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Additional info for Backtesting Value at Risk and Expected Shortfall

Example text

A risk measure is said to be robust if it is quite insensible to measurement errors. Cont et al. [16] observe that measuring the risk of a financial portfolio involves two steps: estimating the loss distribution from empirical data and computing the risk metric ρ which, as usual, is a map assigning a number to each random payoff. They also point out that, even if these two operations have always been studied separately, they are strictly related (at least in applications) and their connection is of great importance when opting for a risk measure, rather than another.

We conclude this subsection by enunciating the fundamental representation theorem for convex measures of risk. Chapter 2. 9. Suppose X is the space of all real-valued functions on a finite set Ω. Then, ρ : X → R is a convex measure of risk if and only if there exists a "penalty function" α : P → (−∞, ∞] such that ρ(X) = sup EQ [−X] − α(Q) Q∈P The function α satisfies α(Q) ≥ −ρ(0) for any Q ∈ P, and it can be taken to be convex and lower semicontinuous on P. Observe that the structure theorem of coherent risk measure is a special case of the above.

14 becomes: n n Li ) > V aRα ( i=1 V aRα (Li ) i=1 Previously, we have stated that VaR is not sub-additive in general. Indeed, whether or not it is the case depends on the properties of the joint loss distribution and the standard situations in which our measure comes out to be sub-additive, are the following: i) The random variables are independent and identically distributed, as well as positively regularly varying. ii) The random variables have an elliptical distribution. iii) The random variables have an Archimedean survival dependence structure.

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