Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations
Abstract
:1. Introduction
2. Systemic Risk Allocations and Their Estimation
2.1. A Class of Systemic Risk Allocations
 1.
 risk contributiontype if ${\varrho}_{j}=\mathbb{E}$;
 2.
 CoVaR type if ${\varrho}_{j}={VaR}_{{\beta}_{j}}$ for ${\beta}_{j}\in (0,1)$;
 3.
 CoRVaR type if ${\varrho}_{j}={RVaR}_{{\beta}_{j,1},{\beta}_{j,2}}$ for $0<{\beta}_{j,1}<{\beta}_{j,2}\le 1$; and
 4.
 CoEStype if ${\varrho}_{j}={ES}_{{\beta}_{j}}$ for ${\beta}_{j}\in (0,1)$.
 (1)
 Risk contributions. If the crisis event is chosen to be ${\mathcal{C}}_{\alpha}^{VaR}$, ${\mathcal{C}}_{{\alpha}_{1},{\alpha}_{2}}^{RVaR}$ or ${\mathcal{C}}_{\alpha}^{ES}$, the allocations of the risk contribution type ${\varrho}_{j}=\mathbb{E}$ reduce to the VaR, RVaR, or ES contributions defined by$$\begin{array}{cc}\hfill {VaR}_{\alpha}(\mathit{X},S)& =\mathbb{E}\left[\mathit{X}\phantom{\rule{4pt}{0ex}}\right\phantom{\rule{4pt}{0ex}}S={VaR}_{\alpha}\left(S\right)],\hfill \\ \hfill {RVaR}_{{\alpha}_{1},{\alpha}_{2}}(\mathit{X},S)& =\mathbb{E}\left[\mathit{X}\phantom{\rule{4pt}{0ex}}\right\phantom{\rule{4pt}{0ex}}{VaR}_{{\alpha}_{1}}\left(S\right)\le S\le {VaR}_{{\alpha}_{2}}\left(S\right)],\hfill \\ \hfill {ES}_{\alpha}(\mathit{X},S)& =\mathbb{E}\left[\mathit{X}\phantom{\rule{4pt}{0ex}}\right\phantom{\rule{4pt}{0ex}}S\ge {VaR}_{\alpha}\left(S\right)],\hfill \end{array}$$
 (2)
 Conditional risk measures. CoVaR and CoES are systemic risk measures defined by$$\begin{array}{cc}\hfill {CoVaR}_{\alpha ,\beta}^{=}({X}_{j},S)& ={VaR}_{\beta}\left({X}_{j}\rightS={VaR}_{\alpha}\left(S\right)),\phantom{\rule{1.em}{0ex}}{CoVaR}_{\alpha ,\beta}({X}_{j},S)={VaR}_{\beta}\left({X}_{j}\rightS\ge {VaR}_{\alpha}\left(S\right)),\hfill \\ \hfill {CoES}_{\alpha ,\beta}^{=}({X}_{j},S)& ={ES}_{\beta}\left({X}_{j}\rightS={VaR}_{\alpha}\left(S\right)),\phantom{\rule{1.em}{0ex}}{CoES}_{\alpha ,\beta}({X}_{j},S)={ES}_{\beta}\left({X}_{j}\rightS\ge {VaR}_{\alpha}\left(S\right)),\hfill \end{array}$$
2.2. Monte Carlo Estimation of Systemic Risk Allocations
 (1)
 Sample from $\mathit{X}$: For a sample size $N\in \mathbb{N}$, generate ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}\underset{}{\stackrel{\mathrm{ind}.}{\sim}}{F}_{\mathit{X}}$.
 (2)
 Estimate the crisis event: If the crisis event $\mathcal{C}$ contains unknown quantities, replace them with their estimates based on ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}$. Denote by $\widehat{\mathcal{C}}$ the estimated crisis event.
