The Analysis And Comparison Of VaR And CVaR Based On Important Sampling

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are two widely used risk measures. They play important roles in risk measurement, portfolio management and regulatory control of financial institutions. The Basel Accord Ⅱ has incorporated the concept of VaR and encourages banks to use VaR for risk management. It has the advantage of being flexible and conceptually simple. However, VaR also has some shortcomings. It provides no information about the amount of loss that investors might suffer beyond the VaR. Moreover, it has also been criticized for not being a coherent risk measure because it is not subadditive by Artzner et.al in 1990. Nevertheless, CVaR always provides information on the potential large losses that an investor can suffer by Rockafellar and Uryasey. It satisfies the subadditivity and is therefore a coherent risk measure. Heyde et.al (2007), however, argued that requiring subadditivity may lead to risk measures that are not robust to the underlying models and data. Because VaR and CVaR both have advantages and disadvantages, it is often difficult to decide which one is better. Risk managers may consider both of them at same time to complement each other.There are typically three approaches to estimate VaR and CVaR:the variance-covariance approach, the historical simulation approach and the Monte Carlo simulation approach. Among the three, the Monte Carlo simulation approach is frequently used. However, the traditional Monte Carlo simulation approach is often time consuming and the low efficiency. Therefore, variance reduction techniques are often used to increase the efficiency of the estimation. Among all variance reduction techniques, important sampling (IS) is a nice choice, because it can allocate more samples to the tail of the distribution that is most relevant to the estimation of VaR and CVaR. There is most important to choice a distribution that guarantees to reduce the variances of the estimators. Furthermore, we show that the technique of exponential twisting guarantees to find a distribution that reduces the variances of the VaR and CVaR at the sameIn this paper, we choose AR model to estimate VaR and CVaR, and we using traditional Monte Carlo simulation approach and important sampling method to estimate the variances for the estimators of VaR and CVaR. We also drew the probability density function diagram for the estimators of VaR and CVaR. The simulation results show that important sampling method is indeed reduces the variances of VaR and CVaR, and increases the efficiency of the estimation. Especially in the confidence level is 0.05, the effect is more outstanding. Finally, we select the gold return rate as the empirical data. We use the ARMA model in fitting the random process. According the historical data to simulation the random number of the target time, used traditional Monte Carlo simulation approach and important sampling method to estimate VaR and CVaR. It is prove that important sampling method is more effective than traditional Monte Carlo simulation approach.