Personality Recognition in Conversation (PRC) aims to identify the personality traits of speakers through textual dialogue content. It is essential for providing personalized services in various applications of Human-Computer Interaction (HCI), such as AI-based mental therapy and companion robots for the elderly. Most recent studies analyze the dialog content for personality classification yet overlook two major concerns that hinder their performance. First, crucial implicit factors contained in conversation, such as emotions that reflect the speakers' personalities are ignored. Second, only focusing on the input dialog content disregards the semantic understanding of personality itself, which reduces the interpretability of the results. In this paper, we propose Affective Natural Language Inference (Affective-NLI) for accurate and interpretable PRC. To utilize affectivity within dialog content for accurate personality recognition, we fine-tuned a pre-trained language model specifically for emotion recognition in conversations, facilitating real-time affective annotations for utterances. For interpretability of recognition results, we formulate personality recognition as an NLI problem by determining whether the textual description of personality labels is entailed by the dialog content. Extensive experiments on two daily conversation datasets suggest that Affective-NLI significantly outperforms (by 6%-7%) state-of-the-art approaches. Additionally, our Flow experiment demonstrates that Affective-NLI can accurately recognize the speaker's personality in the early stages of conversations by surpassing state-of-the-art methods with 22%-34%.