Virtual Standard Currency for Approximating Foreign Exchange Rates

Abstract

ABSTRACTIn the global financial and business world, the wealth that an entity owns usually is composed of various assets or measured by different currencies, the value of which is dependent on the fluctuation of the foreign exchange rates. The analysis of dynamics associated with the foreign exchange rates is one of the important concerns to market strategists, financial planners, or risk managers such that a fundamental problem is induced as how to measure the wealth objectively. In this paper, a concept of the virtual standard currency (VSC) is proposed as a methodology to measure the wealth in a currency portfolio objectively. The VSC is regarded as a virtual base currency such that any real foreign exchange rate matrix is approximated by a rank one matrix consisting of two virtual exchange rate vectors. The existence of the VSC is proved through an optimal solution to the basic rank one approximation problem. The evaluation of wealth in a currency portfolio is free from the buying or selling operations in real currencies so that the currency portfolio is kept invariant during the measurement. The VSC can eliminate uncertainties arising from the choice of a specific real currency and the interactive effects across different kinds of currencies. Also, the modified power method is designed to search for the virtual exchange rates numerically, the convergence of which is also established. Furthermore, some practical examples are presented to verify the feasibility and efficiency of the modified power method in approximating a foreign exchange rate matrix.