Assessing continuous common-shock risk through matrix distributions

TL;DR

This paper introduces a new class of continuous-time bivariate distributions based on Markov processes to model dependencies caused by common shocks, providing analytical tools and estimation methods.

Contribution

The paper develops a novel continuous-time bivariate distribution model for common-shock dependencies, with tractable properties and efficient parameter estimation techniques.

Findings

Effectively captures common-shock dependencies in data.

Analytical expressions for joint distributions and risk measures.

Successful application to real-world data.

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.