The most successful approach to Neural Machine Translation (NMT) when only monolingual training data is available, called unsupervised machine translation, is based on back-translation where noisy translations are generated to turn the task into a supervised one. However, back-translation is computationally very expensive and inefficient. This work explores a novel, efficient approach to unsupervised NMT. A transformer, initialized with cross-lingual language model weights, is fine-tuned exclusively on monolingual data of the target language by jointly learning on a paraphrasing and denoising autoencoder objective. Experiments are conducted on WMT datasets for German-English, French-English, and Romanian-English. Results are competitive to strong baseline unsupervised NMT models, especially for closely related source languages (German) compared to more distant ones (Romanian, French), while requiring about a magnitude less training time.