Jim Gustafsson

Local Transformation Kernel Density Estimation of Loss Distributions

We develop a tailor made semiparametric asymmetric kernel density estimator for the estimation of actuarial loss distributions. The estimator is obtained by transforming the data with the generalized Champernowne distribution initially fitted to the data. Then the density of the transformed data is estimated by use of local asymmetric kernel methods to obtain superior estimation properties in the tails. We find in a vast simulation study that the proposed semiparametric estimation procedure performs well relative to alternative estimators. An application to operational loss data illustrates the proposed method.

Using External Data in Operational Risk

We present a method to combine expert opinion on the likelihood of under-reporting with an operational risk data set. Under-reporting means that not all losses are identified and therefore an incorrect distributional assumption may be made, and ultimately an incorrect assessment made of capital required. Our approach can be applied to help insurers and other financial services companies make better assessments of capital requirements for operational risk using either external or internal sources. We conclude that operational risk capital evaluation can be significantly biased if under-reporting is ignored. The Geneva Papers (2007) 32, 178–189. doi:10.1057/palgrave.gpp.2510129

Multivariate Latent Risk: A Credibility Approach

We investigate a concept of multivariate pricing, which includes claim history for more than one line of business and is a generalization of the Buhlmann-Straub model. The multivariate credibility model is extended to allow for the age of claims to influence the estimation of future claims. The model is applied to data from a portfolio of commercial lines of business.

Multidimensional Credibility With Time Effects - An Application to Commercial Business Lines

This article considers Danish insurance business lines for which the pricing methodology has been dramatically upgraded recently. A costly affair, but nevertheless, the benefits greatly exceed the costs; without a proper pricing mechanism, you are simply not competitive. We show that experience rating improves this sophisticated pricing method as much as it originally improved pricing compared with a trivial flat rate. Hence, it is very important to take advantage of available customer experience. We verify that recent developments in multivariate credibility theory improve the prediction significantly, and we contribute to this theory with new robust estimation methods for time (in-)dependency.

Quantifying Operational Risk Guided by Kernel Smoothing and Continuous Credibility: A Practitioners View

This paper considers the benefits of applying sophisticated statistical techniques to challenges faced in the quantification of operational risk. The evolutionary nature of operational risk modelling to establish capital charges is recognised emphasizing the importance of capturing tail behaviour. Nonparametric smoothing techniques are considered along with a parametric base with a particular view to comparison with extreme value theory. This is presented without detailed proofs in the aim of demonstrating to practitioners the practical benefits of such techniques. The smoothed estimators embedded in a credibility approach supports analysis from pooled data across lines of business or across risk types/regions.

A Mixing Model for Operational Risk

External data can often be useful in improving estimation of operational risk loss distributions. This paper develops a systematic approach that incorporates external information into internal loss distribution modelling. The standard statistical model resembles bayesian methodology or credibility theory in the sense that prior knowledge (external data) has more weight when internal data is sparse than when internal data is abundant.

Nonparametric Estimation of Operational Risk Losses Adjusted for Underreporting

Not all claims are reported when a financial operational risk data base is created. The probability of reporting increase with the size of the operational risk loss and approaches one for very big losses. Operational risk losses comes from many different sources and can be expected to follow a wide variety of distributional shapes. Therefore, an approach to operational risk modelling based on one or two favourites of parametric models are deemed to fail. In this paper we introduce a semiparametric approach to operational risk modelling that is able to take underreporting into account and that allows itself to be guided by prior knowledge of distributional shape.

Adding prior knowledge to quantitative operational risk models

Our approach is based on the study of the statistical severity distribution of a single loss. We analyze the fundamental issues that arise in practice when modeling operational risk data. We address the statistical problem of estimating an operational risk distribution, both abundant data situations and when our available data is challenged from the inclusion of external data or because of underreporting. Our presentation includes an application to show that failure to account for underreporting may lead to a substantial underestimation of operational risk measures. The use of external data information can easily be incorporated in our modeling approach. The paper builds on methodology developed in Bolance et al. (2012b). 1. Quantifying Operational Risk Guided by Prior Knowledge Operational risk is one of the risks that are incorporated in the Basel II regulatory framework for financial institutions and in the Solvency II regulatory framework for insurance companies (Gatzert and Wesker, 2012 and Ashby, 2011), hence the importance of the modelization and quantification of this risk. Also, operational risk is important in the context of Enterprise Risk Management (Hoyt and Liebenberg, 2011 and Dhaene et al. 2012). One major issue addressed in Bolance et al (2012b) is how to incorporate prior knowledge into operational risk models. Such prior knowledge can come in many disguises. One being prior knowledge of parametric shapes of distributions, another being prior knowledge of the frequency of underreporting and a third could be prior knowledge arising from external data sources. The fundamental principles of mixing internal and external operational risk data was originally published in this journal in Gustafsson and Nielsen (2008) and Guillen et al. (2008). Bolance et al. (2012b) take these originally ideas and put them into a broader context, see also the following recent papers proposing alternative methods to quantify operational risk (Cope, E.W., 2012, Cavallo et al., 2012, Feng et al., 2012 and Horbenko et al., 2011). In this paper we show, with a simple example, the effect of incorporating two different types of prior knowledge into the calculation of Value-at-Risk (VaR) and Tail Value-at Risk (TVaR): external operational risk data and expert information about underreporting probability. We 1 We thank the Spanish Ministry of Science / FEDER grant ECO2010-21787-C0301 and Generalitat de Catalunya SGR 1328. Corresponding author: jens.nielsen.1@city.ac.uk 2 We thank the Spanish Ministry of Science / FEDER grant ECO2010-21787-C0301 and Generalitat de Catalunya SGR 1328. Corresponding author: jens.nielsen.1@city.ac.uk

Combining Underreported Internal and External Data for Operational Risk Measurement

Operational risk data sets have two types of sample selection problems: truncation below a given threshold due to data that are not recorded and random censoring above that level caused by data that are not reported. This paper proposes a model for operational losses that improves the internal loss distribution modelling by combining internal and external operational risk data. It also considers the possibility that internal and externa l data have been collected with a different truncation threshold. Moreover, the model is able to cope with unreported losses by means of an estimated underreporting function.