Linear Programming Application: Foreign Exchange Management | FIN 203 | Exams Finance

LINEAR PROGRAMMING APLICATION:

Foreign Exchange Management

Deac International has manufacturing and sales operations in five major trading

countries: United States, United Kingdom, Canada, Germany and Australia. Because of

the different cash needs in the different regions at various times, it is often necessary to

move available funds from one region to another. In general, there will be numerous

ways to exchange available currencies satisfy the requirements. On this particular

Tuesday the divisions in The United Kingdom and Canada are short of cash: 4 million

GBP and 6 million CAD is required. The divisions in the United States, Germany and

Australia have excess cash of 2 million dollars, 3 million Euros and 8 million AUDs.

The prevailing exchange rates are as follows:

USD GBP CAD EUR AUD

10.503796 1.023410 0.6336675 1.075746

1.953200 12.033450 1.2160491 2.097731

0.959232 0.491107 10.6161569 0.608742

1.546790 0.791930 1.612530 11.680435

0.915700 0.468820 0.954620 0.5919960 1

We will refer to the rate in row i and column j as aij which is the bid price of

currency i in units of currency j. For instance a person with a British pound will receive

$1.9532 in exchange. $1.9532 is then the bid price for a unit of GBP in US dollars. On the

other hand, if a person wants to buy a GBP he will receive 0.503796 GBP for every US

dollar. This means that it takes 1/0.503796 = $1.984930 to purchase one GBP. This is the ask

price of GBP to a person with US dollars. The slight difference between the bid and ask

prices represents the transaction cost of trading in the market. Normally the bid and ask prices

are such that there is no arbitrage opportunity-- if one kept exchanging money from one

denomination to another and back again one would lose money

Because there are many ways of redistributing the cash to satisfy the shortages, the

challenge for the manager is to find the most efficient exchanges of currencies. Due to

comparatively short-term high US interest rates, Deac International wants to maximize the

“dollar value” of all the currencies held. For any currency the dollar value is defined to be the

average of the bid and the ask prices. For instance, the dollar value of one GBP is ($1.9532 +

$1.98493) /2 = $1.969065

Questions

1. What is the dollar value of the basket of currencies the company holds currently

before any exchanges are undertaken?

2. Write out a linear programming model to determine the set of exchanges to be

made so that all cash requirements are satisfied and the sum of the dollar value of

all currencies held after the exchanges is maximized.

Hint: Use the decision variables:

Xij= amount of currency i sold to obtain currency j. Notice that this has the

effect of reducing currency i by Xij and increasing currency by aij Xij. For

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LINEAR PROGRAMMING APLICATION:

Foreign Exchange Management Deac International has manufacturing and sales operations in five major trading countries: United States, United Kingdom, Canada, Germany and Australia. Because of the different cash needs in the different regions at various times, it is often necessary to move available funds from one region to another. In general, there will be numerous ways to exchange available currencies satisfy the requirements. On this particular Tuesday the divisions in The United Kingdom and Canada are short of cash: 4 million GBP and 6 million CAD is required. The divisions in the United States, Germany and Australia have excess cash of 2 million dollars, 3 million Euros and 8 million AUDs. The prevailing exchange rates are as follows: USD GBP CAD EUR AUD 1 0.503796 1.023410 0.6336675 1. 1.953200 1 2.033450 1.2160491 2. 0.959232 0.491107 1 0.6161569 0. 1.546790 0.791930 1.612530 1 1. 0.915700 0.468820 0.954620 0.5919960 1 We will refer to the rate in row i and column j as aij which is the bid price of currency i in units of currency j. For instance a person with a British pound will receive $1.9532 in exchange. $1.9532 is then the bid price for a unit of GBP in US dollars. On the other hand, if a person wants to buy a GBP he will receive 0.503796 GBP for every US dollar. This means that it takes 1/0.503796 = $1.984930 to purchase one GBP. This is the ask price of GBP to a person with US dollars. The slight difference between the bid and ask prices represents the transaction cost of trading in the market. Normally the bid and ask prices are such that there is no arbitrage opportunity-- if one kept exchanging money from one denomination to another and back again one would lose money Because there are many ways of redistributing the cash to satisfy the shortages, the challenge for the manager is to find the most efficient exchanges of currencies. Due to comparatively short-term high US interest rates, Deac International wants to maximize the “dollar value” of all the currencies held. For any currency the dollar value is defined to be the average of the bid and the ask prices. For instance, the dollar value of one GBP is ($1.9532 + $1.98493) /2 = $1. Questions

What is the dollar value of the basket of currencies the company holds currently before any exchanges are undertaken?
Write out a linear programming model to determine the set of exchanges to be made so that all cash requirements are satisfied and the sum of the dollar value of all currencies held after the exchanges is maximized. Hint: Use the decision variables: Xij = amount of currency i sold to obtain currency j. Notice that this has the effect of reducing currency i by Xij and increasing currency by aij Xij. For

example, if one were to sell X 12 dollars to buy pounds, one’s dollars would be reduced by X 12 but the GBP balance would increase by a 12 X 12 or by 0. X 12 GBP. Yi = Final balance of currency i held. Notice that the final amount of any currency, i held will be equal to the known beginning holding of that currency, (2 million for dollars, for instance) less the amounts of currency i used to buy other currencies, plus any other currency exchanged into currency i. Use this reasoning to write equations relating Y , final holding variables to X, exchange variables.

Use Solver to determine the best set of currency exchanges.
If the requirement for GBP and CAD did not exist would the best strategy be to stay pat? Or, is there an opportunity to improve the dollar value of the existing basket of currencies with some carefully considered exchanges?
Assume the requirements for GBP is 5 million instead of 4? What effect will this have on the problem and its solution?
Suppose, for a moment, the bid- ask prices were such that if one started with one dollar, exchanged it for some GBPs, exchanged the GBPs for AUDs and finally exchanged the AUDs back into dollars one ended up with more than one dollar. Can such a situation persist for a long time? Explain your reasoning briefly. If you tried to solve the linear programming above in 2 with such exchange rates, what can you say about the solution of the problem?

Linear Programming Application: Foreign Exchange Management | FIN 203, Exams of Finance

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