GLOBAL FINANCIAL REPORTING USING A COMPOSITE CURRENCY:
AN AGGREGATION THEORY PERSPECTIVE
Yuji Ijiri1
Carnegie Mellon University
January 1994
1. Need for a Composite Currency
In foreign currency translation, use of the currency of the parent company's
home country is most common. This is proper only as long as the company's
foreign investments are temporary.
Many multinational corporations invest in foreign countries with intent
to stay in the countries semi-permanently. Truly "global" corporations operate
worldwide with shareholders, employees, customers, and suppliers, spread all
over the world. For these global corporations, use of the home country's currency
in foreign currency translation does not make sense. A simple illustration
will suffice to show the point.
Two global corporations, Corporation X headquartered in Country A and Corporation
Y headquartered in Country B, hold an identical set of assets, let us say,
A$100 (million omitted) of Country A's currency and B$100 (million omitted)
of Country B's currency at the beginning of a period. The two corporations'
assets did not change at all during the period, but the exchange rate between
the A-dollar and the B-dollar changed from A$1 = B$1 at the beginning to A$1
= B$2 at the end. X reports the beginning assets of A$200 and the ending assets
of A$150 (as B$100 is now worth only A$50), or a loss of A$50. Y reports the
beginning assets of B$200 and the ending assets of B$300 (as A$100 is now
worth B$200), or a gain of B$100. The two economically identical companies
report differently, often drastically differently, just because they are headquartered
in different countries.
Use of a single currency cannot solve the difficulty whether it is domestic,
foreign in which investment is made, or an independent third country currency.2
A composite currency can solve or at least mitigate the above difficulty.
The use of a composite currency in foreign currency translation and, more
generally, for financial reporting of global corporations has been reported.
3 However, such a use is very much limited in the past and
almost exclusively for internal use only. It is quite possible that in the
future as such composite currencies as European Currency Unit (ECU)4
or Special Drawing Rights (SDR)5 become popular, the use
of a composite currency as a reporting currency may become widely accepted
for global corporations.
As an aggregate of multiple currencies, use of a composite currency
requires some understanding of theory of aggregation. This paper relates practical
issues of using a composite currency for financial reporting purposes with
theoretical constructs of aggregation theory. Its aim is to offer key points
one should be concerned with in the actual use of composite currencies based
on some theoretical results.6
2. Numerical Illustrations
Example 1: Let us use the above example involving the A-dollar and
the B-dollar with the exchange rate of A$1 = B$1 at the beginning of the year
and A$1 = B$2 at the end of the year. Let us also consider that there is a
composite currency, a global-dollar or G-dollar, which consists of A$1 and
B$1, i.e.:
(1) G$1 = (A$1, B$1).
Any one tendering A$x and B$x can get G$x and vice versa.7
The composition of G$1 remains unchanged regardless of the fluctuation in
the exchange rate between the two currencies. A company, like X or Y in the
above example, holds a portfolio of currencies, (A$100, B$100), which is denoted
by H:
(2) H = (A$100, B$100)
The holding remained unchanged throughout the year.8
However, due to the change in the exchange rate, the following loss, gain,
or none is reported depending upon which of A-dollar, B-dollar, or G-dollar
is used as the reporting currency. This is shown in Table 1 and Figure 1 below.
Note that, at the end of the year, B$100 is worth A$50 and A$100 is worth
B$200, as mentioned before, and that the value of the holding in the global
currency is unchanged because the holding can be tendered for an exchange
with G$100 at the beginning of the year as well as at the end of the year.
At the beginning of the year, the exchange rate was A$1 = B$1, hence G$1
= A$2 = B$2. At the end of the year, the exchange rate was A$1 = B$2, hence
G$1 = A$1.50 = B$3. To see the latter point, at the exchange rate of A$1 =
B$2, the holder of G$1 can exchange it for A$1 and B$1 and sell the B$1 for
A$0.5, obtaining A$1.50 in total. Hence, G$1 = A$1.50. Likewise, the holder
can obtain A$1 and B$1 and then sell the A$1 for B$2, obtaining B$3 in total.
