Risk contributions: duality and sensitivity
研究了投资组合分解中风险贡献的对偶性，证明了对偶分解与原始分解给出相同风险贡献的条件，并探讨了风险贡献对风险体制变化的敏感性。
Given a decomposition of a portfolio P as a sum of K components, practitioners commonly decompose the risk of P as a corresponding sum of risk contributions. In this paper, we prove two theorems about risk contributions. The first theorem concerns a form of duality identified in Grinold [J. Portfolio Manage. 2011, 37(2), 15–30], which may be described as follows. When we view a portfolio decomposition as a coordinate representation of the portfolio with respect to a given vector-space basis, then there is a natural dual basis with respect to which there is an alternative decomposition, here referred to as the dual decomposition. The dual decomposition gives the same contributions to risk as the original decomposition. The first theorem gives necessary and sufficient conditions for a change of basis to preserve risk contributions, and shows that all such changes of basis can be explained in terms of dual decompositions. The second theorem explores sensitivity of portfolio risk to a risk regime change and indicates that large risk contributions or large risks of the components of a decomposition may be harbingers of high sensitivity. This provides a motivation for the practice of reporting both the risk contributions and the risks of the components in a decomposition.