Stressing Dynamic Loss Models

36 Pages Posted: 18 Nov 2022 Last revised: 9 Mar 2023

See all articles by Emma Kroell

Emma Kroell

University of Toronto

Silvana M. Pesenti

University of Toronto

Sebastian Jaimungal

University of Toronto - Department of Statistics

Date Written: November 6, 2022


Stress testing, and in particular, reverse stress testing, is a prominent exercise in risk management practice. Reverse stress testing, in contrast to (forward) stress testing, aims to find an alternative but plausible model such that under that alternative model, specific adverse stresses (i.e. constraints) are satisfied. Here, we propose a reverse stress testing framework for dynamic models. Specifically, we consider a compound Poisson process over a finite time horizon and stresses composed of expected values of functions applied to the process at the terminal time. We then define the stressed model as the probability measure under which the process satisfies the constraints and which minimizes the Kullback-Leibler divergence to the reference compound Poisson model.

We solve this optimization problem, prove existence and uniqueness of the stressed probability measure, and provide a characterization of the Radon-Nikodym derivative from the reference model to the stressed model. We find that under the stressed measure, the intensity and the severity distribution of the process depend on time and state, and hence the stressed model is not a compound Poisson process. We illustrate the dynamic stress testing by considering stresses on VaR and both VaR and CVaR jointly and provide illustrations of how the stochastic process is altered under these stresses. We generalize the framework to multivariate compound Poisson processes and stresses at times other than the terminal time. We illustrate the applicability of our framework by considering "what if" scenarios, where we answer the question: What is the severity of a stress on a portfolio component at an earlier time such that the aggregate portfolio exceeds a risk threshold at the terminal time? Furthermore, for general constraints, we propose an algorithm to simulate sample paths under the stressed measure, thus allowing to compare the effects of stresses on the dynamics of the process.

Keywords: Reverse Stress Testing, Compound Poisson Processes, KL divergence, Value-at-Risk, Conditional Value-at-Risk

Suggested Citation

Kroell, Emma and Pesenti, Silvana M. and Jaimungal, Sebastian, Stressing Dynamic Loss Models (November 6, 2022). Available at SSRN: or

Emma Kroell (Contact Author)

University of Toronto ( email )

105 St George Street
Toronto, Ontario M5S 3G8


Silvana M. Pesenti

University of Toronto ( email )

700 University Avenue 9F
Toronto, Ontario

Sebastian Jaimungal

University of Toronto - Department of Statistics ( email )

100 St. George St.
Toronto, Ontario M5S 3G3

HOME PAGE: http://http:/

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