## Value of a Commodity Swap

Consider an agreement where party A receives spot price for N units of a commodity each period while paying a fixed amount X per unit for N units. If the agreement is made for M periods, the net cash flow stream received by A is (S\ — X, So — X, 53 — X, ..., 5a/ — X) multiplied by the number of units N, where 5/ denotes the spot price of the commodity at time r.

We can value this stream using the concepts of forward markets At time zero the forward price of one unit of the commodity to be received at time i is Ff\ This means that we are indifferent between receiving 5/ (which is currently uncertain) at i and receiving A at /, By discounting back to time zero we conclude that the current value of receiving 5/ at time / is d(0, r) A> where d(0, /) is the discount factor at time zero for cash received at i.

If we apply this argument each period, we find that the total value of the stream is

Hence the value of the swap can be determined from the series of forward prices, Usually X is chosen to make the value zero, so that the swap represents an equal exchange,

Example 10.6 (A gold swap) Consider an agreement by an electronics firm to receive spot value for gold in return for fixed payments. We assume that gold is in ample supply and can be stored without cost—which implies that the swap formula takes an

10.6 BASICS OF FUTURES CONTRACTS 275

almost trivial form. In thai case we know that the forward price is F) = Su/d{(), i) Therefore (10 5) becomes

The summation is identical to the value of the coupon payment stream of a bond Using this fact, it is easy to convert the value formula to where B(M, C') denotes the price (relative to 100) of a bond of maturity M and coupon C per period Any value of C can be used. (See Exercise 8.)

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