 (3)
 Sample from the conditional distribution of $\mathit{X}$ given $\widehat{\mathcal{C}}$: Among ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}$, determine ${\tilde{\mathit{X}}}^{\left(n\right)}$ such that ${\tilde{\mathit{X}}}^{\left(n\right)}\in \widehat{\mathcal{C}}$ for all $n=1,\dots ,N$.
 (4)
 Construct the MC estimator: The MC estimate of ${A}_{j}^{{\varrho}_{j},\mathcal{C}}$ is ${\varrho}_{j}\left({\widehat{F}}_{\tilde{\mathit{X}}}\right)$ where ${\widehat{F}}_{\tilde{\mathit{X}}}$ is the empirical cdf (ecdf) of the ${\tilde{\mathit{X}}}^{\left(n\right)}$’s.
3. MCMC Estimation of Systemic Risk Allocations
3.1. A Brief Review of MCMC
Algorithm 1 Metropolis–Hastings (MH) algorithm. 
Require: Random number generator of the proposal density q(x,·) for all x ∈ E, x^{(0)} ∈ supp(π) and the ratio π(y)/π(x) for x, y ∈ E, where π is the density of the stationary distribution. 
Input: Sample size $N\in \mathbb{N}$, proposal density q, and initial value ${\mathit{X}}^{\left(0\right)}={\mathit{x}}^{\left(0\right)}$. 
Output: Sample path ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}$ of the Markov chain. 
for $n:=0,\dots ,N1$ do 
(1) Generate ${\tilde{\mathit{X}}}^{\left(n\right)}\sim q({\mathit{X}}^{\left(n\right)},\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}})$. 
(2) Calculate the acceptance probability
$$\begin{array}{c}\hfill {\alpha}_{n}:=\alpha ({\mathit{X}}^{\left(n\right)},{\tilde{\mathit{X}}}^{\left(n\right)})=min\left(\right)open="\{"\; close="\}">\frac{\pi \left({\tilde{\mathit{X}}}^{\left(n\right)}\right)q({\tilde{\mathit{X}}}^{\left(n\right)},{\mathit{X}}^{\left(n\right)})}{\pi \left({\mathit{X}}^{\left(n\right)}\right)q({\mathit{X}}^{\left(n\right)},{\tilde{\mathit{X}}}^{\left(n\right)})},1.\end{array}$$

(3) Generate $U\sim \mathcal{U}(0,1)$ and set ${\mathit{X}}^{(n+1)}:={1}_{[U\le {\alpha}_{n}]}{\tilde{\mathit{X}}}^{\left(n\right)}+{1}_{[U>{\alpha}_{n}]}{\mathit{X}}^{\left(n\right)}$. 
end for 
3.2. MCMC Formulation for Estimating Systemic Risk Allocations
 the ratio ${f}_{\mathit{X}}\left(\mathit{y}\right)/{f}_{\mathit{X}}\left(\mathit{x}\right)$ can be evaluated for any $\mathit{x},\mathit{y}\in \mathcal{C}$, and that
 the support of ${f}_{\mathit{X}}$ is ${\mathbb{R}}^{d}$ or ${\mathbb{R}}_{+}^{d}$.
3.3. Estimation with Hamiltonian Monte Carlo
3.3.1. Hamiltonian Monte Carlo with Reflection
Algorithm 2 Leapfrog method for Hamiltonian dynamics. 
Input: Current states $\left(\mathit{x}\right(0),\mathit{p}(0\left)\right)$, stepsize $\u03f5>0$, gradients $\nabla U$ and $\nabla K$. 
Output: Updated position $\left(\mathit{x}\right(\u03f5),\mathit{p}(\u03f5\left)\right)$. 
(1) $\mathit{p}\left(\right)open="("\; close=")">\frac{\u03f5}{2}$. 
(2) $\mathit{x}\left(\u03f5\right)=\mathit{x}\left(0\right)+\u03f5\nabla K(\mathit{p}\left(\right)open="("\; close=")">\frac{\u03f5}{2})$. 
