equitable matrix for money exchange

This example shows how equitable matrices arise in money exchange.Consider $n$ currencies $C_{1},C_{2},\\ldots,C_{n}$ , where $C_{i}$ stays for somename like “dollar”, “euro”, “pound”, etc. Denote the exchange ratebetween currencies $C_{i}$ and $C_{j}$ as $a_{ij}>0$ , i.e.

1\\ C_{i}\\longrightarrow a_{ij}\\ C_{j}.

(1)

We will call $A=(a_{ij})_{i,j=1}^{n}$ an exchange rates matrix.Suppose, $A$ is anequitable matrix, i.e.

a_{ij}=a_{ik}\\cdot a_{kj},\\quad i,j,k=1,\\ldots,n.

(2)

Let us discuss consequences of this. First of all, there is no loss when exchangingbetween 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}\\longrightarrow u\\cdot a_{ij}\\ C_{j}\\longrightarrow u\\cdot(a_{ij}\\cdota%_{ji})\\ C_{i}

and desired conjecture follows from the fact that

a_{ij}\\cdot a_{ji}=1,\\quad i,j=1,\\ldots,n,

(3)

which can be proven by putting $j=i$ in (2) and using diagonal property( $a_{ii}=1$ , for all $i=1,\\ldots,n$ ).But equitable property (2) suggests in fact more than just (3), i.e. morethan just no loss by changing from one currency to another and back. For illustration consideran example.

Example 1.

Let us take three currencies $C_{1}$ , $C_{2}$ , $C_{3}$ with the following exchange rates

$\\displaystyle 1\\ C_{1}$	$\\displaystyle\\longrightarrow$	$\\displaystyle 2\\ C_{2},$
$\\displaystyle 1\\ C_{1}$	$\\displaystyle\\longrightarrow$	$\\displaystyle 3\\ C_{3},$
$\\displaystyle 1\\ C_{2}$	$\\displaystyle\\longrightarrow$	$\\displaystyle 2\\ C_{3}.$

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=\\left(\\begin{array}[]{ccc}1&2&3\\\\1/2&1&2\\\\1/3&1/2&1\\end{array}\\right).

Now assume one has 100 $C_{3}$ units and wants to exchange them to $C_{1}$ . If one does thisdirectly, one will obtain 300 $C_{1}$ units. But if one exchanges first to $C_{2}$ and thento $C_{1}$ , 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 thento $C_{j}$ . Indeed,

\\begin{array}[]{ccccc}&&u\\ C_{i}&\\longrightarrow&u\\cdot a_{ij}\\ C_{j}\\\\u\\ C_{i}&\\longrightarrow&u\\cdot a_{ik}\\ C_{k}&\\longrightarrow&u\\cdot(a_{ik}%\\cdot a_{kj})\\ C_{j}\\end{array}

and the final result is the same due to the equitable property (2). Note, that the matrixfrom the example is not equitable

a_{23}\eq a_{21}\\cdot a_{13}.

The above consideration shows that equitable property does not allow making moneyjust by exchanging currencies. If (2) does not hold for someindexes, for example

a_{ij}<a_{ik}\\cdot a_{kj},

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}\\longrightarrow u\\cdot a_{ik}\\ C_{k}\\longrightarrow u\\cdot(a_{ik}\\cdota%_{kj})\\ C_{j}\\longrightarrow u\\cdot\\left(\\frac{a_{ik}\\cdot a_{kj}}{a_{ij}}%\\right)\\ C_{i}.

If we denote $q:=\\frac{a_{ik}\\cdot a_{kj}}{a_{ij}}>1$ , then after making $N$ suchexchanges instead of $u$ one would have $u\\cdot q^{N}$ units — the capital wouldincrease like geometric progression! If there is an opposite inequality

a_{ij}>a_{ik}\\cdot a_{kj},

then such advantage have those with $C_{j}$ currency. So, condition (2) guaranteesthat 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 showswhy exchange rates matrices are useful in general.

Consider a company which operates on the international market (e.g., a company which makesfurniture/cars/household equipment/etc, and sells their products to more than one country) and, thus, obtainsmoney in different currencies $C_{1},C_{2},...,C_{n}$ . From time to time,for such a company the natural question arises: what is thetotal amount of money we have? Specifically, at a given time the company obtainedthe following money: $u_{1}\\ C_{1},\\;u_{2}\\ C_{2},...,\\;u_{n}\\ C_{n}$ . With given exchange rates $a_{ij}$ , the total amount of money in the currency $C_{i}$ is $\\sum_{j=1}^{n}u_{j}\\cdot a_{ji}$ .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 $u\\cdot A$ 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:

$\\displaystyle C_{1}$	$\\displaystyle=$	$\\displaystyle\\mbox{``euro''=``EUR''},$
$\\displaystyle C_{2}$	$\\displaystyle=$	$\\displaystyle\\mbox{``USA dollar''=``USD''},$
$\\displaystyle C_{3}$	$\\displaystyle=$	$\\displaystyle\\mbox{``British pound''=``GBP''},$
$\\displaystyle C_{4}$	$\\displaystyle=$	$\\displaystyle\\mbox{``Yapanese yen''=``JPY''},$
$\\displaystyle C_{5}$	$\\displaystyle=$	$\\displaystyle\\mbox{``Swiss frank''=``CHF''}.$

The corresponding exchange rates matrix $A$ is presented in Table 1.The values were first collected from\\htmladdnormallinkWikipediahttp://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, andcorresponding amount of money are gathered in Table 2. The last column givescomponents of the row-vector $u$ in our example. To answer the question what is the totalamount of money are obtained by “Peaut”, one needs to compute $u\\cdot A$ . 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”.