On the inadequacy of VaR-based risk management: VaR, CVaR, and nonlinear interactions

In the financial industry, risk has been traditionally managed by the imposition of value-at-risk or VaR constraints on portfolio risk exposure. Motivated by recent events in the financial industry, we examine the role that risk-seeking traders play in the accumulation of large and possibly infinite risk. We proceed to show that when traders employ a conditional value-at-risk (CVaR) metric, much can be said by studying the interaction between value-at-risk (VaR) (a non-coherent risk measure) and CVaR (a coherent risk measure based on VaR). Resolving this question requires characterizing the optimal value of the associated stochastic, and possibly nonconvex, optimization problem, often a challenging problem. Our study makes two sets of contributions. First, under general asset distributions on a compact support, traders accumulate finite risk with magnitude of the order of the upper bound of this support. Second, when the supports are unbounded, under relatively mild assumptions, such traders can take on an unbounded amount of risk despite abiding by this VaR threshold. In short, VaR thresholds may be inadequate in guarding against financial ruin.