Selective Classification, wherein models can reject low-confidence predictions, promises reliable translation of machine-learning based classification systems to real-world scenarios such as clinical diagnostics. While current evaluation of these systems typically assumes fixed working points based on pre-defined rejection thresholds, methodological progress requires benchmarking the general performance of systems akin to the \mathrm{AUROC} in standard classification. In this work, we define 5 requirements for multi-threshold metrics in selective classification regarding task alignment, interpretability, and flexibility, and show how current approaches fail to meet them. We propose the Area under the Generalized Risk Coverage curve ( \mathrm{AUGRC}), which meets all requirements and can be directly interpreted as the average risk of undetected failures. We empirically demonstrate the relevance of \mathrm{AUGRC} on a comprehensive benchmark spanning 6 data sets and 13 confidence scoring functions.