In this paper, a novel two-stage stochastic planning model is proposed based on graph theory for hybrid AC-DC distribution networks. In the proposed model, all possible AC-DC arrangements of the network are investigated to determine the optimal configuration. The Monte Carlo simulation method is applied to account for uncertainties in the output power of distributed generators (DGs) and load demands. In the first stage of the optimization, the planning problem is modeled as a mixed-integer programming (MIP) problem. Moreover, the structure of the AC-DC network is determined using a genetic algorithm (GA) to minimize the system investment costs. The second stage of the optimization is a non-linear programming (NLP) problem. In this step, the optimal power flow (OPF) model is created, and the lifetime of the converters for the determined AC-DC structure is maximized. Moreover, PLECS is used to mathematically model the converter lifetime based on its thermal cycles and fit the converter efficiency curve according to its input current function. Finally, the proposed method is applied for designing a 13-bus system, including various distributed generation sources and AC-DC loads. The evaluation of the results in several scenarios indicates that the proposed model is efficient in improving the converter lifetime and reducing the system costs.