Large Homogeneous Portfolio Research Articles

Unveiling Mispricing Risks: Nonlarge Homogeneous Portfolio Factor Copula Models for Enhanced Valuation of Subordinated Loan Securitization

AbstractThis paper presents an innovative factor copula model for collateralized loan obligation (CLO) tranche valuation, incorporating non‐Gaussian distributions and dynamic correlations without relying on the large homogeneous portfolio (LHP) assumption. Through numerical analyses and comparisons with LHP models, I find that non‐LHP models produce higher tranche spreads, especially for lower‐rated tranches. Sensitivity analysis reveals varying sensitivities to changes in the number of collaterals, risk‐free rate, average collateral ratings, recovery rates, and time to maturity. The non‐LHP one‐factor copula models, including stochastic correlation and random factor loading models, outperform LHP models in root mean squared errors when calibrated to market data. The results underscore the importance of considering model limitations in CLO tranche pricing and highlight potential mispricing of spread risk in higher‐rated tranches using LHP models. The proposed models contribute to a more comprehensive understanding of CLO tranche pricing by accounting for various factors and assumptions influencing fair premiums.

Modeling diversification and spillovers of loan portfolios' losses by LHP approximation and copula

This paper suggests a top-down method for aggregating the economic capital of an entire banking system and decomposing it into loan sectors according to their risk contributions. We model the individual losses of loan sectors by large homogeneous portfolio (LHP) approximation based on multi-factor skew normal credit worthiness and combine them by applying static and dynamic copulas to reflect diversification effects and spillovers across loan sectors. Our method is more efficient and practically useful than typical multi-factor models using numerical integration due to the latency of risk factors in that losses are directly generated by Monte Carlo simulation using copula without knowing any risk factors.As a result of our empirical study on charge-off rates of the U.S. commercial banking system, we find that the residential real estate loan sector is the most affecting as its default risk spills over to the rest of the banking system, and hence its risk contribution to the entire banking system is large. However, the commercial real estate loan and business loan sectors are revealed to be affected sectors whose risk contributions are large, but the losses are mainly due not only to their large exposure size, but also to default contagion from others. The risk contributions of credit cards and other consumer loans as default risk affecting sectors become larger in terms of the recent conditional dependence. Lastly, using time-varying correlation analysis, we find that the subprime mortgage crisis is a systemic event that affects the entire-banking-system, while the commercial real estate and the dotcom bubble crises are sector-wide systemic events.

A Comparative Analysis of Correlation Approaches in Finance

This paper analyses the most widely applied correlation approaches in finance. We first discuss primarily bottom-up approaches, as correlating Brownian motions (Heston 1993), the binomial correlation approach (Lucas 1995), Copulas (Sklar 1959, Li 2000), lattice models with dynamic copulas (Albanese et al. 2005, 2007, 2010), conditionally independent models (Vasicek 1987 and extensions), and the contagion correlation approach (Davis and Lo 1999, Jarrow and Yu 2001). New insights on the Gaussian copula, partly blamed for the global financial crisis 2007/2008 are given. We then discuss top-down approaches as Vasicek’s 1987 large homogeneous portfolio (LHP), inducing correlation in Markov chain models (Hurd and Kuznetsov 2006a, 2006b, Schonbucher 2006), and the top-down contagion model of Giesecke et al 2009. This paper addresses the mathematical properties of the correlation approaches. A follow up paper will empirically test the correlation approaches using default data of the US banking sector before and during the 2007/2008 crisis.

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.

The Crash-NIG factor model

It is well known that the one-factor copula models are very useful for risk management and measurement applications involving the generation of scenarios for the complete universe of risk factors and the inclusion of CDO structures in a portfolio context. For this objective, it is necessary to have a simple and fast model that is also consistent with the scenario simulation framework. In this paper we present three extensions of the NIG one-factor copula model which jointly have not been considered so far: (1) tranches with different maturities modeled in a consistent way, (2) a portfolio with different rating buckets, relaxing the assumption of a large homogeneous portfolio, and (3) different correlation regimes. The regime-switching component of the proposed Crash-NIG factor model is especially important in view of the current credit crisis. We also introduce liquidity premiums into the Crash-NIG factor model and show that the actual credit crisis is substantially driven by liquidity effects.

