Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations

Abstract

We propose a novel framework of estimating systemic risk measures and risk allocations based on a Markov chain Monte Carlo (MCMC) method. 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 methods often provide inefficient estimates due to the rare-event character of 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. Efficiency of the MCMC estimators are demonstrated in a series of numerical experiments.

Design and Implementation of Machine Learning Modelling through Adaptive Hybrid Swarm Optimization Techniques for Machine Management

“…The MultiSwarm PSO (MSPSO) algorithm was utilised to get the optimum feature subset. Along with feature selection, the parameters of the SVM are improved as well 23) . To enhance the categorization, MSPSO and support vectors with F-measure have been employed in conjunction.…”

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Modality for Scenario Analysis and Maximum Likelihood Allocation

“…However, this modification distorts the distribution of X | {S = K} and the resulting estimates of risk allocations are biased. To overcome this issue, Koike and Minami (2019) and Koike and Hofert (2020) proposed MCMC methods for exact simulation from X | {S = K}. Although MCMC methods improve sample efficiency and the resulting estimates are unbiased, their performance highly depends on distributional properties of X | {S = K}, in particular on its modality and heavy-tailedness; see Appendix D for more details.…”

“…r(λ)} for λ ∈ {0, 1} d . In the HMC method, a candidate is proposed according to the so-called Hamiltonian dynamics, and the chain reflects at the boundaries {x ∈ R d : λ (x , K − 1 d x ) = r(λ)}, λ ∈ {0, 1} d so that it does not violate the support constraint; see Koike and Hofert (2020) for details.…”

“…Among the N MC = 10 6 samples, 2000 samples were contained in K d (K, δ) and only 189 samples fell in K C d (K, δ; r). Therefore, this crude simulation method is not efficient since 99.98% of the unconditional samples are discarded.Instead, we conduct an MCMC simulation to generateN MCMC = 10 4 samples directly from X | {X ∈ K C d (K; r)}.Hyperparameters of the HMC method are estimated based on the 189 MC samples; seeKoike and Hofert (2020). The resulting stepsize and integration time are ε = 0.105 and T = 24, respectively.…”