Chapter 3: Commodity Forwards and Futures

Transcription

1 Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique economic characteristic that must be understood in order to appreciate forward pricing in that market. In this chapter we will see how commodity forwards and futures differ from, and are similar to, financial forwards and futures. In Chapter 2 we treated forward and futures prices as the same; we continue to ignore the pricing differences in this chapter Pricing commodity forwards As with forward prices on financial assets, commodity forward prices are the result of a present value calculation. To understand this, it is helpful to consider synthetic commodities. Just as we could create a synthetic stock with a stock forward contract and a zerocoupon bond, we can also create synthetic commodity by combining a forward contract with a zero-coupon bond. Consider the following investment strategy: Enter into a long commodity forward contract at the price F 0 and buy a zero-coupon bond that pays F 0 at time T. Since the forward contract is costless, the cost of this rt investment strategy at time 0 is just the cost of the bond, that is e F 0. At time T the strategy pays S T F 0 + F 0 = S T, where S T is the time T price of the commodity. This investment strategy creates a synthetic commodity, in that it has the same value as a unit of the commodity at time T. Let now E 0 (S T ) denote the expected time-t price as of time 0, and let α denote the appropriate discount rate for a time-t cash flow of S T. Then the present value is T E0 ST e. This expression represents the same value with the cost of the previous strategy. Both reflect what you would pay today to receive one unit of the commodity at time T. Equating the two expressions we have rt T e F0 E0 ST e Rearranging terms we can write the forward price as: (r )T F0 E0 ST e (1) Equation (1) demonstrates the link between the expected commodity price and the forward price. As with financial forward (see Chapter 2), the forward price is a biased estimate of the expected spot price, with the bias due to the risk premium on the commodity, r α. Equation (1) is a general formula that, as we have seen, can be applied to both financial and commodity forwards and futures. However, for commodity forwards the storage of the underlying asset (whether the commodity can be stored and, if so, how costly it is to store) plays an important role for its pricing. If a commodity cannot be physically stored, the no-arbitrage pricing principles discussed in Chapter 2 cannot be used to obtain a forward price. Without storage, equation (1) determines the forward price. However, it is difficult to implement this formula, which requires forecasting the expected future spot price and estimating α. Moreover, even when physically 1

2 possible, storage may be costly. Given the difficulties of pricing commodity forwards, our goal will be to interpret forward prices and to understand the economics of different commodity markets Pricing commodity forwards by arbitrage We consider the very simple, hypothetical example of a forward contract for pencils. Suppose that pencils cost $0.2 today and for certain will cost $0.2 in 1 year. There is nothing inherently inconsistent about assuming that the pencil price is expected to stay the same. However, before we proceed, note that a constant price would not be a reasonable assumption about the price of a stock. A stock must be expected to appreciate, or else no one would own it. At the outset, there is an obvious difference between this commodity and a financial asset. One way to describe this difference between the pencil and the stock is to say that stocks and other financial assets must be held by investors, or stored. This is why the stock price appreciates on average; appreciation is necessary for investors to willingly store the stock. The pencil, by contrast, need not be stored. The price equals the marginal production cost. 1 This distinction between storage and production equilibrium is a central concept in our discussion for commodities. Also suppose that the risk-free interest rate is 10%. What is the forward price for a pencil to be delivered in 1 year? One obvious possible answer to this question, drawing on our discussion of financial 0.1 forwards, is that the forward price should be 0.2 e $ However, common sense suggests that this cannot be the correct answer. You know that the pencil price in one year will be $0.2. Therefore, you would not enter into a forward contract to buy a pencil for $ Common sense also rules out the forward price being less than $0.2. Consider the forward seller. No one would agree to sell a pencil for a forward price of less than $0.2 knowing that the price will be $0.2. Thus, it seems as if both the buyer and seller perspective lead us to the conclusion that the forward price must be $0.2. If the forward price is $0.2, is there an arbitrage opportunity? Suppose you believe that the $0.2 forward price is too low. Following the logic of Chapter 2, you would want to buy the pencil forward and short-sell a pencil. Table 1 depicts the cash flows in this reverse cash-and-carry arbitrage. The result seems to show that there is an arbitrage opportunity. However, Table 1 has a problem. The arbitrage assumes that you can short-sell a pencil buy borrowing it today and returning it in a year. However, recall that pencils cost $0.2 today and will cost $0.2 in a year. Borrowing one pencil and returning one pencil in a year is an interest-free loan of $0.2. No one would lend you the pencil without charging you an additional fee. Thus the apparent arbitrage in the above table has nothing to do with forward contracts on pencil. 1 If the price of a pencil is greater than the cost of production, more pencils are produced, driving down the market. If the price falls, fewer pencils are produced, and the price rises. The market price of pencils thus reflects the cost of production. 2

