Risk contributions, information and reverse stress testing

Abstract

In this paper we describe methods of decomposing risk into subcomponents such as contributing instruments, subportfolios or underlying risk factors e.g., equity, foreign exchange, economy-wide systematic and interest rate risk factors. The Euler allocation principle for allocation of instrument and subportfolio risk contributions to the total portfolio risk is widely applied in practice for portfolio managers risk budgeting and economic capital allocations. However, the decomposition of risk factors into summable contributions is in general not possible and we describe a non-parametric method of extracting relative information of risk factors that is valid for general non-linear portfolios. The measure is based on the Kullback information theory and while it does not constitute a decomposition it can be used to judge the relative importance of risk factors in determining profit and loss. The measure is also useful for understanding the relevance of so-called reverse stress tests where a certain loss level is identified with certain risk factor values. The use of the information measure is illustrated with an application to a portfolio of derivatives.