A Markov-switching generalized additive model for compound Poisson processes, with applications to operational loss models

Abstract

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.

Keywords:

• GAMLSS
• Distributional regression
• Generalized Pareto distribution
• Hidden Markov model
• Operational losses
• Compound Poisson process

JEL Classification:

• C14
• C22
• C24
• C58

Acknowledgements

The authors warmly thank an anonymous referee for her/his detailed comments and suggestions. It has led to improve the clarity of the present manuscript. J. Hambuckers thanks V. Chavez-Demoulin as well as the participants of HEC Lausanne seminars for the fruitful discussions that helped improve earlier versions of the present manuscript. The authors acknowledge F. Piacenza for providing them the data.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed at https://doi/10.1080/14697688.2017.1417625.

Notes

† This approximation is particularly convenient because it also ensures that a linear fit remains unpenalized.

‡ In the present paper, we rely on the fminunc function of the Optimization Toolbox in MatLab, using the quasi-Newton lm-line-search algorithm. This algorithm is an implementation of the one presented in Fletcher (1987). See also Nocedal and Wright (2006) for more details. Gradient and Hessian updates are based on finite difference procedures.

† Values in the grid were based on the ones used in Langrock et al. (2017) and a trial-and-error phase.

‡ Here, we use cubic splines that are twice continuously differentiable, ensuring a visually smooth fit. The number of basis function has been chosen to obtain a balance between flexibility and computational feasibility. Indeed, since we have 3 distribution parameters and two states, the use of an additional basis function leads to 6 more basis weights to estimate. 11 is in line with, e.g. the default number of knots in the popular mgcv R package.

† Notice that this threshold is close to the 90% empirical quantile of all external fraud losses registered by the bank during the considered period.

† As suggested by a reviewer, an interesting alternative would be to consider e.g. VSTOXX instead of VIX as an explanatory variable for , since UniCredit has its core business in Europe. In the supplementary material, the interested reader can find the results obtained from estimating such a model on the present data-set. Results are almost identical and can be explained by today’s financial globalization (Mendoza and Quadrini 2010), causing VIX and VSTOXX time series to be highly similar: their correlation coefficient on the considered period is 0.96. To keep the discussion concise, and because VIX is a more popular measure of market uncertainty, only the results obtained with the VIX are discussed.

Additional information

Funding

T. Kneib was financially supported by DFG [grant number KN 922/4-2]. J. Hambuckers acknowledges the support of the Research Training Group 1644 Scaling Problems in Statistics funded by the German Research Foundation (DFG).