a network model, we analyze the systemic risk within the banking sector. As a result, we measured the time series of credit risk reflecting the financial status of individual financial institutions. We also estimate the banking sector systemic risk using various methods such as correlation coefficient, conditional default probability, joint default probability, and Granger causality method. The analysis method of this paper provides a variety of systemic risk measures to central banks that analyze the stability of the system in the banking sector and may prove a useful means of supervising the market at all times. In conclusion, this study is meaningful in that it provides both the characteristics of the market as a whole and the micro characteristics of individual banks connected with system risk.

A Simple Sovereign Default Model

A very influential model in the determination of capital requirements for credit exposures is Vacisek’s model. This model underpins the Basel framework for credit exposures of financial institutions. Vacisek’s model uses information on asset correlation and the unconditional mean of the probability of default to derive the conditional probability of default under stress. However, the concept of asset values applies best to corporations rather than to sovereign exposures. We adapt Vacisek’s model to capture the event of a sovereign default that results from a country hitting certain thresholds of economic distress. A reasonable assumption is that a significant drop in foreign reserve assets triggers the default event in a developing country as observed during the 1980s debt crisis, in which highly indebted Latin American countries and other developing regions were unable to repay the debt.

Systemic risk in clearing houses: Evidence from the European repo market

We study how crises affect Central Clearing Counterparties (CCPs). We focus on a large and safe segment of the CCP-cleared repo market during the Eurozone sovereign debt crisis. We develop a simple model to infer CCP stress, which is measured as repo rates’ sensitivity to sovereign credit default swaps (CDS) spreads and jointly captures (1) the effectiveness of haircut policies, (2) CCP-member default risk (conditional on sovereign default), and (3) CCP default risk (conditional on both sovereign and CCP-member default). During 2011, repo rates strongly respond to sovereign risk, particularly for Greece, Italy, Ireland, Portugal and Spain (GIIPS): Repo investors behaved as if the conditional probability of CCP default was substantial.

Primary-firm-driven portfolio loss

Many financial institutions provide loans to secondary firms, whose economic survival depends on the economic condition of primary firms. Even if loans from primary firms are not held in the loan portfolio, the financial distress of primary firms can adversely affect the loan portfolio of a financial institution. This paper describes a simple model that can be used for risk management. Our model directly incorporates the dependence of the conditional probability of default and loss given default of secondary firms on primary firms. Two simple examples show that failure to account for such dependence can result in the value-at-risk and the expected shortfall being greatly underestimated.

Contingent Claims Analysis of Sovereign Default Risk in the Eurozone

We study sovereign default risk as measured by credit default swap (CDS) spreads of Eurozone member states between 2008 and 2016. Applying a structural credit risk model we analyze to what extent contingent claims analysis can explain spreads given fundamental balance sheet information. First results confirm that market implied default risk is hardly explainable by available fundamentals. We therefore model the asset value with a jump-diffusion process, which we calibrate to market implied default probabilities. The resulting jump intensities are substantial for both, distressed as well as non-distressed countries. By extending the jump-diffusion model and considering the first-passage time default possibility, our model is able to represent more realistic proportions of jump and pure diffusion risk. Including the first-passage time feature and analyzing conditional default probabilities via correlated jump-diffusion processes further contributes to the analysis of credit contagion and systemic risk. Our approach documents that the largest systemic risk component of the periphery comes from Italy, the smallest from Ireland.

Implied fractional hazard rates and default risk distributions

Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional probability density functions (FPD), however, are not in general conventional probability density functions (Tapiero and Vallois, Physica A,. Stat. Mech. Appl. 462:1161–1177, 2016). As a result, a fractional FPD does not define a fractional hazard rate. However, a fractional hazard rate implies a unique and conventional FPD. For example, an exponential distribution fractional hazard rate implies a Weibull probability density function while, a fractional exponential probability distribution is not a conventional distribution and therefore does not define a fractional hazard rate. The purpose of this paper consists of defining fractional hazard rates implied fractional distributions and to highlight their usefulness to granular default risk distributions. Applications of such an approach are varied. For example, pricing default bonds, pricing complex insurance contracts, as well as complex network risks of various granularity, that have well defined and quantitative definitions of their hazard rates.

Forecasting Cross-Sections of Frailty-Correlated Default

We propose a novel econometric model for estimating and forecasting cross-sections of time-varying conditional default probabilities. The model captures the systematic variation in corporate default counts across e.g. rating and industry groups by using dynamic factors from a large panel of selected macroeconomic and flnancial data as well as common unobserved risk factors. All factors are statistically and economically signiflcant and together capture a large part of the time-variation in observed default rates. In this framework we improve the out-of-sample forecasting accuracy associated with conditional default probabilities by about 10-35% in terms of Mean Absolute Error, particularly in years of default stress.

