  We propose a novel framework of estimating systemic risk measures and risk
allocations based on Markov chain Monte Carlo (MCMC) methods. We consider a
class of allocations whose jth component can be written as some risk measure of
the jth conditional marginal loss distribution given the so-called crisis
event. By considering a crisis event as an intersection of linear constraints,
this class of allocations covers, for example, conditional Value-at-Risk
(CoVaR), conditional expected shortfall (CoES), VaR contributions, and range
VaR (RVaR) contributions as special cases. For this class of allocations,
analytical calculations are rarely available, and numerical computations based
on Monte Carlo (MC) methods often provide inefficient estimates due to the
rare-event character of the crisis events. We propose an MCMC estimator
constructed from a sample path of a Markov chain whose stationary distribution
is the conditional distribution given the crisis event. Efficient constructions
of Markov chains, such as Hamiltonian Monte Carlo and Gibbs sampler, are
suggested and studied depending on the crisis event and the underlying loss
distribution. The efficiency of the MCMC estimators is demonstrated in a series
of numerical experiments.
