This paper studies the dependence structure between Islamic and conventional stocks in five countries (USA, UK, Japan, Malaysia and Pakistan) using the copula CoVaR approach for the period 2004–2016. Results from the copula models show that, for four countries, the Islamic stocks are upper tail dependent and lower tail independent with conventional stocks. While for one country, Islamic stocks are lower tail dependent with the conventional stocks without having upper dependence. We also find that dependence structure between Islamic and conventional stocks is time varying for all the countries. We also estimate the CoVaR for conventional stocks conditional on Islamic stocks being at their VaR. The results show the diversification benefits of the Islamic stocks and investors can use Islamic finance instruments for hedging. We then test for the asymmetry of upside and downside risks and find that there is an asymmetry between upside and downside risks.