In order to meet the objectives set by the Paris Climate Agreement, global greenhouse gas emissions must be drastically reduced. One way to achieve this goal is to set an effective carbon price. Although beneficial for the climate, a rapid increase in this price can have a significant financial impact on corporate firms. Based on the 2018 Intergovernmental Panel on Climate Change (IPCC) scenarios, we study the credit risk sensitivity of 795 international companies. We develop a bottom-up approach and analyze how probabilities of default within each sector might evolve in both the medium (2023) and long term (2060). We find that energy, materials and utilities sectors would be the most affected. Moreover, the risk materializes earlier and is more heterogeneous for utilities. From a policy perspective, the prices associated with a scenario limiting global warming to 2°C have a limited impact on global credit risk. Such a scenario therefore seems achievable without generating substantial financial losses. From these results, we propose a new indicator, the carbon price threshold, that takes the economic and capital structure of the firm into account in measuring carbon risk.