The Expected-Based Method of Value-at-Risk Prediction - Jacob Stevy Seleky; Lina Cahyadi Cahyadi; Sausan Ramadhani (Jambura Journal of Mathematics, 2025)
Value-at-Risk (VaR) remains a fundamental risk measure in financial risk management, providing an indicator for managing capital allocation and avoiding worst-case risk scenarios. Traditionally its defined as a quantile of the loss distribution. However, its computation depends critically on the existence and tractability of the inverse cumulative distribution function (CDF), which may not be available in closed form for complex or empirical distributions. This paper proposes an expectation-based simulation framework for VaR estimation that avoids explicit inversion of the CDF. The method approximates VaR by taking the expectation of order statistics from repeated sampling, effectively constructing a variance-reduced Monte Carlo estimator of the quantile. We provide a rigorous theoretical foundation for the proposed approach, including strong consistency, asymptotic normality, and a bias–variance decomposition. In particular, we show that the estimator achieves variance reduction proportional to the number of simulations while remaining consistent with the classical definition of VaR. Furthermore, under heavy-tailed distributions, the method demonstrates enhanced stability compared to traditional historical simulation, which is known to exhibit high tail variability. Extensive simulation studies confirm the theoretical findings, showing significant improvements in mean squared error and backtesting performance. Overall, the proposed framework provides a flexible alternative for VaR estimation in settings where conventional inversion-based methods are infeasible or unreliable.