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Browsing Mathematics by Subject "ARIMA-GARCH"
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Item Open Access APPLICATION OF 4-DIMENSIONAL COPULAS IN CALCULATING VALUE-AT-RISK FOR THE PORTFOLIO OF 4 SP500 COMPANIES(School of Sciences and Humanities, 2023) Bolatbekov, KairzhanPortfolio risk management is a process aimed at maintaining profit streams and reducing uncertainties in investment decisions. Value-at-Risk (VaR) is a widely used metric to quantify the potential loss of profits. Although historical simulations and Gaussian distribution are common methods for estimating VaR, modelling the joint multivariate distribution of portfolio investments can be challenging. Copula models offer a solution to these challenges for joint distributions. In this study, we calculated VaR and Conditional Value-at-Risk (CVaR) for a portfolio consisting of the four least correlated stocks among the 15 largest companies in the SP 500 using historical simulations and copula models. We evaluated portfolio based on equal weighting. The optimal ARIMA-GARCH model was selected using Akaike Information Criteria (AIC) values. Furthermore, the performance of the VaR estimations was compared and analyzed using goodness-of-fit tests.Item Open Access CALCULATIONS OF VALUE AT RISK FOR THE PORTFOLIO OF 5 S&P 500 STOCKS USING 5-DIMENSIONAL COPULA FUNCTIONS(Nazarbayev University School of Sciences and Humanities, 2024-04-19) Kakenov, AssanaliThe purpose of this project is to compare copula estimations of Value at Risk (VaR) for a portfolio of 5 S&P 500 stocks to historical, normal distribution, and Monte Carlo methods employing dependence measures and ARIMA-GARCH time series models. This study will provide interpretations of financial data between 2019-2024 in a scope of 5 equations: Gaussian, Clayton, t-Copula, Gumbel, and Frank copulas. Correlations between closing prices of the largest 30 S&P 500 companies by market capitalization were calculated, and the portfolio was constructed by selecting 5 stocks with the least average correlation. The Markowitz portfolio optimization model was utilized to estimate the weights of the assets in the portfolio. Log returns, skewness, kurtosis, Shapiro-Wilk, and ADF were measured to describe stationarity and normality of the data. Data autocorrelation was assessed using ACF and PACF for volatility before ARIMA-GARCH modeling. All methodology was followed by appropriate hypothesis tests. Finally, 5-dimensional copulas were used for the VaR estimations for different confidence intervals. While AIC and BIC showed that t-copula was the best fit, the Clayton copula passed the goodness-of-fit test with the largest p-value. Subsequently, the Clayton copula generated VaR estimations closest to the historical data. The method used in this study can be extended for more than five assets without theoretical obstacles.