Measuring the contribution of a bank or an insurance company to the overall systemic risk of the market is an important issue, especially in the aftermath of the 2007-2009 Financial Crisis and the financial downturn of 2020. Specifically, the Office of the Superintendent of Financial Institutions (OSFI) in Canada recommends a multifaceted approach to address this problem, with two major components being the size of the company and its interconnectedness within the market.

One appropriate quantitative measure of systemic risk contribution is the marginal expected shortfall (MES), introduced by Acharya et al. (2017) and known to actuarial scholars as the risk capital allocation rule based on the expected shortfall risk measure since at least Panjer and Jing (2001). The MES risk measure has attracted considerable attention and has been evaluated under various assumptions on the joint probability distribution of companies’ risks in the market of interest. However, in reality, the univariate probability distributions of distinct companies in the market can be quite distinct, and the dependence structure between the companies is rarely known.

In this paper, we derive the upper and lower bounds of MES under the assumption of full information on the marginal distributions of the companies’ risks and an unknown dependence structure. Furthermore, we also derive improved bounds for the MES risk measure when partial information on companies’ risk exposures–and hence their dependence–is available. We employ two well-known factor models, the multiplicative and additive background risk models, to describe such partial information.

Reference:

Acharya, V. V., L. H. Pedersen, T. Philippon, and M. Richardson (2017). Measuring systemic risk. The review of financial studies 30 (1), 2–47.

Panjer, H. H. and J. Jing (2001). Solvency and capital allocation. University of Waterloo, Institute of Insurance and Pension Research.