(3) $\mathit{p}\left(\u03f5\right)=\mathit{p}(\u03f5/2)+\frac{\u03f5}{2}\nabla U\left(\mathit{x}\left(\u03f5\right)\right)$. 
Algorithm 3 Hamiltonian Monte Carlo to simulate $\pi $. 
Require: Random number generator of F_{K}, x^{(0)} ∈ supp(π), π(y)/π(x), x, y ∈ E and F_{K}(p’)/F_{K}(p), p, p’ ∈ ℝ^{d}. 
Input: Sample size $N\in \mathbb{N}$, kinetic energy density ${f}_{K}$, target density $\pi $, gradients of the potential and kinetic energies $\nabla U$ and $\nabla K$, stepsize $\u03f5>0$, integration time $T\in \mathbb{N}$ and initial position ${\mathit{X}}^{\left(0\right)}={\mathit{x}}^{\left(0\right)}$. 
Output: Sample path ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}$ of the Markov chain. 
for $n:=0,\dots ,N1$ do 
(1) Generate ${\mathit{p}}^{\left(n\right)}\sim {F}_{K}$. 
(2) Set $({\tilde{\mathit{X}}}^{\left(n\right)},{\tilde{\mathit{p}}}^{\left(n\right)})=({\mathit{X}}^{\left(n\right)},{\mathit{p}}^{\left(n\right)})$. 
(3) for $t:=1,\dots ,T$,
$$\begin{array}{c}\hfill ({\tilde{\mathit{X}}}^{(n+t/T)},{\tilde{\mathit{p}}}^{(n+t/T))}=\phantom{\rule{4pt}{0ex}}\mathbf{Leapfrog}({\tilde{\mathit{X}}}^{(n+(t1)/T)},{\tilde{\mathit{p}}}^{(n+(t1)/T)},\u03f5,\nabla U,\nabla K).\end{array}$$

end for 
(4) ${\tilde{\mathit{p}}}^{(n+1)}={\mathit{p}}^{(n+1)}$. 
(5) Calculate ${\alpha}_{n}=min\left(\right)open="\{"\; close="\}">\frac{\pi \left({\tilde{\mathit{X}}}^{(n+1)}\right){f}_{K}\left({\tilde{\mathit{p}}}^{(n+1)}\right)}{\pi \left({\mathit{X}}^{\left(n\right)}\right){f}_{K}\left({\mathit{p}}^{\left(n\right)}\right)},1$. 
(6) Set ${\mathit{X}}^{(n+1)}:={1}_{[U\le {\alpha}_{n}]}{\tilde{\mathit{X}}}^{(n+1)}+{1}_{[U>{\alpha}_{n}]}{\mathit{X}}^{\left(n\right)}$ for $U\sim U(0,1)$. 
end for 
3.3.2. Choice of Parameters for HMC
Algorithm 4 Heuristic for determining the stepsize $\u03f5$ and integration time T. 
Input: MC presample ${\mathit{X}}_{1}^{\left(0\right)},\dots ,{\mathit{X}}_{{N}_{0}}^{\left(0\right)}$, gradients $\nabla U$ and $\nabla K$, target acceptance probability $\underline{\alpha}$, initial constant ${c}_{\u03f5}>0$ and the maximum integration time ${T}_{\mathrm{max}}$ (${c}_{\u03f5}=1$ and ${T}_{\mathrm{max}}=1000$ are set as default values). 
Output: Stepsize $\u03f5$ and integration time T. 
(1) Set ${\alpha}_{\mathrm{min}}=0$ and $\u03f5={c}_{\u03f5}{d}^{1/4}$. 
(2) while ${\alpha}_{\mathrm{min}}<\underline{\alpha}$ 
(21) Set $\u03f5=\u03f5/2$. 
(22) for $n:=1,\dots ,{N}_{0}$ 
(221) Generate ${\mathit{p}}_{n}^{\left(0\right)}\sim {F}_{K}$. 
