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Regulatory arbitrage of risk measures

We introduce the regulatory arbitrage of risk measures, one of the key considerations in choosing a suitable risk measure to use in banking regulation. A regulatory arbitrage is the amount of capital requirement reduced by splitting a financial risk into several fragments, regulated via a risk measure separately. Coherent risk measures by definition are free of regulatory arbitrage; dividing risks will not reduce the total capital requirement under a coherent risk measure. However, risk measures in practical use, such as the Value-at-Risk (VaR), are often not coherent and the magnitude of their regulatory arbitrage is then of significant importance. We quantify the regulatory arbitrage of risk measures in a rigorous mathematical framework, and categorize risk measures into three classes: free of regulatory arbitrage, of limited regulatory arbitrage, and of infinite regulatory arbitrage. We provide explicit results to characterize the regulatory arbitrage for general classes of risk measures, including distortion risk measures and convex risk measures. Several examples of risk measures of limited regulatory arbitrage are illustrated, as possible alternatives for coherent risk measures.