Forward and futures contracts are not securities but, rather, trade agreements that enable both buyers and sellers of an underlying commodity or security to lock in the eventual price of their transaction. As such, they typically require no front-end payment from either the long or short position to motivate the other’s participation and, consequently, the contract’s initial market value usually is zero. Once the terms of the agreement are set, however, any change in market conditions will likely increase the value of the contract to one of the participants. For example, an obligation made in November to purchase soybeans in March for $220 per ton is surely quite valuable in January if soybean prices in the spot market are already $250 and no additional har- vest is anticipated in the next two months. A description of the valuation of these agreements, which is different for futures and forward contracts, follows.
Suppose that at Date 0 you had contracted in the forward market to buy Q ounces of gold at Date T for F0,T. At Date t, prior to the maturity date T, you decide that this long position is no longer nec- essary for your portfolio and you want to get rid of the future price risk it entails. Accordingly, you want to unwindyour original obligation. One way to do this is to take a short position in a Date t forward contract designed to offset the terms of the first. That is, at Date t you would agree to sell Q ounces of gold at Date T for the price of Ft,T. This is shown in Panel A of Exhibit 22.4.
Notice that because you now have contracts to buy and sell Q ounces of gold, you have no expo- sure to gold price movements between Dates t and T. The profit or loss on this pair of forward contracts is (Q)[Ft,T– F0,T], or the difference between the selling and purchase prices multiplied by the quantity involved. However, this amount would not be received (if Ft,T>F0,T) or paid until Date T, meaning that the value of the original long forward position when it is sold on Date t (i.e., its unwind value) would be the present value of (Q) [Ft,T– F0,T], or
➤22.4 Vt,T=(Q) [Ft,T– F0,T] ÷(1 +i)(T – t) where:
i=the appropriate annualized discount rate
Equation 22.4 expresses the Date t value of a long forward contract maturing at Date t. Notice two things about this amount. First, Vt,Tcan be either positive or negative depending on whether Ft,Tis greater or less than the original contract price, F0,T. This means that any forward contract carries the potential for symmetric payoffs to both participants. Second, the value of the short side of the same contract is just (Q) [F0,T– Ft,T] ÷(1 +i)(T – t), reinforcing the fact that forward contracts are zero-sum games since whatever the long position gains, the short position loses, and vice versa. For example, if you had originally agreed to a long position in a six-month gold for- ward at F0.05=$400, and after three months the new forward contract price is F0.25, 0.5=$415, the value of your position would be $1,464.68 (=(100) (415 – 400) ÷(1.1)0.25), assuming a 10 per- Valuing Forwards
and Futures
9Some have questioned whether regression-based hedge ratios are stable enough to be useful in practice. Recent work, however, has concluded that they are stationary. See Robert Ferguson and Dean Leistikow, “Are Regression Approach Futures Hedge Ratios Stationary?” Journal of Futures Markets 18, no. 7 (October 1998): 851–866.
cent discount rate. Conversely, the value of the original short position would then have to be –$1,464.68. Finally, notice that as Date t approaches Date T, the value of the contract simply becomes (Q)(FT,T– F0,T).
Valuing a futures contract is conceptually similar to valuing a forward contract with one important difference. As we saw earlier, futures contracts are marked to market on a daily basis, and this settlement amount was not discounted to account for the temporal difference between Dates t and T. That is, the Date t value of the futures contract is simply the undiscounted differ- ence between the futures prices at the origination and unwind (or cover) dates, multiplied by the contract quantity, as shown in panel B of Exhibit 22.4. Thus, the forward contract valuation equation can be adapted for futures as
➤22.5 V*t,T=(Q)(F*t,T– F*0,T)
where:
* =the possibility that forward and futures prices for the same commodity at the same point in time might be different
Cox, Ingersoll, and Ross showed that F*0,Tand F0,Twould be equal if short-term interest rates (i in Equation 22.4) are known but need not be the same under other circumstances.10
Typically, for commodities and securities that support both forward and futures markets, dif- ferences between F*0,Tand F0,Texist but are relatively small. For instance, Cornell and Rein- ganum established few economically meaningful differences between forward and futures prices in the foreign exchange market, while Park and Chen found that certain agricultural and precious metal futures prices were significantly higher than the analogous forward prices. More recently, Grinblatt and Jegadeesh documented that the historical differences in prices for Eurodollar for- ward and futures contracts are due to a mispricing of the latter, although this mispricing has been eliminated over time.11 Finally, note once again that V*t,T can be either positive or negative depending on how contract prices have changed since inception.
