Determining contributions to overall portfolio risk is an important topic in risk management. For positions (instruments and sub-portfolios), this problem has been well studied, and a significant theory built, around the calculation of marginal contributions. We consider the problem of determining the contributions to portfolio risk of risk factors. This cannot be addressed through an immediate extension of techniques for position contributions, since the portfolio loss is a nonlinear function of the risk factors. We employ the Hoeffding decomposition of the portfolio loss into a sum of terms depending on the factors. This decomposition restores linearity, but includes terms arising from joint effects of groups of factors. These cross-factor terms provide information to risk managers, since they can be viewed as best hedges of the portfolio loss involving instruments of increasing complexity. We illustrate the technique on multi-factor portfolio credit risk models, where systematic factors represent industries, geographical sectors, etc.

Modern insurance products are becoming increasingly complex, offering various guarantees, surrender options and bonus provisions. A case in point are the with-profits insurance policies offered by UK insurers. While these policies have been offered in some form for centuries, in recent years their structure and management have become substantially more involved. The products are particularly complicated due to the wide discretion they afford insurers in determining the bonuses policyholders receive. In this paper, we study the problem of an insurance firm attempting to structure the portfolio underlying its with-profits fund. The resulting optimization problem, a non-linear program with stochastic variables, is presented in detail. Numerical results show how the model can be used to analyze the alternatives available to the insurer, such as different bonus policies and reserving methods.