The Value-at-Risk (VaR) measurements are widely applied to estimate exposure to market risks. The traditional methods to VaR computations do not provide satisfactory evaluation of possible losses. In this thesis we propose an approximation formula for the cumulative distribution of the weighted sum of mutually independent distributed random variables. This method is mainly based on the Fourier-cosine expansion, which was proposed to price European options without normality assumption by Fang and Oosterlee (2008). The error analysis on this method is done, and corresponding theoretical results are proved. We use the proposed formula to compute the VaR of a single asset and of a portfolio under the condition that the rates of returns are assumed to come from stable family. A series of numerical results based on empirical data are showed to support the efficiency and feasibility of this method.