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Abstract:
The Basel II Capital Accord requires banks to determine the capital charge to account for operational losses. Compound Poisson process with Lognormal losses is suggested for this purpose. The paper examines the impact of possibly censored and/or truncated data on the estimation of loss distributions. A procedure on consistent estimation of the severity and frequency distributions based on incomplete data samples is presented. It is also demonstrated that ignoring the peculiarities of available data samples leads to inaccurate Value-at-Risk estimates that govern the operational risk capital charge.

Operational risk, censored and truncated cata, EM-Algorithm

Abstract:
We compare in a backtesting study the performance of univariate models for Value-at-Risk (VaR) and expected shortfall based on stable laws and on extreme value theory (EVT). Analyzing these different approaches, we test whether the sum - stability assumption or the max - stability assumption, that respectively imply alpha-stable laws and Generalized Extreme Value (GEV) distributions, is more suitable for risk management based on VaR and expected shortfall. Our numerical results indicate that alpha-stable models tend to outperform pure EVT-based methods (especially those obtained by the so-called block maxima method) in the estimation of Value-at-Risk, while a peaks-over-threshold method turns out to be preferable for the estimation of expected shortfall. We also find empirical evidence that some simple semiparametric EVT-based methods perform well in the estimation of VaR.

VaR, expected shortfall, stable Paretian laws, extreme value theory

Abstract:
In credit risk management, migration or transition matrices are major inputs for risk management, Credit Value-at-Risk or derivative pricing. After reviewing distance measures for migration matrices we propose some new directed difference indices to measure changes in migration behavior in a more risk-sensitive way. We quantify the changes of the classical distance measures and the new distance indices based on Moody's credit migration history from 1982-2001. We study the relationship between the difference indices, business cycle effects and changes in risk capital for exemplary credit portfolios. Our findings strongly support the usefulness of the derived distance indices and its capability to indicate cyclical behavior or changes in credit VaR. We recommend the use of the indices in risk management for internal credit and loan portfolios.

Transition Matrices, Matrix Norms, Difference Indices, Rating Migration, Credit VaR

Abstract:
After briefly recalling the definition and main properties of stable laws, we discuss issues of parameters estimation and numerical methods for computer simulation of stable random variables. We overview the basic properties of stable processes, in particular of Levy and fractional stable motions. Finally, we discuss the application of the stable assumption in two classical problems from statistics and mathematical finance: regression with stable errors, and option pricing with stable distributed returns.

Abstract:
We investigate the term structure of credit spreads and credit default swaps for different rating categories. It is well-known quite that for issuers with lower credit quality higher spreads can be observed in the market and vice versa. However, empirical results on spreads for bonds with the same rating but different maturities are rather controversial. We provide empirical results on the term structure of credit spreads based on a large sample of Eurobonds and domestic bonds from EWU-countries. Further we investigate maturity effects on credit default swaps and compare the results to those of corporate bonds. We find that for both instruments a positive relationship between maturity and spreads could be observed for investment grade debt. For speculative grade debt the results are rather ambiguous. We also find that spreads for the same rating class and same maturity exhibit very high variation.

Credit Spreads, Credit Default Swaps, Maturity Effects, Reduced Form Models

Abstract:
Transition matrices are an important determinant for risk management and VaR calculations in credit portfolios. It is well known that rating migration behavior is not constant through time. It shows cyclical behavior and significant changes over the years. We investigate the effect of changes in migration matrices on credit portfolio risk in terms of Expected Loss and Value-at-Risk figures for exemplary loan portfolios. The estimates are based on historical transition matrices for different time horizons and a continuous-time simulation procedure. We further determine confidence sets for the probability of default (PD) in different rating classes by a bootstrapping methodology. Our findings are substantial changes in VaR as well as for the width of estimated PD confidence intervals.

Credit VaR, Transition Matrices, Rating Migration, Business Cycle, Continuous-time Modeling, PD estimation

Abstract:
We investigate the main properties of high-frequency exchange rate data in the setting of stochastic subordination and stable modeling, focusing on heavy-tailedness and long memory, together with their dependence on the sampling period. We show that the the instrinsic time process exhibits strong long-range dependence and has increments well described by a Weibull law, while the return series in intrinsic time has weak long memory and is well approximated by a stable Levy motion. We also show that the stable domain of attraction offers a good fit to the returns in physical time, which leads us to consider as a realistic model for exchange rate data a process subordinated to an alpha-stable Levy motion (possibly fractional stable) by a long-memory intrinsic time process with Weibull distributed increments.

