Recent algorithms of time-series anomaly detection have been evaluated by applying a Point Adjustment (PA) protocol. However, the PA protocol has a problem of overestimating the performance of the detection algorithms because it only depends on the number of detected abnormal segments and their size. We propose a novel evaluation protocol called the Point-Adjusted protocol with decay function (PAdf) to evaluate the time-series anomaly detection algorithm by reflecting the following ideal requirements: detect anomalies quickly and accurately without false alarms. This paper theoretically and experimentally shows that the PAdf protocol solves the over- and under-estimation problems of existing protocols such as PA and PA\%K. By conducting re-evaluations of SOTA models in benchmark datasets, we show that the PA protocol only focuses on finding many anomalous segments, whereas the score of the PAdf protocol considers not only finding many segments but also detecting anomalies quickly without delay.