Paper: arXiv 2511.04412
Authors: Mark-Oliver Wolf
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.
Complexity vs Empirical Score
- Math Complexity: 6.5/10
- Empirical Rigor: 7.0/10
- Quadrant: Holy Grail — high math complexity, high empirical rigor
Why this score: The paper provides a rigorous mathematical framework for proving the equivalence of direct and indirect estimation methods for life insurance own funds under Solvency II. It introduces an innovative family of mixed estimators and control variates, supported by publicly available source code and empirical testing across multiple ALM models.
Research Flowchart
flowchart TD
A[Research Goal: Evaluate Direct vs. Indirect Own Funds Estimation & Reduce Variance] --> B[Data/Inputs: 3 Life Insurance ALM Models with Varying Asset-Liability Coupling]
B --> C[Methodology: Prove Equivalence & Develop $2^T$ Mixed Estimator Family]
C --> D[Computational Process: Integrate Control Variate Framework & Include Market Frictions]
D --> E[Outcome 1: Model Coupling Dictates Performance; Strong Coupling Favors Indirect Method]
D --> F[Outcome 2: Control Variates Reduce Variance up to 90% Relative to Direct Estimator]
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- Least- Squares Monte Carlo Methods in the Life Insurance Sector — link
- LSMC Lite Zur SCR Quantifizierung von Modelländerungen
- Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies — link
- A Quantum Engineer ’s Guide to Superconducting Qubits — link
- Estimating Mutual Information — link
- Closed-Form Solutions for Guaranteed Minimum Accumulation Benefits
- Pricing Multi-Asset Derivatives by Variational Quantum Algorithms — link
- Variational Quantum Simulations of Stochastic Differential Equations — link
- Near-Optimal Quantum Tomography: Estimators and Bounds — link
- An Empirical Study on the Lapse Rate : The Cointegration Approach — link
- Quantum Architecture Search via Deep Reinforcement Learning — link
- Unsupervised Quantum Machine Learning for Fraud Detection — link
- Unsupervised Quantum Machine Learning for Fraud Detection — link
- Quantum Annealing for Neural Network Optimization Problems: A New Approach via Tensor Network Simulations — link
- Calculating Expectiles and Range Value-at-Risk Using Quantum Computers — link
- Combining Multi-Asset and Intrinsic Risk Measures — link
- Modelling in Life Insurance – A Management Perspective — link
- Financial Risk Management on a Neutral Atom Quantum Processor — link
- An Equilibrium Pricing Model for Weather Derivatives in a Multi-Commodity Setting — link
- A Multi-Period Equilibrium Pricing Model of Weather Derivatives — link
- Response to " Exponential Challenges in Unbiasing Quantum Monte Carlo Algorithms with Quantum Computers" — link
- Least Squares Monte Carlo for Fast and Robust Capital Projections
- Control Variates — link
- Quantum Hamiltonian Descent — link
- Central Bank Communication and the Yield Curve
- The Bitter Truth about Gate-Based Quantum Algorithms in the NISQ Era — link
- Department of Chemistry , Department of Physics and Astronomy , and Purdue Quantum Science and Engineering Institute Purdue University , West Lafayette , IN 47907, United States
- Ensemble-Learning Variational Shallow-Circuit Quantum Classifiers — link
- Nested Monte Carlo Simulation in Financial Reporting: A Review and a New Hybrid Approach — link
- Nested Monte Carlo Simulation in Financial Reporting: A Review and a New Hybrid Approach — link
- Quantum Monte Carlo Algorithm for Solving Black-Scholes PDEs for High-Dimensional Option Pricing in Finance and Its Proof of Overcoming the Curse of Dimensionality — link
- Quantum Monte Carlo Algorithm for Solving Black-Scholes PDEs for High-Dimensional Option Pricing in Finance and Its Proof of Overcoming the Curse of Dimensionality — link
- A Universal Quantum Circuit Design for Periodical Functions — link
- A Quantum Online Portfolio Optimization Algorithm — link
- Improved Convergence Rate of Nested Simulation with LSE on Sieve — link
- Noise Can Be Helpful for Variational Quantum Algorithms — link
