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Back-translation (BT) is an effective semi-supervised learning framework in neural machine translation (NMT). A pre-trained NMT model translates monolingual sentences and makes synthetic bilingual sentence pairs for the training of the other NMT model, and vice versa. Understanding the two NMT models as inference and generation models, respectively, the training method of variational auto-encoder (VAE) was applied in previous works, which is a mainstream framework of generative models. However, the discrete property of translated sentences prevents gradient information from flowing between the two NMT models. In this paper, we propose the categorical reparameterization trick (CRT) that makes NMT models generate differentiable sentences so that the VAE's training framework can work in an end-to-end fashion. Our BT experiment conducted on a WMT benchmark dataset demonstrates the superiority of our proposed CRT compared to the Gumbel-softmax trick, which is a popular reparameterization method for categorical variable. Moreover, our experiments conducted on multiple WMT benchmark datasets demonstrate that our proposed end-to-end training framework is effective in terms of BLEU scores not only compared to its counterpart baseline which is not trained in an end-to-end fashion, but also compared to other previous BT works. The code is available at the web.