**Title**: Risk contribution framework for non-linear portfolios

**Abstract:**

The computation of risk contributions is a necessary complementary tool to be used in conjunction with the computation of risk measures and stress tests. Even though the theory of risk measures for a generic probability distribution has been well established, so far there seems to be no agreement in the way risk components should be computed for a portfolio with non-linear exposure to the risk drivers. In this paper we go beyond the traditional approach of using Euler’s theorem to define risk contributions and we define a generic framework to compute the risk components for a generic class of risk measures known as severable risk measures. The result has a wide array of applications and matches the standard definition of risk contributions in the specific case of linear dependence from the risk-driver exposures. One of the main highlights of this framework is that the computed risk components can be made compatible with any ex-post portfolio component schema, thus creating a strong link between the worlds of risk management and that of performance measurement. Similarly to the standard case, in the case of a portfolio with a non-linear dependence from the risk-drivers we can express the risk contributions to be proportional to a newly-defined risk exposure and to the marginal risk drivers.

**Keywords: **Risk management, risk measures, risk attribution, risk contribution, non-linear portfolios.