Finite Difference Methods for Estimating Marginal Risk Contributions in Asset Management

Abstract

The decomposition of portfolio risks in terms of the underlying assets, which are extremely important for risk budgeting, asset allocation and risk monitoring, is well described by risk contributions. However, risk contributions cannot be calculated analytically for a considerable number of the risk models used in practice. We therefore study the use of finite difference methods for estimating risk contributions. We find that for practically relevant setups the additional estimation errors of the finite difference formulas are negligibly small. Since finite difference methods work for complex risk models and are independent of decisions about underlying distributions, we suggest the use of finite difference methods as the standard procedure for estimating risk contributions. As an application, we consider a general risk model that fits a kernel density estimation to the historical asset return distribution combined with a finite difference method in order to arrive at the risk contributions. It turns out that this general risk model combined with a finite difference method for calculating risk contributions works well in terms of estimation error.

Keywords: risk contributions, estimation error, expected shortfall (ES).finite difference (FD) methods, kernel density estimation