Abstract:
In this paper we derive closed-form solutions for the cumulative density function and the average value-at-risk for five subclasses of the infinitely divisible distributions: classical tempered stable distribution, Kim-Rachev distribution, modified tempered stable distribution, normal tempered stable distribution, and rapidly decreasing tempered stable distribution. We present empirical evidence using the daily performance of the S&P 500 for the period January 2, 1997 through December 29, 2006.

tempered stable distribution, infinitely divisible distribution, value-at-risk, conditional value-at-risk, average value-at-risk

Abstract:
In this paper, we present a procedure for consistent estimation of the severity and frequency distributions based on incomplete insurance data and demonstrate that ignoring the thresholds leads to a serious underestimation of the ruin probabilities. The event frequency is modelled with a non-homogeneous Poisson process with a sinusoidal intensity rate function. The choice of an adequate loss distribution is conducted via the in-sample goodness-of-fit procedures and forecasting, using classical and robust methodologies.

Natural Catastrophe, Property Insurance, Loss Distribution, Truncated Data, Ruin Probability

Abstract:
In the study of asset returns, the preponderance of empirical evidence finds that return distributions are not normally distributed. Despite this evidence, non-normal multivariate modelling of asset returns does not appear to play an important role in asset management or risk management because of the complexity of estimating multivariate non-normal distributions from market return data. In this paper, we present a new subclass of generalized elliptical distributions for asset returns that is sufficiently user friendly, so that it can be utilized by asset managers and risk managers for modelling multivariate non-normal distributions of asset returns. For the distribution we present, which we call the multi-tail generalized elliptical distribution, we (1) derive the densities using results of the theory of generalized elliptical distributions and (2) introduce a function, which we label the tail function, to describe their tail behaviour. We test the model on German stock returns and find that (1) the multi-tail model introduced in the paper significantly outperforms the classical elliptical model and (2) the hypothesis of homogeneous tail behaviour can be rejected.

Abstract:
A new approach for using Lévy processes to compute value-at-risk (VaR) using high-frequency data is presented in this paper. The approach is a parametric model using an ARMA(1,1)-GARCH(1,1) model where the tail events are modelled using fractional Lévy stable noise and Lévy stable distribution. Using high-frequency data for the German DAX Index, the VaR estimates from this approach are compared to those of a standard nonparametric estimation method that captures the empirical distribution function, and with models where tail events are modelled using Gaussian distribution and fractional Gaussian noise. The results suggest that the proposed parametric approach yields superior predictive performance.

Abstract:
The liberalization of electricity markets has forced the energy producing companies to react to the new situation. The abolishment of monopolies and the launch of open markets have increased the need of calculating costs closer to the profit frontier to be still competitive, not only against the other German but also against foreign suppliers. Thus, an efficient risk management and risk controlling are needed to ensure the financial survival of the company even during bad times. In this work we use the RAROC methodology to develop a Monte Carlo Simulation based model to quantify risks related to wholesale electricity contracts, also called full load contracts. We do not only consider risk due to market price fluctuations but also due to correlation effects between the spot market price and the load curve of a single customer.

Power Markets, Spot Market Prices, Load Contracts, Risk Management, RAROC

Abstract:
Transition matrices are an important determinant in risk management and VAR calculations for credit portfolios. It is well known that rating migration behavior is not constant through time in that it shows cyclicality and significant change over the years. We investigate the effect of changes in migration matrices on credit portfolio risk in terms of expected loss and value-at-risk figures for illustrative loan portfolios. The estimates are based on historical transition matrices for different time horizons and a continuous-time simulation procedure. We further determine confidence sets for the probability of default (PD) in different rating classes by a bootstrapping methodology. Our findings are that there are substantial changes in VAR as well as in the width of estimated PD confidence intervals.

transition matrices, VAR, credit portfolios, migration matrices, value-at-risk, loan portfolios, PD, probability default, bootstrapping