**Abstract**

We define a framework to compute the risk contributions of a portfolio consistently with a given performance-measurement schema. The framework has a wide array of applications, such as risk attribution, and matches the standard risk decomposition when the portfolio has a linear dependence on the factor exposures. We go beyond the traditional risk-decomposition approach, that uses Euler’s theorem, and instead require the risk measure to belong to a newly-defined class of severable risk measures. As such the risk-decomposition framework defined here can also be used in presence of liquidity risk. We show how the framework can also be used to compute the risk-driver contributions for a non-linear portfolio, define a corresponding concept of risk-driver exposure, and show how to hedge portfolios with respect to a given risk measure.

**Keywords**: isk measures, distortion risk measures, standard deviation, severable risk measures, risk decomposition, risk contributions, risk attribution, portfolio hedging, risk exposure, non-linear portfolios, liquidity risk, performance attribution, performance contributions, performance-contribution schema

**Pdf file**: risk-measure-components.pdf