Value-at-Risk (VaR) is the most popular methodology in risk management because it is easy accessibility to common users. The importance of VaR is rapidly increasing because the international agreement in banking industry, the Basel Accord, uses the VaR methodology extensively. However, Bedder (1995) and Hendricks (1996) warned of limitations of the VaR approach in risk management; the VaR methodology requires distributional assumptions for the relevant risk factors. Moreover, the VaR estimate depends on not only the assets class constituting portfolio, but also the model used to estimate the volatility of those assets. In this regard, it is valuable to investigate which volatility model produces superior risk measurement for a given portfolio. In this study we seek to determine the best among various models in estimating 99% VaR and 99.5% VaR for the long and short position of a portfolio in the Korean stock market. Models are evaluated in terms of both the accuracy of probability and the independence of extreme events occurrence. We use the conditional coverage test proposed by Christoffersen (1998) to test those properties jointly. We compare five univariate models and three multivariate models using the hypothetical portfolio consisting of twenty stocks whose market values rank in the top 20th in the Korean stock market; the five univariate models are the Simple Moving Average (SMA) model, the Exponentially Weighted Moving Average (EWMA) model, the GARCH model with a normal distribution, the GARCH model with a t-distribution, and the Historical Simulation (HS) model. The three multivariate GARCH models are the Constant Conditional Correlation (CCC) model, the Dynamic Conditional Correlation (DCC) model, and the Orthogonal GARCH (O-GARCH) model. We can summarize our analysis of empirical results as follows. First, we find that the overall performances of multivariate models are better than those of univariate models in evaluating VaR for the our hypothetical portfolio. Second, the performance of the DCC model is better than that of the other multivariate models such as the CCC model and the O-GARCH model. However, the CCC model as well as the DCC model passes conditional coverage tests for many cases. Third, the SMA model and the HS model, the most commonly used models in Korean financial institutes, fail to pass the tests. This means those models are relatively less appropriate in evaluating VaR for a portfolio similar to our portfolio with respect to the conditional coverage perspective. This might be caused by the fact that those models fail to efficiently incorporate new information into the VaR evaluation. This paper is organized as follows: we review the VaR concepts, which is followed by a review of the various methods to evaluate VaR. After that, we move to the back-testing procedure for the model selection criteria. The empirical result of the models is presented and discussed. Then, we summarize and conclude our research results. This paper is different from existing studies in terms of the number of assets in the portfolio and models. Most of prior research has used a portfolio consisting of two, three, or at most, five stocks. However, in this paper we use a hypothetical portfolio consisting of 20 stocks to test the performances of the multivariate models. In addition, we compare the multivariate VaR methods with the univariate VaR methods in the literature. Especially, we apply the CCC model and the DCC model to evaluate VaR. Overall, the most distinctive point of this paper is the result of this comparison, which allows us to determine the value of conditional correlation estimation in this VaR application.
