Improved Estimation Methods for Value-at-Risk, Expected Shortfall and Risk Contributions with High Precision

The (marginal) risk contribution is very useful for analyzing the concentration risk in a portfolio. However, it is difficult to estimate the risk contributions for value-at-risk (VaR) and expected shortfall (ES) precisely, especially using a Monte Carlo simulation. We applied a saddlepoint approximation to estimate the distribution function, so that the difficulty of estimating the risk contributions for VaR was dissolved. In this paper, we propose new estimation methods for ES and the risk contributions for ES based on the conditional independence and a saddlepoint approximation. Numerical studies confirm that these new methods are much better than existing ones.