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A neural network model for solvency calculations in life insurance

Published online by Cambridge University Press:  01 December 2020

Lucio Fernandez-Arjona Open the ORCID record for Lucio Fernandez-Arjona [Opens in a new window]
Lucio Fernandez-Arjona*
Affiliation:
University of Zurich, 8006Zurich, Switzerland
*
E-mail: [email protected]
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Abstract

Insurance companies make extensive use of Monte Carlo simulations in their capital and solvency models. To overcome the computational problems associated with Monte Carlo simulations, most large life insurance companies use proxy models such as replicating portfolios (RPs). In this paper, we present an example based on a variable annuity guarantee, showing the main challenges faced by practitioners in the construction of RPs: the feature engineering step and subsequent basis function selection problem. We describe how neural networks can be used as a proxy model and how to apply risk-neutral pricing on a neural network to integrate such a model into a market risk framework. The proposed model naturally solves the feature engineering and feature selection problems of RPs.

Keywords

Economic capitalSwiss solvency testSolvency IINeural networksNested Monte CarloReplicating portfolios
Type
Paper
Information
Annals of Actuarial Science , Volume 15 , Issue 2 , July 2021 , pp. 259 - 275
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Institute and Faculty of Actuaries

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References

Adelmann, M., Fernandez Arjona, L., Mayer, J. & Schmedders, K. (2019). A large-scale optimization model for replicating portfolios in the life insurance industry. Working Paper, University of Zurich.Google Scholar
Akaike, H. (1998). Information theory and an extension of the maximum likelihood principle. In Selected Papers of Hirotugu Akaike (pp. 199–213). Springer.CrossRefGoogle Scholar
Bauer, D., Bergmann, D. & Reuss, A. (2010). Solvency II and nested simulations–a least-squares Monte Carlo approach. In Proceedings of the 2010 ICA Congress.Google Scholar
Beutner, E., Pelsser, A. & Schweizer, J. (2013). Fast convergence of regress-later estimates in least squares Monte Carlo. Available at SSRN 2328709.CrossRefGoogle Scholar
Beutner, E., Pelsser, A. & Schweizer, J. (2016). Theory and validation of replicating portfolios in insurance risk management. Available at SSRN 2557368.CrossRefGoogle Scholar
Broadie, M., Du, Y. & Moallemi, C.C. (2015). Risk estimation via regression. Operations Research, 63(5), 1077–1097.CrossRefGoogle Scholar
Burnham, K.P. & Anderson, D.R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edition. Springer, New York.Google Scholar
Cambou, M. & Filipović, D. (2018). Replicating portfolio approach to capital calculation. Finance and Stochastics, 22(1), 181–203.CrossRefGoogle Scholar
Castellani, G., Fiore, U., Marino, Z., Passalacqua, L., Perla, F., Scognamiglio, S. & Zanetti, P. (2018). An investigation of machine learning approaches in the solvency II valuation framework. Available at SSRN 3303296.CrossRefGoogle Scholar
Chai, T. & Draxler, R.R. (2014). Root mean square error (RMSE) or mean absolute error (MAE)?–arguments against avoiding RMSE in the literature. Geoscientific Model Development, 7(3), 1247–1250.CrossRefGoogle Scholar
Chen, W. & Skoglund, J. (2012). Cashflow replication with mismatch constraints. The Journal of Risk, 14(4), 115.Google Scholar
Fernandez-Arjona, L. (2019). ESG simulations and RPDB cash flows (june 2019). Available online at the address http://dx.doi.org/10.17632/6vvzh2w4g4.1. Mendeley Data, v1.CrossRefGoogle Scholar
Gan, G. & Lin, X.S. (2015). Valuation of large variable annuity portfolios under nested simulation: a functional data approach. Insurance: Mathematics and Economics, 62, 138–150.Google Scholar
Glasserman, P. (2013). Monte Carlo Methods in Financial Engineering, vol. 53. Springer Science & Business Media, New York.Google Scholar
Glasserman, P. & Yu, B. (2002). Simulation for american options: regression now or regression later? In Monte Carlo and Quasi-Monte Carlo Methods 2002 (pp. 213–226). Springer.Google Scholar
Hanin, B. & Sellke, M. (2017). Approximating continuous functions by relu nets of minimal width. arXiv preprint arXiv:1710.11278.Google Scholar
Hejazi, S.A. & Jackson, K.R. (2016). A neural network approach to efficient valuation of large portfolios of variable annuities. Insurance: Mathematics and Economics, 70, 169–181.Google Scholar
Hornik, K. (1991). Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2), 251–257.CrossRefGoogle Scholar
Lee, R.D. & Carter, L.R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659–671.Google Scholar
Longstaff, F.A. & Schwartz, E.S. (2001). Valuing american options by simulation: a simple least-squares approach. The Review of Financial Studies, 14(1), 113–147.CrossRefGoogle Scholar
McKinney, W. (2010). Data structures for statistical computing in python. In van der Walt, S. and Millman, J. (Eds.), Proceedings of the 9th Python in Science Conference (pp. 51–56).CrossRefGoogle Scholar
Natolski, J. & Werner, R. (2014). Mathematical analysis of different approaches for replicating portfolios. European Actuarial Journal, 4(2), 411–435.CrossRefGoogle Scholar
Ninomiya, Y., Kawano, S., (2016). AIC for the lasso in generalized linear models. Electronic Journal of Statistics, 10(2), 2537–2560.CrossRefGoogle Scholar
Oliphant, T. (2006). NumPy: a guide to NumPy. Trelgol Publishing, USA [accessed November 2019].Google Scholar
Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M. & Duchesnay, E. (2011). Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12, 2825–2830.Google Scholar
Pelsser, A. & Schweizer, J. (2016). The difference between LSMC and replicating portfolio in insurance liability modeling. European Actuarial Journal, 6(2), 441–494.CrossRefGoogle ScholarPubMed
Risk, J. & Ludkovski, M. (2018). Sequential design and spatial modeling for portfolio tail risk measurement. SIAM Journal on Financial Mathematics, 9(4), 1137–1174.CrossRefGoogle Scholar
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1), 267–288.Google Scholar
Van Der Walt, S., Colbert, S.C. & Varoquaux, G. (2011). The numpy array: a structure for efficient numerical computation. Computing in Science & Engineering, 13(2), 22.CrossRefGoogle Scholar
Vidal, E.G. & Daul, S. (2009). Replication of insurance liabilities. RiskMetrics Journal, 9(1), 79–96.Google Scholar
Willmott, C.J. & Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30(1), 79–82.CrossRefGoogle Scholar
Zou, H., Hastie, T., Tibshirani, R. (2007). On the “degrees of freedom” of the lasso. The Annals of Statistics, 35(5), 2173–2192.CrossRefGoogle Scholar