3 Table 1: Reverse cash-and-carry arbitrage for a pencil Cash flows Time 0 Time 1 Long forward F 0 Short-sell pencil Lend short-sells proceeds Total F 0 If you find someone willing to lend you pencils for years, you should borrow as many as you can and invest the proceeds in risk-free bonds. You will earn the interest rate and pay nothing to borrow the money. Consequently, the pencil borrower must make a payment to the lender to compensate the lender for lost time value of money. The pencil therefore has a lease rate of 10%, since that is the interest rate. With this change the corrected reverse cash-and-carry arbitrage is in Table 2. Table 2: Correct reverse cash-and-carry arbitrage for a pencil Cash flows Time 0 Time 1 Long forward F 0 Short-sell pencil Lend short-sells proceeds Total F 0 When we correctly account for the lease payment, this transaction no longer earns profits when the forward price is $0.2 or greater. A cash-and-carry arbitrage where we buy the pencil and sell it forward, and simultaneously lend the pencil demonstrates that the forward price cannot be greater than $0.2 (see Table 3). Thus the forward price must be $0.2. Table 3: Cash-and-carry arbitrage for a pencil Cash flows Time 0 Time 1 Short forward 0 F Borrow Buy pencil and lend it Total 0 F This result can be also confirmed by equation (1) which holds for all commodities and assets. To apply this equation to the pencil, recognize that the appropriate discount rate α for a pencil is r, the risk-free rate. Hence, we have (r )T F E S e 0.2 e $ T The pencil is obviously a special example, but this discussion establishes the important point that in order to understand arbitrage relationship for commodity forwards, we have to think about the cost of borrowing and income from lending an asset. Borrowing and leasing costs also determine the pricing of financial forwards, but the cash flow associated with borrowing and lending financial assets is the dividend yield, which is readily observable. The commodity analogue to dividend 3

4 yield is the lease rate, which may not be directly observable. We now discuss leasing more generally The commodity lease rate The lease rate definition The discussion of commodity forwards has raised the issue of a lease rate. We now discuss this lease rate in greater detail. Consider again the perspective of a commodity lender, who in the previous discussion required that we pay interest to borrow the pencil. If α is the expected return on a stock that has the same risk as the commodity; α is therefore the appropriate discount rate for S T (the commodity value at time T). The NPV of the investment (the commodity loan) is T NPV E0 ST e S0 Suppose that we expect the commodity price to increase at the rate g, so that gt E0 ST S0e Then the NPV of the commodity loan is (g )T NPV S0e S0 In order for the commodity loan to be a fair deal the NPV must be equal to zero. This can be true if the lender demands the borrower to return e ( g)t units of the commodity for each unit borrowed. This is like a continuous proportional lease payment of α g to the lender. Thus, the lease rate is the difference between the commodity discount rate and the expected growth rate of the commodity price, or ql g With this payment, the NPV of the commodity loan is ( g)t (g )T NPV S0e e S0 0 When the future pencil was certain to be $0.2, the growth rate was zero, the opportunity cost α was the risk-free rate and thus the lease rate was equal to the riskfree rate. Note that if S T were the price of a nondividend-paying stock, its expected growth rate would equal the expected return, so g = α, and no payment would be required for the stock loan to be a fair deal. Commodities however are produced; as with the pencil, their expected price growth need not be equal α Forward prices and the lease rate We can now use the notion of the lease rate in order to calculate the forward price for a commodity. The key insight is that the lease rate payment is a dividend. If we borrow the asset, we have to pay the lease rate to the lender, just as with a dividendpaying stock. If we buy the asset and lend it out, we receive the lease payment. Thus, the formula for the forward price of a commodity is (r q L )T F S e (2) 0 0 4

