Individual Obligors Research Articles

Optimal Loan Portfolio under Regulatory and Internal Constraints

The environment surrounding banks is becoming increasingly severe. Particularly, to prevent the next financial crisis, Basel III requires financial institutions to prepare higher levels of capitals by 1 January 2028, and the financial stability board (FSB) suggests the risk appetite framework (RAF) as their internal risk management. Hence, efficient usage of their own capitals for banks is more important than ever to improve profitability. Under such circumstances, this paper is the first to consider an optimization problem for a typical loan portfolio of international banks under comprehensive risk constraints with realistic profit margins and funding costs to achieve an efficient capital allocation. Concretely, after taking concentration risks on large individual obligors into account, we obtain a loan portfolio that attains the maximum profit under Basel regulatory capital and loan market constraints, as well as internal management constraints, namely, risk limits on business units and industrial sectors. Moreover, we separately calculate credit risk amounts of the internal constraints in terms of regulatory and economic capitals to compare the optimized profits. In addition, considering sharp increases in default probabilities of all obligors as in the global financial crisis, we perform a stress test on the optimization results to investigate the effects of changes in risk amounts and profits. As a result, we propose to unify risk constraints on the business units and industrial sectors by using credit risk amounts in terms of economic capitals.

A transient Cramér–Lundberg model with applications to credit risk

AbstractThis paper considers a variant of the classical Cramér–Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e. the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. As well as an exact analysis (in terms of transforms) of this ruin probability, an asymptotic analysis is performed, including an efficient importance-sampling-based simulation approach.The base model is extended in multiple dimensions: (i) we consider a model in which there may, in addition, be losses that do not correspond to defaults, (ii) then we analyze a model in which the individual obligors are coupled via a regime switching mechanism, (iii) then we extend the model so that between the losses the reserve process behaves as a Brownian motion rather than a deterministic drift, and (iv) we finally consider a set-up with multiple groups of statistically identical obligors.

A dynamic Bayesian approach for probability of default and stress test

Obligor defaults are cross-sectionally correlated as obligors share common economic conditions; in addition obligors are longitudinally correlated so that an economic shock like the IMF crisis in 1998 lasts for a period of time. A longitudinal correlation should be used to construct statistical scenarios of stress test with which we replace a type of artificial scenario that the banks have used. We propose a Bayesian model to accommodate such correlation structures. Using 402 obligors to a domestic bank in Korea, our model with a dynamic correlation is compared to a Bayesian model with a stationary longitudinal correlation and the classical logistic regression model. Our model generates statistical financial statement under a stress situation on individual obligor basis so that the genearted financial statement produces a similar distribution of credit grades to when the IMF crisis occurred and complies with Basel IV (Basel Committee on Banking Supervision, 2017) requirement that the credit grades under a stress situation are not sensitive to the business cycle.

Capital Structure Arbitrage: Model Choice and Volatility Calibration

When identifying relative value opportunities across credit and equity markets, the arbitrageur faces two major problems, namely positions based on model misspecification and mismeasured inputs. Using credit default swap data, this paper addresses both concerns in a convergence-type trading strategy. In spite of differences in assumptions governing default and calibration, we find the exact structural model linking the markets second to timely key inputs. Studying an equally-weighted portfolio of all relative value positions, the excess returns are insignificant when based on a traditional volatility from historical equity returns. However, relying on an implied volatility from equity options results in a substantial gain in strategy execution and highly significant excess returns - even when small gaps are exploited. The gain is largest in the speculative grade segment, and cannot be explained from systematic market risk factors. However, although the strategy may seem attractive at an aggregate level, positions on individual obligors can be very risky.

DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

Approximation of Economic Capital of Loan Portfolio with Guarantors

This paper derives a closed form formula for value at risk (VaR) and expected shortfall (ES) under a widely used Merton type multi-factor model for a credit portfolio with guaranteed loans. Typically used Monte Carlo simulations tend to converge slowly when the default probabilities are low and the confidence level is high, especially when the loans are guaranteed by highly rated guarantors. The problem is even more pronounced when the contributions of individual obligors are considered, and it is difficult to obtain sufficiently accurate and stable incremental VaR or ES of a new loan via Monte Carlo simulations. The formula presented here yields accurate results, and can also be used for calculating risk contributions of individual loans, as well as incremental risks of new loans. The accuracy of the formula is illustrated by comparing the results from the closed form formula to a large number of Monte Carlo simulations.

