Proper Conditioning for Coherent VaR in Portfolio Management

Value at risk (VaR) is a central concept in risk management. As stressed by Artzner et al. (1999, Coherent measures of risk, Math. Finance9(3) 203--228), 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 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 because it corresponds to such poor conditioning information that expected loss appears to be infinite. We then conclude that lack of coherence of decentralized VaR management, that is VaR nonsubadditivity at the richest level of information, should be an exception rather than a rule.