equitable matrix for money exchange

equitable matrix for money exchange

This example shows how

equitable matrices

arise in money exchange.
Consider

n

currencies

C

1

,

C

2

,

…

,

C

n

, where

C

i

stays for some
name like “dollar”, “euro”, “pound”, etc. Denote the

exchange rate

between currencies

C

i

and

C

j

as

a

i

⁢

j

>

0

, i.e.

1

⁢

C

i

⟶

a

i

⁢

j

⁢

C

j

.

(1)

We will call

A

=

(

a

i

⁢

j

)

i

,

j

=

1

n

an

exchange rates matrix

.
Suppose,

A

is an
equitable matrix, i.e.

a

i

⁢

j

=

a

i

⁢

k

⋅

a

k

⁢

j

,

i

,

j

,

k

=

1

,

…

,

n

.

(2)

Let us discuss

consequences

of this. First of all, there is no loss when exchanging
between two currencies: if one exchanges

u

units of

C

i

to

C

j

and then back to

C

i

, one will have again

u

units. Indeed,

u

⁢

C

i

⟶

u

⋅

a

i

⁢

j

⁢

C

j

⟶

u

⋅

(

a

i

⁢

j

⋅

a

j

⁢

i

)

⁢

C

i

and desired conjecture follows from the fact that

a

i

⁢

j

⋅

a

j

⁢

i

=

1

,

i

,

j

=

1

,

…

,

n

,

(3)

which can be proven by putting

j

=

i

in (

2

) and using

diagonal

property
(

a

i

⁢

i

=

1

, for all

i

=

1

,

…

,

n

).
But equitable property (

2

) suggests in fact more than just (

3

), i.e. more
than just no loss by changing from one currency to another and back. For illustration consider
an example.

Example 1

.

Let us take three currencies

C

1

,

C

2

,

C

3

with the following exchange rates

1

⁢

C

1

⟶

2

⁢

C

2

,

1

⁢

C

1

⟶

3

⁢

C

3

,

1

⁢

C

2

⟶

2

⁢

C

3

.

The above

relations

define

a

12

,

a

13

,

a

23

in the exchange rates matrix

A

.
Let us define other

elements

by (

3

). This gives us

A

=

(

1

2

3

1

/

2

1

2

1

/

3

1

/

2

1

)

.

Now assume one has 100

C

3

units and wants to exchange them to

C

1

. If one does this
directly, one will obtain 300

C

1

units. But if one exchanges first to

C

2

and then
to

C

1

, one will obtain 400

C

1

units.

For an equitable exchange rates matrix the above described situation is impossible:
exchanging

u

⁢

C

i

units to

C

j

is the same as first exchanging to

C

k

and then
to

C

j

. Indeed,

u

⁢

C

i

⟶

u

⋅

a

i

⁢

j

⁢

C

j

u

⁢

C

i

⟶

u

⋅

a

i

⁢

k

⁢

C

k

⟶

u

⋅

(

a

i

⁢

k

⋅

a

k

⁢

j

)

⁢

C

j

and the final result is the same due to the equitable property (

2

). Note, that the matrix
from the example is not equitable

a

23

≠

a

21

⋅

a

13

.

The above consideration shows that equitable property does not allow making money
just by exchanging currencies. If (

2

) does not hold for some
indexes, for example

a

i

⁢

j

<

a

i

⁢

k

⋅

a

k

⁢

j

,

then having

u

⁢

C

i

units one can make money just by exchanging them to

C

j

through

C

k

and back

u

⁢

C

i

⟶

u

⋅

a

i

⁢

k

⁢

C

k

⟶

u

⋅

(

a

i

⁢

k

⋅

a

k

⁢

j

)

⁢

C

j

⟶

u

⋅

(

a

i

⁢

k

⋅

a

k

⁢

j

a

i

⁢

j

)

⁢

C

i

.