- An Overview of Gaussian Process Regression for Volatility Forecasting — link
- Stochastic Kriging for Efficient Nested Simulation of Ecpexted Shortfall — link
- Hierarchical Quantum Circuit Representations for Neural Architecture Search — link
- Gaussian Process Models for Quantitative Finance — link
- Statistical Machine Learning for Quantitative Finance — link
- Adaptive Batching for Gaussian Process Surrogates with Application in Noisy Level Set Estimation — link
- Evaluating Gaussian Process Metamodels and Sequential Designs for Noisy Level Set Estimation — link
- A Variational Algorithm for Quantum Neural Networks — link
- A Preprocessing Perspective for Quantum Machine Learning Classification Advantage in Finance Using NISQ Algorithms — link
- Stress and Scenario Tests in the Context of a Romanian Non-Life Insurance Company — link
- Quantum Activation Functions for Quantum Neural Networks — link
- Toward Pricing Financial Derivatives with an IBM Quantum Computer — link
- Quantum Duality in Mathematical Finance — link
- Quantitative Risk Management: Concepts, Techniques and Tools
- Response Surface-Based Simulation Metamodelling Methods — link
- Discretization Error in Discounted Cash Flows for the Available Capital
- A Survey on Quantum Reinforcement Learning — link
- REFUNDABLE INCOME ANNUITIES : FEASIBILITY OF MONEY-BACK GUARANTEES
- A Quantum Hopfield Associative Memory Implemented on an Actual Quantum Processor — link
- Transport und Logistik für den Klimawandel fit machen
- Extracting a Function Encoded in Amplitudes of a Quantum State by Tensor Network and Orthogonal Function Expansion — link
- Pricing Multi-Asset Derivatives by Finite-Difference Method on a Quantum Computer — link
- Quantum Optimization Using Variational Algorithms on Near-Term Quantum Devices — link
- Interpretable Machine Learning — link
- The Subgraph Isomorphism Problem for Port Graphs and Quantum Circuits — link
- Quantum Neuron with Real Weights — link
- Realization of a Quantum Neural Network Using Repeat-until-Success Circuits in a Superconducting Quantum Processor — link
- Transformation of Quantum States Using Uniformly Controlled Rotations — link
- Estimation of Expectations and Variance Components in Two-Level Nested Simulation Experiments — link
- Probabilistic Machine Learning: An Introduction
- Elliptical Slice Sampling — link
- Management of Climate Risks in Agriculture–Will Weather Derivatives Permeate? — link
- Portfolio Effects and the Willingness to Pay for Weather Insurances — link
- Classification vs Regression in Overparameterized Regimes: Does the Loss Function Matter?
- Mathematical Foundation of the Replicating Portfolio Approach
- Mathematical Foundation of the Replicating Portfolio Approach — link
- Mathematical Foundation of the Replicating Portfolio Approach — link
- Does Publication of Interest Rate Paths Provide Guidance? — link
- The Quantum Query Complexity of Approximating the Median and Related Statistics — link
- Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification — link
- An Introduction to Copulas
- Fourier Expansion in Variational Quantum Algorithms — link
- Convergence Rates & Asymptotic Normality for Series Estimators
- Convergence Rates and Asymptotic Normality for Series Estimators — link
- An Evaluation of Hardware-Efficient Quantum Neural Networks for Image Data Classification — link
- Quantum Computation and Quantum Information
- Robust Regression Technique in LSMC Proxy Modeling
- Numerical Optimization
- Physics- Informed Neural Networks ( PINNs ) in Finance — link
- Risk Management in Global Supply Chains : Addressing Vulnerabilities in Shipping and Logistics — link
- Quantum Computing for Finance: Overview and Prospects — link
- Reinforcement Learning for Optimization of Variational Quantum Circuit Architectures — link
- Financial Index Tracking via Quantum Computing with Cardinality Constraints — link
- Das interne Modell der Feldafinger Brandkasse goes R — link
- A Novel Approach in Probabilistic Quantification of Risks within the Context of Maritime Supply Chain: The Case of Extreme Weather Events in the Arctic — link