Hence, G$1 = B$3. The exchange rates is at equilibrium when G$1 = A$1.50 and
G$1 = B$3.
X reports a loss of A$50 because its foreign assets lost value from the
viewpoint of the A-dollar. Similarly, Y reports a gain of B$100 because its
foreign assets gained value from the viewpoint of the B-dollar. These measurements
make sense if X and Y intend to repatriate all investment back to their respective
home country in the near future. If not, the gain or loss is deceiving. More
specifically, if it is the objective of X or Y or both to remain invested
in both countries at the ratio of, say, 50-50, then there should be no gains
or losses from an exchange rate fluctuation between the two currencies because
a gain in one is offset by a loss in the other.9 A measurement
using G-dollar reports the no-gain, no-loss situation correctly, since a gain
in one currency offset equally by a loss in the other currency.
X reports a loss of A$50 but is compensated by the fact that it is now
in the unit of the A-dollar, a strengthened currency relative to the B-dollar.
Y reports a gain of B$100 but is compensated by the fact that it is now in
the unit of the B-dollar, a weakened currency relative to the A-dollar. The
composite currency has neither strengthened nor weakened.
Figure 1 below depicts the three measurements in the A-dollar, B-dollar,
and G-dollar of the portfolio of assets H = (A$100, B$100) both at the beginning
of the year (BOY) when the exchange rate was A$1 = B$1 and at the end of the
year (EOY) when the exchange rate was A$1 = B$2.
Table 1: Valuation of (A$100, B$100) in A$, B$, and G$
Reporting Currency
Holding A-dollar B-dollar G-dollar
Beg of year A$1 = B$1 (A$100, B$100) A$200 B$200 G$100
End of year A$1 = B$2 (A$100, B$100) A$150 B$300 G$100
Gain or loss A$50 loss B$100 gain None
Figure 1: Valuation of (A$100, B$100) in A$, B$, and G$
In Figure 1, there are two "iso-value lines" are drawn, one for the exchange
rate at the beginning of the year and the other for the exchange rate at the
end of the year. An iso-value line means that, at the given exchange rate,
any portfolio of currencies lying on the line has an equal value with any
others lying on the same line. Any of (A$100, B$100), (A$150, B$50), or (A$50,
B$150) lies on the line connecting A$200 BOY and B$200 BOY, hence they all
have the same value at the beginning exchange rate of A$1 = B$1. Any of (A$100,
B$100), (A$150, B$0), or (A$50, B$200) lies on the line connect ing A$150
EOY and B$300 EOY, hence they all have the same value at the ending exchange
rate of A$1 = B$2.
Note that there is no gains or losses for the year when measured in G-dollar
because both iso-value lines cut across the G-dollar line at exactly the same
spot valued at G$100. On the other hand, measured in A$ (X-axis) or in B$
(Y-axis), the two iso-value lines cross the axes at different points, creating
a A$50 loss in the A$ measurement and a B$100 gain in the B$ measurement.
Generally speaking, a performance measurement must be congruent with the
objective of the business; hence whether it is a good performance measure
or not must be judged in relation to the objective. If it is the objective
of the company to repatriate all foreign investments into Country A, the A-dollar
should be used as the reporting currency. If it is the objective of the company
to repatriate all foreign investment into Country B, the B-dollar should be
used as the reporting currency. Then, it logically follows that if the objective
is to maintain investments in both countries at the 50-50 ratio, then the
G-dollar should be used as the reporting currency.
3. Perfect and Imperfect Reporting Currencies
In the above example, the mix of currencies held was in exact proportion
as in the portfolio of currencies used in defining the G-dollar. In the next
example, let us consider one in which this is not the case.
Example 2: All the details in Example 2 are exactly the same as
those in Example 1 above, except that the portfolio of currencies held is
changed from H = (A$100, B$100) as in Example 1 to:
(3) H' = (A$160, B$40).
The corporate objective is, as before, to maintain a 50-50 investment
portfolio in both countries. The new valuations using the three currencies,
A$, B$, and G$, are shown in Table 2 and Figure 2 below. In addition, both
in Table 2 and Figure 2, a new global currency G'-dollar is added. While G$1
= (A$1, B$1), the new composite currency G'$1 consists of:
(4) G'$1 = (A$1.00, B$0.25).
