A Bootstrap Method with Importance Resampling to Evaluate Value-at-Risk

Abstract

計算投資組合之風險值，蒙地卡羅分析至今是最有用的方法，然而最大的缺點是計算的時間長。在本篇論文中，我們假設風險因子為一個多維常態分配和一個多維t分配，提供一個有效的方法，一個重點抽樣拔靴演算法估計投資組合損失的機率，然後計算投資組合的風險值。在拔靴法的模擬研究與敏感度分析上，我們指出對於分位數與尾部機率的估計比蒙地卡羅方法更有效率。然後我們也觀察對於多維常態常態分配與t分配估計，分位數與尾部機率估計是不敏感的。最後對於本論文所建議的方法，我們對台灣兩個股票之投資組合，彰化銀行典中鋼，做一個實證的分析。

To evaluate a portfolio value-at-risk (VaR), Monte Carlo analysis is by far the most powerful method. However, the biggest drawback of this method is its computational time. In this paper, we model the return of risk factors with a multivariate normal as well as a multivariate t distribution, and provide an efficient method, a bootstrap algorithm with importance resampling, to estimate portfolio loss probability and portfolio value-at-risk. In the simulation study and sensitivity analysis of the bootstrap method, we first note that the estimate for the quantile and tail probability with importance resampling is more efficient than the naive Monte Carlo method. Next, we observe that the estimates of the quantile and the tail probability are not sensitive to the estimated parameters for the multivariate normal and the multivariate t distribution. As an illustration of our proposed methods, we report an empirical study based on two stock index returns in Taiwan, the Chang Hwa Bank and the China Steel Corporation.