Financial contagion and systemic risk measures are commonly derived from conditional quantiles by using imposed model assumptions such as a linear parametrization. In this paper, we provide model free measures for contagion and systemic risk which are independent of the specification of conditional quantiles and simple to interpret. The proposed systemic risk measure relies on the contagion measure, whose tail behavior is theoretically studied. To emphasize contagion from extreme events, conditional quantiles are specified via hierarchical Archimedean copula. The parameters and structure of this copula are simultaneously estimated by imposing a non-concave penalty on the structure. Asymptotic properties of this sparse estimator are derived and small sample properties illustrated using simulations. We apply the proposed framework to investigate the interconnectedness between American, European and Australasian stock market indices, providing new and interesting insights into the relationship between systemic risk and contagion. In particular, our findings suggest that the systemic risk contribution from contagion in tail areas is typically lower during times of financial turmoil, while it can be significantly higher during periods of low volatility.