Bounds on Aggregate Assets

Abstract

Aggregating financial assets together to form a portfolio, commonly referred to as "asset pooling", is a standard practice in the banking and insurance industries. Determining a suitable probability distribution for this portfolio with each underlying asset is a challenging task unless several distributional assumptions are made. On the other hand, imposing assumptions on the distribution inhibits its ability to capture various idiosyncratic behaviors. It limits the model's usefulness in its ability to provide realistic risk metrics of the true portfolio distribution. In order to conquer this limitation, we propose two methods to model a pool of assets with much less assumptions on the correlation structure by way of finding analytical bounds. Our first method uses the Fréchet-Hoeffding copula bounds to calculate model-free upper and lower bounds for aggregate assets evaluation. For the copulas with specific constraints, we improve the Fréchet- Hoeffding copula bounds by providing bounds with narrower range. The improvements proposed are very robust for different types of constraints on the copula function. However, the lower copula bound does not exist for dimension three and above. Our second method tackles the open problem of finding lower bounds for higher dimensions by introducing the concept of Complete Mixability property. With such technique, we are able to find the lower bounds with specified constraints. Three theorems are proposed. The first theorem deals with the case where all marginal distributions are identical. The lower bound defined by the first theorem is sharp under some technical assumptions. The second theorem gives the lower bound in a more general setup without any restriction on the marginal distributions. However the bound achieved in this context is not sharp. The third theorem gives the sharp lower bound on Conditional VaR. Numerical results are provided for each method to demonstrate sharpness of the bounds. Finally, we point out some possible future research directions, such as looking for a general sharp lower bound for high dimensional correlation structures.