This issue of The Journal of Risk addresses problems in the estimation and minimization of the popular value-at-risk (VaR) risk measure and the closely related expected shortfall (ES; also known as conditional value-at-risk (CVaR)). Further illustrations of their versatility are also provided: namely, new perspectives on hedging in incomplete markets and on funding liquidity are offered. According to several sources, one of the main challenges when estimating cumulative VaR and ES losses is the limitation of empirical data to their marginal distributions. In the first paper in this issue, "Asymptotic equivalence of conservative value-at-risk- and expected shortfall-based capital charges", Giovanni Puccetti and Ludger Rüschendorf show that, under certain restrictions, the worst estimates of these two metrics are close to each other when the number of sources is large. They also show that this asymptotic equivalence is in fact robust in many practical instances where the restrictions are not satisfied.
While VaR, in contrast to CVaR, has been known to violate coherence axioms, it has also been shown to be more robust. However, the minimization of VaR has long been a challenge and in the issue's second paper, "A gradual nonconvexification method for minimizing value-at-risk", Jiong Xi, Thomas F. Coleman, Yuying Li and Aditya Tayal address this through the development of an efficient computational method. The authors demonstrate the effectiveness of their approach both theoretically and empirically.
In our third paper, "Conditional value-at-risk-based optimal partial hedging", Jianfa Cong, Ken Seng Tan and Chengguo Weng extend the conceptual applicability of CVaR. Specifically, they develop an efficient approach, driven by CVaR, that results in the partial hedging of derivatives in incomplete markets. In particular, they show that their strategy can be viewed as a bull call spread on the risk.
In the fourth and final paper in the issue, "Optimal hedging of funding liquidity risk", Wei Chen and Jimmy Skoglund show how the VaR and CVaR measures can be extended to unify two standard and complementary approaches to hedging liquidity risk. They illustrate that their resulting recommended optimal liquidity hedging strategy conforms to the Basel III framework.
Warrington College of Business Administration,
University of Florida