(222) for $t:=1,\dots ,{T}_{\mathrm{max}}$ 
(2221) Set ${\mathit{Z}}_{n}^{\left(t\right)}=\mathbf{Leapfrog}({\mathit{Z}}_{n}^{(t1)},\u03f5,\nabla U,\nabla K)$ for ${\mathit{Z}}_{n}^{(t1)}=({\mathit{X}}_{n}^{(t1)},{\mathit{p}}_{n}^{(t1)})$. 
(2222) Calculate
$$\begin{array}{c}\hfill \phantom{\rule{45.5244pt}{0ex}}{\alpha}_{n,t}=\alpha ({\mathit{Z}}_{n}^{(t1)},{\mathit{Z}}_{n}^{\left(t\right)})\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\Delta}_{t}=\left\right{\mathit{X}}_{n}^{\left(t\right)}{\mathit{X}}_{n}^{\left(0\right)}\left\right\left\right{\mathit{X}}_{n}^{(t1)}{\mathit{X}}_{n}^{\left(0\right)}\left\right.\end{array}$$

(2223) if ${\Delta}_{t}<0$ and ${\Delta}_{t1}>0$, break and set ${T}_{n}^{*}=t1$. 
end for 
end for 
(23) Compute ${\alpha}_{\mathrm{min}}=min\left({\alpha}_{n,t}\phantom{\rule{4pt}{0ex}}\right\phantom{\rule{4pt}{0ex}}t=1,2,\dots ,\phantom{\rule{4pt}{0ex}}{T}_{n}^{*},\phantom{\rule{4pt}{0ex}}n=1,\dots ,{N}_{0})$. 
end while 
(3) Set $T=\lfloor \frac{1}{{N}_{0}}{\sum}_{n=1}^{{N}_{0}}{T}_{n}^{*}\rfloor $. 
3.4. Estimation with Gibbs Sampler
3.4.1. True Gibbs Sampler for Estimating Systemic Risk Allocations
3.4.2. Choice of Parameters for GS
Algorithm 5 Random scan Gibbs sampler (RSGS) with heuristic to determine $({p}_{1},\dots ,{p}_{d})$. 
Require: Random number generator of π_{j−j} and x^{(0)} ∈ supp(π). 
Input: MC presample ${\tilde{\mathit{X}}}_{1}^{\left(0\right)},\dots ,{\tilde{\mathit{X}}}_{{N}_{0}}^{\left(0\right)}$, sample size $N\in \mathbb{N}$, initial state ${\mathit{x}}^{\left(0\right)}$, sample size of the prerun ${N}_{\mathrm{pre}}$ and the target autocorrelation $\rho $ (${N}_{\mathrm{pre}}=100$ and $\rho =0.15$ are set as default values). 
Output:N sample path ${\mathit{X}}^{\left(1\right)},\dots ,{\mathit{X}}^{\left(N\right)}$ of the Markov chain. 
(1) Compute the sample covariance matrix $\widehat{\Sigma}$ based on ${\tilde{\mathit{X}}}_{1}^{\left(0\right)},\dots ,{\tilde{\mathit{X}}}_{{N}_{0}}^{\left(0\right)}$. 
(2) Set ${p}_{j}\propto {\widehat{\Sigma}}_{j,j}{\widehat{\Sigma}}_{j,j}{\widehat{\Sigma}}_{j,j}^{1}{\widehat{\Sigma}}_{j,j}$ and ${\mathit{X}}^{\left(0\right)}={\mathit{X}}_{\mathrm{pre}}^{\left(0\right)}={\mathit{x}}^{\left(0\right)}$. 
(3) for $n:=1,\dots ,{N}_{\mathrm{pre}}$ 
(31) Generate $J=j$ with probability ${p}_{j}$. 
