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

TL;DR

This paper compares direct and indirect methods for estimating own funds in life insurance, introduces a novel family of mixed estimators with variance reduction, and evaluates their performance considering market frictions and asset-liability coupling.

Contribution

It introduces a new family of mixed estimators within a control variate framework, extending to market frictions, and analyzes their performance in different asset-liability coupling scenarios.

Findings

Indirect method performs better with strong asset-liability coupling.

Control variates can reduce variance to one-tenth of the direct estimator.

Performance depends on model-specific asset-liability dynamics.

Abstract

The Solvency Capital Requirement (SCR) calculation is computationally intensive, relying on the market-consistent estimation of own funds. While Solvency II prioritizes the direct valuation method, it theoretically yields the same value as the indirect method. This paper evaluates their practical performance within a risk-neutral pricing framework. First, we present a simplified proof that direct and indirect estimators converge to the same value. For $T$ being the number of time steps in the simulation, we then introduce a novel family of $2^T$ mixed estimators including both methods as edge cases, integrating them into a control variate framework for significant variance reduction. This framework is further extended to incorporate market frictions for real-world applicability. Evaluating these estimators on three life insurance asset-liability management models demonstrates that their performance is fundamentally driven by the degree of asset-liability coupling. While stronger coupling in realistic settings consistently favors the indirect method, neither baseline estimator is universally superior. Furthermore, this coupling directly impacts the success of our proposed control variates. They can dramatically reduce variance to one-tenth of the standard direct estimator, but their efficacy remains model-dependent. The source code is publicly available on https://gitlab.cc-asp.fraunhofer.de/itwm-fm-lv-public/wolf-estimation-of-own-funds.