Importance sampling for risk contributions of credit portfolios

Value-at-Risk is often used as a risk measure of credit portfolios, and it can be decomposed into a sum of risk contributions associated with individual obligors. These risk contributions play an important role in risk management of credit portfolios. They can be used to measure risk-adjusted performances of subportfolios and to allocate risk capital. Mathematically, risk contributions can be represented as conditional expectations, which are conditioned on rare events. In this paper, we develop a restricted importance sampling (IS) method for simulating risk contributions, and devise estimators whose mean square errors converge in a rate of n^-1. Furthermore, we combine our method with the IS method in the literature to improve the efficiency of the estimators. Numerical examples show that the proposed method works quite well.