5 Equation (2) is similar to equation (3) of Chapter 2 except that the dividend yield is replaced by the lease rate. Also equation (2) holds whether or not the commodity can be, or is stored. One of the implications of equation (2) is that the lease rate has to be consistent with the forward price. Thus when we observe the forward price, we can infer what the lease rate would have to be. Specifically, if the forward price is F 0 the annualized lease rate is 1 F 0 ql r ln (3) T S0 The behavior of forward prices can vary over time. Two terms often used by commodity traders are contango and backwardation. If one given date the forward curve is upward-sloping i.e., forward prices more distant in time are higher then we say the market is in contango. 2 If the forward curve is downward sloping we say the market is in backwardation. Equation (2) implies that contango occurs when the lease rate is less than the risk-free rate. Backwardation occurs when the lease rate exceeds the risk-free rate Storage costs Storage costs and forward prices Sometimes it makes sense for a commodity to be stored, at least temporarily. Storage is also called carry, and a commodity that is stored is said to be in a carry market. One reason for storage is seasonal variation in either supply or demand, which causes a mismatch between the time at which the commodity is produced and the time at which it is consumed. When feasible, storage is almost always costly. Suppose a commodity merchant who owns one unit of the commodity. He can sell it today, receiving S 0, or take a short position at a forward contract and time T and receive F 0. In the second case you must store the commodity up to time T. If we denote the future value of storage costs for one unit of the commodity from time 0 to T as U, then the standard arbitrage argument requires rt S0 e F0 U Rearranging terms we have that: rt F0 S0e U In the special case where storage costs are paid continuously and are proportional to the value of the commodity, storage cost is like a continuous negative dividend of u, and we can write the forward price as (r u)t F S e (4) The set of prices for different expiration dates for a given asset is called the forward curve for that date. 5

6 When there are no storage costs (u = 0), equation (4) reduce to our familiar forward pricing formula from Chapter 2. When there are storage costs, the forward price is higher. This is because the selling price must compensate the commodity merchant for both the financial cost of storage (interest) and the physical cost of storage. With storage costs, the forward curve can rise faster than the interest rate. We can view storage costs as a negative dividend in that, instead of receiving cash flow for holding the asset, you have to pay to hold the asset. Keep in mind that just because a commodity can be stored does not mean that it should (or will) be stored. Pencils were not stored because storage was not economically necessary: A constant new supply of pencils was available to meet pencil demand. Thus, equation (4) describes the forward price when storage occurs. Whether and when a commodity is stored are peculiar to each commodity Storage costs and the lease rate Suppose that there is a carry market for a commodity, so that its forward price is given by equation (4). What is the lease rate in this case? Again we must think as commodity lenders. If we lend the commodity, we are saved from having to pay storage cost. Thus, the lease rate should equal the negative of the storage cost. In other words, the lender will pay the borrower this storage cost. In effect, the commodity borrower is providing storage for the commodity lender, who receives back the commodity at a point in the future. The lender making a payment to the borrower generates a negative dividend The convenience yield The discussion of commodities to this point has ignored business reasons for holding commodities. For example, suppose you are a food manufacturer for whom corn is an essential output. You will hold an inventory of corn. If you end up holding too little and run out of corn, you must stop producing. Your physical inventory of corn has a value since it provides insurance that you can keep producing in case there is a disruption in the supply for corn. In this situation, corn holdings provide an extra return that is sometimes referred to as the convenience yield. What are the implications of the convenience yield for the forward price? Suppose that someone approached you to borrow a commodity from which you derived a convenience yield c proportional to the value of the commodity. The commodity lender saves u c by not physically storing the commodity; hence the commodity borrower pays c u, compensating the lender for convenience yield less storage cost. Using an argument identical to that of Table 2 we conclude that (r u c)t F0 S0e This is the restriction imposed by a reverse cash-and-carry, in which the arbitrageur borrows the commodity and goes long the forward. Now consider what happens if you perform a cash-and-carry, buying the commodity and selling it forward. If you are an average investor, you will not earn the convenience yield (it is earned only by those with a business reason to hold the commodity). You could try to lend the commodity, reasoning that the borrower could 6

