On the Estimation of Own Funds for Life Insurers: A Study of Direct, Indirect, and Control Variate Methods in a Risk-Neutral Pricing Framework

Mark-Oliver Wolf

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Abstract: The Solvency Capital Requirement (SCR) calculation under Solvency II is computationally intensive, relying on the market-consistent estimation of own funds. While regulation mandates the direct estimation method, it results in the same estimate as the indirect method under consistent assumptions as shown by Girard (2002). This paper studies their comparative properties to offer new insights into their practical performance. First, we provide a straightforward proof that the direct and indirect estimators for own funds converge to the same value. Second, we introduce a novel family of mixed estimators that encompasses the direct and indirect methods as its edge cases. Third, we leverage these estimators to develop powerful variance reduction techniques, constructing a single control variate from the direct and indirect estimators and a multi-control variate framework using subsets of the mixed family. These techniques can be combined with existing methods like Least-Squares Monte Carlo. We evaluate the estimators on three simplified asset-liability management models of a German life insurer, Bauer's model MUST and IS case by Bauer, Kiesel et al. (2006), and openIRM by Wolf et al. (2025). Our analysis confirms that neither the direct nor indirect estimator is universally superior, though the indirect method consistently outperforms the direct one in more realistic settings. The proposed control variate techniques show significant potential, in some cases reducing variance to one-tenth of that from the standard direct estimator. However, we also identify scenarios where improvements are marginal, highlighting the model-dependent nature of their efficacy. The source code is publicly available on https://gitlab.cc-asp.fraunhofer.de/itwm-fm-lv-public/wolf-estimation-of-own-funds.

Date: 2025-11, Revised 2025-11
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