Proper Conditioning for Coherent VaR in Portfolio Management

Abstract

Value at Risk (VaR) is a central concept in risk management. As stressed by Artzner et al. (1999), VaR may not possess the subadditivity property required to be a coherent measure of risk. The key idea of this paper is that, when tail thickness is responsible for violation of subadditivity, eliciting proper conditioning information may restore VaR rationale for decentralized risk management. The argument is threefold. First, since individual traders are hired because they possess a richer information on their specific market segment than senior management, they just have to follow consistently the prudential targets set by senior management to ensure that decentralized VaR control will work in a coherent way. The intuition is that if one could build a fictitious conditioning information set merging all individual pieces of information, it would be rich enough to restore VaR subadditivity. Second, in this decentralization context, we show that if senior management has access ex-post to the portfolio shares of the individual traders, it amounts to recovering some of their private information. These shares can be used to improve backtesting in order to check that the prudential targets have been enforced by the traders. Finally, we stress that tail thickness required to violate subadditivity, even for small probabilities, remains an extreme situation since it corresponds to such poor conditioning information that expected loss appears to be infinite. We then conclude that lack of coherency of decentralized VaR management, that is VaR non-subadditivity at the richest level of information, should be an exception rather than a rule.

Multivariate dependence and portfolio optimization algorithms under illiquid market scenarios

“…Our modeling framework belongs to the portfolio optimization and risk measurement literature pioneered by the seminal works of Markowitz (1952) and (Morgan, 1996), and is broadly linked to the studies by Rockafellar and Uryasev (2002), Garcia et al (2007), andHo et al (2008), where portfolio optimization with respect to variance, VaR and CVaR is conducted under normal distributions. It also connects to the studies by Campbell et al (2001), and Baptista (2004, 2008) who consider downside risk and VaR contraints, instead of the mean-variance framework.…”

VaR- and CVaR-Minimal Futures Hedging Strategies: An Analytical Approach

“…In addition, recent studies report that cases in which VaR α is not subadditive are only relevant to a limited extent in practice (Garcia et al, 2007). By the linearity of the conditional expectation, it holds that…”

External Risk Measures and Basel Accords

“…Adrian and Brunnermeier [3] point out that risk measures based on contemporaneous observations, such as the Basel II risk measure (16), are procyclical; i.e., risk measurement obtained by such risk measures tends to be low in booms and high in crises, which impedes effective regulation. Gordy and Howells [20] examine the procyclicality of Basel II from the perspective of market discipline.…”

“…The additional capital requirements based on stressed VaR help reduce the procyclicality of the original risk measure (16). In addition to the capital charge specified in (17), the Basel III Accord requires banks to hold additional incremental risk capital charge (IRC) against potential losses resulting from default risk, credit migration risk, credit spread risk, etc., in the trading book, which are incremental to the risks captured by the formula (17) (Basel Committee on Banking Supervision [7,8]).…”