We propose an approach to calibrate the conditional value-at-risk (CoVaR) of financial institutions based on neural network quantile regression. Building on the estimation results we model systemic risk spillover effects across banks by considering the marginal effects of the quantile regression procedure. We adopt a dropout regularization procedure to remedy the well-known issue of overfitting for neural networks, and we provide empirical evidence for the favorable out-of- sample performance of a regularized neural network. We then propose three measures for systemic risk from our fitted results. We find that systemic risk increases sharply during the height of the financial crisis in 2008 and again after a short period of easing in 2011 and 2015. Our approach also allows identifying systemically relevant firms during the financial crisis.