7 be a commercial user to whom you would pay storage cost less the convenience yield. But those who earn the convenience yield likely already hold an inventory of the commodity. There may be no way for you to earn the convenience yield when performing a cash-and-carry. Thus, for an average investor, the cash-and-carry has the cash flows (r u)t (r u)t F0 ST ST S0e F0 S0e This expression implies that (r u)t F0 S0e if there is to be no cash-and-carry arbitrage. In summary, the price range within which there is no arbitrage is (r u c)t (r u)t Se 0 F0 Se 0 (5) The convenience yield produces a no-arbitrage region rather a no-arbitrage price. The observed lease rate will depend upon both storage costs and convenience yield. The difficulty with the convenience yield in practice is that convenience is hard to observe. The concept of the convenience yield serves two purposes: 1. It explains patterns in storage for example, why a commercial user might store the commodity when the average will not. 2. It provides additional parameter to better explain the forward curve Examples of commodity futures We will now examine particular commodities to illustrate the concepts from the previous sections Gold futures Gold is durable, relatively inexpensive to store (compared to its value), widely held, and actively produced through gold mining. Because of transportation costs and purity concerns, gold often trades in certificate form, as a claim to physical gold at a specific location. Figure 1 is a newspaper listing for the NYMEX gold futures contract. Figure 2 graphs the futures prices for all available gold futures contract, the forward curves, for different days. What is interesting about the gold forward curve is that it is steadily increasing with the time to maturity. From our previous discussion, the forward price implies a lease rate. Short-sales and loans of gold are common in the gold market, and gold borrowers have to pay the lease rate. On the lending size, large gold holders (including some central banks) put gold on deposit with brokers, in order that it may be loaned to short-sellers. The gold lenders earn the lease rate. This can be calculated in practice using equation (3). Table 4 shows the 6-month and 1-year lease rates for the four gold forward curves depicted in Figure 2, computed using equation (3). 7

8 Figure 1: NYMEX gold futures contract on July 21, Figure 2: Forward curves for gold for four dates. If you wish to hold gold as part of an investment portfolio, you can do so by holding physical gold or synthetic gold (holding bonds and going long gold futures). Which should you do? If you hold physical gold without lending it, and if the lease rate is positive you forgo the lease rate. You also bear the storage costs. Thus, synthetic gold is generally the preferable way to obtain gold price exposure. Table 4 shows that the 6-month gold lease rate is 1.46% in June Thus, by holding physical gold instead of synthetic gold, an investor would lose the 1.46% return. In June 2003 and 2004, however, the lease rate was about -0.1%. If storage costs are about 0.1%, an investor would be indifferent between holding physical and synthetic gold. This lease rate could also include the convenience yield obtained by a nonfinancial holder of gold who use it for manufacturing purposes. 8

