Value‐at‐Risk bounds with two‐sided dependence information

Abstract

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.

Inflation and Portfolio Management

“…Then various models of portfolio management balancing return and risk have been proposed, such as mean-semivariance model (Markovitz, 1959), mean-absolute deviation model (Konno & Yamazaki, 1991), value-at-risk model (Jorion, 1996), mean-risk curve model (Huang, 2008), mean-semivariance-CVaR model (Najafi & Mushakhian, 2015), etc. These models have been widely used and extended (Estrada, 2007;Wei, 2018;Lux & Rüschendorf, 2019).…”

Robust distortion risk measures

“…Specifically, the upper bound on a distortion risk measure is always at least as large as its value under the assumption of a comonotonic dependence, a situation that is arguably very extreme. Papers that study bounds with additional dependence constraints or by imposing modeling structures include Bernard et al (2017), Lux and Rüschendorf (2019), and Wang et al (2019). Another stream of the literature is concerned with deriving risk bounds of the aggregate loss 𝑆 under partial knowledge of its moments.…”

“…Papers that study bounds with additional dependence constraints or by imposing modeling structures include Bernard et al. (2017), Lux and Rüschendorf (2019), and Wang et al. (2019).…”