Simulating Risk Contributions of Credit Portfolios

The 2007–2009 financial turmoil highlighted the need for more active management of credit portfolios. After measuring portfolio credit risk, an important step toward active risk management is to measure risk contributions of individual obligors to the overall risk of the portfolio. In practice, value-at-risk is often used as a risk measure for credit portfolios, and it can be decomposed into a sum of the risk contributions of individual obligors. Estimation of these risk contributions is computationally challenging, mainly because they are expectations conditioned on a rare event. In this paper, we tackle this computational problem by developing a restricted importance sampling (RIS) method for a class of conditional-independence credit risk models, where defaults of obligors are conditionally independent given an appropriately chosen random vector. We propose fast estimators for risk contributions and their confidence intervals. Furthermore, we study the incorporation of traditional importance sampling methods into the RIS method to further improve its efficiency for the widely used Gaussian copula model. Numerical examples show that the proposed method works well.

Economic Capital Modeling Closed Form Approximation for Real-Time Applications

Economic capital (ECap) modeling is a fundamental part of Pillar II of the Basel framework. Indeed, 'sophisticated' financial institutions need to have in place internal models for the assessment of the level of the overall capital buffer which is deemed sufficient to cover the risk of their business activities. On top, ECap models are also frequently used for pricing purposes on an ex-ante basis: financial institutions need to know the incremental economic capital (IECap), i.e. the size by which the overall capital buffer needs to be increased after addition of e.g. a single new loan to the existing portfolio. This is important in order to be able to price such additional loan accordingly. Finally, ECap contributions (ECapC) are also required ex-post in order to break down the overall capital buffer to the individual obligors, products etc. within the portfolio. Simulation of IECap and ECapC can be computationally expensive and unstable, but it appears that closed form approximations provide accurate, consistent and quick solutions in many cases. The formula introduced here is based on the multi-factor approximation from [Pykhtin, 2004] applicable to a default-mode Merton type model. As such, default correlations between obligors (stemming from a multi-factor-model) are taken into account, but the formula also captures specific amortization schedules and loss given default (LGD) values for each individual position -- a feature which is of practical relevance, but often neglected in standard default-mode models. For the time being, credit-risk mitigants such as the existence of guarantors on individual loans, are not captured by the formula, neither are credit migrations or correlations between default probabilities and LGDs. Our formula allows for approximation of all ECap contributions without extra computational cost. After calculation of the ECap of the original portfolio, IECap can be computed within few seconds and more accurately than in standard linear approximations based on ECap contributions.

Multiperiod Corporate Default Prediction with the Partially-Conditioned Forward Intensity

The forward-intensity model of Duan, {et al} (2012) is a parsimonious and practical way for predicting corporate defaults over multiple horizons. However, it has a noticeable shortcoming because default correlations through intensities are conspicuously absent when the prediction horizon is more than one data period. We propose a new forward-intensity approach that builds in correlations among intensities of individual obligors by conditioning all forward intensities on the future values of some common variables, such as the observed interest rate and/or a latent frailty factor. The new model is implemented on a large sample of US industrial and financial firms spanning the period 1991-2011 on the monthly frequency. Our findings suggest that the new model is able to correct the structural biases at longer prediction horizons reported in Duan et al (2012). Not surprisingly, default correlations are also found to be important in describing the joint default behavior.

An approximation for credit portfolio losses

Mixture models play an important role in the modeling of portfolio losses. In these models the risk of default of individual obligors (indexed by i ∈ {1, . . . , m}) depends on an underlying set of common economic factors, denoted Ψ. Given these factors, the losses due to default li of individual obligors are assumed to be stochastically independent. Dependence between different obligors stems only from dependence of the individual default probabilities on the set of factors. These models are used for both risk management of credit portfolios and valuation of multi-name credit derivatives. The current article investigates both issues. The numerical evaluation of the portfolio loss distribution is usually based on the two-stage structure of mixture models. For instance, in order to sample from the loss distribution by standard Monte Carlo, one generates first a realization of the systematic factor variable Ψ. In a second step one generates a sequence of independent variates li, 1 ≤ i ≤ m, according to the conditional distribution of (li)1≤i≤m given Ψ. Standard Monte Carlo can be quite slow, and so various numerical techniques for estimating the distribution of the total loss of a portfolio in mixture models have been developed. In this paper we focus on the the second stage, i.e. the conditional loss distribution given the underlying factors, and propose an alternative way for evaluating the conditional distribution of the total loss L = ∑m i=1 li. Our approximation is based on a central limit theorem; error bounds can be derived from the Berry-Esseen-inequality. We compare the numerical performance of our method relative to the standard Vasicek-approximation and the true loss distribution obtained by standard Monte Carlo methods. It turns out that our suggested approximation technique often provides more accurate results than the Vasicek-approximation while being computationally less expensive

