Some Notes on Forward Prices

Some Notes on Forward Prices

A contract for delivery of an asset at some future date for a price agreed upon now and payable at is a forward contract for the underlying asset. The price you pay at date is the forward price, denoted . The underlying asset may be a commodity such as corn or crude oil, a financial asset such as a stock or bond, or a rate such as an interest rate or an exchange rate.

Forwards can be used to create a synthetic version of the underlying asset. Assume there is a zero coupon bond maturing at date paying $1.00 face value with certainty. The price today of this zero is denoted . Enter into a long forward contract for the asset at the forward price and buy zeros. Since there is no cash flow involved in entering the forward contract, the cost of this strategy is

. (1)

At date , the zeros pay you which is used to settle the long forward position and you have the asset. The strategy initiated at date is a synthetic version of the asset, a prepaid forward contract and (1) gives the prepaid forward price for the asset.

There is another way to view this prepaid forward price using DCF. Let denote the value of the asset at date and let denote the expectation of this value at date . The date value of one unit of the asset at date is a standard calculation in DCF. Let denote the appropriate risk-adjusted discount rate for this asset. Then the date value of receiving one unit of the asset at date is

. (2)

It is important to note that (1) and (2) give the same value, i. e., the date value of one unit of the asset to be received at date . Letting denote the continuously compounded risk-free rate for the period from to and equating (1) and (2) gives

. (3)

Rearranging (3) gives

, (4)

which expresses the forward price as a biased estimate of the expected future spot price . The bias is due to the risk premium .

Expression (3) deserves some emphasis. The date forward price for the asset discounted to the present at the risk-free rate is the present value of one unit of the asset received at time . If we can observe (or theoretically know) the forward price , the prepaid forward price gives us this present value without having to estimate the risk-adjusted required rate or the expected price . This is useful in calculating the present value of expected future cash flows that depend on future commodity prices. As a simple example, consider the Kingsley Solomon gold mine example on pp. 305-308 in (BMA). In that example, the production plan for extracting gold calls for extracting .1 million ounces of gold per year for the next ten years. The value of this plan now is , where is the present value of the price of gold in year . If now the ten year forward curve for gold is , and now the ten year term structure is , then and the value of the production plan is . This is the principle behind the calculations in the Kingsley Solomon example in (BMA), who then use a simplified "spot/futures parity" result to determine the forward curve . A more complete picture of this forward curve formation follows.

This is important even if the forward market for the particular commodity does not exist if we have a "no arbitrage theory" of the forward price if a market did exist. If a forward market did exist and the commodity could be leased, as gold can, what would be the "no arbitrage theory" of the forward price? The commodity lender would be giving up one unit of the commodity now worth , the known spot price of the commodity. In return, at date the lender gets back units of the commodity worth the random amount , where is the continuously compounded commodity lease rate. The NPV of this investment is

NPV =

= . (5)

Suppose the price of the commodity is expected to grow at the continuously compounded rate of over the period from to so that

.

Then the NPV of the loan in (5) is

NPV = . (6)

The commodity loan is a fair deal for the lender if the NPV in (6) is zero. This condition occurs when the lease rate is set equal to , the income yield on the underlying asset. This competitive lease rate is

. (7).

Note that this lease rate may be positive if the required rate is greater than the expected growth rate, e. g., if the underlying has a cash yield like a dividend-paying stock. It may be zero, e. g., if the underlying is a nondividend-paying stock then , here shorting the stock requires only its return as payment. It may be negative for commodities that have high storage costs. The lease rate is what Hull refers to as net convenience yield, the convenience yield net of storage costs. See the reference to Hull.

What is the forward price for an underlying with an active lease market? The lease rate is like a dividend. If we borrow the asset, we have to pay the lender the lease rate just as if we borrow a dividend-paying stock. By analogy the forward price reflects this lease rate in that

. (8)

This result is spot/futures parity (see p.734 in (BMA) for a version with discrete compounding).

Expression (8) gives (BMA) the forward curve for gold in the Kingsley Solomon example. There (BMA) assume that gold is like a non-dividend paying stock (footnote 6, page 306), and therefore that the lease rate on gold is . Consequently they get that and then using (3) they get that . The assumption that the lease rate on gold is zero is just a simplification. Gold can be leased for a positive lease rate. (See the last two sentences in footnote 6, p 306 in (BMA)). More on this in the American Barrick case.) (8) can be used to infer the lease rate from observed forward prices. If you know the lease rates then (8) can be used to calculate the forward curve.

Reference:

Brealey, R. A., Myers, S. C., and Allen, F. 2008. Principles of Corporate Finance, 9th ed., McGraw- Hill/Irwin, Inc.

McDonald, R. L., 2006. Derivatives Markets, 2nd ed. Boston: Addison Wesley. The material here was taken from chapter 6.

Hull, J. C., 2003. Options, Futures, and Other Derivatives, 5th ed. Prentice Hall. (3.21) p. 60.

Cases and Readings in Corporate Finance. 5 Prof. D. C. Nachman