UNWIND VALUES FOR FORWARD AND FUTURES CONTRACTS
Date 0 Date t Date T
(Origination) (Unwind) (Maturity)
A. Forward Contract
• Long Forward (F0,T) • Short Forward (Ft,T)
• Contract Unwind Value: Vt,T= (Q) [Ft,T– F0,T] ÷(1 +i)(T – t) B. Futures Contract
• Long Futures (F*0,T) • Short Futures (F*t,T)
• Contract Unwind Value: V*t,T= (Q) [F*t,T– F*0,T] EXHIBIT 22.4
10John Cox, Jonathan Ingersoll, and Stephen Ross, “The Relation between Forward Prices and Futures Prices,” Journal of Financial Economics 9, no. 4 (December 1981): 321–346.
11Bradford Cornell and Marc R. Reinganum, “Forward and Futures Prices: Evidence from Foreign Exchange Markets,”
Journal of Finance 36, no. 5 (December 1981): 1035–1045; H. Y. Park and Andrew H. Chen, “Differences between For- ward and Futures Prices: A Further Investigation of Marking to Market Effects,” Journal of Futures Markets 5, no. 7 (February 1985): 77–88; and Mark Grinblatt and Narasimhan Jegadeesh. “Relative Pricing of Eurodollar Futures and Forward Contracts,” Journal of Finance 51, no. 4 (September 1996): 1499–1522.
In many respects, the relationship between the spot and forward prices at any moment in time is a more challenging question than how the contract is valued. We can understand the intuition for this relationship with an example: You have agreed at Date 0 to deliver 5,000 bushels of corn to your counterparty at Date T. What is a “fair” price (F0,T) to charge? Recognizing that the contract price can be anything that two parties agree to, one way to look at this question is to consider how much it will cost you to fulfill your obligation. If you wait until Date T to purchase the corn on the spot market, you have a speculative position since your purchase price (ST) will be unknown when you commit to a selling price.
Alternatively, suppose you buy the corn now for the current cash price of S0per bushel and store it until you have to deliver it at Date T. Under this scheme, the forward contract price you would be willing to commit to would have to be high enough to cover (1) the present cost of the corn and (2) the cost of storing the corn until contract maturity. In general, these storage costs, denoted here as SC0,T, can involve several things, including commissions paid for the physical warehousing of the commodity (PC0,T) and the cost of financing the initial purchase of the underly- ing asset (i0,T) but less any cash flows received (D0,T) by owning the asset between Dates 0 and T.
Thus, in the absence of arbitrage opportunities, the forward contract price should be equal to the current spot price plus the cost of carrynecessary to transport the asset to the future delivery date:
➤22.6 F0,T=S0+SC0,T=S0+(PC0,T+i0,T– D0,T)
Notice that even if the funds needed to purchase the commodity at Date 0 are not borrowed, i0,T
accounts for the opportunity cost of committing one’s own financial capital to the transaction.
This cost of carry model is useful in practice because it applies in a wide variety of cases. For some commodities, such as corn or cattle, physical storage is possible but the costs are enor- mous. Also, neither of these assets pays periodic cash flows in the traditional sense of the term.