- QReLU and M- QReLU : Two Novel Quantum Activation Functions to Aid Medical Diagnostics
- Active Learning Methods for Biophysical Parameter Estimation — link
- Control-Variate Estimation Using Estimated Control Means — link
- Quantum- Inspired Tensor Neural Networks for Option Pricing — link
- Vecchia Approximated Bayesian Heteroskedastic Gaussian Processes — link
- Equivalence Checking of Quantum Circuits With the ZX-Calculus — link
- Hedging Effectiveness of Weather Derivatives in Arable Farming – Is There a Need for Mixed Indices? — link
- The Difference between LSMC and Replicating Portfolio in Insurance Liability Modeling — link
- Quantum Boltzmann Machines : Applications in Quantitative Finance — link
- Quantum Boltzmann Machines : Applications in Quantitative Finance — link
- Generalization despite Overfitting in Quantum Machine Learning Models — link
- Matrix Cookbook — link
- Modeling and Simulation in Engineering , 2nd Edition — link
- Adaptive Designs of Experiments for Accurate Approximation of a Target Region — link
- Quantum Machine Learning for Finance ICCAD Special Session Paper — link
- Quasiprobability Decompositions with Reduced Sampling Overhead — link
- Essays on Valuation and Risk Management for Insurers — link
- Essays on Valuation and Risk Management for Insurers — link
- A Fair Pricing Approach to Weather Derivatives — link
- Variational Quantum Amplitude Estimation — link
- Quantum-State Preparation with Universal Gate Decompositions — link
- Demonstration of Algorithmic Quantum Speedup — link
- Solvency II Standard Formula Model - All Change — link
- Enhancing Sustainability through Weather Derivative Option Contracts : A Risk Management Tool in Greek Agriculture — link
- Understanding Deep Learning — link
- Optimal Design of Experiments | SIAM Publications Library — link
- Study on Diversification in Internal Models
- Quantum Annealing : An Overview — link
- Gaussian Processes for Machine Learning
- Improving the Least-Squares Monte-Carlo Approach — link
- Improving the Least-Squares Monte-Carlo Approach — link
- A Domain-agnostic , Noise-resistant , Hardware-efficient Evolutionary Variational Quantum Eigensolver — link
- Quantum Computational Finance: Monte Carlo Pricing of Financial Derivatives — link
- Quantum Computational Finance: Quantum Algorithm for Portfolio Optimization — link
- Quantum Computational Finance: Martingale Asset Pricing for Incomplete Markets — link
- Identifying the Determinants of Lapse Rates in Life Insurance: An Automated Lasso Approach — link
- Identifying the Determinants of Lapse Rates in Life Insurance: An Automated Lasso Approach — link
- Mathematical Statistics Lecture
- Regression and Time Series Analysis
- Bericht Über Die Solvabilität Und Finanzlage ( SFCR )
- Pricing Weather Derivatives — link
- A Performance Characterization of Quantum Generative Models — link
- Sequential Design and Spatial Modeling for Portfolio Tail Risk Measurement — link
- Multi-Year Analysis of Solvency Capital in Life Insurance — link
- Multi-Year Analysis of Solvency Capital in Life Insurance — link
- Advanced Macroeconomics
- Convergence Properties of Sequential Bayesian \mkbibemph D -Optimal Designs — link
- The Design and Analysis of Computer Experiments — link
- Instruction Scheduling Based on Subgraph Isomorphism for a High Performance Computer Processor
- Credit Risk Meets Large Language Models : Building a Risk Indicator from Loan Descriptions in P2P Lending — link
- DESIGN OF SUPPLY CHAIN CONTRACTS UNDER WEATHER RISK
- On the Regularity of the Positive Part of Functions — link
- The Standard Formula of Solvency II : A Critical Discussion — link
- Maß Und Integral : Eine Einführung Für Bachelor-Studenten
- MEASURES , INTEGRALS AND MARTINGALES
- Biased Control-Variate Estimation — link
- Mortality Modeling: Machine Learning and Mortality Shocks
- Effect of Data Encoding on the Expressive Power of Variational Quantum-Machine-Learning Models — link
- Is Quantum Advantage the Right Goal for Quantum Machine Learning? — link
