CASE 1: Foreign Exchange at HiT ech

HiTech, Inc., has manufacturing and sales operations in five major trading countries: United States, United Kingdom, France, Germany, and Japan. Due to the different cash needs in the various countries at various times, it is often necessary to move available funds from one country and denomination to another. In general, there will be numerous ways to rearrange funds to satisfy cash requirements out of availabilities. On this particular morning the divisions in France and Japan are short of cash. Specifically, the requirements are, respectively, 7 million francs and 1040 million yen. The divisions in the United States, Britain, and Germany are long on cash. They have surpluses of 2 million dollars, 5 million pounds, and 3 million marks. Since there are many possible ways of redistributing the cash to satisfy the shortages out of the surpluses, the issue to be addressed is how one compares the possible conversion strategies. Because of high short-term U.S. interest rates, the firm has decided to evaluate its final cash position by this measure: the equivalent total dollar value of its final cash holdings.

On this morning, as usual, at 7:00 A.M. Jack Walker, the corporate treasurer, and Ezra Brooks, VP for overseas operations, meet at corporate headquarters to determine what funds, if any, should be moved. Refer to Exhibits I and 2 as you go through the dialog. The conversation proceeds as follows:

Good morning, Jack. I have something to show you. I've asked Fred to set this exchange model up on a spreadsheet. I think it will make our lives considerably easier.

I like the idea, but you'll have to explain the model to me.

Sure, Jack. It contains all the usual information, but let's go through it step by step. The figures in the rectangle defined by C3 through G7 are the exchange rates. If we let \(a_{i j}\) be the rate in row i and column \(j\), then one unit of currency \(i\) will exchange for \(a_{i j}\) units of currency \(j\). In fact, these data reflect the bid-ask prices. For example, if we sell one British pound we get $1.665. That is, 1.665 is the bid price, in dollars, for a pound. On the other hand, if we sell one dollar we will receive 0.591 pounds. This means we can buy a pound for \(1 / 0.591 = 1.692\) (the asking price, in dollars, for a pound is 1.692). Hence the bid-ask spread is 1.665, 1.692. You can see that if we start with $1 and buy as many pounds as possible and then use those pounds to buy dollars we end up with \(0.591 \times 1.665= 0.9810\) dollars-we lose money.

That's the transaction cost. So obviously we want to minimize these transaction costs by not moving more money around than we have to. But where does this model say something about our cash needs today?

Our current cash holdings are shown in column C rows 17 to 21 . All figures are in millions; we have 2 million dollars, 5 million pounds, and 3 million marks. Our requirements appear in column \(G\) in the same rows. You can see that we need 7 million French francs and 1040 million yen. As you know, our policy is to satisfy requirements in such a way that the dollar value of final holdings is maximized.

1. Write out the LP model for the foreign exchange problem. In your model use the following notation for the data given in the problem description:
   \(a_{i j}=\) exchange rate from currency \(i\) into currency \(j\) (i.e., 1 unit of currency \(i\) will exchange for \(a_{i j}\) units of currency \(j\) )
   \(C_{i}=1 / 2\left(a_{i j}+1 / a_{i j}\right)=\) "average dollar value" of currency \(i\)
   \(b_{i}=\) initial holding in currency \(i\)
   \(L_{i}=\) minimum amount of currency \(i\) required as final holding
   Denote the decision variables, as follows:
   \(X_{i j}=\) amount of currency \(i\) changed into currency \(j, j \neq i\)
   \(Y_{i}=\) final holding in currency \(i\)
2. In the dialog, Jack says, "We want to minimize these transaction costs by not moving more money around than we have to." Suppose we define \(O V_{1}=\) maximum "average dollar value" that can be generated from initial holdings.
   In a case like this, a more highly constrained LP cannot have a better $O V$ than a less highly constrained LP, so it must always be true that \(O V_{2} \leq O V_{1}\). Let us define, for the problem,
   Transactions Costs \(=O V_{1}-O V_{2}\)
   Using this definition, is the statement by Jack correct? That is, does the optimized solution in Exhibit 3 minimize transaction costs?
3. Recall the spreadsheet presented in Exhibit 1. Use this spreadsheet to answer the following questions:

1. Suppose that the exchange rates for two currencies (say the franc and the mark) are such that if we start with 1 franc and execute the trade 1 franc \(\rightarrow\) marks \(\rightarrow\) francs we end up with more than 1 franc. What would the optimal value of the objective function be under these circumstances? What economic term is used to describe this condition?
2. Comment on the following statement: If HiTech has no specific cash requirements, the optimal solution would be to stand pat (i.e., in order to maximize "average dollar value" of final holdings, one should do no trading).

Price: $28.86

Solution: The downloadable solution consists of 12 pages, 1686 words.
Deliverable: Word Document