In such situations, it is quite likely that F0,T>S0and the market is said to be in contango. On the other hand, common stock is costless to store but often pays a dividend. The presence of this cash flow sometimes makes it possible for the basis to be positive (i.e., F0,T<S0), meaning that SC0,T
can be negative. There is another reason why SC0,Tmight be less than zero. For certain storable commodities that do not pay a dividend, F0,T< S0can occur when there is effectively a “pre- mium” placed on currently owning the commodity. This premium, called a convenience yield, results from a small supply of the commodity at Date 0 relative to what is expected at Date T after, say, a crop harvest. (Oats are a commodity that sometimes satisfies this condition, as indi- cated in Exhibit 22.3.) Although it is extremely difficult to quantify, the convenience yield can be viewed as a potential negative storage cost component that works in a manner similar to D0,T. A futures market in which F0,T<S0is said to be backwardated.
An immediate implication of Equation 22.6 is that there should be a direct relationship between contemporaneous forward and spot prices; indeed, this positive correlation is the objec- tive of any well-designed hedging strategy. A related question involves the relationship between F0,Tand the spot price expected to prevail at the time the contract matures (i.e., E(ST)). There are three possibilities. First, the pure expectations hypothesis holds that, on average, F0,T=E(ST), so that futures prices serve as unbiased forecasts of future spot prices. When this is true, futures prices serve an important price discovery function for particiants in the applicable market. Conversely, F0,Tcould be less than E(ST), a situation that Keynes and Hicks argued would arise whenever short hedgers outnumber long hedgers.12In that case, a risk premium in the form of a lower contract price would be necessary to attract a sufficient number of long speculators.
The Relationship between Spot and Forward Prices
914 CHAPTER 22 FORWARD AND FUTURESCONTRACTS
12John Maynard Keynes, A Treatise on Money (London: Macmillan, 1930); and John Hicks, Value and Capital (Oxford:
Claredon Press, 1939).
13See Avraham Kamara, “The Behavior of Futures Prices: A Review of Theory and Evidence,” Financial Analysts Jour- nal 40, no. 4 (July–August 1984): 68–75; Tim Krehbiel and Roger Collier, “Normal Backwardation in Short-Term Inter- est Rate Markets,” Journal of Futures Markets 16, no. 8 (December 1996): 899–913; and Robert Brooks, Interest Rate Modeling and the Risk Premiums in Interest Rate Swaps (Charlottesville, Va.: Research Foundation of the Institute of Chartered Financial Analysis, 1997).
For reasons that are not entirely clear, this situation is termed normal backwardation. Finally, a normal contango market occurs when the opposite is true, specifically, when F0,T>E(ST).
The existence of a risk premium in the futures market is hotly debated. Kamara surveyed the early literature on the subject and found the evidence from the commodity markets to be mixed.
He concluded that although the normal backwardation hypothesis was supported, futures mar- kets are mainly driven by risk-averse hedgers who have been able to acquire “cheap” insurance.
Krehbiel and Collier examined the price behavior in the Eurodollar and Treasury bill futures markets and found evidence consistent with the existence of risk premia that were necessary to balance net hedging and net speculative positions. Finally, Brooks documented that the risk pre- mia priced into Eurodollar futures contracts have a substantial impact on other financial securi- ties as well. Specifically, he showed that prices for interest rate swaps—which can be viewed as portfolios of Eurodollar contracts—are biased upward, causing borrowers who use swaps to con- vert their variable-rate loans into synthetic fixed-rate debt to make higher payments, on average, than if they had not hedged.13
FINANCIAL FORWARDS AND FUTURES: APPLICATIONS AND STRATEGIES
Originally, forward and futures markets were organized largely around trading agricultural com- modities, such as corn and wheat. Although markets for these products remain strong, the most significant recent developments in this area have involved the use of financial securities as the asset underlying the contract. In fact, Exhibit 22.5 shows that 9 of the 10 most heavily traded derivative contracts in the United States are based on financial securities. In this section, we take a detailed look at three different types of financial forwards and futures: interest rate, equity index, and foreign exchange.
