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Traditional exposure measures focus on direct exposures to evaluate the losses an institution is exposed to upon the default of a counterparty. Since the Global Financial Crisis of '07/'08, indirect exposures via common asset holdings are increasingly recognized too. Yet direct and indirect exposures fail to capture the losses that result from shock propagation and amplification following the counterparty's default. In this paper, we introduce the concept ``higher-order exposures'' to refer to these spill-over losses and propose a way to formalize and quantify them. Using granular data of the South African financial system and a contagion model that captures the most commonly studied contagion channels and their interactions, we show that higher-order exposures make up a significant part of exposures – particularly during times of financial distress when exposures matter most. We also show that higher-order exposures cannot simply be extrapolated from direct or indirect exposures, since they depend strongly on the network structure and the robustness of individual institutions. Our findings suggest that higher-order exposures should inform the design and calibration of those tools in the regulators' arsenal where exposures matter -- including large exposure limits, capital requirement calibration, stress test design and resolution. Failure to do so may result in both lax ex-ante regulation and ill-informed ex-post handling of financial crises.