Risk Factor Contributions in Portfolio Credit Risk Models

Determining contributions to overall portfolio risk is an important topic in financial risk management. At the level of positions (instruments and subportfolios), this problem has been well studied, and a significant theory has been built, in particular around the calculation of marginal contributions. We consider the problem of determining the contributions to portfolio risk of risk factors, rather than positions. This problem cannot be addressed through an immediate extension of the techniques employed for position contributions, since, in general, the portfolio loss is not a linear function of the risk factors. We employ the Hoeffding decomposition of the loss random variable into a sum of terms depending on the factors. This decomposition restores linearity, at the cost of including terms that arise from the joint effect of more than one factor. The resulting cross-factor terms provide useful information to risk managers, since the terms in the Hoeffding decomposition can be viewed as best quadratic hedges of the portfolio loss involving instruments of increasing complexity. We illustrate the technique on multi-factor models of portfolio credit risk, where systematic factors may represent different industries, geographical sectors, etc.