Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk

6.3.5 VALUE-AT-RISK

Although the VaR is not subadditive and differentiable in general, the derivatives of VaR_α [⋅] to calculate risk allocations (6.76),

(6.85)

may exist for some risk collections. Under some technical conditions, it can be calculated as,

(6.86)

For precise conditions when this is true, see Tasche (1999). Here we just note that it is easy to verify that these contributions add up to the total risk,

In the case of VaR, the Euler allocation can be difficult to estimate using the Monte Carlo sample, because Pr[X = VaR_α [X]] = 0 in the case of continuous distributions. To handle this problem, the condition X = VaR_α [X] can be replaced by |X – VaR_α [X]| < ε for some ε > 0 large enough to have Pr[|X – VaR_α [X]| < ε] > 0. However, this condition will be satisfied by only a few Monte Carlo simulations and important sampling techniques are needed to get an accurate estimation (see Glasserman 2005).

It can be somewhat easier to calculate the Euler allocations using the finite difference approximation,

(6.87)

with some small suitable Δ ≠ 0. Note that the choice of Δ depends ...