Abstract
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.
Highlights
Recent theoretical (Allen and Carletti 2006; Allen and Gale 2005; Allen et al 2012; De Young 2012; Ibragimov et al 2011; van Oordt 2014; Wagner 2010) and empirical studies (Yang et al 2020; Bégin et al 2019; Slijkerman et al 2013; De Jonghe 2010; Olibe et al 2008)have addressed the problem of diversification and systemic risk
If we model the financial positions of the banking system by R = ν1 + ν2, in the same way that an individual bank would aggregate its risks, and we decide to measure the systemic risk by the value at risk, invoking (7) once more we obtain: VaRα (ν1 + ν2 ) = VaRα (1 − r1 + r2 ) x + (1 − r2 + r1 )y
One of the controversies of diversification is that, it may be beneficial for a bank in reducing the risk of its portfolio, it comes at the expense of increasing the likelihood of systemic risk
Summary
Recent theoretical (Allen and Carletti 2006; Allen and Gale 2005; Allen et al 2012; De Young 2012; Ibragimov et al 2011; van Oordt 2014; Wagner 2010) and empirical studies (Yang et al 2020; Bégin et al 2019; Slijkerman et al 2013; De Jonghe 2010; Olibe et al 2008). The theoretical model that Wagner (2010) presents shows how diversification decreases the probability of failure for each bank individually, but it does not decrease the probability of default of the banking system as a whole (i.e., systemic risk). One of the aims of this work is to extend Wagner’s theoretical analysis by including portfolios with short positions. We show that if we consider the generic case depicted, that when each bank diversifies by including short positions on the other bank’s portfolio; the probability of joint default decreases, even though the probability of each bank defaulting individually may increase. This paper can be seen as an extension of (Cadenas et al 2020); where short positions are not considered and there is no reference to Gaussian returns
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