The application of Hermite polynomials to risk allocation

We investigate a practical and fast analytic framework for portfolio modeling and tail risk allocation using Hermite polynomials. This framework was first discussed in "An analytical framework for credit portfolio risk measures" by Mikhail Voropaev in 2011. Here, we further develop this analytic approach, removing some issues with the original derivations and generalizing the results. In particular, we present a revised set of terms for the second-order value-at-risk and expected shortfall adjustments and associated risk contributions. We also compute third-order terms in full. We finish with an application to loan portfolios and some empirical examples.