Value at Risk of Stock Market Returns Using a Hidden Markov–LSTM Hybrid Model

Value at Risk (VaR) remains a central measure for downside market risk and is widely used in regulatory and internal risk management frameworks. Accurate VaR forecasting is challenging because equity returns exhibit volatility clustering, heavy tails, and structural changes associated with shifts between calm and turbulent market conditions. This study proposes a hybrid VaR forecasting framework that combines a Gaussian Hidden Markov Model (HMM) for latent regime identification with a Long Short Term Memory (LSTM) network for conditional volatility forecasting. The model is estimated using daily adjusted closing prices for Microsoft Corporation (MSFT) over 2016 to 2022 with a strict out of sample evaluation, where 2016 to 2019 is used for training and 2020 to 2022 is used for testing. The LSTM predicts a realized volatility proxy computed from rolling return dispersion, while the HMM regime sequence is used as an additional explanatory input to capture regime dependent dynamics. Tail risk is handled by fitting a Student t distribution to standardized residuals from the training period and constructing parametric VaR and Expected Shortfall (ES) forecasts at the 1 percent level. Out of sample backtesting using the Kupiec unconditional coverage test indicates that the observed violation frequency is close to the nominal rate and is not rejected, supporting that regime information combined with nonlinear volatility learning improves VaR calibration for equity returns.