A risk measure that satisfies a set of essential properties for a position or a basket of assets, namely: monotonicity, subadditivity, homogeneity, and translational invariance. Subadditivity, as a property of a coherent risk measure, implies that adding together two risky position (e.g., portfolios) will not increase the measure of risk. Specifically, a coherent risk measure is monotonic, subadditive, positive homogeneous and cash invariant.
A coherent risk measure for a basket of options, for example, implies that the above properties will not spin out of normal limits for the entire set or when additional components are added to the basket.
Generally speaking, a coherent risk measure is used to quantify the risk of a position or set of assets subject to the above identifying conditions. It represents a spectrum of certain expected values over a so-called risk envelope.