The discussion of this composite currency will be introduced shortly.
Now for the new holding H' = (A$160, B$40), a loss of A$20 is reported
under the A-dollar reporting currency. Under the objective of ultimately recovering
all investments in A-dollars, the year-end holding can provide only A$180
(A$160 plus A$20 from exchanging B$40 at the year-end exchange rate of A$1
= B$2), whereas at the beginning the company could have recovered A$200 (A$160
plus A$40 from exchanging B$40 at the year-beginning exchange rate of A$1
= B$1). Similarly, for the new holding, a gain of B$160 is reported under
the B-dollar reporting currency which can be likewise verified.
The same approach can be used in understanding the meaning of the G$20
gain under the G-dollar reporting currency. It is the objective of the company
is to end up with a 50-50 investment portfolio even if actual holding may
depart from it temporarily. Hence, at the beginning of the year, the holding
of (A$160, B$40) is valued at G$100. This is because a 50-50 portfolio of
G$100 = (A$100, B$100) is obtainable from the holding of (A$160, B$40) at
the prevailing exchange rate of A$1 = B$1 by selling A$60 for B$60. At the
end of the year, due to the new exchange rate of A$1 = B$2, the same holding
of (A$160, B$40) can now generate G$120 = (A$120, B$120) by exchanging A$40
for B$80.
Under the new G'-dollar currency, however, no gains or losses are reported.
As shown in Figure 2, both iso-value lines (BOY and EOY) path through a point
on the G'-dollar line. Hence, the same total value is reported at the beginning
and at the end of the year, resulting in no gains or losses.
Table 2: Valuation of (A$160, B$40) in A$, B$, G$, and G'$
Reporting Currency
Holding A-dollar B-dollar G-dollar G'-dollar
BOY A$1=B$1 (A$160, B$40) A$200 B$200 G$100 G'$160
EOY A$1=B$2 (A$160, B$40) A$180 B$360 G$120 G'$160
Gain or loss A$20 loss B$160 gain G$20 gain None
Figure 2: Valuation of (A$160, B$40) in A$, B$, G$, and G'$
This means that if it is the objective of the company to maintain a portfolio
of assets between the two countries at the ratio of 80-20, then the G'-dollar
is the right reporting currency to use. But if the objective is to maintain
a 50-50 ratio, the G'-dollar distorts the improvement in the portfolio.
A general principle may now become evident. For an objective of a 50-50
portfolio, the G-dollar is a "perfect reporting currency," as it informs the
user exactly the progress made along the 50-50 line of the portfolio mix.
When an actual holding departs from this line of growth, the departure is
eliminated by a hypothetical exchange of currencies using the exchange rate
then prevailing. With respect to this objective, the G'-dollar is an "imperfect
reporting currency," as it can distort the performance under some conditions.
The notion of perfect reporting currency is an example of a perfect aggregation
defined in the aggregation theory. A perfect reporting currency is not unique,
since G$1 = (A$1.00, B$1.00) or (A$10.00, B$10.00) or (A$0.50, B$0.50) can
all share the same property of a perfect reporting currency when the objective
is to maintain a 50-50 portfolio of investments.
The composite currency G'$1 = (A$1.00, B$0.25) is not perfect with respect
to the 50-50 portfolio objective (although it is perfect with respect to the
80-20 portfolio objective). However, not all imperfect composite currencies
are alike as some are closer to being perfect than others. Figure 2 provides
a hint that, for a composite currency G-dollar to be perfect, the angle of
the G-dollar line must coincide with the angle of the objective portfolio
line and that the closeness of the angles of the two lines may be an indication
of the composite currency close to being perfect. This is indeed true.
The degree of imperfectness can be defined and measured by the so-called
aggregation coefficient10 r, which is, for a two-currency
case with the composition of the two currencies being C = (1, Q) and C' =
(1, Q'), measured by:
(5)
with 0 < p <= 1, when Q > 0 and Q' >= 0, and with p = 1 (or p
= -1, when liabilities are introduced, in which case -1 <= p <= 1) indicating
a perfect aggregation.
