A Reconciliation of the Top-Down and Bottom-Up Approaches to Risk Capital Allocations: Proportional Allocations Revisited

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.