Virtual Standard Currency for Approximating Foreign Exchange Rates

ABSTRACT

In 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.

KEY WORDS AND PHRASES:

• Foreign exchange rate
• virtual standard currency
• virtual exchange rates
• wealth measures
• forex
• currency portfolio
• rank one approximation
• modified power method

Acknowledgment

We thank the editors and three referees for their insights and suggestions. The research of the first author was supported by the China Scholarship Council for him to visit the Department of Statistics at the University of Wisconsin–Madison. He expresses his deep gratitude to the department and its faculty, whose support is crucial for the successful completion of this project. He is also supported in part by the National Natural Science Foundation of China under Project 71031005/G0103. The research of the second author is supported by the U.S. National Science Foundation under Project DMS-1505367.

Supplementary Materials

The online appendix contains several supplementary tables, properties, and proofs, which are referred in the paper.

Notes

^1. Data are retrieved from the website (http://srh.bankofchina.com/search/whpj/search.jsp) of the market maker, Bank of China, at CST 11:35:38 am on February 15, 2016.

Additional information

Notes on contributors

Hongxuan Huang

HONGXUAN HUANG ([email protected]; corresponding author) is Associate Professor of Operations Research and Statistics in the Department of Industrial Engineering at Tsinghua University, Beijing, China. He holds a Ph.D. degree in Management Science and Engineering from Beihang University, China. He is the author of two books on mathematical programming. He is also a member of the editorial board of the Optimization Letters and the author of several book chapters, including Encyclopedia of Optimization, as well as of a number of papers in journals and conference proceedings. Dr. Huang has received several grants and consulted for a number of major corporations.

Zhengjun Zhang

ZHENGJUN ZHANG ([email protected]) is Full Professor of Statistics in the Department of Statistics at University of Wisconsin-Madison. He received his Ph.D. degrees in Management Engineering and Statistics from Beihang University and the University of North Carolina at Chapel Hill, respectively. Dr. Zhang’s main research areas include the Big Data structure and inference, particularly in extreme value analysis for interdependent critical risk variables in finance, climate, and medical sciences, in stochastic optimizations in large and complex systems. Some of his selected journal publications are Annals of Statistics, Journal of Royal Statistical Society, Series B, Journal of American Statistical Association, Journal of Econometrics, Journal of Banking and Finance, Extremes, and Automatica.