We investigate the risk–return trade-off on the US and European stock markets. We investigate the non-linear risk–return trade-off with a special eye to the tails of the stock returns using quantile regressions. We first consider the US stock market portfolio. We find that the risk–return trade-off is significantly positive at the upper tail (0.9 quantile), where the upper tail is large positive excess returns. The positive trade-off is as expected from asset pricing models. For the lower tail (0.1 quantile), that is for large negative stock returns, the trade-off is significantly negative. Additionally, for the median (0.5 quantile), the risk–return trade-off is insignificant. These results are recovered for the US industry portfolios and for Eurozone stock market portfolios.