The Solvency II directive asks insurance companies to derive their solvency capital requirement from the full loss distribution over the coming year. While this is in general computationally infeasible in the life insurance business, an application of the Least-Squares Monte Carlo (LSMC) method offers a possibility to overcome this computational challenge. We outline in detail the challenges a life insurer faces, the theoretical basis of the LSMC method and the necessary steps on the way to a reliable proxy modeling in the life insurance business. Further, we illustrate the advantages of the LSMC approach via presenting (slightly disguised) real-world applications.

Under the Solvency II regime, life insurance companies are asked to derive their solvency capital requirements from the full loss distributions over the coming year. Since the industry is currently far from being endowed with sufficient computational capacities to fully simulate these distributions, the insurers have to rely on suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. In this paper, we present and analyze various adaptive machine learning approaches that can take over the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over generalized linear model (GLM) and generalized additive model (GAM) methods to multivariate adaptive regression splines (MARS) and kernel regression routines. We justify the combinability of their regression ingredients in a theoretical discourse. Further, we illustrate the approaches in slightly disguised real-world experiments and perform comprehensive out-of-sample tests.

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that models the insurer’s loss with respect to the risk factors the insurance business is exposed to. After giving a mathematical introduction to feed forward neural networks and describing the involved hyperparameters, we apply this popular form of neural networks to a slightly disguised data set from a German life insurer. Thereby, we demonstrate all practical aspects, such as the hyperparameter choice, to obtain our candidate neural networks by bruteforce, the calibration (“training”) and validation (“testing”) of the neural networks and judging their approximation performance. Compared to adaptive OLS, GLM, GAM and FGLS regression approaches, an ensemble built of the 10 best derived neural networks shows an excellent performance. Through a comparison with the results obtained by every single neural network, we point out the significance of the ensemble-based approach. Lastly, we comment on the interpretability of neural networks compared to polynomials for sensitivity analyses.

Practitioners sometimes suggest to use a combination of Sobol sequences and orthonormal polynomials when applying an LSMC algorithm for evaluation of option prices or in the context of risk capital calculation under the Solvency II regime. In this paper, we give a theoretical justification why good implementations of an LSMC algorithm should indeed combine these two features in order to assure numerical stability. Moreover, an explicit bound for the number of outer scenarios necessary to guarantee a prescribed degree of numerical stability is derived. We embed our observations into a coherent presentation of the theoretical background of LSMC in the insurance setting.

In August 2020 we published “Comprehensive Internal Model Data for Three Portfolios” as an outcome of our work for the committee “Actuarial Data Science” of the German Actuarial Association. The data sets include realistic cash-flow models outputs used for proxy modelling of life and health insurers. Using these data, we implement the hitherto most promising model in proxy modeling consisting of ensembles of feed-forward neural networks and compare the results with the least squares Monte Carlo (LSMC) polynomial regression. To date, the latter represents—to our best knowledge—the most accurate proxy function productively in use by insurance companies. An additional goal of this publication is a more precise description of “Comprehensive Internal Model Data for Three Portfolios” for other researchers, practitioners and regulators interested in developing solvency capital requirement (SCR) proxy models.