Interest rate forwards and futures were among the first derivatives to specify a financial security as the underlying asset. The earliest versions of these contracts were designed to lock in the for- ward price of a particular fixed-coupon bond, which in turn locks in its yield. As we will see in Chapter 24, this market has progressed to where such contracts as forward rate agreements and interest rate swaps now fix the desired interest rate directly without reference to any specific underlying security. To understand the nuances of the most popular exchange-traded instru- ments, it is useful to separate them according to whether they involve long- or short-term rates.
Treasury Bond and Note Contract Mechanics The U.S. Treasury bond and note con- tracts at the Chicago Board of Trade (CBT) are among the most popular of all the financial futures contracts; in fact, Exhibit 22.5 shows that the T-bond contract has historically been one of the most frequently traded futures contract of any kind. A smaller T-bond contract—one-half the delivery amount—is also available at the CBT. Delivery dates for both note and bond futures fall in March, June, September, and December. Exhibit 22.6 shows a representative set of quotes for these contracts.
Both the T-bond and the longer-term T-note contracts traded at the CBT call for the delivery of $100,000 face value of the respective instruments. For the T-bond contract, any Treasury bond Long-Term
Interest Rate Futures Interest Rate Forwards and Futures
that has at least 15 years to the nearest call date or to maturity (if noncallable) can be used for delivery. Bonds with maturities ranging from 6.5 to 10 years and 4.25 to 5.25 years can be used to satisfy the 10-year and 5-year T-note contracts, respectively. Delivery can take place on any day during the month of maturity, with the last trading day of the contract falling seven business days prior to the end of the month.
Mechanically, the quotation process for T-bond and T-note contracts work the same way. For example, the settlement price of 103–10, for the March 2002 T-bond contract on the CBT rep- resents 10310⁄32percent of the face amount, or $103,312.50. The contract price went down by 12 ticks(–12) from the previous day’s settlement, meaning that the short side had its margin account increased by 12⁄32percent of $100,000—or $375—where each 1⁄32movement in the bond’s price equals $31.25 (i.e., 1,000 ÷32).
916 CHAPTER 22 FORWARD AND FUTURESCONTRACTS
EXHIBIT 22.6 TREASURY BOND AND NOTE FUTURES QUOTATIONS
Source: The Wall Street Journal, 8 February 2002. Printed with permission from The Wall Street Journal, Dow Jones
& Co. Inc.
LEADING U.S. DERIVATIVE CONTRACTS RANKED BY TRADING VOLUME
CONTRACT, EXCHANGE, & COUNTRY 2000 VOLUME 2001 VOLUME
3-mo Eurodollar futures, CME 90,401,234 149,631,019
3-mo Eurodollar options, CME 23,019,049 69,292,451
U.S. Treasury bond futures, CBT 53,995,803 47,946,921
10-yr T-note futures, CBT 38,873,890 45,625,926
E-mini S&P 500 futures, CME 15,695,292 31,882,399
Crude oil futures, NYM 31,437,747 31,865,663
E-mini NASDAQ 100, CME 7,859,834 26,573,778
5-yr T-note futures, CBT 19,429,630 24,386,253
S&P 500 Index options, CBOE 18,530,998 19,150,977
S&P 500 Index futures, CME 18,617,638 18,662,040
Exchange-traded funds, CBOE — 18,238,503
EXHIBIT 22.5
Source: Futures Industry Association Reprinted with permission.
Although T-bond and T-note futures contracts are called interest rate futures, what the long and short positions actually agree to is the price of the underlying bond. Once that price is set, however, the yield will be locked in. When a yield is quoted, it is for reference only and typi- cally assumes a coupon rate of 6 percent and 20 years to maturity. For the March 2002 bond con- tract, the settlement yield would be 5.7198 percent, which can be established by solving for the internal rate of return in the following “bond math” problem
This pricing formula takes into account the fact that Treasury bonds pay semiannual interest. So, a 20-year, 6 percent bond makes 40 coupon payments of 3 percent each. Thus, the long position in this contract has effectively agreed in February to buy a 20-year T-bond in March priced to yield 5.72 percent. If, in March, the actual yield on the 20-year bond is below 5.72 percent (i.e., the bond’s price is greater than $103,312.50), the long position will have made a wise deci- sion. Thus, the long position in this contract gains as prices rise and rates decrease and loses as increasing rates lead to lower bond prices.
