Diversification can control probability of default or risk, but not both

One of the controversies of diversification is that it may not be beneficial to banks, as it tends to increase systemic risk. Recent theoretical and empirical work have addressed this problem. We argue, from a theoretical perspective, that this controversy ultimately depends on how risk is assessed or measured. In particular, we observe that when one talks about random losses (risk) there are two intertwined approaches. On the one hand, one can fix the loss level and ask with what probability does that occur. On the other, one can fix a confidence level (or probability of loss) and ask, for example, what is the smallest loss with that probability. In a banking system, a systemic crisis occurs when all banks default simultaneously. Using the theoretical work of Wagner, where he proposed a simple model of a banking system in which a systemic crisis increases with diversification, we extend his analysis to show that if one allows for short positions; then the probability of default decreases, but the risk, measured by the value at risk (a non-coherent risk measure) increases. This brings up an interesting methodological question for risk management: Should we consider the probability of a given (acceptable) loss or, should we consider the minimum loss with an acceptable probability? We show that, within Wagner's model and depending on which question is asked, a different answer can be obtained. This, in turn, lead us to discuss some implications of these results for risk managers and regulators.