- Is Quantum Advantage the Right Goal for Quantum Machine Learning? — link
- The Quest for a Quantum Neural Network — link
- Supervised Quantum Machine Learning Models Are Kernel Methods — link
- Richtlinie Der Schweizerischen Aktuarvereinigung Zur Bestimmung Ausreichender Technischer Rückstellungen Leben Gemäss FINMARundschreiben 2008/43 " Rückstellungen Lebensversicherung "
- Active Learning Literature Survey
- QAOA with $ N \textbackslash cdot P\textbackslash geq 200$ — link
- How Do Firms Perceive and React to Extreme Weather Risk in Their Supply Bases? — link
- A Hybrid Classical-Quantum Algorithm for Digital Image Processing — link
- A Systematic Review of Packages for Time Series Analysis — link
- Modelling Longevity Risk for Solvency II
- An Example of Use of Variational Methods in Quantum Machine Learning — link
- Optimal Experimental Design — link
- Efficient State Preparation for a Register of Quantum Bits — link
- Efficient State Preparation for a Register of Quantum Bits — link
- Option Pricing Using Quantum Computers — link
- Option Pricing Using Quantum Computers — link
- Towards Quantum Advantage in Financial Market Risk Using Quantum Gradient Algorithms — link
- Monte Carlo Theory, Methods and Examples — link
- Quantum Optical Neural Networks — link
- Effectiveness of Weather Derivatives as a Risk Management Tool in Food Retail : The Case of Croatia — link
- Surrogate Models for Uncertainty Quantification: An Overview — link
- Active Learning in Approximately Linear Regression Based on Conditional Expectation of Generalization Error
- Active Learning in Approximately Linear Regression Based on Conditional Expectation of Generalization Error — link
- Active Learning in Approximately Linear Regression Based on Conditional Expectation of Generalization Error — link
- Machine Learning in Non-Stationary Environments : Introduction to Covariate Shift Adaptation — link
- Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis — link
- Credit Risk Modeling Using Transfer Learning and Domain Adaptation — link
- Amplitude Estimation without Phase Estimation — link
- An Artificial Neuron Implemented on an Actual Quantum Processor — link
- Quantum Implementation of an Artificial Feed-Forward Neural Network — link
- Stochastic Gradient Line Bayesian Optimization for Efficient Noise-Robust Optimization of Parameterized Quantum Circuits — link
- Sign-Problem Free Quantum Stochastic Series Expansion Algorithm on a Quantum Computer — link
- Amplitude Estimation via Maximum Likelihood on Noisy Quantum Computer — link
- Quantum Computation for Pricing Caps Using the LIBOR Market Model — link
- Capital Allocation for Credit Portfolios with Kernel Estimators — link
- Weather Derivatives and Hedging the Weather Risks — link
- Exponential Concentration and Untrainability in Quantum Kernel Methods — link
- Variational Learning of Inducing Variables in Sparse Gaussian Processes — link
- Testing for (in)Finite Moments — link
- Dropout- Based Active Learning for Regression — link
- Stuart M . Turnbull — link
- Quantum Topological Data Analysis with Linear Depth and Exponential Speedup — link
- Framework for Understanding Quantum Computing Use Cases from a Multidisciplinary Perspective and Future Research Directions — link
- Asymptotic Statistics — link
- ZX-calculus for the Working Quantum Computer Scientist — link
- Efficient State Preparation for Quantum Amplitude Estimation — link
- Efficient State Preparation for Quantum Amplitude Estimation — link
- eSPA +: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems — link
- Estimation and Model Identification for Continuous Spatial Processes — link
- The Quantum House Of Cards — link
- Exploration of Quantum Neural Architecture by Mixing Quantum Neuron Designs : ( Invited Paper ) — link
- High-Dimensional Regression in Practice: An Empirical Study of Finite-Sample Prediction, Variable Selection and Ranking — link
- The Milliman VALUES Report Is Available for Purchase. For Further Information, Please Contact:
- Quantum Amplitude Arithmetic
