Cascade Sensitivity Measures

Abstract

In risk analysis, sensitivity measures quantify the extent to which the probability distribution of a model output is affected by changes (stresses) in individual random input factors.For input factors that are statistically dependent, we argue that a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed cascade sensitivity, defined as a derivative of a risk measure applied on the output, in the direction of an input factor. The derivative is taken after suitably transforming the random vector of inputs, thus explicitly capturing the direct impact of the stressed input factor, as well as indirect effects via other inputs. Furthermore, alternative representations of the cascade sensitivity measure are derived, allowing us to address practical issues, such as incomplete specification of the model and high computational costs. The applicability of the methodology is illustrated through the analysis of a commercially used insurance risk model.

Sensitivity Measures Based on Scoring Functions

“…Examples of sensitivity measures include variance-based (Saltelli and Tarantola 2002), moment-independent (Borgonovo 2007), and quantile-based sensitivities (Browne et al 2017). Alternative approaches include those based on divergence measures (Gamboa et al 2018, Pesenti et al 2019, Fort et al 2021, Pesenti 2021) and differential sensitivity measures, see Tsanakas and Millossovich (2016) and Pesenti et al (2021) in a risk management context. However, as argued in Borgonovo et al (2021), the choice of a sensitivity measure should be intimately tied to the functional of interest T via the notion of strictly consistent scoring functions and moreover reflect the information value of risk factors.…”

A theory of multivariate stress testing

“…How does the stressing mechanism η i impact the distribution of the whole vector X? One way to characterize the post-stress distribution of X is via inverse Rosenblatt transforms, as considered by Rüschendorf and de Valk (1993) and Pesenti et al (2021). We can always represent the risk factors by…”

“…In this way, stressing can be seen to represent a contamination of the marginal distributions with respect to heavier tailed ones, as is often done in the study of model uncertainty (e.g. Cont et al, 2010, Pesenti et al, 2021.…”

“…As is not untypical when dealing with complex computational models (e.g. Pesenti et al, 2021), this model is largely a black box to us. We do not have a parametric form for the distributions of X i , which are themselves outputs of sub-models.…”

“…Defining stresses via measure changes affords numerical benefits, as it does not require repeated simulation under alternate model assumptions. However, these literatures do not explicitly integrate dependence considerations into the design of stress tests and the few papers that take a genuinely multivariate perspective, e.g., McNeil and Smith (2012), Pesenti et al (2021), deploy rather different frameworks.…”