In this paper, we propose a novel multivariate shortfall risk measure to evaluate the systemic risk of a financial system, where the allocation weight is scenario-dependent and optimally chosen from a predetermined feasible set, and examine its properties such as (quasi-)convexity and translation invariance. To compute the proposed risk measure, we reformulate it as a two-stage stochastic program. When the underlying risk is discretely distributed, the second-stage program is a finite convex program while for the continuous case, is a semi-infinite program. To tackle the latter, we use the polynomial decision rule to approximate it and reformulate it as a tractable optimization program via the standard sums-of-squares techniques. Some convergence results are established for the approximation scheme. Moreover, we apply the proposed risk measure to the risk capital allocation problem and introduce the scenario-dependent allocation strategy. In contrast to the existing allocation methods, the new approach considers losses of all scenarios and minimizes the systemic risk. We then carry out some numerical tests on the proposed model and computational schemes for a continuous system, a discrete system, and a risk capital allocation problem in life insurance. The results show that our allocation strategy performs better than the Euler allocation rule based on the expected shortfall and the method by Armenti et al., 2018, and is robust to the (un-)systemic changes of the considered dataset. Finally, we extend our model by incorporating the cost of risk capital and investigate its impact on the optimal total amount of risk capital.

在本文中，我们提出了一种新颖的多变量短缺风险度量来评估金融系统的系统风险，其中分配权重取决于场景并从预定的可行集中最佳选择，并检查其属性，例如（准）凸性和平移不变性。为了计算建议的风险度量，我们将其重新表述为两阶段随机程序。当基础风险离散分布时，第二阶段程序是有限凸程序，而对于连续情况，是半无限程序。为了解决后者，我们使用多项式决策规则对其进行近似，并通过标准平方和技术将其重新表述为易于处理的优化程序。为近似方案建立了一些收敛结果。而且，我们将所提出的风险措施应用于风险资本分配问题，并引入了情景相关的分配策略。与现有的配置方法相比，新方法考虑了所有情景的损失，并将系统性风险降至最低。然后，我们针对连续系统、离散系统和人寿保险中的风险资本分配问题对所提出的模型和计算方案进行了一些数值测试。结果表明，我们的分配策略比基于预期短缺的欧拉分配规则和 Armenti 等人，2018 年提出的方法表现更好，并且对所考虑数据集的（非）系统变化具有鲁棒性。最后，我们通过纳入风险资本成本来扩展我们的模型，并研究其对最佳风险资本总量的影响。