Calibration of Credit Default Probabilities in Discrete Default Intensity and Logit Models

We consider discrete default intensity based and logit type reduced form models for conditional default probabilities for corporate loans where we develop simple closed form approximations to the maximum likelihood estimator (MLE) when the underlying covariates follow a stationary Gaussian process. In a practically reasonable asymptotic regime where the default probabilities are small, say $1-3\%$ annually, the number of firms and the time period of data available is reasonably large, we rigorously show that the proposed estimator behaves similarly or slightly worse than the MLE when the underlying model is correctly specified. For more realistic case of model misspecification, both estimators are seen to be equally good, or equally bad. Further, beyond a point, both are more-or-less insensitive to increase in data. These conclusions are validated on empirical and simulated data. The proposed approximations should also have applications outside finance, where logit-type models are used and probabilities of interest are small.

Systemic risk in the European sovereign and banking system

We investigate the systemic risk of the European sovereign and banking system during 2008–2013. We utilize a conditional measure of systemic risk that reflects market perceptions and can be intuitively interpreted as an entity’s conditional joint probability of default, given the hypothetical default of other entities. The measure of systemic risk is applicable to high dimensions and not only incorporates individual default risk characteristics but also captures the underlying interdependent relations between sovereigns and banks in a multivariate setting. In empirical applications, our results reveal significant time variation in systemic risk spillover effects for the sovereign and banking system. We find that systemic risk is mainly driven by risk premiums coupled with a steady increase in physical default risk.

CoRisk: Measuring Systemic Risk Through Default Probability Contagion

We propose a novel systemic risk measurement model, based on stochastic processes, correlation networks and conditional probabilities of default.For each country we consider three different spread measures, one for each sector of the economy (sovereigns, corporates, banks), and we model each of them as a linear combination of two stochastic processes: a country-specific idiosyncratic component and a common systematic factor. We then build a partial correlation network model, and by combining it with the spread measures we derive the conditional default probabilities of each sector. Comparing them with the unconditional ones, we obtain the CoRisk, which measures the variation in the probability of default due to contagion effects. Our measurement model is applied to understand the time evolution of systemic risk in the economies of the European monetary union, in the recent period. The results show that, overall, the sovereign crisis has increased systemic risks more than the financial crisis. In addition, peripheral countries turn out to be exporters, rather than importers of systemic risk, and, conversely, core countries.

Systemic Risk in Clearing Houses: Evidence from the European Repo Market

How do crises affect Central clearing Counterparties (CCPs)? We focus on CCPs that clear and guarantee a large and safe segment of the repo market during the Eurozone sovereign debt crisis. We start by developing a simple framework to infer CCP stress, which can be measured through the sensitivity of repo rates to sovereign CDS spreads. Such sensitivity jointly captures three effects: (1) the effectiveness of the haircut policy, (2) CCP member default risk (conditional on sovereign default) and (3) CCP default risk (conditional on both sovereign and CCP member default). The data show that, during the sovereign debt crisis of 2011, repo rates strongly respond to movements in sovereign risk, in particular for GIIPS countries, indicating significant CCP stress. Our model suggests that repo investors behaved as if the conditional probability of CCP default was very large. JEL Classification: E58, E43, G01, G21

A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. We apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors, and select parameter values by minimizing the Kullback–Leibler divergence of the resulting parametric family from the ideal (zero-variance) importance density. Optimal parameter values are shown to satisfy intuitive moment-matching conditions, and the asymptotic behaviour of large portfolios is used to approximate the requisite moments. In a sense we generalize the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models. We show how to implement our algorithm in the t copula model and compare its performance there to the algorithm developed by Chan and Kroese (2010). We find that our algorithm requires substantially less computational time (especially for large portfolios) but is slightly less accurate. Our algorithm can also be used to estimate more general risk measures, such as conditional tail expectations, whereas Chan and Kroese (2010) is specifically designed to estimate loss probabilities.

A General Importance Sampling Algorithm for Estimating Portfolio Loss Probabilities in Linear Factor Models

This paper develops a novel importance sampling algorithm for estimating the probability of large portfolio losses in the conditional independence framework. In a sense we modify the algorithm of Glasserman and Li (2005) so that it can be applied in a wider variety of models, including the t copula. The basic idea is to apply exponential tilts to (i) the distribution of the natural sufficient statistics of the systematic risk factors and (ii) conditional default probabilities, given the simulated values of the systematic risk factors. Motivated by the cross-entropy method we determine optimal values of the tilt parameters by minimizing the Kullback-Liebler divergence of the candidate family from the ideal (i.e. zero-variance) importance sampling measure, leading to an intuitive moment-matching characterization. Finally, by exploiting the large portfolio approximation to the portfolio loss, we develop efficient and accurate approximations to the requisite moments. In the context of the Gaussian copula the proposed algorithm offers comparable performance to the Glasserman and Li (2005) algorithm, which is known to be asymptotically optimal. In the context of the t copula the proposed algorithm offers comparable performance for substantially less computational time than the Chan and Kroese (2010) algorithm, which also has good asymptotic properties.

