Portfolio credit derivatives, including the basket credit default swaps, are designed to facilitate the transfer of credit risk amongst market participants. Investors consider them as cheap tools to hedge a portfolio of credits, instead of individual hedging of the credits. The prime aim of this work is to model the hazard rate process using stochastic default intensity models, as well as extend the results to the pricing of basket default swaps. We focused on the  $n$th-to-default swaps whereby the spreads are dependent on the $n$th default time, and we estimated the joint survival probability distribution functions of the intensity models under the risk-neutral pricing measure, for both the homogeneous and the heterogeneous portfolio. This work further employed the Monte-Carlo method, under the one-factor Gaussian copula model to numerically approximate the distribution function of the default time, and thus, the numerical experiments for pricing the $n$th default swaps were made viable under the two portfolio types. Finally, we compared the effects of different swap parameters to various $n$th-to-default swaps.