We have cash reserves in ten different currencies in the amounts given in the second column of the

following table $($in millions$).$

$2$

Initial Position Desired Position

USD $1524$

EUR $7065$

GBP $32$

JPY $1532$

CHF $23$

CAD $76$

AUD $86$

NZD $118$

HKD $3227$

SGD $1822$

Our objective is to convert these amounts in order to have reserves at least equal to the amounts given

in the last column of the table. The second table contains the conversion rates.

USD EUR GBP JPY CHF CAD AUD NZD HKD SGD

USD $10\mathrm{.}76120\mathrm{.}6016104\mathrm{.}2630\mathrm{.}9191\mathrm{.}08641\mathrm{.}07051\mathrm{.}19247\mathrm{.}75021\mathrm{.}2492$

EUR $1\mathrm{.}313710\mathrm{.}79033136\mathrm{.}95851\mathrm{.}207271\mathrm{.}42711\mathrm{.}40621\mathrm{.}566310\mathrm{.}18061\mathrm{.}6409$

GBP $1\mathrm{.}662121\mathrm{.}26531173\mathrm{.}2971\mathrm{.}52751\mathrm{.}80571\mathrm{.}77921\mathrm{.}981912\mathrm{.}88152\mathrm{.}0763$

JPY $0\mathrm{.}00960\mathrm{.}00730\mathrm{.}005810\mathrm{.}00880\mathrm{.}01040\mathrm{.}01030\mathrm{.}01140\mathrm{.}07430\mathrm{.}012$

CHF $1\mathrm{.}08810\mathrm{.}82830\mathrm{.}6546113\mathrm{.}44611\mathrm{.}18211\mathrm{.}16481\mathrm{.}29748\mathrm{.}4331\mathrm{.}3592$

CAD $0\mathrm{.}92050\mathrm{.}70070\mathrm{.}553895\mathrm{.}97150\mathrm{.}84610\mathrm{.}98541\mathrm{.}09767\mathrm{.}13371\mathrm{.}1499$

AUD $0\mathrm{.}934190\mathrm{.}71120\mathrm{.}56297\mathrm{.}40150\mathrm{.}85861\mathrm{.}014911\mathrm{.}11397\mathrm{.}24031\mathrm{.}167$

NZD $0\mathrm{.}83860\mathrm{.}63840\mathrm{.}504687\mathrm{.}61250\mathrm{.}77010\mathrm{.}91110\mathrm{.}897716\mathrm{.}51\mathrm{.}0476$

HKD $0\mathrm{.}1290\mathrm{.}09820\mathrm{.}077613\mathrm{.}45320\mathrm{.}11860\mathrm{.}14020\mathrm{.}13810\mathrm{.}153910\mathrm{.}1612$

SGD $0\mathrm{.}80050\mathrm{.}60940\mathrm{.}481683\mathrm{.}4650\mathrm{.}73570\mathrm{.}86960\mathrm{.}85690\mathrm{.}95466\mathrm{.}20391$

The numbers in this table should be read as follows: to buy $1$ U$.$S$.$ dollar for Euro requires $0\mathrm{.}7612$ e

$($the element in the first row and the second column$),1$ U$.$S$.$ dollar costs $0\mathrm{.}6016$ British pounds $($GBP$),$

etc. If we want to buy $1$ Euro for U$.$S$.$ dollars we have to pay $ $1\mathrm{.}3137($the element in the EUR row and

USD column$),$ etc.

Design the conversion plan to meet the desired goals and to maximize the USD value of the obtained

positions.

$2\mathrm{.}2$ The Linear Programming Model

Let us denote by n the number of currencies considered $($n $=10$ in our case$).$ Let a j $,$ j $=1,...,$ n

denote the initial positions and let bi $,$ i $=1,...,$ n$,$ denote the desired positions. Finally, we denote by

ri j the cost of currency i in currency j $($the element in row i and column j of the rates table$).$

We introduce the decision variables xi j representing the amounts of currency i purchased for cur$-$

rency j$,$ i$,$ j $=1,...,$ n$.$ For i $=$ j the variable xii represents the amount kept in currency i$.$

$3$

The decision variables have to satisfy the following equations and inequalities, called constraints:

n$$

i$=1$

ri j xi j $=$ a j $,$ j $=1,...,$ n $($amount of currency j spent$)$

n$$

j$=1$

xi j $>=$ bi $,$ i $=1,...,$ n$,($amount of currency i obtained$)$

xi j $>=0,$ i $=1,...,$ n$,$ j $=1,...,$ n$.$

In the first equation we assume that only the original amounts can be converted, not the amounts obtained

from other conversions.

The set X of conversion plans x satisfying the constraints is called the feasible set.

The value of the obtained positions can be expressed as the dollar value the amounts obtained, that is

f $($x$)=$

n$$

i$=1$

$1$

r$1$i

n$$

j$=1$

xi j $.$

It is called the objective function. The problem is to maximize the objective function over x in the

feasible set X $,$ that is to find a point $$x in X $($the optimal solution$)$ such that

f $($x$)>=$ f $($x$)$ for all x in X