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Individual loss reserving methods have undergone substantial development in the past decade, driven by increased access to granular-level insurance claims data. This research aims to propose a micro loss reserving model tailored for multi-coverage insurance policies, where a single insurance claim might trigger payments from multiple coverage types. A copula-based multivariate regression approach is employed to jointly model the settlement time and loss amount, effectively capturing the dependence among various types of loss amounts and their correlation with the settlement time. We stress the importance of considering both types of dependence for accurate reserving prediction and uncertainty quantification. Furthermore, we propose computationally efficient algorithms for parameter estimation and dynamic prediction. Through numerical experiments and real data analysis, we demonstrate the effectiveness of the proposed multivariate predictive model in loss reserving applications.