Modeling and Risk Analysis Using Parametric Distributions with an Application in Equity-Linked Securities

Abstract

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.

Investment Modelling Using Value at Risk Bayesian Mixture Modelling Approach and Backtesting to Assess Stock Risk

“…The selection of MNAR and MLAR models was measured using DIC with those being the smallest DIC is considered the best. Table 3 presents the best model of MNAR, namely MNAR (2;[2], [2,4]); and MLAR, namely MLAR(2;[2], [4]). The DIC of those models are -3293.10 and -3698.88 respectively.…”

“…The DIC of those models are -3293.10 and -3698.88 respectively. The DIC of the MLAR(2;[2], [4]) model is smaller than the MNAR(2;[2], [2,4]); however, the isolated model cannot be said to be the best model for estimating VaR; instead, both models can be used to calculate VaR.…”

“…Value at Risk Modelling and Backtesting Using Kupiec Test Value at Risk (VaR) calculated using MNAR (2;[2], [2,4]) and MLAR(2;[2], [4]) approach can be seen in Table 4. The best VaR was obtained by performing the POF Kupiec test as shown in Table 5.…”

“…The Kupiec test used the null hypothesis where the = ̂ and the alternative hypothesis of ≠ . The result shows that VaR MLAR (2;[2], [4]) approach is accurate at 5% and 1% quantiles. To be precise, = ̂ because the LR value is less than .…”

“…VaR is generally estimated by assuming returns that have a normal distribution. However, this may not be appropriate because, in reality, return distribution is not always normal [4]. Some results of stock return visualization indicate that the data distribution is leptokurtic, which tends to resemble Laplace distribution.…”