Efforts to predict stock market outcomes have yielded limited success due to the inherently stochastic nature of the market, influenced by numerous unpredictable factors. Many existing prediction approaches focus on single-point predictions, lacking the depth needed for effective decision-making and often overlooking market risk. To bridge this gap, we propose a novel model, RAGIC, which introduces sequence generation for stock interval prediction to quantify uncertainty more effectively. Our approach leverages a Generative Adversarial Network (GAN) to produce future price sequences infused with randomness inherent in financial markets. RAGIC's generator includes a risk module, capturing the risk perception of informed investors, and a temporal module, accounting for historical price trends and seasonality. This multi-faceted generator informs the creation of risk-sensitive intervals through statistical inference, incorporating horizon-wise insights. The interval's width is carefully adjusted to reflect market volatility. Importantly, our approach relies solely on publicly available data and incurs only low computational overhead. RAGIC's evaluation across globally recognized broad-based indices demonstrates its balanced performance, offering both accuracy and informativeness. Achieving a consistent 95% coverage, RAGIC maintains a narrow interval width. This promising outcome suggests that our approach effectively addresses the challenges of stock market prediction while incorporating vital risk considerations.