As the environment changes, the stationarity assumption in hydrological analysis has become questionable. If nonstationarity of an observed time series is not fully considered when handling climate change scenarios, the outcomes of statistical analyses would be invalid in practice. This study established bivariate time-varying copula models for risk analysis based on the generalized additive models in location, scale, and shape (GAMLSS) theory to develop the nonstationary joint drought management index (JDMI). Two kinds of daily streamflow data from the Soyang River basin were used; one is that observed during 1976-2005, and the other is that simulated for the period 2011-2099 from 26 climate change scenarios. The JDMI quantified the multi-index of reliability and vulnerability of hydrological drought, both of which cause damage to the hydrosystem. Hydrological drought was defined as the low-flow events that occur when streamflow is equal to or less than Q80 calculated from observed data, allowing future drought risk to be assessed and compared with the past. Then, reliability and vulnerability were estimated based on the duration and magnitude of the events, respectively. As a result, the JDMI provided the expected duration and magnitude quantities of drought or water deficit.