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 complement the recent work of Giles and Goda (2019) and give 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 Squares 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 potential of this numerical method and analyse in particular the effect of the portfolio allocation on the SCR at future dates.

本文研究了对条件期望最大值的期望的多级蒙特卡洛估计器。这个问题在考虑很多压力测试时自然会出现，并且出现在SCR标准公式的利率模块的计算中。我们获得了理论收敛结果，这些结果补充了 Giles 和 Goda（2019）最近的工作，并通过以某种方式描述最大值周围的规律性属性的参数提供了一些额外的易处理性。然后，我们使用人寿保险 ALM 储蓄业务的标准公式，将 MLMC 估算器应用于未来日期的 SCR 计算。我们将其与使用最小二乘蒙特卡罗或神经网络获得的估计量进行比较。我们发现 MLMC 估计器在计算上更有效，并且具有避免回归问题的主要优势，这在保险公司由于路径依赖性而预测资产负债表的情况下尤为重要。最后，我们讨论了这种数值方法的潜力，并特别分析了未来日期投资组合分配对 SCR 的影响。