In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of prespecified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to be able to do so at the lowest possible cost and to identify the corresponding portfolios, or, equivalently, their payoffs. We study the existence and uniqueness of such optimal payoffs as well as their behavior under a perturbation or an approximation of the underlying capital position. This behavior is naturally linked to the continuity properties of the set‐valued map that associates to each capital position the corresponding set of optimal eligible payoffs. Upper continuity can be ensured under fairly natural assumptions. Lower continuity is typically less easy to establish. While it is always satisfied in a polyhedral setting, it generally fails otherwise, even when the reference risk measure is convex. However, lower continuity can often be established for eligible payoffs that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.

中文翻译：

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合格资产的最佳收益的存在，唯一性和稳定性

在资本充足率框架中，风险度量用于确定金融机构为了通过给定的资本充足率测试而必须筹集和投资于预先指定的合格资产组合的最小资本量。从资本效率的角度来看，重要的是要以尽可能低的成本做到这一点，并确定相应的投资组合，或等效地，确定其收益。我们研究了这种最优收益的存在和唯一性，以及在基本资本头寸的扰动或近似下的行为。这种行为自然与设置值映射的连续性属性相关，该设置值映射将与每个资本头寸相关的一组最佳合格收益相关联。在相当自然的假设下可以确保较高的连续性。较低的连续性通常较难建立。虽然在多面体环境中始终可以满足要求，但即使参考风险度量是凸面的，也通常会失败。但是，通常可以为接近最佳的合格收益建立较低的连续性。除了资本充足率外，我们的结果在定价，对冲和资本分配问题上有多种自然应用。