If we denote

q

:=

a

i

⁢

k

⋅

a

k

⁢

j

a

i

⁢

j

>

1

, then after making

N

such
exchanges instead of

u

one would have

u

⋅

q

N

units — the capital would
increase like geometric progression! If there is an opposite inequality

a

i

⁢

j

>

a

i

⁢

k

⋅

a

k

⁢

j

,

then such advantage have those with

C

j

currency. So, condition (

2

) guarantees
that no one can speculate on currency exchange, thus motivating the name
“equitable” — “to be fair”.

Let us give an

interpretation

of the matrix-vector

multiplication

for exchange rates matrices.
This interpretation is not

connected

with the equitable property (

2

) and shows
why exchange rates matrices are useful in general.

Consider a company which operates on the international market (e.g., a company which makes
furniture/cars/household equipment/etc, and sells their

products

to more than one country) and, thus, obtains
money in different currencies

C

1

,

C

2

,

…

,

C

n

. From time to time,
for such a company the natural question arises: what is the
total amount of money we have? Specifically, at a given time the company obtained
the following money:

u

1

⁢

C

1

,

u

2

⁢

C

2

,

…

,

u

n

⁢

C

n

. With given exchange rates

a

i

⁢

j

, the total amount of money in the currency

C

i

is

∑

j

=

1

n

u

j

⋅

a

j

⁢

i

.
That’s why we have

if matrix

A

is the exchange rates matrix for currencies

C

1

,

…

,

C

n

,

u

=

(

u

1

,

…

,

u

n

)

is a row-vector with

components

expressing amount of units in each currency,
then components of the vector

u

⋅

A

express the total amount of money in each currency.

The following example gives illustration to this.

Example 2

.

Imagine a company located in Germany, let’s call it “Peaut”, which makes auto “Leoptera”.
It sells this auto to five different countries: Germany (its home country), USA, United Kingdom,
Japan, and Switzerland. Thus, it needs to operate with the following currencies:

C

1

=

“euro”=“EUR”

,

C

2

=

“USA dollar”=“USD”

,

C

3

=

“British pound”=“GBP”

,

C

4

=

“Yapanese yen”=“JPY”

,

C

5

=

“Swiss frank”=“CHF”

.

The corresponding exchange rates matrix

A

is presented in Table

1

.
The values were first collected from

\htmladdnormallink

Wikipediahttp://en.wikipedia.org/
at the middle of 2005, and then they were modified such that the resulting matrix is equitable.

EUR

USD

GBP

JPY

CHF

EUR

1

1.25

0.625

125

1.5626

USD

0.8

1

0.5

100

1.25

GBP

1.6

2

1

200

2.5

JPY

0.008

0.01

0.005

1

0.0125

CHF

0.64

0.8

0.4

80

1

Table 1:

Exchange rate matrix used in the example.

The price of “Leoptera” in each country, number of sold cars during a year, and
corresponding amount of money are gathered in Table

2

. The last column gives
components of the row-vector

u

in our example. To answer the question what is the total
amount of money are obtained by “Peaut”, one needs to compute

u

⋅

A

. The result is presented in Table

3

.

Country where the cars were sold

Price for one car

Amount of

sold cars

(in thousands)

Obtained money

(in milliards)

Germany

21.000 EUR

100

2.1 EUR

USA

32.000 USD

35

1.12 USD

UK

14.000 GBP

40

0.56 GBP

Japan

3.1 mln JPY

8

24.8 JPY

Switzerland

34.000 CHF

5

0.17 CHF

Table 2:

Statistic of selling “Leoptera” during a year.

Country’s currency

The total amount of money in different currencies

(in milliards)

EUR

4.1992

USD

5.249

GBP

2.6245

JPY

524.9

CHF

6.56146

Table 3:

Total amount of money obtained by “Peaut”.

Title

equitable matrix for money exchange

Canonical name

EquitableMatrixForMoneyExchange

Date of creation

2013-03-22 14:59:25

Last modified on

2013-03-22 14:59:25

Owner

mathforever (4370)

Last modified by

mathforever (4370)

Numerical id

9

Author

mathforever (4370)

Entry type

Example

Classification

msc 91-00

Classification

msc 91B28

Classification

msc 15-00

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