(32) Update ${X}_{\mathrm{pre},J}^{\left(n\right)}\sim {\pi}_{JJ}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right{\mathit{X}}_{\mathrm{pre}}^{(n1)})$ and ${\mathit{X}}_{\mathrm{pre},J}^{\left(n\right)}={\mathit{X}}_{\mathrm{pre},J}^{(n1)}$. 
end for 
(4) Set
$$\begin{array}{c}\hfill T={argmin}_{h\in {\mathbb{N}}_{0}}\left(\right)open="\{"\; close="\}">\mathrm{estimated}\phantom{\rule{4.pt}{0ex}}\mathrm{autocorrelations}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}{\mathit{X}}_{\mathrm{pre}}^{\left(1\right)},\dots ,{\mathit{X}}_{\mathrm{pre}}^{\left({N}_{\mathrm{pre}}\right)}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\phantom{\rule{4.pt}{0ex}}\mathrm{lag}\phantom{\rule{4.pt}{0ex}}h\phantom{\rule{4pt}{0ex}}\le \rho .\end{array}$$

(5) for $n:=1,\dots ,N$, $t:=1,\dots ,T$ 
(51) Generate $J=j$ with probability ${p}_{j}$. 
(52) Update ${X}_{J}^{(n1+t/T)}\sim {\pi}_{JJ}\left(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\right{\mathit{X}}^{(n1+(t1)/T)})$ and ${\mathit{X}}_{J}^{(n1+t/T)}={\mathit{X}}_{J}^{(n1+(t1)/T)}$. 
end for 
4. Numerical Experiments
4.1. Simulation Study
4.1.1. Model Description
 (M1)
 generalized Pareto distributions (GPDs) with parameters $({\xi}_{j},{\beta}_{j})=(0.3,1)$ and survival Clayton copula with parameter $\theta =2$ so that Kendall’s tau equals $\tau =\theta /(\theta +2)=0.5$;
 (M2)
 multivariate Student t distribution with $\nu =5$ degrees of freedom, location vector $\mathbf{0}$, and dispersion matrix $\Sigma =\left({\rho}_{i,j}\right)$, where ${\rho}_{j,j}=1$ and ${\rho}_{i,j}=ij/d$ for $i,j=1,\dots ,d$, $i\ne j$.
4.1.2. Results and Discussions
4.2. Empirical Study
4.3. Detailed Comparison of MCMC with MC
 (A)
 VaR${}_{0.99}$, RVaR${}_{0.95,0.99}$, and ES${}_{0.99}$ contributions are estimated by the MC, HMC, and GS methods for dimensions $d\in \{4,6,8,10\}$. Note that the GS is applied only to RVaR and ES contributions, not to VaR contributions (same in the other scenarios).
 (B)
 For $d=5$, VaR${}_{{\alpha}^{VaR}}$, RVaR${}_{{\alpha}_{1}^{RVaR},{\alpha}_{2}^{RVaR}}$ and ES${}_{{\alpha}^{ES}}$ contributions are estimated by the MC, HMC, and GS methods for confidence levels ${\alpha}^{VaR}\in \{0.9,0.99,0.999,0.9999\}$, $({\alpha}_{1}^{RVaR},{\alpha}_{2}^{RVaR})\in \{(0.9,0.9999),(0.9,0.99),(0.99,0.999),(0.999,0.9999)\}$ and ${\alpha}^{ES}\in \{0.9,0.99,0.999,0.9999\}$.
 (C)
 For $d=5$, VaR${}_{0.9}$, RVaR${}_{0.9,0.99}$ and ES${}_{0.9}$ contributions are estimated by the MC and HMC methods with the parameters $({\u03f5}_{\mathrm{opt}},{T}_{\mathrm{opt}})$ (determined by Algorithm 4) and $(\u03f5,T)\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}\{(10{\u03f5}_{\mathrm{opt}},2{T}_{\mathrm{opt}}),\phantom{\rule{4pt}{0ex}}(10{\u03f5}_{\mathrm{opt}},{T}_{\mathrm{opt}}/2),\phantom{\rule{4pt}{0ex}}({\u03f5}_{\mathrm{opt}}/10,2{T}_{\mathrm{opt}}),\phantom{\rule{4pt}{0ex}}({\u03f5}_{\mathrm{opt}}/10,{T}_{\mathrm{opt}}/2)\}$.
