VaR Forecasting for Financial Asset Series Based on Truncated Cauchy Distribution

Against the backdrop of increasing volatility in financial markets, traditional Value at Risk (VaR) models have shown significant shortcomings in estimating tail risks. This paper introduces a truncated Cauchy distribution for constructing a Value-at-Risk (VaR) model for the first time. This model mitigates the effect of extreme values by truncating the Cauchy distribution over a finite interval while maintaining thick-tailed properties. The model uses the median and the median absolute deviation (MAD) to determine the location and scale parameters, thereby improving model stability. An empirical study based on daily return data of the French CAC40 index from 1995 to 2022 was conducted. The rolling window method was used to calculate VaR at the 95% confidence level and compare it with classical models, such as the Normal distribution, the Student's t-distribution, and the RiskMetrics model. Results show that the truncated Cauchy model passes the POF test at multiple confidence levels and significantly outperforms other models in forecasting performance, particularly during extreme market periods, such as the 2008 financial crisis and the 2020 COVID-19 pandemic. This study expands the theoretical foundation of VaR modeling and provides effective tools for tail risk control and robustness measures in financial markets.