Vector-Valued Tail Value-at-Risk and Capital Allocation

Multivariate TVaR-Based Risk Decomposition for Vector-Valued Portfolios

“…Finally, using the projection methods developed in Section 4, we obtain optimal values s * Table 1 illustrates that the orthogonal projection based on the multivariate lower orthant VaR always provides smaller results compared to the orthogonal projection based on the multivariate lower orthant TVaR, which is consistent with previous results in Cossette et al [13]. Moreover, as the dependence parameter increases, one sees that allocation values also increase, which shows consistency for practical applications.…”

“…Therefore, we refer the reader to Cossette et al [12] and Cossette et al [13] for more details, graphical illustrations and demonstrations related to bivariate lower and upper orthant TVaR.…”

“…The latter situation legitimates the use of multivariate risk measures on which to base capital allocation, as presented in the next sections. See Cossette et al ( [12,13]) for a review of multivariate risk measures.…”

“…In Section 2, desirable properties of risk contributions are set, the univariate TVaR-based allocation principle is described and multivariate extensions, lower and upper orthant VaR and TVaR, discussed in Cossette et al ( [12,13]), are presented. Section 3 defines lower and upper orthant risk decomposition methods for each risk and group of aggregated risks.…”

Multivariate geometric expectiles

“…From an actuarial perspective, multivariate risk measures generalizing VaR are treated in Embrechts and Puccetti [2006], Cossette et al [2012], Cousin and Di Bernardino [2013] and Torres et al [2015]. Multivariate versions of TVaR have been defined in Cousin and Di Bernardino [2014] and Cossette et al [2015]. Maume-Deschamps et al [2017] also introduce a multivariate extension of expectiles.…”