5. Conclusion, Limitations and Future Work
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
i.i.d.  Independent and identically distributed 
Probability distribution function  
cdf  Cumulative distribution function 
ecdf  Empirical cdf 
GPD  Generalized Pareto distribution 
MSE  Mean squared error 
LLN  Law of large numbers 
CLT  Central limit theorem 
VaR  ValueatRisk 
RVaR  Range VaR 
ES  Expected shortfall 
MES  Marginal expected shortfall 
CoVaR  Conditional VaR 
CoES  Conditional ES 
MC  Monte Carlo 
SMC  Sequential Monte Carlo 
MCMC  Markov chain Monte Carlo 
ACR  Acceptance rate 
ACP  Autocorrelation plot 
MH  Metropolis–Hastings 
GS  Gibbs sampler 
MGS  Metropolized Gibbs sampler 
DSGS  Deterministic scan GS 
RPGS  Random permutation GS 
RSGS  Random scan GS 
HMC  Hamiltonian Monte Carlo 
RBHMC  Rollback HMC 
RMHMC  Riemannian manifold HMC 
ALAE  Allocated loss adjustment expense 
P&L  Profit and loss 
Appendix A. Hamiltonian Dynamics with Boundary Reflection
Algorithm A1 Leapfrog method with boundary reflection. 
Input: Current state $\left(\mathit{x}\right(0),\mathit{p}(0\left)\right)$, stepsize $\u03f5>0$, gradients $\nabla U$ and $\nabla K$, and constraints $({\mathit{h}}_{m},{v}_{m})$, $m=1,\dots ,M$. 
Output: Updated state $\left(\mathit{x}\right(\u03f5),\mathit{p}(\u03f5\left)\right)$. 
(1) Update $\mathit{p}(\u03f5/2)=\mathit{p}\left(0\right)+\u03f5/2\nabla U\left(\mathit{x}\right(0\left)\right)$. 
(2) Set $(\mathit{x},\mathit{p})=\left(\mathit{x}\right(0),\mathit{p}(\u03f5/2\left)\right)$, ${\u03f5}_{\mathrm{temp}}=\u03f5$. 
(3) while ${\u03f5}_{\mathrm{temp}}>0$ 
(31) Compute
$$\begin{array}{cc}\hfill {\mathit{x}}^{*}& =\mathit{x}+{\u03f5}_{\mathrm{temp}}\nabla K\left(\mathit{p}\right),\hfill \\ {t}_{m}& \hfill =({v}_{m}{\mathit{h}}_{m}^{\top}\mathit{x})/\left(\u03f5{\mathit{h}}_{m}^{\top}\mathit{p}\right),\phantom{\rule{1.em}{0ex}}m=1,\dots ,M.\end{array}$$

(32) if ${t}_{m}\in [0,1]$ for any $m=1,\dots ,M$, 
(321) Set
$$\begin{array}{cc}\hfill {m}^{*}& =argmin\left\{{t}_{m}\phantom{\rule{4pt}{0ex}}\right\phantom{\rule{4pt}{0ex}}0\le {t}_{m}\le 1,\phantom{\rule{4pt}{0ex}}m=1,\dots ,M\},\hfill \\ \hfill {\mathit{x}}_{r}^{*}& ={\mathit{x}}^{*}2\frac{{\mathit{h}}_{{m}^{*}}^{\top}{\mathit{x}}^{*}{v}_{{m}^{*}}}{{\mathit{h}}_{{m}^{*}}^{\top}{\mathit{h}}_{{m}^{*}}}{h}_{{m}^{*}},\hfill \\ \hfill {\mathit{p}}_{r}& =\frac{{\mathit{x}}^{*}\mathit{x}{t}_{{m}^{*}}\u03f5\mathit{p}}{\u03f5(1{t}_{{m}^{*}})}.\hfill \end{array}$$

(322) Set $(\mathit{x},\mathit{p})=({\mathit{x}}_{r}^{*},{\mathit{p}}_{r})\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{\u03f5}_{\mathrm{temp}}=(1{t}_{{m}^{*}}){\u03f5}_{\mathrm{temp}}$. 
