Efficient Computation of Solvency Capital Requirement using Multilevel Monte Carlo Methods

Many insurance companies are struggling to overcome the computational challenges involved in computing the Solvency Capital Requirement (SCR) under the Solvency II regime. Market Standard approaches such as Least-Square Monte-Carlo and Replication Portfolios can be difficult to calibrate and validate in practice but also lead to deficient outcomes if not calibrated properly. We show how the Multilevel Monte-Carlo method is a relevant alternative to compute risk-based capital requirements as it does not rely on any proxy assumptions.

These types of problem involving simulations within simulations can be framed in the so-called Nested Simulation setting where outer scenarios are used to project the portfolio risk factors up to the risk horizon under the real-world probability, then inner simulations are necessary to compute the portfolio value conditionally on each primary scenario. This brute force approach is too time consuming to be used in a real insurance business case.

We introduce the Multilevel Monte Carlo methods (MLMC) developed by Giles (2008), Giles et Al. (2019) along with their associated refinement strategies from Lemaire & Pagès (2017). These methods rely on a smart allocation of a given computational budget between inner and outer scenarios spread across different levels to obtain an optimal tradeoff between variance reduction and bias correction. In the literature, these MLMC techniques and their variants were applied mostly to pricing of simplified financial portfolio models. Cherchali et al. (2021) applied successfully MLMC techniques to SCR computation with a Standard Formula approach. Our contribution is their application to an Internal Model approach.

The talk will be structured follows: At first, we introduce the standard MLMC methods and describe refinements that are relevant for SCR quantile estimation, then in a second part we illustrate the performances of different algorithms on numerical experiments on a simplified insurance balance sheet / portfolio to compute the SCR. A comparison of computational efficiency of a crude nested Monte-Carlo estimator against multilevel Monte-Carlo estimators will be presented.

As a main outcome, we will exhibit the advantages for insurance companies of the MLMC estimators which avoid the usage of a proxy while reducing the computational burden of a crude nested simulation setting, thus providing insurers with a relevant alternative to proxy methods.

References :

- Michael B Giles. Multilevel Monte-Carlo path simulation. Operations research, 56(3):607–617, 2008.

- Michael B. Giles and Abdul-Lateef Haji-Ali. Multilevel nested simulation for efficient risk estimation. SIAM/ASA J. Uncertain. Quantif., 7(2):497–525, 2019

- Lemaire, V., & Pagès, G. (2017). Multilevel Richardson–Romberg extrapolation. Bernoulli, 23(4A), 2643-2692.

- Alfonsi, A., Cherchali, A., & Acevedo, J. A. I. (2021). Multilevel Monte-Carlo for computing the SCR with the standard formula and other stress tests. Insurance: Mathematics and Economics, 100, 234-260.