In Table 2, between the G-dollar, which is (A$1, B$1) or Q = 1, and the
G'-dollar, which is (A$1, B$1/4) or Q' = 1/4, p is 0.8575 and between the
G-dollar and the A-dollar, which is (A$1, B$0) or Q' = 0, it 0.7071. The aggregation
coefficient indicates that while the G'-dollar is not perfect with respect
to the 50-50 growth line (the G-dollar line), it is "better" than the A-dollar
as the aggregation coefficient for G' is closer to 1 than that for A.
If we represent by q the angle between the G-dollar line and the G'-dollar
line, r may be written as11:
(6)
In terms of the degree of closeness to being perfect, p^2, the aggregation
effectiveness coefficient is most useful. Here, p^2 between the G-dollar line
and the G'-dollar line is 25/34 = 0.7353 and that between the G-dollar line
and the A-dollar line is 1/2 = 0.5000. This is because p^2 indicates the fraction
of the sum of squared errors that is eliminated by using the surrogate aggregation
-- nearly 3/4 of the sum of squared errors are eliminated when the G'-dollar
is used while only 1/2 is eliminated when the A-dollar is used. A more general
discussion will follow shortly. However, some further illustration using above
numerical examples may clarify the matter before general formulations are
discussed.
For any asset holding H = (hA, hB), we may consider a "rate vector," RZ
= (rA, rB) such that RZH* = (rA, rB)(hA, hB)* = rAhA + rBhB, where Z represents
the particular reporting currency and * is the transpose of the vector, represents
the total value of the holding using the prevailing ex- change rate. Table
3 below shows the rate vectors for all four reporting cur- rencies, A$, B$,
G$, and GÕ$, using exchange rates at the beginning of the year, at
the end of the year, and for the net change during the year.
Table 3: Rate Vectors for Four Reporting Currencies
Reporting Currency Z = A-dollar B-dollar G-dollar G'-dollar
Composition of Unit (A$, B$) (1, 0) (0, 1) (1, 1) (1, 0.25)
Rate Vectors RZ = (A$, B$)
Beginning of Year (A$1=B$1) (1, 1) (1, 1) (1/2, 1/2) (4/5, 4/5)
End of Year (A$1 = B$2) (1, 1/2) (2, 1) (2/3, 1/3) (8/9, 4/9)
Net Change During Year (0, -1/2) (1, 0) (1/6, -1/6) (4/45,-16/45)
Composition of Unit shows the content of each reporting currency, A-dollar
and B-dollar consisting each one unit of the respective currency, G-dollar
consisting of A$1 and B$1, and G'-dollar consisting of A$1 and B$0.25.
Components of rate vectors show the value of A$1 and the value of B$1,
respectively, expressed in the unit of each reporting currency, value being
determined by the prevailing exchange rate shown on the left. At the beginning
of the year, A$1 = B$1, hence, (1, 1) in both the A-dollar and B-dollar columns,
while A$1 = B$1 = G$0.5, hence, (1/2, 1/2) in the G-dollar column and A$1
= B$1 = G'$0.8 (since G'$1 can be converted into A$1.25 or B$1.25), hence
(4/5, 4/5) in the G'-dollar column.
At the end of the year, A$1 = B$2, hence, (1, 1/2) and (2, 1) in the A-dollar
and the B-dollar columns. Similarly, A$1 = B$2 = G$1.5, hence, (2/3, 1/3)
in the G-dollar column. Finally, A$1 = B$2 = G$1.125, hence (8/9, 4/9) in
the G'-dollar column. The last row was obtained by subtracting the beginning
of the year row from the end of the year row.
It may be seen that Tables 1 and 2 are both obtained when the asset holding
H = (A$100, B$100) for Table 1 or H' = (A$160, B$40) for Table 2 is multiplied
by the rate vectors in Table 3. For example, Table 1, Beginning of Year, A-dollar:
A$200 = (A$100, B$100)(1, 1)* and Table 2, Gain, G-dollar: G$20 gain = (A$160,
B40)(1/6, -1/6)* = G$120/6.
