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TEST FOR CHANGES IN THE MODELED SOLVENCY CAPITAL REQUIREMENT OF AN INTERNAL RISK MODEL

Published online by Cambridge University Press:  06 August 2021

Daniel Gaigall
Daniel Gaigall*
Affiliation:
Institute of Probability and Statistics, Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany, E-Mail: gaigall@stochastik.uni-hannover.de House of Insurance, Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany, E-Mail: daniel.gaigall@insurance.uni-hannover.de Group Risk Management, HDI Service AG, HDI-Platz 1, 30659 Hannover, Germany, E-Mail: daniel.gaigall@hdi.de
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Abstract

In the context of the Solvency II directive, the operation of an internal risk model is a possible way for risk assessment and for the determination of the solvency capital requirement of an insurance company in the European Union. A Monte Carlo procedure is customary to generate a model output. To be compliant with the directive, validation of the internal risk model is conducted on the basis of the model output. For this purpose, we suggest a new test for checking whether there is a significant change in the modeled solvency capital requirement. Asymptotic properties of the test statistic are investigated and a bootstrap approximation is justified. A simulation study investigates the performance of the test in the finite sample case and confirms the theoretical results. The internal risk model and the application of the test is illustrated in a simplified example. The method has more general usage for inference of a broad class of law-invariant and coherent risk measures on the basis of a paired sample.

