This paper provide a large-deviations approximation of the tail distribution of total financial losses on a portfolio consisting of many positions. Applications include the total default losses on a bank portfolio, or the total claims against an insurer. The results may be useful in allocating exposure limits, and in allocating risk capital across different lines of business. Assuming that, for...

We analyse the mathematical structure of portfolio credit risk models with particular regard to the modelling of dependence between default events in these models. We explore the role of copulas in latent variable models (the approach that underlies KMV and CreditMetrics) and use non-Gaussian copulas to present extensions to standard industry models. We explore the role of the mixing distributi...

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasi...

Dynamic reduced form models of portfolio credit risk can be distinguished by the way in which the intensity of the default process is specified. In a bottom up model, the portfolio intensity is an aggregate of the constituent intensities. In a top down model, the portfolio intensity is specified without reference to the constituents. This expository article contrasts these modeling approaches. ...

The impact of undiversified idiosyncratic risk on value-at-risk and expected shortfall can be approximated analytically via a methodology known as granularity adjustment (GA). In principle, the GA methodology can be applied to any risk-factor model of portfolio risk. Thus far, however, analytical results have been derived only for simple models of actuarial loss, i.e., credit loss due to defaul...

We consider an asset allocation problem in a continuous-time model with stochastic volatility and (possibly correlated) jumps in both, the asset price and its volatility. First, we derive the optimal portfolio for an investor with constant relative risk aversion. One main finding is that the demand for jump risk now also includes a hedging component, which is not present in models without jumps...

CreditRisk+ is an influential and widely implemented model of portfolio credit risk. As a close variant of models long used for insurance risk, it retains the analytical tractability for which the insurance models were designed. Value-at-risk can be obtained via a recurrence-rule algorithm, so Monte Carlo simulation can be avoided. Little recognized, however, is that the algorithm is fragile. U...

In the Black-Cox model, a firm makes default when its value hits an exponential barrier. Here, we propose an hybrid model that generalizes this framework. The default intensity can take two different values and switches when the firm value crosses the barrier. Of course, the intensity level is higher below the barrier. We get an analytic formula for the Laplace transform of the default time and...

We review the pricing of synthetic CDO tranches from the point of view of factor models. Thanks to the factor framework, we can handle a wide range of well-know pricing models. This includes pricing approaches based on copulas, but also structural, multivariate Poisson and affine intensity models. Factor models have become increasingly popular since there are associated with efficient semi-anal...