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Applications of the geometric mean to international banking data

Online Publication Date: 01 May 1990

Mathematical Education in Science and Technology,21:3,499 — 503

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INT. J. MATH. EDUC. SCI. TECHNOL., 1990, VOL. 21, NO. 3, 499-503

Department of Mathematics, Utica College of Syracuse University,

Applications of the geometric mean are relatively rare. However, in the field of

international banking this measure of central tendency has gained popularity in

recent years on account of its reciprocity property. Two simple examples are

discussed. The first involves the computation of an average of daily exchange

rates of a currency in a given period and the second relates to a widely used

calculation of a simple effective exchange rate index for a currency relative to a

base period.

The arithmetic mean is more important than the geometric mean (or harmonic

mean) in advanced statistical work because of its superior mathematical tractability

and certain sampling properties. In addition, it is generally easier to find applications

example, in the determination of the average rate of growth of a population over a

given time period. However, in the field of international banking, the desirability for

the use of a consistent exchange rate average provides for elegant applications of the

geometric mean.

The basic reciprocity property of a mean (required for consistency reasons) is

discussed in general terms, followed by two simple examples relating to the

computation of

(i) The average of daily exchange rates for a currency over a given period,

(ii) An effective exchange rate index for a currency relative to a base period.

The reciprocity property desired of a mean for exchange rate computations may

currency JF.

such that

M,M,= 1 (1)

means of the a;-variate so that

respectively. It is then easy to show that only the geometric mean possesses the

reciprocity property, i.e.

For the arithmetic mean and harmonic mean, the relationships can be given in the

for any positive-valued variate [1]. Thus

Information about the daily fluctuations of the official exchange rate of a home

is valuable to the international banking community, businessmen, researchers,

potential investors and traders. This information is usually made available by the

central or reserve bank of the country concerned either at the end of a period (day,

week, month, etc.) rate or as an average rate for a period (week, month, quarter, etc.).

rate of the standard currency (say, U.S. dollar) in terms of the home currency, i.e.

The use of the geometric mean would ensure that on average, using the notation

price of a unit of foreign currency in terms of the domestic currency (or vice versa).

In the conduct of exchange rate policy, a country might wish to determine whether

its currency is, in a certain sense, overvalued or undervalued. If it is overvalued, the

price of a unit of foreign currency is relatively low and this could lead to excessive

imports and have an adverse impact on the foreign exchange reserves of the country.

If the currency is undervalued, the price of a unit of foreign currency is relatively

high, and this may tend to discourage imports of needed raw materials and

machinery and slow the growth of the economy.

As a measure of the extent of appreciation or depreciation of a domestic currency

depending on the objectives of the monetary authorities [2]. A commonly used

nominal exchange rate index is one that measures the movements of the currencies of

all the trading partners of a country relative to a base period, taking into account the

weight of each currency as reflected by the volume of trade conductd in each foreign

currency.

currency at time / in terms of currency i. The two exchange rates are linked by the

equality

either the #-variate or they-variate. The geometric (GEER) and arithmetic (AEER)

effective exchange rate indices are defined as follows:

to verify that

so that only the geometric nominal effective exchange rate index possesses, in

general, the reciprocity property defined by (1). As shown by (3)

In practice, indices are typically computed by using representative monthly,

the corresponding base period averages.

1986. (A good source of actual data on exchange rates is the International Financial

Statistics, a monthly publication of the International Monetary Fund, based in

Washingron, D.C.)

Assume for simplicity that the country trades only in two foreign currencies Fl

and F2 and that the weights of these two currencies in 1986 are given by Wj = 1/4 and

attributable to each currency in total 1986 trade transactions. Suppose further that

exchange rates averaged for the year in 1986 and 1987 are as follows:

1986 1987

Then computation of the geometric and arithmetic indices yields

^ 65-80 GEERj,= 151-97

The value of 65-80 for GEER^ can be interpreted to indicate in 1987 a depreciation

consistent, on account of their reciprocity property.

the foreign currency in 1987 = 0-658 times the domestic currency value of 1 unit of

the foreign currency in 1986.

value of 1 unit of the domestic currency in 1987 = (1/0-658) times the foreign

currency value of 1 unit of the domestic currency in 1986.

period. Movements in nominal rates do not however imply anything about the

purchasing power of a currency, nor do they indicate the degree to which the

competitiveness of a country's tradable goods may have changed over time. For

example, suppose that a particular country's currency has remained unchanged in

partners relative to a base period. Then one unit of the country's currency will buy

relatively more in its trading partner countries compared to the base period, and the

country's export goods will have become more expensive relative to the export

goods' prices of its trading partners. To measure the extent to which the purchasing

power of a currency has changed over time, it is necessary to calculate a real effective

exchange rate index. This can be done by deflating the nominal effective exchange

rate index by a measure of relative price movements [3]. A geometic real effective

exchange rate index (GREER) for a domestic currency can be defined by computing

a weighted average of the deflated partner country exchange rates as follows:

The international banking applications discussed here would hopefully provide

additional motivation for the study of the geometric mean. The presentation in a

classroom can be brief if attention is focused on some specific numerical examples.

(New York: Hafner), pp. 36-37.

major conceptual and methodological issues. IMF Staff Papers, September, Vol. 30.