Decomposing total risk of a portfolio into the contributions of individual assets

Analysing the concentration risk in the portfolio is one of the important issues for the risk management of financial institutions. Among several measures proposed in order to quantify the concentration risk, we consider the risk contribution (RC) of asset j, defined by RCj ≡ aj∂Rp/∂aj, where aj is the holding amount of asset and Rp is the total risk of the portfolio. RCj can satisfy the additivity; the sum of RCs of all assets are equal to the total risk of the portfolio Rp. We can select many famous risk measures as Rp such as standard deviation, Value at Risk (VaR) and Expected Shortfall (ES), however, the accurate and robust estimation of RC is very difficult. One of the hopeful methods is “hybrid method.” In this method, assuming that the future prices of individual assets are conditionally independent with respect to the risk factors, we generate some scenarios of the risk factors by the Monte Carlo simulation, and calculate the conditional distribution of the future value of the portfolio and RCs of individual assets analytically by using the saddlepoint approximation. The unconditional estimates are obtained as the expectation of conditionals. The hybrid method gives much more reliable estimates of VaR and RCs to VaR than the ordinary Monte Carlo simulation, however, the accuracy of the estimates of ES and RCs to ES is not so good. In this article, we summarize the hybrid method in more general setting, and propose a more accurate estimation method of ES and RCs for ES. Since this new method is based on the universal mathematical relation between VaR and ES, it can be applicable to many risk evaluation models. We show some numerical examples to confirm some merits of our method.