Estimation methods for expected shortfall

Accurate Evaluation of Expected Shortfall for Linear Portfolios with Elliptically Distributed Risk Factors

“…For instance, a key element of a recent proposal by the Basel Committee on Banking Supervision [6] is moving the quantitative risk metrics system in regard to trading book capital requirement policies from 99% VaR to 97.5% ES. The surge in interest in ES estimation methods also has been reflected in the recent and extensive survey by Nadarajah et al [2], which emphasizes many new developments and covers over 140 references on the subject. In this context, using elliptically distributed risk factors emerges as an appealing choice in multivariate settings, because elliptical distributions can model heavy-tailed, and thus riskier, financial return distributions flexibly while remaining analytically tractable.…”

“…This later error more than offsets the missing scaling factor of 1/2, implying that fixing both errors will increase estimates of ES for portfolios of risk factors that are distributed according to a multivariate Student t distribution. Both the linear and nonlinear errors are propagated in the survey paper Nadarajah et al [2] and are confounded by additional numerical errors in [1] in the tabulated values for ES in the multivariate Student t case. More specifically, we find that the inaccurate analytical expression derived in [1] does not match the reported numerical values in [1] for ES in the multivariate Student t case and neither one of them matches the correct expression and values we derive.…”

“…2 In particular, while Lu's Ph.D. thesis [36] contains a stand-alone derivation of the correct analytical expression for ES in the multivariate Student t case, it still reports the inaccurate ES expression in [1] for the general elliptical case; it also does not correct any of the numerical inaccuracies in [1] for the multivariate Student t case. Similarly, [18] provides a stand-alone derivation of the correct ES expression only for the multivariate Student t case, addressing neither the general elliptical case nor the need to correct numerical results in [1].…”

“…1 These benefits, however, 1 Imposing any particular parametric distributional assumptions can be avoided at the cost of sacrificing analytical tractability based on non-parametric approaches to ES estimation such as Scaillet [7] and [8], among many others. See Nadarajah et al [2] for further details.…”

“…Including the recent survey Nadarajah et al [2], the inaccurate results in [1] have been referenced in at least 30 subsequent English publications without correction [2][3][4]. 2 In particular, while Lu's Ph.D. thesis [36] contains a stand-alone derivation of the correct analytical expression for ES in the multivariate Student t case, it still reports the inaccurate ES expression in [1] for the general elliptical case; it also does not correct any of the numerical inaccuracies in [1] for the multivariate Student t case.…”