극단치 분포와 시계열 변동성모형을 이용한 VaR추정

Estimation of value at risk based on extreme value distributions and time series volatility models

The risks which arise in financial market may include market risks, credit risks, liquidity risks, operational risks, and sometimes legal risks. Recently, Value at risk(VaR) model was developed to deal with one aspect of financial risk, market risk. In this paper, we use extreme value theory(EVT) to compute VaR since VaR is affected by extreme returns or spikes and EVT is particularly appropriate for measuring extreme risks. There are three types of extreme value distributions which are Frechet, Gumbel and Weibull Distribution. Based on extreme value distributions, we analyzed foreign exchange data in Korea. We regard foreign exchange returns follow a certain fat-tailed distribution. In order to compare the VaR values before IMF and after IMF in Korea, we divided the whole data into two groups, say data A for “befor IMF” and data B for “after IMF”. Based on the graphs of quantile functions and Q-Q plot for the data, we concluded that Gumbel distributrion is most adequate for the underlying distribution of the data. However, we found that the estimated VaR value based on Gumbel distribution seems to underestimate the true VaR value for data A in view of the empirical VaR value. In order to overcome such deficiency, we also considered the time series volatility model. Based on Box-Jenkins model identification method, we selected the two models such as MA(1)-GARCH(1,1) model for data A and AR(1)-GARCH(1,1) model for data B. From the VaR analysis based on the extreme value distributions and the time series volatility models, we regard that MA(1)-GARCH(1,1) model and Gumbel distribution are appropriate for data A and data B, respectively. The estimated VaR value based on the time series volatility model for data A were used as an optional model selection method, and we wish to identify the time series volatility model more clearly for further study.