Emergent Systemic Risk Horizon (ESRH)

• Emergent Systemic Risk Horizon (ESRH) is a quantitative metric that identifies the critical time-scale T* when contagion-induced losses rapidly accelerate.
• It leverages a PD–contagion model and multi-period Monte Carlo simulations to capture evolving loss distributions and default propagation in financial networks.
• ESRH informs supervisory policy by establishing explicit criteria based on mean-loss acceleration and tail-probability growth for effective stress testing.

The Emergent Systemic Risk Horizon (ESRH) is a quantitative systemic-risk metric introduced to detect the critical time-scale $T^*$ at which contagion-driven losses in a financial network begin to accelerate beyond a selected threshold rate. Built on the Probability of Default–contagion (PD–contagion) model by Petrone and Latora, ESRH provides temporal resolution of systemic fragility, offering supervisors an explicit criterion for the time available before risk dynamics enter a regime of rapid loss growth. The ESRH leverages Monte Carlo simulations on multi-period loss-distributions as well as explicit modeling of default propagation through exposures, incorporating both statistical and network-theoretical features intrinsic to banking systems (Petrone et al., 2016).

1. PD–Contagion Model Dynamics

The foundational framework for ESRH is the PD–contagion model, which describes the evolution of each bank’s default probability %%%%1%%%% as a function of its exposure to counterparts and the systemic propagation of failure. Each of $N$ banks is connected by exposures $w_{ij}(t)$, where $w_{ij}(t) = a_{ij}(t)\,LGD_j(t)$ and $LGD_j(t)$ is the loss-given-default. Default events are encoded by indicator variables $$\delta_j(t)=1_{\{\text{bank }j\text{ defaulted at }t\}}$$, and the impact of defaults on bank $i$ at time $t$ is

$I_i(t) = \sum_{j=1}^N w_{ij}(t)\,\delta_j(t).$

Asset and capital dynamics respond by reducing according to

$$E_i(t+\Delta t) = E_i(t) - I_i(t),\qquad A_i(t+\Delta t) = A_i(t) - I_i(t).$$

The PD updates incorporate the feedback from contagion via:

• Merton update:

$$PD_i(t+\Delta t) = 1 - \Phi \bigg( \frac{\ln[A_i(t) - I_i(t)] - \ln B_i - \frac{1}{2}\sigma_i^2\Delta t}{\sigma_i\sqrt{\Delta t}} \bigg),$$

with $B_i = A_i(0) - E_i(0)$, $\sigma_i$ as volatility, and $\Phi$ the normal CDF.

• Linear update:

$$PD_i(t+\Delta t) = \min\Big\{1,\, PD_i(t) + \frac{(1-PD_i(t))\,I_i(t)}{E_i(t)} \Big\}.$$

If the impact $I_i(t)$ ever exceeds the capital $E_i(t)$, $PD_i$ is set to 1, reflecting immediate default.

2. Monte Carlo Procedure and Loss Quantification

Systemic risk assessment is performed via multi-period Monte Carlo simulation. Over a horizon $T = M\Delta t$:

1. At each time-step, generate vectors $X(t)\sim \mathcal{N}(0,\rho)$ using a prescribed default-correlation matrix $\rho$.
2. Set $\delta_i(t) = 1 \iff X_i(t) < \Phi^{-1}(PD_i(t))$ for each $i$.
3. Compute $I_i(t)$, update $(E_i, A_i, PD_i)$, and tally single-step losses $L(t) = \sum_{i=1}^N A_i(t)\,LGD_i(t)\,\delta_i(t)$.
4. Discount and accumulate total loss: $L_{\rm tot}(T) = \sum_{t=1}^M D(t)\,L(t)$.
5. Repeat for $R \gg 1$ scenarios to obtain empirical distributions of $L_{\rm tot}(T)$.

Statistics derived include mean loss $\overline{L}(T)$, variance, and tail quantiles $q_\alpha(T)$. These measurements are integral to ESRH evaluation.

3. Formal Definition of ESRH

ESRH $T^*$ is operationally defined as the minimal time $T$ for which losses (mean or tail probabilities) show acceleration above regulatory thresholds. Two definitions are given:

(A) Mean-loss acceleration:

$$T^* = \min\left\{T : \frac{d}{dT}\overline{L}(T) \ge \gamma\right\},$$

where $\gamma > 0$ is the critical-loss-rate.

(B) Tail-probability growth:

$$T^* = \min\left\{T : \frac{d}{dT}P[L_{\rm tot}(T) > L_0] \ge \eta\right\},$$

with $L_0$ as the critical loss level and $\eta$ the tail growth-rate threshold.

Both definitions use derivatives to identify periods of rapid system-wide risk escalation.

4. Numerical and Analytical Procedures for Computing $T^*$

Estimation of ESRH requires discretization and scenario analysis. Numerically, for a predefined grid $T_n = n\Delta t$ ($n=1,\dots,M$):

• For each $T_n$, run $R$ Monte Carlo scenarios and compute $\overline{L}(T_n)$ or $P[L_{\rm tot}(T_n)>L_0]$.
• Finite-difference approximations compute the relevant slope:

$$\Delta\overline{L}_n = \frac{\overline{L}(T_n) - \overline{L}(T_{n-1})}{\Delta t}$$

• Identify $T^*$ as the smallest $T_n$ where the acceleration criteria are met.

Analytic solutions are tractable for small systems (e.g., two banks) via Markov-chain methods, where transition probabilities $\pi_s(t)$ can be solved and the critical slope for the “all-default” state detected.

5. Dependence of ESRH on Network and Financial Parameters

The systemic risk horizon is functionally sensitive to several network and bank-specific parameters:

• Exposures ($w_{ij}$) or LGDs $\uparrow$: Accelerates contagion, yielding smaller $T^*$.
• Capital buffers ($E_i$) $\uparrow$ or asset volatilities ($\sigma_i$) $\downarrow$: Slow contagion feedback, increasing $T^*$.
• Default correlation ($\rho$) regimes:
  • In the typical regime, $\rho \uparrow$ increases likelihood of simultaneous defaults and decreases $T^*$.
  • In the strong-contagion regime, lower $\rho$ produces more early single defaults and stronger PD feedback, paradoxically decreasing $T^*$.

A plausible implication is that under certain conditions—especially strong contagion—standard diversification heuristics may fail, as loss concentration can increase with declining inter-bank correlation.

6. ESRH in Supervisory Policies and Stress Testing

The financial interpretation of $T^*$ is the time remaining, under current capital and exposure settings, before losses due to contagion undergo dangerous acceleration. ESRH thereby offers supervisory authorities a model-driven quantitative target for setting stress-test horizons and capital buffers. By identifying the systemic risk horizon, policy can be formulated to delay $T^*$ beyond regulatory benchmarks, mitigating potential system-wide instability as modeled in dynamic credit-risk networks (Petrone et al., 2016).