Asset correlations and credit portfolio risk: an empirical analysis

In credit risk modelling, the correlation of unobservable asset returns is a crucial component for the measurement of portfolio risk. In this paper, we estimate asset correlations from monthly time series of Moody's KMV asset values for around 2,000 European firms from 1996 to 2004. We compare correlation and value-atrisk (VaR) estimates in a one-factor or market model and a multi-factor or sector model. Our main finding is a complex interaction of credit risk correlations and default probabilities affecting total credit portfolio risk. Differentiation between industry sectors when using the sector model instead of the market model has only a secondary effect on credit portfolio risk, at least for the underlying credit portfolio. Averaging firm-dependent asset correlations on a sector level can, however, cause a substantial underestimation of the VaR in a portfolio with heterogeneous borrower size. This result holds for the market as well as the sector model. Furthermore, the VaR of the IRB model is more stable over time than the VaR of the market model and the sector model, while its distance from the other two models fluctuates over time. --Asset correlations,sector concentration,credit portfolio risk

Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth model

The Bertalanffy–Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * ma − q * mb. The special case using the von Bertalanffy exponent-pair a = 2/3 and b = 1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy–Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder–Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, it is known that for most fish data any exponent a < 1 could be used to model growth without affecting the fit to the data significantly (when the other parameters were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma − q * ln(m) * ma. Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy–Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage

Beta-expansions in algebraic function fields over finite fields

In the present paper, we define a new kind of digit system in algebraic function fields over finite fields. There are striking analogies of these digit systems to the well known β-expansions defined in R+. Results corresponding to classical theorems as well as open problems will be proved. In orde

Complexity and effective dimension of discrete Lévy

Discretisation methods to simulate stochastic differential equations belong to the main tools in mathematical finance. For Itô processes, there exist several Euleror Runge-Kutta-like methods which are analogues of well known approximation schemes in the non stochastic case. In the multidimensional case, there appear several difficulties, caused by the mixed second order derivatives. These mixed terms (or more precisely their differences) correspond to special random variables called Lévy stochastic area terms. In the present paper, we compare three approximation methods for such random variables with respect to computational complexity and the so called effective dimension

Efficient simulation of Lévy areas

Discretization methods to simulate stochastic differential equations belong to the main tools in mathematical finance. For Itô processes, there exist several Euleror Runge-Kutta-like methods which are analogues of well known approximation schemes in the non stochastic case. In the multidimensional case, there appear several difficulties, caused by the mixed second order derivatives. These mixed terms (or more precisely their differences) correspond to special random variables called Lévy stochastic area terms. In the present paper, we compare three approximation methods for such random variables with respect to computational complexity and the so called effective dimension