The pricing puzzle : the default term structure of collateralised loan obligations

Ambivalence in the regulatory definition of capital adequacy for credit risk has recently stirred the financial services industry to collateral loan obligations (CLOs) as an important balance sheet management tool. CLOs represent a specialised form of Asset-Backed Securitisation (ABS), with investors acquiring a structured claim on the interest proceeds generated from a portfolio of bank loans in the form of tranches with different seniority. By way of modelling Merton-type risk-neutral asset returns of contingent claims on a multi-asset portfolio of corporate loans in a CLO transaction, we analyse the optimal design of loan securitisation from the perspective of credit risk in potential collateral default. We propose a pricing model that draws on a careful simulation of expected loan loss based on parametric bootstrapping through extreme value theory (EVT). The analysis illustrates the dichotomous effect of loss cascading, as the most junior tranche of CLO transactions exhibits a distinctly different default tolerance compared to the remaining tranches. By solving the puzzling question of properly pricing the risk premium for expected credit loss, we explain the rationale of first loss retention as credit risk cover on the basis of our simulation results for pricing purposes under the impact of asymmetric information. Klassifikation: C15, C22, D82, F34, G13, G18, G2

The pricing of correlated default risk: evidence from the credit derivatives market

In order to analyze the pricing of portfolio credit risk – as revealed by tranche spreads of a popular credit default swap (CDS) index – we extract risk-neutral probabilities of default (PDs) and physical asset return correlations from single-name CDS spreads. The time profile and overall level of index spreads validate our PD measures. At the same time, the physical asset return correlations are too low to account for the spreads of index tranches and, thus, point to a large correlation risk premium. This premium, which covaries negatively with current realized correlations and positively with future realized correlations, sheds light on market perceptions of and attitude towards correlation risk. -- Das Portfoliokreditrisiko setzt sich aus drei Hauptkomponenten zusammen: der Ausfallwahrscheinlichkeit (probability of default, PD), der Verlustquote (loss given default, LGD) und der Wahrscheinlichkeitsverteilung für gemeinsame Ausfälle. Mit der rasanten Entwicklung innovativer Produkte im Bereich der strukturierten Finanzierung ist die Bedeutung der dritten Komponente zusehends gestiegen. Allerdings herrscht keine Einigkeit darüber, wie die Marktteilnehmer diese schätzen. Im vorliegenden Arbeitspapier schlagen wir zunächst einen auf CDSMarktdaten beruhenden Ansatz zur Ableitung der Wahrscheinlichkeitsverteilung für gemeinsame Ausfälle vor. Mit diesem Ansatz werden risikoneutrale PDs und physische Asset-Return-Korrelationen aus der Höhe der Preise und dem Gleichlauf (Co-movement) von Single-name-CDS-Spreads abgeleitet. Anschließend benutzen wir diese Schätzungen in einer konkreten Anwendung unseres Ansatzes zur Berechnung von Prognosen für Tranchenspreads eines bekannten CDS-Index (Dow Jones CDX North America Investment Grade Index) und vergleichen diese mit empirischen Spreads am CDS-Indexmarkt.Portfolio credit risk,Correlation risk premium,CDS index,Tranche spread,Copula

CDO Pricing with Copulae

Modeling the portfolio credit risk is one of the crucial issues of the last years in the financial problems. We propose the valuation model of Collateralized Debt Obligations based on a one- and two-parameter copula and default intensities estimated from market data. The presented method is used to reproduce the spreads of the iTraxx Europe tranches. The two-parameter model incorporates the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependency among the assets from the same group is described with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. Our approach outperforms the standard market pricing procedure based on the Gaussian distribution.CDO, CDS, multifactor models, multivariate distributions, Copulae, correlation smile

CDO and HAC

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on copula functions with up to three parameters, with default intensities estimated from market data and with a random loss given default that is correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. We apply hierarchical Archimedean copulae (HAC) whose construction allows for the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependence among the assets from the same group is specified with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. The copula with two and three parameters models the relation between the loss given default and the default times. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution.CDO, CDS, multivariate distributions, Copulae, correlation smile, loss given default

On Correlation Effects and Default Clustering in Credit Models

We establish Markovian models in the Heath, Jarrow and Morton paradigm where the credit spreads curves of multiple firms and the term structure of interest rates can be represented analytically at any point in time in terms of a finite number of state variables. The models make no restrictions on the correlation structure between interest rates and credit spreads. In addition to diffusive and jump-induced default correlations, default events can impact credit spreads of surviving firms. This feature allows a greater clustering of defaults. Numerical implementations highlight the importance of taking interest rate-credit spread correlations, credit-spread impact factors and the full credit spread curve information into account when building a unified model framework that prices any credit derivative.

Securitisation of Mezzanine Capital in Germany

A recent trend in the German Asset Backed Securities (ABS) market is the securitisation of subordinated loans and profit participation agreements (PPAs) granted to medium-sized enterprises (MEs). This paper provides an overview of this growing market and analyses the benefits of such transactions for the portfolio companies as well as originators and potential investors. Simulations of ten recent transactions indicate that despite of relatively low interest rates charged on obligors, originators and investors can earn attractive returns at fairly low risk. In particular, the junior tranches of these securitisations exhibit quite attractive risk-return profiles.securitisation, middle market transactions, mezzanine loans, medium-sized enterprises, junior tranche

Asset pricing and investor risk in subordinated asset securitisation

As a sign of ambivalence in the regulatory definition of capital adequacy for credit risk and the quest for more efficient refinancing sources collateral loan obligations (CLOs) have become a prominent securitisation mechanism. This paper presents a loss-based asset pricing model for the valuation of constituent tranches within a CLO-style security design. The model specifically examines how tranche subordination translates securitised credit risk into investment risk of issued tranches as beneficial interests on a designated loan pool typically underlying a CLO transaction. We obtain a tranchespecific term structure from an intensity-based simulation of defaults under both robust statistical analysis and extreme value theory (EVT). Loss sharing between issuers and investors according to a simplified subordination mechanism allows issuers to decompose securitised credit risk exposures into a collection of default sensitive debt securities with divergent risk profiles and expected investor returns. Our estimation results suggest a dichotomous effect of loss cascading, with the default term structure of the most junior tranche of CLO transactions (“first loss position”) being distinctly different from that of the remaining, more senior “investor tranches”. The first loss position carries large expected loss (with high investor return) and low leverage, whereas all other tranches mainly suffer from loss volatility (unexpected loss). These findings might explain why issuers retain the most junior tranche as credit enhancement to attenuate asymmetric information between issuers and investors. At the same time, the issuer discretion in the configuration of loss subordination within particular security design might give rise to implicit investment risk in senior tranches in the event of systemic shocks. JEL Classifications: C15, C22, D82, F34, G13, G18, G2

Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives