In the current reality of prudent risk management, the problem of determining aggregate risk capital in financial entities has been intensively studied. As a result, canonical methods have been developed and even embedded in regulatory accords. Though applauded by some and questioned by others, these methods provide a much desired standard benchmark for everyone. The situation is very different when the aggregate risk capital needs to be allocated to the business units (BUs) of a financial entity. That is, there are overwhelmingly many ways to conduct the allocation exercise, and there is arguably no standard method to do so on the horizon. Two overarching approaches to allocate the aggregate risk capital stand out. These are the top-down allocation (TDA) approach that entails that the allocation exercise be imposed by the corporate center, and the bottom-up allocation (BUA) approach that implies that the allocation of the aggregate risk to BUs is informed by these units. Briefly, the TDA starts with the aggregate risk capital that is then replenished among the BUs according to the views of the center, thus limiting the inputs from the BUs. The BUA does start with the BUs but it is, as a rule, too granular and so may lead to missing the wood for the trees. Irrespective of whether the TDA or the BUA is assumed, it is the proportional contribution of the riskiness of a stand-alone BU to the aggregate riskiness of the financial entity that is of central importance, and it is routinely computed nowadays as the quotient of the allocated risk capital due to the BU of interest and the aggregate risk capital due to the financial entity. For instance, in the simplest case when the mathematical expectation plays the role of the risk measure that generates the allocation rule, the desired proportional contribution is just a quotient of two means. Clearly, in general, this quotient of means does not concur with the mean of the quotient random variable that captures the genuine stochastic proportional contribution of the riskiness of the BU of interest. Inspired by this observation, herein we reenvision the way in which the allocation problem is tackled in the state of the art. As a by-product, we unify the TDA and the BUA into one encompassing approach.

在当前审慎风险管理的现实中，金融实体的风险总资本的确定问题得到了深入研究。因此，规范的方法已经被开发出来，甚至被嵌入到监管协议中。尽管有人为之鼓掌，有人质疑，但这些方法为每个人提供了一个非常理想的标准基准。当需要将总风险资本分配给金融实体的业务单位 (BU) 时，情况就大不相同了。也就是说，有很多方法可以进行分配练习，并且可以说没有标准的方法可以做到这一点。两种分配总风险资本的总体方法脱颖而出。这些是自上而下的分配 (TDA) 方法，需要由企业中心进行分配，自下而上的分配 (BUA) 方法意味着将总风险分配给 BU 的信息由这些单位提供。简而言之，TDA从总风险资本开始，然后根据中心的意见在BU之间进行补充，从而限制了BU的投入。BUA 确实从 BU 开始，但通常它过于细化，因此可能会导致树木不见木头。无论假设 TDA 还是 BUA，独立 BU 的风险对金融实体总风险的比例贡献才是最重要的，现在通常计算为分配给利息 BU 的风险资本和给金融实体的总风险资本。例如，在最简单的情况下，当数学期望扮演生成分配规则的风险度量的角色时，所需的比例贡献只是两个均值的商。显然，一般来说，该均值商与捕捉感兴趣的 BU 风险的真正随机比例贡献的商随机变量的均值不一致。受这一观察的启发，我们在此重新设想了在现有技术中解决分配问题的方式。作为一个副产品，我们将 TDA 和 BUA 统一为一个包罗万象的方法。该均值商与捕获感兴趣的 BU 风险的真正随机比例贡献的商随机变量的均值不一致。受这一观察的启发，我们在此重新设想了在现有技术中解决分配问题的方式。作为一个副产品，我们将 TDA 和 BUA 统一为一个包罗万象的方法。该均值商与捕获感兴趣的 BU 风险的真正随机比例贡献的商随机变量的均值不一致。受这一观察的启发，我们在此重新设想了在现有技术中解决分配问题的方式。作为一个副产品，我们将 TDA 和 BUA 统一为一个包罗万象的方法。