Because the bond and note futures contracts allow so many different instruments to qualify for delivery, the seller would naturally choose to deliver the least expensive bond if there were no adjustments made for varying coupon rates and maturity dates. To account for this, the CBT uses conversion factorsto correct for the differences in the deliverable bonds. The conversion factor is based on the price of a given bond if its yield is 6 percent at the time of delivery and the face value is $1. For example, the March 2002 conversion factor for the 9 percent T-bond matur- ing in November 2018 would be 1.3115, calculated as
The actual delivery price, or invoice price, for that Treasury bond would be the quoted futures price, $103,312.50, times the conversion factor, $1.3115, for a total of $135,494.34 (plus accrued interest). The buyer must pay more than $103,312.50 because the seller is delivering “more valu- able” bonds since their coupon rate exceeds 6 percent.
The conversion factors used by the CBT are technically correct only when the Treasury yield curve is flat at 6 percent. Therefore, there usually will be a cheapest to deliver bond that maxi- mizes the difference between the invoice price (the amount received by the short) and the cash market price (the amount paid by the short to acquire the delivery bond). Market participants always know which bond is the cheapest to deliver. Therefore, the T-bond futures contract trades as if this particular security were the actual underlying delivery bond. In fact, the cheapest to deliver security usually is the T-bond with the longest duration when yields are above 6 percent, and the one with the shortest duration for yields less than 6 percent.
A Duration-Based Approach to Hedging In Chapter 19, we stressed that the main ben- efit of calculating the duration statistic was its ability to link interest rate changes to bond price changes by the formula
∆P ∆
P D i n
i n
≈ − +
+
( / ) ( / )
1 1
1 3115 0 045
1 0 03
1 1 0 03 33
1 33
. .
( . ) ( . )
= + +
= +
∑ t
t
$ , . $
( / )
$ , ( / )
1 033 125 30
1 2
1 000 1 2 40
1
= 40
+ +
= +
∑ i t i t
We also saw that a more convenient way to write this expression is:
where:
Dmod=the bond’s modified duration, combining the Macaulay duration and its periodic yield into a single measure
Earlier in this chapter, we noted that the objective of hedging was to select a hedge ratio (N) such that ∆S – ∆F(N) =0, where S is the current spot price of the underlying asset and F is the cur- rent futures contract price. Rewriting this leaves
Using the modified duration relationship, this optimal hedge ratio can now be expanded as follows:
or
➤22.7
where:
ai= the “yield beta”
The yield beta is also called the ratio of changes in the yields applicable to the two instruments where n is the number of payment periods per year (e.g., n=2 for semiannual coupon bonds).14 As a general example of the duration-based approach to setting hedge ratios, consider the fol- lowing fixed-income securities, each making annual payments (i.e., n=1):
INSTRUMENT COUPON MATURITY YIELD
A 8% 10 years 10%
B 10% 15 years 8%
N D
D
S F
S F
*= mod = i×
mod
β
N S
F S S
F F
S F
D i n
D i n
S F
S S
F F
* ( / )
( / )
= =
× = − ×
− × ×
∆
∆
∆
∆
∆
∆
mod mod
N S
*= ∆F
∆
∆P ∆ ∆
P
D
i n i n D i n
≈ − +
+ = −
( / ) ( / ) ( / )
1 1 mod
918 CHAPTER 22 FORWARD AND FUTURESCONTRACTS
14In an early study on the topic, Gerald Gay, Robert Kolb, and Raymond Chiang, “Interest Rate Hedging: An Empirical Test of Alternative Strategies,” Journal of Financial Research 6, no. 3 (Fall 1983): 187–197, tested the duration-based hedge ratio against several other more naive approaches and found that it reduced the risk of the underlying bond posi- tion by the greatest amount.