Abstract
The expected shortfall or ES shares the basic properties of the VaR given in the previous chapter: it is negative; it scales for positive multiples of positions; it vanishes for hedges; it is not additive; etc. For normally distributed PnLs, it is again a mere multiple of the standard deviation, and the scaling factor of the 2.5%-ES is, with 2.34.., very close to the 2.33.. of the 1%-VaR.
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Notes
- 1.
See, for example, “Seven Proofs for the Subadditivity of Expected Shortfall” at https://people.math.ethz.ch/~embrecht/ftp/Seven_Proofs.pdf. Also note that the idea here remains true even in case of freak positive ES values; we would just have to phrase it more awkwardly.
- 2.
Verify, along a similar line of thought as given at the beginning of this chapter, that
$$\displaystyle \begin{aligned} |{\mathrm{cES}}[\alpha|\varOmega]| \leqslant |{\mathrm{ES}}[\alpha]| . \end{aligned}$$
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Auer, M. (2018). Properties of ES. In: Hands-On Value-at-Risk and Expected Shortfall. Management for Professionals. Springer, Cham. https://doi.org/10.1007/978-3-319-72320-4_13
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DOI: https://doi.org/10.1007/978-3-319-72320-4_13
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