- Seaborn.Kdeplot — Seaborn: Statistical Data Visualization — link
- Machine Learning for Financial Forecasting, Planning and Analysis: Recent Developments and Pitfalls — link
- Encoding Patterns for Quantum Algorithms — link
- An Aspect of Optimal Regression Design for LSMC — link
- Asymptotic Theory for Econometricians
- Learning in the Presence of Concept Drift and Hidden Contexts — link
- Floating Point Representations in Quantum Circuit Synthesis — link
- Quantum Risk Analysis — link
- Automated Function Implementation via Conditional Parameterized Quantum Circuits with Applications to Finance — link
- On the Estimation of Own Funds for Life Insurers : A Study of Direct , Indirect , and Control Variate Methods in a Risk-Neutral Pricing Framework — link
- openIRM : Publicly Accessible Internal Risk Model of an Artificial Life Insurer for Analyzing and Benchmarking Actuarial Methods in the Solvency II Setting — link
- A Quantum Algorithm for Pricing Asian Options on Valuation Trees — link
- A Study of Direct & Indirect Method and the Construction of Control Variates for Estimating the Available Capital
- Understanding Pathologies of Deep Heteroskedastic Regression — link
- Active Learning for Regression Using Greedy Sampling — link
- Financial Modeling , Actuarial Valuation and Solvency in Insurance — link
- Statistical Foundations of Actuarial Learning and Its Applications — link
- Variational Decoupling of Quantum Dynamics
- ADTCD : An Adaptive Anomaly Detection Approach Toward Concept Drift in IoT — link
- Quantum Ridgelet Transform : Winning Lottery Ticket of Neural Networks with Quantum Computation — link
- Nonlinear Quantum Neuron : A Fundamental Building Block for Quantum Neural Networks — link
- Spectral Analysis of Product Formulas for Quantum Simulation — link
- Gaussian Process Regression with Local Explanation — link
- Bayesian Neural Networks — link
- Power and Limitations of Single-Qubit Native Quantum Neural Networks — link
- Quantum Complexity of Permutations — link
- Optimal (Controlled) Quantum State Preparation and Improved Unitary Synthesis by Quantum Circuits with Any Number of Ancillary Qubits — link
- A Least Squares Monte Carlo Approach to Liability … - Barrie & Hibbert — link
- Quantum Multi-Agent Meta Reinforcement Learning — link
- Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment — link
- Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment — link
- Risk-Neutral Valuation of Participating Life Insurance Contracts in a Stochastic Interest Rate Environment — link
- Weather Derivatives and Weather Insurance : Concept , Application , and Analysis . — link
- Differentiable Quantum Architecture Search — link
- Quantum State Preparation with Optimal Circuit Depth : Implementations and Applications — link
- Sample Recycling for Nested Simulation with Application in Portfolio Risk Measurement — link
- Sample Recycling for Nested Simulation with Application in Portfolio Risk Measurement — link
- Stock Price Prediction Based on a Hybrid Quantum-Classical Neural Network Model — link
- Efficient Two-Stage Gaussian Process Regression Via Automatic Kernel Search and Subsampling — link
- A Review of Quantum Neural Networks : Methods , Models , Dilemma
- Hedging Crop Yield with Exchange-Traded Weather Derivatives — link
- A Brief Survey of Quantum Architecture Search — link
- Coherent Modeling of the Risk in Mortality Projections : A Semi-Parametric Approach
- Global Sensitivity Analysis for Stochastic Simulators Based on Generalized Lambda Surrogate Models — link
- Supply Chain Risk Management and Credit Assurance for Enterprises in Extreme Weather — link
- Quantum Encoding and Analysis on Continuous Stochastic Process — link
- Lapse Rate Models in Life Insurance and a Practical Method to Foresee Interest Rate Dependencies
- Quantum Generative Adversarial Networks for Learning and Loading Random Distributions — link
- Variational Quantum Algorithm for Unconstrained Black Box Binary Optimization: Application to Feature Selection — link
1 of 657 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.