Multivariate TVaR-Based Risk Decomposition for Vector-Valued Portfolios

Abstract

Abstract:In order to protect stakeholders of insurance companies and financial institutions against adverse outcomes of risky businesses, regulators and senior management use capital allocation techniques. For enterprise-wide risk management, it has become important to calculate the contribution of each risk within a portfolio. For that purpose, bivariate lower and upper orthant tail value-at-risk can be used for capital allocation. In this paper, we present multivariate value-at-risk and tail-value-at-risk for d ≥ 2, and we focus on three different methods to calculate optimal values for the contribution of each risk within the sums of random vectors to the overall portfolio, which could particularly apply to insurance and financial portfolios.

A consistent estimator to the orthant-based tail value-at-risk

“…As both TVaR α (X) and TVaR α (X) are infinite collections of points, from a decision making standpoint, a strategy for establishing the appropriate allocation is necessary. Here, we refer to Mailhot and Mesfioui [15] for their discussion on capital allocation based on multivariate lower and upper orthant TVaR. In particular, we focus on VaR and TVaR-based orthogonal projections in the bivariate case.…”

“…The lower (upper) orthant VaR was introduced by Embrechts and Puccetti [10], in the bivariate context, as a level set generated by the cdf (survival function (sf)) of a random vector X. Both the orthant VaR and TVaR were later developed by Cossette et al [4,5], respectively, with multivariate extensions of each discussed in [15]. In the univariate context, estimators of VaR and TVaR can be written…”

Risk allocation through shapley decompositions, with applications to variable annuities

“…The manuscripts of Dhaene et al (2012), Albrecht (2014) and Guo et al (2018) survey multiple other existing allocation methods. For instance, Parker (1997) and Christiansen and Helwich (2008) sequentially condition on some of the risk factors to obtain a risk decomposition. Another recent alternative approach to risk allocation consists in applying multivariate risk measures, see for instance Mailhot and Mesfioui (2016) or Cossette et al (2016).…”

Remarks on a copula‐based conditional value at risk for the portfolio problem

“…It should be noted that copula‐based approach to MVaR has so far been developed under various other terminologies such as the conditional tail expectation (Cousin & Di Bernardino, 2014; Huerlimann, 2014) or the TCE (Brahim et al, 2018; Zhu & Li, 2012), the tail VaR (Bargés et al, 2009; Maihot & Mesfioui, 2016), and so on (He & Gong, 2009); for a review and basic properties of these notions, we refer to Rüschendorf (2012), and concerning other related works, we refer for instance to the references cited in these references. Our extends the multivariate CVaR ( ) introduced by Lee and Prékopa (2013), by which our current research is motivated.…”