Pricing CDO tranches with stochastic correlation and random factor loadings in a mixture copula model

In this paper, we introduce a mixture distribution of Gaussian and Variance Gamma distribution. Then we use the one-factor double mixture copula model to solve the problem of CDO pricing. Two cases of stochastic correlation and random factor loadings are considered. In each case, the unconditional characteristic function of accumulated loss is calculated and the loss distribution can therefore be derived by using the fast Fourier transform. The loss distribution of a large homogeneous portfolio is also derived. Furthermore, we analyze the numerical results.

The Crash-NIG copula model: Risk measurement and management of credit portfolios

The one-factor copula models became very popular for modelling dependence in credit portfolios and collateralised debt obligation (CDO) valuation owing to their simplicity. Still, it is also well known that they are too simple for an exact pricing. Nevertheless, it is possible to extend the model in various ways so that it is possible to describe historical correlation behaviour realistically. Such an extension of the one-factor copula model, called the Crash-NIG copula model, is proposed by the authors with the following characteristics: (i) more tail dependence than in the Gaussian case, (ii) consistent term structure dimension, (iii) different rating buckets, relaxing the assumption of a large homogeneous portfolio, and (iv) different correlation regimes. Here the authors demonstrate how to apply this model for generating rating transition and default scenarios of a credit portfolio together with the other relevant risk factors. Repricing of instruments on the simulated scenario paths, that is the most difficult problem for such complex instruments as CDO tranches, can also be done efficiently fast using the same model. Finally, portfolio optimisation can be performed on the derived profit and loss distributions that are shown to be very different from normal.

A generic framework for stochastic Loss-Given-Default

In this document a method is discussed to incorporate stochastic Loss-Given-Default (LGD) in factor models, i.e. structural models for credit risk. The general idea exhibited in this text is to introduce a common dependence of the LGD and the probability of default (PD) on a latent variable, representing the systemic risk. Though our theory can be applied to any arbitrary firm-value model and any underlying distribution for the LGD, provided its support is a compact subset of [ 0 , 1 ] , special attention is given to the extension of the well-known cases of the Gaussian copula framework and the shifted Gamma one-factor model (a particular case of the generic one-factor Lévy model), and the LGD is modeled by a Beta distribution, in accordance with rating agency models and the Credit Metrics model. In order to introduce stochastic LGD, a monotonically decreasing relation is derived between the loss rate L , i.e. the loss as a percentage of the total exposure, and the standardized log-return R of the obligor’s asset value, which is assumed to be a function of one or more systematic and idiosyncratic risk factors. The property that the relation is decreasing guarantees that the LGD is negatively correlated to R and hence positively correlated to the default rate. From this relation, expressions are then derived for the cumulative distribution function (CDF) and the expected value of the loss rate and the LGD, conditionally on a realization of the systematic risk factor(s). It is important to remark that all our results are derived under the large homogeneous portfolio (LHP) assumption and that they are fully consistent with the IRB approach outlined by the Basel II Capital Accord. We will demonstrate the impact of incorporating stochastic LGD and using models based on skew and fat-tailed distributions in determining adequate capital requirements. Furthermore, we also skim the potential application of the proposed framework in a credit risk environment. It will turn out that both building blocks, i.e. stochastic LGD and fat-tailed distributions, separately, increase the projected loss and thus the required capital charge. Hence, the aggregation of a model based on a fat-tailed underlying distribution that accounts for stochastic LGD will lead to sound capital requirements.

A note on the large homogeneous portfolio approximation with the Student-t copula

We extend the Large Homogeneous Portfolio (LHP) approximation to the case of the Student-t copula, and provide analytic formulae for the density and the cdf of the portfolio loss distribution. We compare the Value-at-Risk implied by the Student-t copula to that obtained using the Gaussian as well as two prominent members of the Archimedean family, namely the Clayton and the Gumbel copulae.