Assessing continuous common-shock risk through matrix distributions

TL;DR

This work introduces the common-shock continuous PH (CSPH) distribution to model explicit dependencies induced by a single triggering event in a bivariate setting. The construction uses identical pre-shock evolution under a shared CTMC until a common shock, after which components evolve independently, yielding analytically tractable joint distributions and risk measures via matrix-exponential methods. The authors establish the denseness of CSPH, develop a master formula for transformed moments, and derive a suite of risk measures (Pearson, entropic, CV@R_CS, MTCE_CS, MTCov_CS) that can be computed efficiently. They demonstrate the framework on synthetic data and real Danish fire-insurance data, highlighting CSPH's ability to capture common-shock dependencies and to quantify conditional dependence, with implications for risk aggregation and pricing in insurance and finance.

Abstract

We introduce a class of continuous-time bivariate phase-type distributions for modeling dependencies from common shocks. The construction uses continuous-time Markov processes that evolve identically until an internal common-shock event, after which they diverge into independent processes. We derive and analyze key risk measures for this new class, including joint cumulative distribution functions, dependence measures, and conditional risk measures. Theoretical results establish analytically tractable properties of the model. For parameter estimation, we employ efficient gradient-based methods. Applications to both simulated and real-world data illustrate the ability to capture common-shock dependencies effectively. Our analysis also demonstrates that common-shock continuous phase-type distributions may capture dependencies that extend beyond those explicitly triggered by common shocks.

Figures (10)

• Figure 5.1: Entropic risk measure for the marginal distributions of the CSPH model in Example \ref{['ex:risk_measure']}, computed for varying values of $\vartheta$ and conditional on the common shock exceeding a threshold $a = 0, 1.5, 3, 4.5, 6, 7.5$.
• Figure 5.2: CV@R for the marginal distributions of the CSPH model in Example \ref{['ex:risk_measure']} plotted as a function of the threshold parameter $a$ affecting the common-shock component $\tau_{1,2}$.
• Figure 5.3: $\hbox{MTCE}^{\text{CS}}_a$ (left) and $\hbox{MTCov}^{\text{CS}}_a$ (right) for the CSPH model in Example \ref{['ex:risk_measure']} plotted for different values of the threshold $a$.
• Figure 6.1: Conditional means, $\mathbb{E}[\tilde{\tau}_i \mid \tau_{1,2}=t]$, $i =1, 2$, for the CSPH model in Example \ref{['ex:risk_measure']} plotted as a function of $t$.
• Figure 7.1: Fit quality assessment for the synthetic data example. Center: Scatter plot of 2,000 simulated bivariate observations from a CSPH model with $|\mathcal{E}| = 3$ transient states and $|\mathcal{S}| = 2$ post-shock states, overlaid with contour curves of the fitted CSPH joint density. Top and right: Histograms of the simulated margins along with the corresponding marginal densities of the fitted CSPH model. The close alignment between fitted and true distributions indicates successful parameter recovery.

Theorems & Definitions (16)

• Definition 3.1: Bivariate Common-Shock $\hbox{PH}$ Distribution
• Theorem 4.1: Joint Density
• proof
• Theorem 4.2: Joint Cumulative Distribution Function
• proof
• Theorem 4.3: Joint Moment Generating Function
• proof
• Theorem 5.1
• proof
• Remark 5.1