Measuring risk is a crucial aspect of the portfolio optimization problem in finance, and of capital adequacy assessment in risk management. Expected Shortfall (ES) has been proposed as a coherent risk measure, by contrast with Value-at-Risk (VaR) and the standard-deviation-type of measures. Based on a coherent risk measure, for instance ES, we can discuss a coherent capital allocation for the purpose of internal risk management and performance measure, if ES is used for economic capital held by financial firms as a cushion to absorb the unexpected losses. Properly allocating risk capital down to the business level is important for the purpose of risk management and portfolio performance measurement. Even if there is a doubt about the reason for allocating ES, instead of VaR, the statistical properties of the statistic, marginal ES, from the proposed coherent allocation rule, are still of interest, because it is exactly the sensitivity of the target portfolio's ES.The idea of a coherent capital allocation rule by using a cost sharing rule, the Aumann-Shapley value in game theory, proposed by Denault (2001), happens to result in the same formula as proposed by Tasche (2000), who independently develops the "suitable" allocation rule based on the discussion of risk-adjusted returns. The fact, that two aspects of the concerns are satisfied by the same allocation formula, brings two fields together in an integrated way, so that a systematic risk management in a banking system seems very promising. Fundamental statistical issues arise in several places in a coherent risk management system. Primary interests will be, and are always, in modeling the profit/loss (P/L) distributions. Statistical modeling is receiving more and more attention currently, as well as economic modeling. For our purpose, we place more emphasis on the estimation and inference of ES and allocation statistics (marginal contribution of ES) under different situations. We also modify the back-testing rules based on ES. We propose a collection of weighted test statistics aiming at detecting the underestimated ES. Asymptotic properties of the test statistics are offered. The power of the tests in the context of an exponential family and the local alternatives is provided and the optimal weighting scheme is discussed.
