When the two legs of a spread are highly correlated and, therefore, the opportunity for profit from price divergence is of short duration (less than 1 or 2 days), the trade is-called an arbitrage. True arbitrage has, theoretically, no trading risk; however, it is offset by smalls-profits and limited opportunity. For example, a spacial arbitrageur using the interbank market might call one bank in Tokyo and another in Frankfurt to find their rates on the" Mexican peso. If they differ, the trader would buy the peso from one bank and sell the peso at another provided:
1. The price difference was greater than the bid-asked spread, representing the cost of converting the currencies.
2. The arbitrageur has proper credit established with both banks.
3- The transaction can be performed simultaneously (by telephone). This requires one trader with a telephone in each ear or two traders working side-by-side.
Large-scale arbitrage has become the domain of major financial institutions who employ many traders, each provided with high-tech computer displays, sophisticated analytic software, and lots of telephones. These traders specialize in specific interest rate markets, foreign exchange, individual stock selection, or less often, precious metals. They constantly scan quotes from across the world to find price differences, then act quickly using cash, forward and futures markets. They trade large quantities to profit from small variations. For the interest rate markets, there are computer programs that compare the various types of coupons and maturities to identify an opportunity quickly. Such operations have become an integral part of the banking industry; they keep rates in-line with other banks and generate steady profits.
The relationship of one futures market delivery month to the spot price of that market is different according to the type of product, which has already been described in general terms. The mathematics of some of these relationships can become very complex, and the reader is referred to texts that deal specifically with these subjects.2 The following sections describe the most important features of these relationships.
Storable commodities can be purchased in the cash market, stored, and sold at a later time. They can also be delivered on a futures contract, held in storage, and redelivered against another contract. This puts an upper limit on the amount of the carrying charges that can be added to futures prices. The difference in cost between holding the physical commodity and buying it on the futures market are:
1. The financing cost involved in the purchase of the physical commodity:
added interest cost = spot price ([1 + interest rate}"1" "if"'""" _ j)
Where life of futures contract is expressed in years. v
2. The cost of storage, if any.
3. A convenience cost for not buying the physical product, and the ability to be able to sell it at any time.
These three costs are added to the futures spot price to get the fair value of the futures price at the time of delivery. The strategies that keep the spot and futures prices aligned, where F is the futures price, 5 is the spot price, r is the annualized interest rate, t is the life of the futures contract (in years) from today to the time of delivery, and k is the net annual storage cost (expressed as a percentage of the spot price) are:
Method 1: Buy the futures contract for F; take delivery at expiration. Margin cost can be collateralized. Net cost is F. y
Method 2: Borrow the cost of the spot commodity 5 and buy the physical product; pay the interest, 5([ 1 + r]' - 1), and the storage net-of-convenience cost, Sx k x t, until the corresponding futures market delivery.
1 See Aswath Damodaran, Investment Valuation (John Wiley & Sons, New York, 1996, pp. 448-458), which provided the basis for these formulas; also see Marsha Stigum, Money Market Calculations (Dow jfones-Irwin, 1981).
The two methods must have the same net cost; otherwise, everyone would choose the cheaper alternative. Therefore the two strategies are equal, and they form the basic arbitrage relationship between futures and spot prices:
This relationship represents the ideal case. If you add more realistic features, including separate borrowing and lending rates, rb and ra, where rh > ra, and assume that the short seller cannot recover the saved storage costs and must pay transaction costs, ts, as well, you get a normal range in which futures prices can fluctuate:
When futures prices move outside this range, there is a possibility of arbitrage.
One well-known, second-order arbitrage combines foreign exchange forward rates with interest rate parity. Consider the following: A U.S. corporation would like to invest SI million for the next 6 months. The current U.S. T-bill return for the next 6 months is lower than the rate in West Germany, and the inflation rate is about the same. The corporation is faced with the decision of whether to convert U.S. dollars to Deutschemarks and invest in West German time deposits or accept the lower U.S. rates. The decision is made easier if the corporation purchases goods from West Germany, since it must eventually convert U.S. dollars to Deutschemarks to satisfy payments; the conversion cost will then exist with either choice.
What if the value of the Deutschemark loses 1% against the U.S. dollar during the 6-month investment period? A corporation whose payment is stated in Deutschemarks suffers a 1% loss in the total interest received. If the 6-month return was 4%, interest received is now valued at $400 less than the $40,000 total, a small amount for the corporation making payment in Deutschemarks. A speculator would face a different problem because the entire return of $1,040,000 would be reduced by 1% to $1,029,600, netting a return of only 2.96%, less the additional cost of conversion. For the speculator, shifts in exchange rates often overwhelm the relative improvement in interest rate return.
The interest rate parity theorem will normally explain the differences between the foreign exchange rates and the relative interest rates of countries. It states that the forward rate of a currency is equal to its present value plus the interest earned in that country for the period of the forward rate.3
Using the futures or interbank market for the forward rate (DM, yr is 1 year forward) and the spot rate for the current value (DMspo,), the annual interest rate in West Germany (Ia„) is applied to obtain the relationship
Because the forward value of the U.S. dollar can be expressed similarly as
Compute the implied interest rate/forward rate parity by dividing the second equation by the first:
3 James E. Higgens and Allan M. Loosigian, "Foreign Exchange Futures," in Handbook of Futures Markets, Perry ]. Kaufman (ed.) (John Wiley & Sons, New York, 1984).
For example, if U.S. interest rates are 8%, West German rates are 6%, and the U.SJDMipo! is 0.5000, the 1-year forward rate would be
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