else 
(323) Set $(\mathit{x},\mathit{p})=({\mathit{x}}^{*},\mathit{p})$ and ${\u03f5}_{\mathrm{temp}}=0$. 
end if 
end while 
(4) Set $\mathit{x}\left(\u03f5\right)=\mathit{x}$ and $\mathit{p}\left(\u03f5\right)=\mathit{p}+\frac{\u03f5}{2}\nabla U\left(\mathit{x}\right)$. 
Appendix B. Other MCMC Methods
Appendix B.1. RollBack HMC
Appendix B.2. Riemannian Manifold HMC
Appendix B.3. Metropolized Gibbs Samplers
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MC  HMC  

Estimator  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{3}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{3}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$ 
(I) GPD + survival Clayton with VaR crisis event: $\{S={VaR}_{0.99}\left(S\right)\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{VaR}]$  9.581  9.400  9.829  9.593  9.599  9.619 
Standard error  0.126  0.118  0.120  0.007  0.009  0.009 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  12.986  12.919  13.630  13.298  13.204  13.338 
Standard error  0.229  0.131  0.086  0.061  0.049  0.060 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  13.592  13.235  13.796  13.742  13.565  13.768 
Standard error  0.647  0.333  0.270  0.088  0.070  0.070 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  14.775  13.955  14.568  14.461  14.227  14.427 
Standard error  0.660  0.498  0.605  0.192  0.176  0.172 
(II) GPD + survival Clayton with RVaR crisis event: $\{{VaR}_{0.975}\left(S\right)\le S\le {VaR}_{0.99}\left(S\right)\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{RVaR}]$  7.873  7.780  7.816  7.812  7.802  7.780 
Standard error  0.046  0.046  0.046  0.012  0.012  0.011 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  11.790  11.908  11.680  11.686  11.696  11.646 
Standard error  0.047  0.057  0.043  0.053  0.055  0.058 
${RVaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  12.207  12.382  12.087  12.102  12.053  12.044 
Standard error  0.183  0.197  0.182  0.074  0.069  0.069 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  13.079  13.102  13.059  12.859  12.791  12.713 
Standard error  0.182  0.173  0.188  0.231  0.218  0.187 
MC  GS  

Estimator  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{3}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{3}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$ 
(III) GPD + survival Clayton with ES crisis event: $\{{VaR}_{0.99}\left(S\right)\le S\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{ES}]$  15.657  15.806  15.721  15.209  15.175  15.190 
Standard error  0.434  0.475  0.395  0.257  0.258  0.261 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  41.626  41.026  45.939  45.506  45.008  45.253 
Standard error  1.211  1.065  1.615  1.031  1.133  1.256 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  49.689  48.818  57.488  55.033  54.746  54.783 
Standard error  4.901  4.388  4.973  8.079  5.630  3.803 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  104.761  109.835  97.944  71.874  72.588  70.420 
Standard error  23.005  27.895  17.908  4.832  4.584  4.313 
(IV) Multivariate t with RVaR crisis event: $\{{VaR}_{0.975}\left(S\right)\le S\le {VaR}_{0.99}\left(S\right)\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{RVaR}]$  2.456  1.934  2.476  2.394  2.060  2.435 
Bias  0.019  −0.097  0.038  −0.043  0.029  −0.002 
Standard error  0.026  0.036  0.027  0.014  0.023  0.019 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  4.670  4.998  4.893  4.602  5.188  4.748 
Standard error  0.037  0.042  0.031  0.032  0.070  0.048 
${RVaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  5.217  5.397  5.240  4.878  5.717  5.092 
Standard error  0.238  0.157  0.145  0.049  0.174  0.100 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  5.929  5.977  5.946  5.446  6.517  6.063 
