# Application of Bayesian penalized spline regression for internal modeling in life insurance

## Abstract

Solvency 2 requires insurance companies to compute a Best-Estimate of their Liabilities (BEL) as well as a Solvency Capital Requirement (SCR). Life insurance companies being in the business of selling participating contracts with financial guarantees have to rely on a Monte-Carlo approach to appropriately value their BEL, which is the source of a first Monte-Carlo statistical error. In addition, several insurance companies rely on a (partial) internal model to derive their SCR. In this context, insurance companies rely again on a Monte-Carlo approach to value their SCR, which is the source of a second Monte-Carlo statistical error. These later computations require evaluating the BEL several thousand times, which is not possible in practice, since one single Monte-Carlo BEL evaluation can be a computational burden. The BEL has therefore to be approximated by an analytic proxy function, which introduces an additional source of numerical approximation error. In this paper, we show how these three sources of error (statistical and numerical) are intrinsically related to one another. We show that to obtain the best possible SCR accuracy, the computing power invested in assessing the Monte-Carlo SCR should be directly related to that invested in computing the Monte-Carlo BEL. Interestingly, and to achieve these results, we introduce a novel proxy method, which is highly practical, modular, smooth and naturally relates the approximation errors to the Monte-Carlo statistical errors. Furthermore, our approach allows insurance companies to naturally and transparently start reporting confidence levels on their prudential reporting, which is not disclosed so far by insurance companies and would be a relevant information within solvency disclosures for the industry.

## Keywords

Life Insurance Mathematics Bayesian penalized spline regression Solvency Capital Requirement## Notes

### Acknowledgements

This work has been supported by the French National Agency for Research and Technology (ANRT) CIFRE 614/2015 and by PwC’s Risk and Value Measurement Services. The author gratefully acknowledges Prof. Agathe Guilloux, Prof. Olivier Lopez, in particular Dr. Jean Baptiste Monnier and Didier Riche for their time and discussions who helped develop this methodology. These collaborations led to a successful implementation of the ALM simulator. Special thanks also go to two anonymous referees for their insightful comments and valuable suggestions that significantly improve the paper.

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