# Estimation of Risk Contributions with MCMC

## TL;DR

This paper introduces a Markov Chain Monte Carlo (MCMC) based estimator for risk contributions in financial portfolios, improving efficiency and accuracy over traditional methods, especially in high-dimensional, rare-event scenarios.

## Contribution

The paper presents a novel MH algorithm-based estimator for VaR contributions that is more sample-efficient and accurate than existing estimators, applicable to complex, high-dimensional risk models.

## Key findings

- MH estimator has smaller bias and mean squared error than existing methods.
- The method is consistent and asymptotically normal.
- Effective in high-dimensional, inhomogeneous risk models.

## Abstract

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.