Jul 07

Sub-additive property of risk measure (paper)

by David Harper, CFA, FRM, CIPM

FRM |

Here is a tight summary by Kay Giesecke of Standford et all, courtesy of www.defaultrisk.com, on alternative measures to value at risk (VaR). An FRM 2008 Quant AIM says,

  • Discuss the coherence property of risk measures

About VaR, the author says, "the ubiquitous value at risk, which does not account for the size of the losses exceeding the value at risk. Equally bothersome, value at risk may even penalize diversification. Average value at risk, which is also known as expected shortfall, does somewhat better – however not perfectly. Less well known but superior is utility-based shortfall risk"

Wilmott shares four coherence properties.

  • Sub-additive
  • Monotonicity
  • Positive homogeneity
  • Translation invariance

For our purposes, the most important is sub-additivity. This is the requirement that the measure not penalize diversification. That is, the risk of the portfolio cannot be greater than the sum of its components. This is also known as the convexity requirement, as its called in the paper (but we have plenty of other meanings for convexity, we don't need to add it here!)

Professor Giesecke gives intuitive examples of each of the coherence properties. And then brief introduction of alternative measures:

  • Average value at risk (a.k.a., expected shortfall) which, unlike VaR, does meet subadditive/convex criteria.
  • Utility-based shortfall risk

And the two EVT distributions that concern the FRM candidate:

  • Local maximum within intervals (block maxima) characterized by GEV
  • Peaks over threshold (POT) characterized by Generalized Pareto Distributions (GPD)

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