An appendix to this paper shows how this notion of "close to being perfect"
can be quantized under a general formulation of aggregation so that its mathematical
property can be understood more fully.
4. Implementation Issues and Accounting Policy Implications12
The fact that performance measurement depends upon the objective of the
corporation may sound odd at first, but it would not take much time for us
to realize that this is the nature of any performance evaluation. This is
because performance measurement is fundamentally a measurement that reflects
the progress toward a given objective. If the objective changes, the measurement
should change as well.
It is not only proper but also necessary for the corporation to change
the composition of the reporting currency as its global objective changes.
For reporting purposes, changes in the reporting currency should be disclosed
with suitable explanations in sufficient details to enable investors to make
reconciliations in a manner analogous to accounting changes.13
It would be idea for a corporation to develop its own reporting currency
for its internal performance evaluation.14 For external
reporting purposes, however, such a highly individualized reporting currency
might reduce comparability of financial data across global corporations. Here,
several major composite currencies might be developed as standard, allowing
corporations to choose one among them that best fits their investment objective.
Choice among them must be dictated at least in part by the degree of their
closeness to being perfect in identifying where on the company's growth path
the company's actual financial position is. Even this limited choice would
be far better than the existing practice of forcing the corporation to use
the home currency, which implicitly assumes eventual repatriation of all foreign
investments in the future.
While the development of a composite currency will be necessary to properly
express financial data of global corporations, the analysis presented in the
paper suggests that, in the interim, efforts should also be directed toward
unbundling of data when there was a major shift in exchange rates. If the
exchange rates used are all at the same point in time, the users of financial
data can accurately concert from one reported currency to another. It is when
exchange rates at more than one point in time are used in aggregating data,
such as the net income in the examples in this paper involving the beginning-
and the end-of-year rates, that creates problems for users.
In conclusion, a choice of a reporting currency is an important accounting
issue. For a global corporation which invests permanently in multiple countries,
a reporting currency should not be the currency of its home country but should
be a composite currency. Furthermore, a composite currency should be chosen
among alternatives in such a way that the composition of individual currencies
in the portfolio of the composite currency closely approximates the target
asset holding of the corporation in the respective country. In this way, the
investors are accurately informed of the progress made by the corporation
along the growth line dictated by its objective.
It is highly recommended that the International Accounting Standards Committee
examine the reporting currency issue for global corporations. Global financial
statements have three key ingredients -- language, currency, and standards.
So far, the IASC's efforts are directed solely toward harmonizaiton of accounting
standards. Such efforts can be fruitfully directed towards harmonization of
reporting currency and, eventually, toward harmonizaiton of language as well.
Appendix. Linear Aggregation Coefficient
In this appendix, we analyze the essential points derived by numerical
examples in the main text under a more general formulation. In general, the
total value V of an asset holding is the sum of products of prices p = (p1,
p2, ..., pn) and quantities q = (q1, q2, ..., qn), i.e.:
(7)
where * is, as before, for transpose. An aggregation problem occurs
because V is not available but only its surrogate V' aggregated by using a
different set of prices p' = (p1', p2', ..., pn'):
(8)
The question is how useful V' is in determining V.15
If qi's are independent random variables with mean q-i and variance si^2 (i
= 1, 2, ..., n), then an estimate of V, V^, determined by the following will
be the best in the sense that it minimizes the expected value of squared errors.
(9)
, where
(10)
(11)
where all summations go from i = 1 to n. Without information on V',
the best estimate that is available is:
(12)
The use of V' allows the user to reduce the sum of squared errors by
the fraction p^2, a quantity called the aggregation effectiveness coefficient.
Here p, the linear aggregation coefficient is:
(13)
and when all si's are equal, (13) reduces to:
(14)
where, as before, all summations go from i = 1 to n.16,17
In the case of currency translation, each asset expressed in a monetary
amount, si = piqi, is further multiplied by the exchange rate ri, namely the
translated total value W is the inner product of r = (r1, r2, ..., rn) and
s = (s1, s2, ..., sn):
(15)
Here, exchange rates in r = (r1, r2, ..., rn) are stated as the value of
a unit of currency i measured in the reporting currency as shown in Table
3, for example. As may be observed in this table, for any given point in time
at which exchange rates are all determined uniquely, r depends upon the re-
porting currency only up to a scalar multiple. Hence, knowing r and the composition
of reporting currencies, it is possible to identify W from a sur- rogate total
value W' by rescaling it. Gain or loss must, however, be deter- mined by first
identifying the total value of the asset holding at two different points in
time and then making a subtraction to determine the net change.
