Coherent measurement of factor risks

Abstract: We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price of oil, S&P500 index, or the credit spread; - risk contributions and factor risk contributions, which provide a coherent alternative to the sensitivity coefficients. We also propose two particular classes of coherent risk measures called Alpha V@R and Beta V@R, for which all the objects described above admit an extremely simple empirical estimation procedure. This procedure uses no model assumptions on the structure of the price evolution. Moreover, we consider the problem of the risk management on a firm's level. It is shown that if the risk limits are imposed on the risk contributions of the desks to the overall risk of the firm (rather than on their outstanding risks) and the desks are allowed to trade these limits within a firm, then the desks automatically find the globally optimal portfolio.