Determining risk contributions by unit exposures to portfolio-wide economic capital is an important task in financial risk management. Despite its practical demands, computation of risk contributions is challenging for most risk models because it often requires rare-event simulation . In this paper , we address the problem of estimating risk contributions when the total risk is measured by Value-at-Risk (VaR). We propose a new estimator of VaR contributions , that utilizes Markov chain Monte Carlo (MCMC) method. Unlike the existing estimators, our MCMC-based estimator is computed by samples of conditional loss distribution given the rare event of our interest. MCMC method enables to generate such samples without evaluating the density of total risk. Thanks to these features, our estimator has improved sample-efficiency compared with the crude Monte Carlo method. Moreover, our method  is widely applicable to various risk models specified by joint portfolio loss density. In this paper, we show that our MCMC-based estimator has several attractive properties, such as consistency and asymptotic normality. Our numerical experiment also demonstrates that , in various risk models used in practice, our MCMC estimator has smaller bias and MSE compared with these of existing estimators.
Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations . In this study , we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has the consistency and asymptotic normality, and is widely applicable to various risk models having joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models , even those having high-dimensional (approximately 500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error compared with existing estimators.
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