While this is the case when exchange rates are all unique at any given
point in time, this is not always the case in accounting records and reports
since exchange rates used in foreign currency translation need not be limited
to prevailing rates at the end of the year. Average of the year rates as well
as historical rates are also used. Rates may be updated monthly or even daily.
A mixed use of the current rate method and the temporal method further complicates
the matter further.18 The breakdown of assets holding by
the types of currency they are linked to is normally not disclosed.19
In the aggregation theory, the exact forms of the two aggregation functions,
one being a principal and the other, a surrogate, are assumed to be known.
This knowledge is used in determining V from V' or W from W' without knowing
the actual value of the variables q or s. The complexity of exchange rates
and the way they are mixed in the accounting process makes it difficulty to
identify the two aggregation functions. However, a choice of a good reporting
currency improves the situation.
Note that because of symmetry between price and quantity, (7) may be viewed
as defining a function p(q) which takes q as its argument or defining a function
q(p) which takes p as its argument. Likewise, (15) may be viewed as defining
(i) a function r(s) which takes s as its argument or defining (ii) a function
s(r) which takes r as its argument.
For the current problem of choosing a composite currency, the problem may
be viewed as the issue of congruence between two functions s(r) and s'(r),
both functions taking exchange rates as their argument. Then, by choosing
a reporting currency whose currency components, s', are closer to being proportional
to the company's asset holding, s, the aggregation is improved regardless
of how exchange rates fluctuates.
5. References
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Rates, and the Problem of Foreign Currency Translation: An Analytical-Empirical
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Cooper, John. "SDR as an Artificial Currency Unit," Accountancy
(England), 99, May 1987, pp. 72-4.
Edison, H.J. "Is the ECU an Optimal Currency Basket?" International
Finance Discussion Papers #282, Board of Governors of the Federal Reserve
System, Washington, DC, 20551.
Ijiri, Yuji. "The Linear Aggregation Coefficient as the Dual of the
Linear Correlation Coefficient," Econometrica, 36(2), April 1968, pp.
252-59.
Ijiri, Yuji. "Fundamental Queries in the Aggregation Theory," Journal
of American Statistical Association, December 1971, pp. 766-82.
Ijiri, Yuji. Theory of Accounting Measurement. Sarasota, FL:
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Ijiri, Yuji. "Foreign Currency Accounting and Its Transition," in R.J.
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Kirsch, Robert J. and Wayne Johnson. "The Impact of Fluctuating Exchange
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of, Foreign Subsidiaries," International Journal of Accounting, 26(3),
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Klein, Martin and Sigrid M. MYller. "ECU Interest Rates and ECU Basket
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Mueller, Gerhard G., Helen Gernon, and Gary Meek, Accounting: An
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6. Footnotes
1. This paper was prepared under a grant from Carnegie Bosch
Institute on an international accounting project, which is hereby gratefully
acknowledged. The author is Robert M. Trueblood University Professor of Accounting
and Economics, Graduate School of Industrial Administration, Carnegie Mellon
University, Pittsburgh, PA 15213. Helpful comments from Sok-Hyon Kang and Steve
Sung are gratefully acknowledged.
2. The last alternative, the use of a third-country
currency, say, C-dollars, for all corporations, avoids the problem of two
identical set of assets having been differently translated. However, it has
a conceptual problem of what the translated amount means when the corporation
has nothing to do with the third country.
3. Royal Dutch/Shell, as a part of their efforts to
develop "world" accounting principles on their own, is using, for financial
statements for internal use, a composite currency consisting of a dozen currencies
weighted by the amount of Shell's sales in each country. See Mueller, Gernon,
Meek (1991).