Standard error  0.204  0.179  0.199  0.156  0.248  0.344 
(V) Multivariate t with ES crisis event: $\{{VaR}_{0.99}\left(S\right)\le S\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{ES}]$  3.758  3.099  3.770  3.735  3.126  3.738 
Bias  0.017  −0.018  0.029  0.005  0.009  −0.003 
Standard error  0.055  0.072  0.060  0.031  0.027  0.030 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  8.516  8.489  9.051  8.586  8.317  8.739 
Standard error  0.089  0.167  0.161  0.144  0.156  0.158 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  9.256  9.754  10.327  9.454  9.517  9.890 
Standard error  0.517  0.680  0.698  0.248  0.293  0.327 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  11.129  12.520  12.946  11.857  12.469  12.375 
Standard error  0.595  1.321  0.826  0.785  0.948  0.835 
MC  HMC  

Estimator  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{1}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$  ${\mathit{A}}_{\mathbf{2}}^{\mathit{\varrho},\mathcal{C}}\left(\mathit{X}\right)$ 
(I) VaR crisis event: $\{S={VaR}_{0.99}\left(S\right)\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{VaR}]$  842465.497  73553.738  844819.901  71199.334 
Standard error  7994.573  7254.567  6306.836  6306.836 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  989245.360  443181.466  915098.833  428249.307 
Standard error  307.858  24105.163  72.568  20482.914 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  989765.514  500663.072  915534.362  615801.118 
Standard error  4670.966  54576.957  669.853  96600.963 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{VaR})$  990839.359  590093.887  915767.076  761038.843 
Standard error  679.055  75024.692  47.744  31211.908 
(II) RVaR crisis event: $\{{VaR}_{0.975}\left(S\right)\le S\le {VaR}_{0.99}\left(S\right)\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{RVaR}]$  528455.729  60441.368  527612.751  60211.561 
Standard error  3978.477  2119.461  4032.475  2995.992 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  846956.570  349871.745  854461.670  370931.946 
Standard error  1866.133  6285.523  2570.997  9766.697 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  865603.369  413767.829  871533.550  437344.509 
Standard error  5995.341  29105.059  12780.741  21142.135 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{RVaR})$  882464.968  504962.099  885406.811  529034.580 
Standard error  3061.110  17346.207  3134.144  23617.278 
(III) ES crisis event: $\{{VaR}_{0.99}\left(S\right)\le S\}$  
$\mathbb{E}\left[\mathit{X}\right{\mathcal{C}}^{ES}]$  8663863.925  137671.653  2934205.458  140035.782 
Standard error  3265049.590  10120.557  165794.772  14601.958 
${RVaR}_{0.975,0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  35238914.131  907669.462  17432351.450  589309.196 
Standard error  2892208.689  31983.660  443288.649  3471.641 
${VaR}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  56612082.905  1131248.055  20578728.307  615572.940 
Standard error  1353975.612  119460.411  1364899.752  12691.776 
${ES}_{0.99}\left(\mathit{X}\right{\mathcal{C}}^{ES})$  503537848.192  2331984.181  25393466.446  649486.810 
Standard error  268007317.199  468491.127  1138243.137  7497.200 
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Koike, T.; Hofert, M. Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations. Risks 2020, 8, 6. https://doi.org/10.3390/risks8010006
Koike T, Hofert M. Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations. Risks. 2020; 8(1):6. https://doi.org/10.3390/risks8010006
Chicago/Turabian StyleKoike, Takaaki, and Marius Hofert. 2020. "Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations" Risks 8, no. 1: 6. https://doi.org/10.3390/risks8010006