Abstract

Risk measurement and optimization remains an essential aspect of stock market investment as the risk averse investor always seek to know the risk of investment. In this study, the value-at-risk (VaR) is estimated for bi-index stock portfolio investment. The VaR is estimated using bivariate copula models with Alpha power transformed logistic (APTL) marginals in comparison to normal, student’s t, Cauchy and logistic distributed marginals. The APTL distribution applied in this work for bi-index portfolio copula-VaR estimation has been used to estimate quantile value-at-risk for single stocks in the US stock markets but the bi-index portfolio case has not been considered. Copula is used to model the bi-index data. Copula bivariate APTL distribution and density functions that can be used to model different bivariate data sets in different fields like in economics and meteorology are derived using the Gaussian, Student’s t, Clayton, Gumbel and frank copulas. In estimating the copula-value-at-risk for bi-index stock portfolio, some of the world’s biggest stock indices are used. This includes the New York stock Exchange(NYSE), National Association of Securities Dealers Automated Quotations (NASDAQ) stock Exchange, Japan Stock exchange (JPX) and Shanghai stock exchange (SSE). The result of the copula model selection using AIC and BIC showed that the student’s t copula is the most appropriate copula to model the NYSE-NASDAQ and SSE-JPX portfolios. The copula-VaR estimates of the NYSE-NASDAQ and SSE-JPX portfolios were then obtained using the student’s t copula with APTL, APTL-logistic, normal, student’s t , Cauchy and logistic marginal at 99%, 99.5% and 99.9% confidence levels. The results obtained from the data analysis showed that the student’s t copula model with APTL marginal gives the minimum expected loss for equally weighted investment in the NYSE-NASDAQ and SSE-JPX portfolios at 99.5% and 99.9% confidence levels in comparison to the other marginals.