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This paper provides an exact and computable invariant currency value index (ICVI) which is independent of base currency choice. Thus, given a fixed set of currencies, the index of a currency will have the same value, regardless of base currency choice. This currency index can be used as an indicator to assess movements of an individual currency's value in world currency markets. The methodological and mathematical reasoning behind ICVI is formulated in terms of a simple exchange model (SIMEX). To demonstrate one possible application we employ ICVI to construct a currency basket of minimum variance. Utilizing a quadratic optimization framework, we compute optimal weights for currencies and construct a stable aggregate currency (SAC). Comparative empirical analyses of a five-currency SAC and the IMF's Special Drawing Rights (SDR) demonstrates that the SAC has lower volatility and lower correlations with its components than the SDR. In a similar way it is shown that a three-currency SAC has a smaller variance than the world money basket proposed by R. Mundell. Numerous academic and business implications are possible for further study with the use of the indices ICVI and SAC.

It is common practice in international economics and finance to denominate multiple currencies in terms of a base currency or numeraire. One problem with this multi-currency convention is that, depending on the base currency chosen, the resulting time series can dramatically change their dynamics due to fluctuations in currency values over time. For example, the relationship between the yen and pound sterling will be different if the dollar is used as the numeraire from the case when the euro is used as numeraire. This paper provides an exact and computable invariant currency value index (ICVI) which is independent of base currency choice. Thus, given a fixed set of currencies, the index of a currency will have the same value, regardless of base currency choice. As such, ICVIs for the dollar, euro, and yen are independent of a chosen base currency. The conception of the invariant index of a currency's value in exchange is based on a simple exchange model (SIMEX), which describes direct pair-wise exchanges of goods (commodities, services, currencies, etc.). We show that ICVIs could be used to better understand the valuation of currencies and other assets in a global context in which multiple currency participants exist. In this respect they could be used to gain insight into currency value questions such as: Did the U.S. dollar go up or down in world currency markets? To our knowledge, no other work has been published on this potentially valuable multi-currency index. ICVIs could be applied to a variety of empirical problems in international economics and finance. To demonstrate one possible application we employ our ICVI to solve for a minimum variance currency basket. Throughout history, economists have sought a stable numeraire (or benchmark commodity) for the purpose of international trade and finance. Utilizing a variance minimization framework, we compute optimal weights for five hard currencies and construct a stable aggregate currency (SAC). Comparative empirical analysis of SAC and the IMF's Special Drawing Rights (SDR) for the period 1981–1998 demonstrates the low volatility and low correlations with its components of this stable aggregate currency. We therefore conclude that the SAC could be used to harden the SDR. Following recent work by Nobel Laureate Robert Mundell, further analyses consider the problem of constructing a stable world currency. In Mundell's words, “A few economists have recently recognized the merits of and need for a world currency. Whether that can be achieved in the near future will depend on politics as well as economics. But it is, nevertheless, a project that would restore a needed coherence to the international monetary system.” (Mundell and Friedman, 2001, p. 27). A comparative analysis of Mundell's recommended world currency basket to our minimum variance currency basket constructed from the same currencies confirms the low volatility and correlations with its components of SAC. We conclude that the simplicity of SAC could be used to develop a world money that would be easy for businesses in the financial services industry to implement. Our invariant currency value index and stable aggregate currency have major implications to the construction of index numbers for contract settlement. Shiller (1993) has argued that the development of macro markets is dependent on the availability of generally agreed upon indices that are well suited to the settlement of contracts (see Shiller, 1993, p. 208). Price index methods have been applied to a variety of measurement problems, including inflation (i.e., consumer and produced price indices), stock prices (i.e., aggregate market indices), real estate prices (i.e., constant quality indices), etc. In the present paper, ICVI is an index of value that eliminates the problem of base currency. Also, SAC is an index method of constructing minimum variance currency baskets. We believe that these new indices contribute to the effort to create widely agreed upon measures of international currency for use in basket currencies, as well as international prices of financial assets, real goods, and services. In Section 2 a simple exchange model (SIMEX) is outlined. Section 3 derives our invariant currency value index (ICVI). Section 4 applies ICVI to solving for minimum variance currency baskets, which we refer to as a stable aggregate currency (SAC). Section 5 applies SAC to the problem of hardening the IMF's SDR. Section 6 extends the analyses to the problem of stable world money. Section 7 gives conclusions and implications. The Appendix contains further discussion.

