In this study, we model the returns of a stock index using various parametric distribution models. There are four indices used in this study: HSCEI, KOSPI 200, S&P 500, and EURO STOXX 50. We applied 12 distributions to the data of these stock indices-Cauchy, Laplace, normal, Student's t, skew normal, skew Cauchy, skew Laplace, skew Student's t, hyperbolic, normal inverse Gaussian, variance gamma, and general hyperbolic-for the parametric distribution model. In order to choose the best-fit distribution for describing the stock index, we used the information criteria, goodness-of-fit test, and graphical tail test for each stock index. We estimated the value-at-risk (VaR), one of the most popular management concepts in the area of risk management, for the return of stock indices. Furthermore, we applied the parametric distributions to the risk analysis of equity-linked securities (ELS) as they are a very popular financial product on the Korean financial market. Relevant risk measures, such as VaR and conditional tail expectation, are calculated using various distributions. For calculating the risk measures, we used Monte Carlo simulations under the best-fit distribution. According to the empirical results, investing in ELS is more risky than investing in securities, and the risk measure of the ELS heavily depends on the type of security.
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