[Paper Review] Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests

Aurélien Alfonsi, Adel Cherchali+1 more|arXiv (Cornell University)|Oct 23, 2020

Insurance, Mortality, Demography, Risk Management32 references4 citations

TL;DR

This paper proposes a multilevel Monte Carlo (MLMC) estimator for computing the Solvency Capital Requirement (SCR) under the Solvency II standard formula and stress tests, particularly for future dates in life insurance ALM portfolios. It establishes theoretical convergence rates dependent on regularity near the maximum of conditional expectations and demonstrates superior computational efficiency over Least Squares Monte Carlo and neural networks, especially avoiding regression path-dependency issues.

ABSTRACT

This paper studies the multilevel Monte-Carlo estimator for the expectation of a maximum of conditional expectations. This problem arises naturally when considering many stress tests and appears in the calculation of the interest rate module of the standard formula for the SCR. We obtain theoretical convergence results that complements the recent work of Giles and Goda and gives some additional tractability through a parameter that somehow describes regularity properties around the maximum. We then apply the MLMC estimator to the calculation of the SCR at future dates with the standard formula for an ALM savings business on life insurance. We compare it with estimators obtained with Least Square Monte-Carlo or Neural Networks. We find that the MLMC estimator is computationally more efficient and has the main advantage to avoid regression issues, which is particularly significant in the context of projection of a balance sheet by an insurer due to the path dependency. Last, we discuss the potentiality of this numerical method and analyze in particular the effect of the portfolio allocation on the SCR at future~dates.

Motivation & Objective

• To develop a numerically efficient method for computing the Solvency Capital Requirement (SCR) at future dates under the Solvency II standard formula.
• To address the challenge of estimating the expectation of a maximum of conditional expectations arising in stress testing and SCR calculations.
• To provide theoretical convergence guarantees for the multilevel Monte Carlo estimator in this context, particularly under regularity conditions around the maximum.
• To compare the MLMC estimator with alternative methods such as Least Squares Monte Carlo and neural networks in the context of ALM and life insurance portfolios.
• To analyze the impact of portfolio allocation on future SCR values using the proposed numerical framework.

Proposed method

• The paper applies the multilevel Monte Carlo (MLMC) estimator to compute the SCR as the expectation of a maximum of conditional expectations, which arises in stress testing under the standard formula.
• It introduces a theoretical framework that establishes convergence rates for the MLMC estimator under a regularity parameter η, linking convergence speed to the behavior of conditional variances near the maximum.
• The method uses a hierarchical discretization of paths, with coarse and fine levels, to reduce variance in the difference estimator between levels.
• Key components include the use of conditional expectations, indicator functions for sign changes around the maximum, and bounds on the moments of estimation errors.
• Theoretical bounds are derived using properties of conditional expectations, moment inequalities, and the tower property to control variance across levels.
• The approach avoids regression by directly estimating the maximum of conditional expectations through pathwise MLMC sampling, reducing bias from regression approximation.

Experimental results

Research questions

• RQ1 How can the multilevel Monte Carlo method be theoretically justified for computing the SCR when the SCR involves the maximum of conditional expectations?
• RQ2 What regularity conditions on the underlying stochastic processes ensure convergence of the MLMC estimator in this context?
• RQ3 How does the MLMC estimator compare in computational efficiency and accuracy to Least Squares Monte Carlo and neural network-based estimators for future SCR computation?
• RQ4 What is the impact of asset allocation on the projected SCR at future dates, as revealed by the MLMC simulation framework?
• RQ5 Can the MLMC method effectively handle path-dependent cash flows and balance sheet projections in ALM settings without regression-based approximations?

Key findings

• The MLMC estimator achieves a convergence rate of order K^(-1 - η/2) for the variance of the difference estimator, where η quantifies the regularity of the conditional distribution around the maximum.
• Theoretical bounds show that the variance of the MLMC difference estimator decays faster when the conditional variance D^p_{2+η}(X) is well-behaved and the distance to the maximum is not too small.
• In numerical experiments, the MLMC estimator outperforms both Least Squares Monte Carlo and neural network estimators in terms of computational efficiency for SCR projection.
• The MLMC method avoids the path-dependency issues inherent in regression-based methods, making it particularly suitable for long-horizon ALM and balance sheet stress testing.
• The method reveals that portfolio allocation significantly influences the evolution of SCR over time, with implications for strategic asset allocation and cost of capital modeling.
• The framework is robust and scalable, with theoretical guarantees that support its use in practical Solvency II and ORSA (Own Risk and Solvency Assessment) applications.