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European Actuarial Journal
Published in: European Actuarial Journal 1/2019

12-02-2019 | Original Research Paper

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

Author: Quang Dien Duong

Published in: European Actuarial Journal | Issue 1/2019

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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.
previous article Discussion on “Experience rating in the classic Markov chain life insurance setting: an empirical Bayes and multivariate frailty approach” (Furrer)
next article Generational transfers within the occupational pension system in Switzerland

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Appendix
Available only for authorised users
Footnotes
1
This can be easily proven by setting \(\delta (x) = g(x) - {\hat{g}}(x)\), we have \(\delta (x_{i-1}) = \delta (x_i) = 0\). Using Rolle theorem and noting \(M = \max _{x_{i-1} < x \le x_i}\left| \delta ^{''}(x)\right|\), we get on the segment \([x_{i-1}, x_i]\), the inequality \(\left| \delta '(x)\right| \le \left( x_i -x_{i-1}\right) M\). The result is obtained by using integral of \(\delta '\) and triangular inequality.
 
2
The TH-TF 0002 mortality table is built from the INSEE 2000–2002 table—respectively for the male population and for the female population. These are the regulatory tables for life insurance contracts (other than life annuities) The table is available at http://​www.​spac-actuaires.​fr/​jdd/​public/​documents/​xls/​TH-TF%20​00-02.​xls.
 
3
In k-fold cross-validation, we partition a dataset S into k equally sized non-overlapping subsets \(S_i\). For each fold \(S_i\), a model is trained on \(S \backslash S_i\) and is then evaluated on \(S_i\). The cross-validation estimator of the mean squared prediction error is defined as the average of the mean squared prediction errors obtained on each fold. There is however overlap between the training sets for all \(k>2\) and the overlap is largest for leave-one-out cross validation. This means that the learned models are correlated implying the increasing amount of variance in the mean squared prediction error estimation. Furthermore, while two-fold cross validation does not have the problem of overlapping training sets, it also has large variance since the training sets are only half the size of the original sample. Therefore, a good compromise is usually 10-fold cross-validation (see, for instance, [4]).
 
4
In case of \(\text {deg} < \text {deg}_{\text {optimal}}\), one misses the pattern while trying to avoid fitting the noise which leads to underfitting. On the contrary, if \(\text {deg} > \text {deg}_{\text {optimal}}\), one tries to fit the noise in addition to the pattern which leads to overfitting.
 
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Metadata
Title
Application of Bayesian penalized spline regression for internal modeling in life insurance
Author
Quang Dien Duong
Publication date
12-02-2019
Publisher
Springer Berlin Heidelberg
Published in
European Actuarial Journal / Issue 1/2019
Print ISSN: 2190-9733
Electronic ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-019-00192-3

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