4. See, for example, Edison (1986) and Klein and MYller
(1992) for discussions on ECU as a currency basket and related issues on its
composition.
5. See Cooper (1987) on the use of SDR as a currency
unit.
6. See Ijiri (1981) for a comprehensive discussion
on the historical transitions of accounting standards on foreign currency
translation and the fundamental structure of foreign currency translation
problems.
7. This is the same as a portfolio of Standard &
Poor 500 stocks in the index options trading. Not only it can be bought or
sold in the market, but a share of S&P 500 can be exchanged for a basket
of comparable shares of its components stocks and vice versa.
8. Although the asset holdings are here all currencies
for ease of illustration, types of assets are not limited to currencies but
can include any types of assets and even liabilities which will be shown in
negative values. More details will be discussed in a later section of this
paper when general aggregation formulations are discussed.
9. To be strictly correct, a continual shifting of
assets from one country to another is needed so as to maintain a 50-50 portfolio
as the value of the assets changes. The statement in the main text is presented
as a simplification. Although a measurement in G-dollar is always G$100 as
long as it is defined as G$1 = (A$1, B$1), the G-dollar begins to lose its
suitability as an aggregation method for the two companies' asset holding
if one currency continues to deteriorate its value against the other. A discussion
on this subject must, however, be postponed until the next section which introduces
the so-called aggregation error and its magnitude.
10. See Ijiri (1968) for a discussion of the linear
aggregation coefficient which is viewed as the dual of the linear correlation
coefficient by reversing the role of variables and functionals.
11. This may be seen from the fact that, defining
a as the angle between the G-dollar line and the A-dollar line and b as the
angle between the G'-dollar line and the A-dollar line, and using tan a =
Q and tan b = Q', we have tan q = tan(a - b) = (tan a - tan b)/(1 + tan a
tan b) = (Q - Q')/(1 + QQ') which may be substituted in (6) to obtain (5).
12. This section was added based on the suggestion
to add the development issues of the composite currency by two anonymous referees
of this Journal, to whom the author expresses appreciation.
13. Changes in the functional currency of a foreign
subsidiary might have similar characteristics to changes in the reporting
currency but the latter, of course, has an impact on all units of the corporation,
not just a subsidiary.
14. The idea of using the company name as the unit
of the reporting currency of the company was suggested to the author by Mr.
Keiichi Komiya, Senior Executive Vice President of Toshiba Corporation. For
example, if Toshiba corporation develops its own composite currency, it may
be called "Toshiba." The Corporation may report its income as, say, T500,000,000
or 5 million Toshiba. Here, one Toshiba is a basket of fixed amounts of several
major currencies. The use of the company name in the reporting currency seems
to reflect the company-specific nature of the currency well and connotes the
unit of achievement toward the company objective.
15. See Theil (1954) for the early comprehensive discussions
of linear aggregation. See also Rosenblatt (1960). See Ijiri (1968) and (1975)
for a general formulation of aggregation problems, use of generalized inverses
--Penrose (1954)--for linear aggregations, and their applications to accounting.
See also Ijiri (1971) for a survey of over 100 articles in economics and other
social science fields classifying them based on the fundamental queries raised
in the articles.
16. This formula was used in (6) for illustration
using the examples introduced earlier. (Note that each coefficient pi can
be adjusted by multiplying it by si, to use (14) for cases where variances
are known but not all equal.) Since coefficients and variables are stated
most generally, they may assume negative values. In the case of asset holdings,
negative values may be use to represent liabilities to deliver the indicated
amount of goods or services.
17. Note that the formula for the linear aggregation
coefficient is identical to that for the linear correlation coefficient; in
the former the formula is applied to a function space (each point in the space
representing a function) and in the latter, to a variable space.
18. Mehta and Thapa (1991) reports the difficulty
in making comparisons among multinationals because of the flexibility allowed
in the choice of functional currency and frequent switches of functional currency.
Kirsch and Johnson (1991) reports a wide variety of rate alternatives are
used even with the same multinational firm. See also Abdel-Magid and Cheung
(1986).
19. Segment reporting based on geographical regions
of the world can give rough idea about the distribution of assets but not
fine enough to identify the currency linkage.
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