Default Time from Tuberculosis Treatment in the Southern Republic of Benin Using Mixture Cure Model for Survival Analysis

Background: To date, this is the first statistical analysis of default time from tuberculosis (TB) treatment conducted in the republic of Benin. This cohort study assessed the cured fraction, the conditional probability of default (CPD) from treatment course and identified the risk factors predicting its timing. Methods: TB case was a patient with positive culture for the Mycobacterium tuberculosis complex or with two sputum smears positive for acid-fast bacilli. The 2008’s cohort of TB patients was extracted from Benin national TB registry. Active TB Patients who were diagnosed between November 2007 and December 2008 were included and followed up to December, 23 rd 2009 relative to their specific initiation date. Default was defined using the WHO guidelines. Primary endpoint was the time to default from the anti-TB treatment course. Risk factors were assessed with univariate and multivariate Cox regression integrating the mixture cure models. Results: More than 5% (n=64) of defaulters were identified with a cured fraction of 94.1%. About 3% of patients defaulted within the first 2 months (CPD = 0.025 ± 0.004). HIV/AIDS co-infection and TB history were independently associated with default time (log-rank = 35.21; P < 0.0001 and log-rank = 10.11; P = 0.0015 respectively). With Cox proportional hazards (PH) analysis, predictors of default time were HIV/AIDS, TB history and Age (hazard ratio [HR] = 3.82 [2.28; 6.41], P < 0.0001 for HIV-positive, HR = 0.13 [0.03; 0.53], P = 0.0045 for previous-TB infection and Age (HR = 3.49 [1.25; 9.74], P = 0.0170 for 65 years old or more, respectively). With logistic Cox PH mixture cure model HIV/AIDS and Age significantly increased the probability of default (Odds Ratios [OR] = 4.04 [2.64; 7.08], P < 0.0001 for HIV-positive, and OR = 4.23 [1.58; 12.10], P = 0.006 for 65 years old or more, respectively) whereas TB history significantly reduced default probability (OR = 0.04 [0.004; 0.54], P = 0.0148 for previous-TB infection. Conclusion: Whatever the alternative model considered, this study provides the first evidence that HIV/AID, TB history and Age were the major predictive factors of default time from anti-TB treatment in Benin. Therefore, additional efforts to improve the compliance of patients with anti-TB treatment through a better management of the co-infection with HIV/AIDS in accordance with patient’s specific age group may be an important feature of a prospective TB control strategy in the future. This should be emphasized in the early treatment course for these subgroups of TB patients.

Learning Network Structure of Financial Institutions from CDS Data

Default contagion is a concern especially after witnessing the financial crisis. How the financial institutions are connected is crucial to how likely they will default together. However, the information on financial network structure is not available. This paper builds a probabilistic graphical model relating the network structure to observable prices. The financial network structure can be learned using CDS data. Conditional default probability and expected conditional loss can be calculated accordingly. As the information is extracted from the CDS price, all quantities are under the risk-neutral measure, and thus account for the market fear that can trigger bank run. This knowledge can be used to gauge systemic risk and inform policy decisions.

Sovereign Crises and Bank Financing: Evidence from the European Repo Market

How do crises affect Central clearing Counterparties (CCPs)? We focus on CCPs that clear and guarantee a large and safe segment of the repo market during the Eurozone sovereign debt crisis. We start by developing a simple framework to infer CCP stress, which can be measured through the sensitivity of repo rates to sovereign CDS spreads. Such sensitivity jointly captures three effects: (1) the effectiveness of the haircut policy, (2) CCP member default risk (conditional on sovereign default) and (3) CCP default risk (conditional on both sovereign and CCP member default). The data show that, during the sovereign debt crisis of 2011, repo rates strongly respond to movements in sovereign risk, in particular for GIIPS countries, indicating significant CCP stress. Our model suggests that repo investors behaved as if the conditional probability of CCP default was very large.

The large homogeneous portfolio approximation with a two-factor Gaussian copula and random recovery rate

In this paper we consider the large homogeneous portfolio (LHP) approximation with a two-factor Gaussian copula and random recovery rate. In addition, we assume that the earlier the default occurs, the less the asset recovers; in other words, random recovery rate and individual default times have a positive rank correlation. Under the LHP assumption, the conditional cumulative loss of the reference portfolio is approximated by the product of loss given default and conditional default probability. In order to derive semi-analytic formulas for the loss distribution and the expected tranche loss, we use a Gaussian two-factor model and assume that the recovery rate depends on one systematic factor. In addition, we consider stochastic correlation for a better fit to credit default swap index tranches. The derived semi-analytic formula only involves integration with respect to the standard normal density and can be computed by Gauss–Hermite quadrature. Numerical tests show that the two-factor model with stochastic correlation and random recovery fits iTraxx tranche premiums better than other correlation or recovery assumptions under the Gaussian LHP framework. We also apply our model to credit risk assessment such as value-at-risk of the loss distribution.