When Value at Risk Is and Is Not Subadditive

Summary

The document raises the risk-measure question of whether Value at Risk (VaR) satisfies subadditivity: whether the risk of a combined position is no greater than the sum of the risks of its components. It notes that VaR can fail this property in general, while elliptical distributions are associated with subadditivity, and asks which distributions are elliptical and whether that family is the only setting where the property holds.

The text provides no resolution, worked example, or empirical evidence; it frames questions rather than presenting a complete analysis. It mentions normal and Student t distributions as possible elliptical examples and asks about stable and hyperbolic distributions, but does not establish their status or describe conditions involving dependence or correlation. Readers should treat it as a prompt to examine distributional assumptions and portfolio dependence when assessing VaR, not as a sufficient characterization of when VaR is subadditive.

Key ideas

• VaR may violate subadditivity, so portfolio risk can exceed the sum of standalone VaR measures.
• The document identifies elliptical distributions as a setting in which VaR subadditivity is expected to hold.
• It raises, but does not answer, which distribution families are elliptical.
• The effect of dependence and other conditions on VaR subadditivity remains unresolved in the document.

Tags

# subadditivity of VaR


# subadditivity of VaR












It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e.

$VaR(X)+VaR(Y) \le VaR(X+Y)$

But for elliptical distributions subadditivity is true. Questions:

(1) Which distributions are elliptical? I guess its the (multi)normal and t-distributions...Are stable and hyperbolic distributions elliptical,too?

(2) Is subadditvity only fulfilled for elliptical distributions? Are there any other conditions (e.g. correlation) which have an impact on subadditivity of VaR?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.