YouTube T4-05: Coherent risk measures and why VaR is not coherent

Nicole Seaman

Director of FRM Operations
Staff member
Coherence requires that a risk measure meets all four of the following conditions unconditionally: 1. Translation invariance (aka, adding cash reduces risk), 2. Positive homogeneity (aka, risk is proportional to size"), 3. Monotonicity (aka, If Y dominates X, then Y is less risky than X), and 4. Subadditivity (aka, the risk measure should not penalize diversification). Value at risk (VaR) is a popular risk measure but VaR is NOT coherent because it is not necessarily sub-additive (instead, VaR is only subadditive if the returns are normally distributed). We can illustrate VaR's lack of subadditivity by observing that the VaR of a single bond can easily be zero, yet when combined into a portfolio of identical bonds, the portfolio VaR is greater than zero. VaR is often not subadditive when such a property is most desired: when the tails are heavy. Lack of subadditivity is of practical significance.

David's XLS is here:

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