We provide an exact and computable solution for the problem of an index (indica- tor) of a currency’s value in exchange, the index being independent of base currency choice. In brief, our invariant currency value index ( ICVI ) is computed by using the geometric mean of all currencies taken into account to normalize the currency’s value. Thus, given a 1xed set of currencies, the invariant index of a currency from the set will have the same value, regardless of base currency choice. The methodological and mathematical reasoning behind the invariant index of currency’s value in exchange is formulated in the framework of a simple exchange model (SIMEX). This model takes into account only pair-wise direct exchange of any two goods (e.g., commodities, ser- vices, currencies, etc.). To demonstrate one possible application we employ our ICVI to construct a min- imum variance currency basket ( MVCB ). Utilizing a variance minimization frame- work, we compute optimal weights for currencies and construct a stable aggregate currency ( SAC ) comprised of 1ve diJerent hard currencies (i.e., German mark, French franc, British pound, Japanese yen, and U.S. dollar). Comparative empirical analyses of our 1ve-currency SAC to the IMF’s Special Drawing Rights ( SDR ) for the period1981–1998 demonstrates that SAC has lower volatility than the SDR as well as its component currencies. Consistent with Adam Smith’s observation, the best numeraire for valuing commodities (and currencies) is a commodity (or currency) that is stable over time. In this regard, based on our analyses, we conclude that SAC could be used to harden the SDR . Additionally, SAC has potential implications to the development of a world currency as proposed by Nobel Laureate Professor Mundell and others. World currency propo- nents have primarily focused on the international policy and structural reforms needed to implement a basket currency, with little or no attention to the mathematical construc- tion of the basket currency index. SAC could be valuable to computational discussions in this area. As an example, we consider Professor Mundell’s recommended currency basket comprised of the dollar, euro, and yen. Comparative analysis of Mundell’s rec- ommended world currency basket to our minimum variance currency basket constructed from the same currencies con1rms the low volatility and correlation properties of SAC . We conclude that SAC could be used to develop a world money that would be easy for businesses in the 1nancial services industry to implement, as well as impersonal and not subject to political authority. Finally, as pointed out by an anonymous referee, a major implication of our in- variant currency value index and stable aggregate currency is their contribution to the construction of index numbers for contract settlement. As observed by Shiller (1993, p. 209), while consumer price indices, real estate valuation indices, stock price in- dices, and many others began as theoretical constructs of interest to a small audience of interested economists, they later became widely accepted by the public as common tools for establishing settlement claims in risk management contracts. Emphasizing the importance of indices, he commented that “A major barrier to the establishment of many macro markets today is a shortage of indices that are agreed upon as well suited to be the basis of cash settlement of contracts.” (Shiller, 1993, p. 208). In the present paper ICVI is an index of value that eliminates the problem of base currency. Also, SAC is a tool for constructing minimum variance currency baskets. We believe that these new indices contribute to the eJort to create widely agreed upon measures of international currency, as well as in analyses involving international prices of 1nancial assets, real goods, and services. We infer that optimal currency baskets have potential applications to a number of academic and practical issues in business, including exchange rate risk measurement, portfolio analysis of multi-currency assets, international capital budgeting decisions with multi-currency cash Iows, the practice of reporting constant currency accounting numbers on 1nancial statements, etc. Future research is recommended to investigate these and other potential applications as well as to increase our understanding of currency invariant indices and minimum variance currency basket indices.