9 Table 4: 6-month and 12-month lease rates for four dates Natural gas Natural gas is a market in which seasonality and storage costs are important. Figure 3 shows a newspaper listing for natural gas futures. Natural gas has several interesting characteristics. First, gas is costly to transport internationally, so prices and forward curves vary regionally. Second, once a given well has begun production, gas is costly to store. Third, demand for gas is highly seasonal, with peak demand arising from heating in winter months. Thus there is a steady stream of production with variable demand, which leads to large and predictable price swings. Figure 4 displays 3-year (2002) and 6-year (2001, 2003, and 2004) forward curve for different days. Figure 3: NYMEX natural gas futures contract, on July 21,

10 Figure 4: Forward curves for natural gas for four days. Gas storage is costly and demand for gas is highest in winter. The steady rise of the forward curve during the fall months suggests that storage occurs just before the heaviest demand. Table 5 shows prices for October through December. Table 5: June natural gas futures prices for October through December. The monthly increase in gas prices over these months ranges from $0.13 to $0.23. Assuming that the interest rate is about 0.15% per month and that you use the equation rt F0 S0e U storage cost U in November 2004 would satisfy e U implying an estimated storage cost of U = $ Crude oil Both oil and natural gas produce energy and are extracted from wells, but the different physical characteristics and uses of oil lead to a different forward curve than that of 10

11 gas. Oil is easier to transport than gas. Transportation of oil takes time, but oil has a global market. Oil is easier to store than gas. Thus, seasonals in the price of crude oil are relatively unimportant. Figure 5 shows a newspaper listing for oil futures. The NYMEX forward curve on four dates is plotted in Figure 6. Figure 5: NYMEX crude oil futures contract, on July 21, Figure 6: Forward curves for crude oil for four days. On the four dates in the figure, near-term oil prices range from $25 to $40, while the 7-year forward price in each case is between $22 and $30. The long-run forward price is less volatile than the short-run forward price, which makes economic sense. In the short-run, an increase in demand will cause a price increase since supply is fixed. In the long-run, however, both supply and demand have time to adjust to price changes with the result that prices movements are attenuated. 11

12 Exercises 1. The current price of oil is $32.00 per barrel. Forward prices for 6, 9 and 12 months are $31.37, $30.75, $30.14, and $ Assuming a 2% annual risk-free rate, what is the annualized lease rate for each maturity? Is this an example of contango or backwardation? 2. Suppose that the pencils cost $0.20 today and the lease rate for pencils is 5%. The risk-free rate is 10%. The pencil price in 1 year is uncertain and pencils can be stored without any cost. (a) If you short-sell a pencil for 1 year, what payment do you have to make for the pencil lender? Would it make sense for a financial investor to store pencils in equilibrium? (b) Show that the equilibrium forward price is $ (c) Explain what ranges of forward prices are ruled out by arbitrage in the four cases where pencils can and cannot be short-sold and can and cannot be loaned. 3. Suppose that the gold spot price is $300/oz., the 1-year forward price is and the risk-free rate is 5%. (a) What is the lease rate? (b) What is the return on a cash-and-carry strategy in which gold is not loaned? (c) What is the return on a cash-and-carry strategy in which gold is loaned, earning the lease rate? 4. Consider the following forward price curve for widgets: Dec Mar Jun Sep Dec Mar Jun Assume that the risk-free rate is 6% and the storage cost of widgets is $0.03 quarterly. (a) What are some possible explanations for the shape of this forward curve? (b) What annualized rate of return do you earn on a cash-and-carry entered into December 2004 and closed in March 2005? Is your answer sensible? (c) What annualized rate of return do you earn on a cash-and-carry entered into December 2004 and closed in September 2005? Is your answer sensible? Remark: Do not borrow money to conduct a cash-and-carry. Go and directly buy the widgets. You will have a cash outflow at time 0 and a cash inflow at time T. Measure the return of this investment. Source: Mc Donald, R., Derivatives Markets, 2 nd edition, Addison Wesley 12