Measuring marginal risk contributions in credit portfolios

We consider the problem of decomposing the credit risk in a portfolio into a sum of risk contributions associated with individual obligors or transactions. For some standard measures of risk – including value-at-risk and expected shortfall – the total risk can be usefully decomposed into a sum of marginal risk contributions from individual obligors. Each marginal risk contribution is the conditional expected loss from that obligor, conditional on a large loss for the full portfolio. We develop methods for calculating or approximating these conditional expectations. Ordinary Monte Carlo estimation is impractical for this problem because the conditional expectations defining the marginal risk contributions are conditioned on rare events. We develop three techniques to address this difficulty. First, we develop importance sampling estimators specifically designed for conditioning on large losses. Next, we use the analysis underlying the importance sampling technique to develop a hybrid method that combines an approximation with Monte Carlo. Finally, we take this approach a step further and develop a rough but fast approximation that dispenses entirely with Monte Carlo. We develop these methods in the Gaussian copula framework and illustrate their performance in multifactor models.

An approximation method for analysis and valuation of credit correlation derivatives

This paper presents a model for approximating the value of a basket of default-correlated assets and analyzes subordinate tranches in securitized debt obligations. The model is calibrated to an intensity-based simulation of correlated defaults and represents an alternative computation method to full Monte Carlo simulation. Timing of individual obligor defaults are driven by intensity processes and collateral value is modeled with a jump-diffusion process where the number of jumps corresponds to the total number of defaults in the asset pool. This approach allows decomposition of subordinate obligations in terms of a collection of simpler securities and yields useful risk management information.

Measuring Marginal Risk Contributions in Credit Portfolios

We consider the problem of decomposing the credit risk in a portfolio into a sum of risk contributions associated with individual obligors or transactions. For some standard measures of risk - including value-at-risk and expected shortfall - the total risk can be usefully decomposed into a sum of marginal risk contributions from individual obligors. Each marginal risk contribution is the conditional expected loss from that obligor, conditional on a large loss for the full portfolio. We develop methods for calculating or approximating these conditional expectations. Ordinary Monte Carlo estimation is impractical for this problem because the conditional expectations defining the marginal risk contributions are conditioned on rare events. We develop three techniques to address this difficulty. First, we develop importance sampling estimators specifically designed for conditioning on large losses. Next, we use the analysis underlying the importance sampling technique to develop a hybrid method that combines an approximation with Monte Carlo. Finally, we take this approach a step further and develop a rough but fast approximation that dispenses entirely with Monte Carlo. We develop these methods in the Gaussian copula framework and illustrate their performance in multifactor models.

Copula-Dependent Defaults in Intensity Models

In this paper we present a new approach to incorporate dynamic default dependency in intensity-based default risk models. The model uses an arbitrary default dependency structure which is specified by the Copula of the times of default, this is combined with individual intensity-based models for the defaults of the obligors without loss of the calibration of the individual default-intensity models. The dynamics of the survival probabilities and credit spreads of individual obligors are derived and it is shown that in situations with positive dependence, the default of one obligor causes the credit spreads of the other obligors to jump upwards, as it is experienced empirically in situations with credit contagion. For the Clayton copula these jumps are proportional to the pre-default intensity. If information about other obligors is excluded, the model reduces to a standard intensity model for a single obligor, thus greatly facilitating its calibration. To illustrate the results they are also presented for Archimedean copulae in general, and Gumbel and Clayton copulae in particular. Furthermore it is shown how the default correlation can be calibrated to a Gaussian dependency structure of CreditMetrics-type.