Keywords

BootstrapEmpirical processFunctional Delta methodHadamard differentiabilityPaired sample
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 3 , September 2021 , pp. 813 - 837
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Albers, W. Löhnberg, P. (1984). An approximate confidence interval for the difference between quantiles in a bio-medical problem. Statistica Neerlandica 38, 20–22.CrossRefGoogle Scholar
Bank of England (2016). Monitoring model drift and standard formula SCR reporting for firms with an approved internal model. Supervisory Statement 15/16. https://www.bankofengland.co.uk/prudential-regulation/publication/2016/solvency2-monitoring-model-drift-and-standard-formula-scr-reporting-ss Google Scholar
Baklizi, A. (2018). Interval Estimation of Quantile Difference in the Two-Parameter Exponential Distributions. Journal of Testing and Evaluation 46, 2654–2660.CrossRefGoogle Scholar
Beutner, E. Zähle, H (2010). A modified functional delta method and its application to the estimation of risk functionals. Journal of Multivariate Analysis 101, 2452–2463.CrossRefGoogle Scholar
Beutner, E. Zähle, H (2016). Functional delta-method for the bootstrap of quasi Hadamard differentiable functionals. Electronic Journal of Statistics 10, 1181–1222.CrossRefGoogle Scholar
Bristol, D.R. (1990). Distribution-free confidence intervals for the difference between quantiles. Statistica Neerlandica 44, 87–90.CrossRefGoogle Scholar
Casarano, G., Castellani, G., Passalacqua, L., Perla, F., Zanetti, P. (2017). Relevant applications of Monte Carlo simulation in Solvency II. Soft Computing 21, 1181–1192.CrossRefGoogle Scholar
Chakraborti, S., Desu, M.M. (2008). A Distribution–Free Confidence Interval For The Difference Between Quantiles With Censored Data. Statistica Neerlandica 40, 93–98.CrossRefGoogle Scholar
Christiansen, M. C., Niemeyer, A. (2014). The fundamental definition of the Solvency Capital Requirement in Solvency II. ASTIN Bulletin 44, 501–533.CrossRefGoogle Scholar
Culver, Q., Heitmann, D., Weiß, C. (2018). The Influence of Seed Selection on the Solvency II Ratio. Der Aktuar 1/2018.Google Scholar
Cox, T.F., Jaber, K. (1985). Testing the Equality of two Normal Percentiles. Communication in Statistics- Simulation and Computation 14, 345–356.CrossRefGoogle Scholar
Casarano, G., Castellani, G., Passalacqua, L., Perla, F., Zanetti, P. (2017). Relevant applications of Monte Carlo simulation in Solvency II. Soft Computing 21, 1181–1192.CrossRefGoogle Scholar
Dacorogna, M. (2017). Approaches and Techniques to Validate Internal Model Results. SSRN Electronic Journal.CrossRefGoogle Scholar
Danielsson, J., James, K.R., Valenzuela, M., Zer, I. (2016). Model Risk of Risk Models. Journal of Financial Stability 23, 79–91.CrossRefGoogle Scholar
Devroye, L. (1990). A note on Linnik’s distribution. Statistics and Probability letters 9, 305–306.CrossRefGoogle Scholar
Dudley, R. M. (1984). A course on empirical processes. Lecture Notes in Mathematics 1097, 1–142. Springer, New York.CrossRefGoogle Scholar
Gänßler, P., Ziegler, K. (1994). A Uniform Law of Large Numbers for Set-Indexed Processes with Applications to Empirical and Partial-Sum Processes. Probability in Banach Spaces 9, 385–400.CrossRefGoogle Scholar
Guo, H., Krishnamoorthy, K. (2005). Comparison Between Two Quantiles: The Normal and Exponential Cases. Communication in Statistics- Simulation and Computation 34, 243–252.CrossRefGoogle Scholar
Huang, L.-F., Johnson, R.A. (2006). Confidence regions for the ratio of percentiles. Statistics & Probability Letters 76, 384–392.CrossRefGoogle Scholar
Huang, L.-F. (2016). Approximated non parametric confidence regions for the ratio of two percentiles. Communication in Statistics- Theory and Methods 46, 4004–4015.CrossRefGoogle Scholar
Huang, L.-F. (2016). Adjusted Wilcoxon signed rank test tables for ratio of percentiles. Communication in Statistics- Simulation and Computation 46, 5763–5771.CrossRefGoogle Scholar
Li, X., Tian, L., Wang, J., Muindi, J.R. (2012). Comparison of quantiles for several normal populations. Computational Statistics & Data Analysis 56, 2129–2138.CrossRefGoogle Scholar
Kang, S.-G., Kim, D.-H., and Lee, W.-D. (2007). Noninformative Priors for the Difference of Two Quantiles in Exponential Models. Communications for Statistical Applications and Methods 14, 431–442.CrossRefGoogle Scholar
Kosorok, M.R. (1999). Two-Sample Quantile Tests under General Conditions. Biometrika 86, 909–921.CrossRefGoogle Scholar
Krätschmer, V., Schied, A., Zähle, H. (2015). Quasi-Hadamard differentiability of general risk functionals and its application. Statistics & Risk Modeling 32, 25–47.CrossRefGoogle Scholar
Malekzadeh, A., Jafari, A.A. (2018). Testing equality of quantiles of two-parameter exponential distributions under progressive Type II censoring. Journal of Statistical Theory and Practice 12, 776–793.CrossRefGoogle Scholar
Malekzadeh, A., Kharrati-Kopaei, M. (2020). Simultaneous confidence intervals for the quantile differences of several two-parameter exponential distributions under the progressive type II censoring scheme. Journal of Statistical Computation and Simulation 90, 1–20.CrossRefGoogle Scholar
Malekzadeh, A., Mahmoudi, S.M. (2020). Constructing a confidence interval for the ratio of normal distribution quantiles. Monte Carlo Methods and Applications 26, 325–334.CrossRefGoogle Scholar
Ozturk, O., Balakrishnan, N. (2009). Exact two-sample nonparametric test for quantile difference between two populations based on ranked set samples. Annals of the Institute of Statistical Mathematics 61, 235–249.CrossRefGoogle Scholar
Prihoda, T.J. (1981). A generalized approach to the two sample problem: the quantile approach. Dissertation, Texas A&M University.CrossRefGoogle Scholar
Stahl, G. (2016). Model Uncertainty in a Holistic Perspective. Kallsen J., Papapantoleon A. (eds) Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189. Springer, Cham, 189–215.CrossRefGoogle Scholar
The Comprehensive R Archive Network. (2018). https://cran.r-project.org/web/packages/copula/copula.pdf Google Scholar
VaG (2016). Gesetz über die Beaufsichtigung der Versicherungsunternehmen.Google Scholar
van der Vaart, A., Wellner, J.A. (1996). Weak convergence and empirical processes. Springer, New York.CrossRefGoogle Scholar
van der Vaart. (1998). Asyptotic Statistics. Cambridge.Google Scholar
Ziegler, K. (1997). Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions. Journal of Multivariate Analysis 62, 233–272.CrossRefGoogle Scholar
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