**Abstract**

We describe a method, suitable for numerical simulations, to compute the risk contributions for a generic portfolio of asset. The standard results in the covariance framework provide risk components computed from the derivative of the risk function with respect to the asset exposure. Those results are generalized to a generic positively-homogeneous risk measure, but cannot be easily applied to value at risk because of the resulting instabilities. We show how introducing a new risk measure, the unbiased average value at risk, it is possible to split exactly the portfolio risk into stable additive components. The results obtained in this paper are general, stable, and can be used in portfolios containing products belonging any asset class. The risk-decomposition results are generalised to the computation of risk components at segment levels. Finally, we show a real-world application of the described method with a numerical example.

**Keywords**: risk, risk contributions, risk attribution,risk components,portfolio risk, risk measures, value at risk, var,risk decomposition, marginal risk, portfolio segments

**Pdf file**: portfolio-risk-decomposition.pdf