## 定义

${r}_{t}=\mu +{\epsilon }_{t}$ 是资产收益率的对数数列，其中 $\mu$ 是期望收益率，${\epsilon }_{t}$ 是一个均值为零的白噪声过程。时间 $t$ 和时间 $t+k$ 之间的对数收益率之和可以写作：$rt,t+k=∑i=1krt+k$资产的 $k$日前-$\mathrm{VaR}$ 的标准定义为收益率 ${r}_{t,t+k}$ 的分布的1%或5%分位数。V-Lab的长期风险度量采用基于模拟的方法来计算 $k=30$$k=365$ 时的 $\mathrm{VaR}$

## 参考文献

Engle, Robert F., The Risk that Risk Will Change. Journal Of Investment Management (JOIM), Fourth Quarter 2009. https://www.joim.com/article/the-risk-that-risk-will-change/

Engle, R. F. and J. G. Rangel, 2008. The Spline-GARCH Model for Low-Frequency Volatility and Its Global Macroeconomic Causes. Review of Financial Studies 21(3): 1187-1222. https://www.jstor.org/stable/40056848

Glosten, L. R., R. Jagannathan, and D. E. Runkle, 1993. On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks. Journal of Finance 48: 1779-1801. https://doi.org/10.1111/j.1540-6261.1993.tb05128.x

Zakoian, J. M., 1994. Threshold Heteroscedastic Models. Journal of Economic Dynamics and Control 18: 931-955. https://doi.org/10.1016/0165-1889(94)90039-6