Explainable Deep Learning for Solvency Capital Requirement Calculation in Life Insurance

The European Union Directive 2009/138/EC -- better known as Solvency II requires that, to be solvent, insurance and reinsurance undertakings should hold their own funds able to cover losses in excess to expected ones at the 99.5% confidence level over a one-year period. For the insurance company that adopts the internal model, the measurement of this Solvency Capital Requirement (SCR) requires deriving the full probability distribution of the Net Asset Value (NAV), the value of assets minus the value of liabilities, over a one-year period. To comply with the market-consistent evaluation principles, the calculation of this distribution follows a two-step procedure. In the first step, one needs to generate a large number of future risk scenarios over a one-year horizon according to a multivariate stochastic model. It is generally performed by simulating, under the real-world probability measure, trajectories of the relevant risk factors driving the assets and liabilities cash flows via Monte Carlo simulations. In the second step, assets and liabilities have to be evaluated to assess the company's balance sheet in each outer scenario. The market-consistent value of the liabilities generally cannot be expressed in closed form due to the complex financial structure of insurance contracts containing embedded options and guarantees. In that case, the standard approach for obtaining an estimate consists of simulating, under risk-neutral probability measures, trajectories of the risk drivers over the policy lifetime and evaluating the expected value of the discounted liability cash flows. When simulations are used in both inner and outer steps, the solvency requirement measurement leads to nested simulations that present prohibitive computational times, even using distributed computation with several hundreds of cores. The Least Square Monte Carlo (LSMC) is one of the most popular methods used for reducing the computational costs of full nested simulations. The main idea consists of finding an approximation for the conditional expectation function to obtain an estimate of the NAV in each scenario, avoiding the second stage of simulations. The traditional version of the LSMC method makes use of an approximation based on orthogonal polynomials. Recently, Deep Neural Networks (DNNs) have become very popular in actuarial science and insurance since their ability to extend and generalize traditional actuarial models and produce more accurate predictions. They have been successfully applied to several actuarial tasks, such as pricing, reserving and mortality forecasting. Furthermore, DNN-based extensions of the LSMC for solvency calculation appear in the literature that overperform the classical LSMC method. Despite the huge potential of DNNs, the lack of explainability and interpretability affecting these models, sometimes defined as black boxes, limits their adoption in the insurance industry, which is a highly regulated sector.

We introduce an extension of the LSMC method based on modern explainable deep learning models called “localGLMnet”. More specifically, the proposed method combines the flexibility of the Neural Networks with the explainable structure of the Generalised Linear Model. Instead of using the DNNs to directly compute the quantity of interest, the localGLMnet computes the regression coefficients for each data point to put into a GLM. The inspection of these attention coefficients allows us to explain on which basis the predictions are formulated and derive useful insights about the risk exposure of the company, e.g., identifying the most relevant risk drivers and understanding the relationship between the risk drivers and the value of the insurance liabilities.

Numerical results performed on realistic insurance portfolio data show that our method is able to obtain accurate results and provide useful insights about the impact of the risk drivers on the NAV of the portfolio.