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CHAPTER 3 Futures Prices In this chapter, we discuss how futures contracts are priced. This chapter is organized into the following sections: 1. Reading Futures Prices 2. The Basis and Spreads 3. Models of Futures Prices 4. Futures Prices and Expectations 5. Future Prices and Risk Aversion 6. Characteristics of Futures Prices Chapter 3 1 Reading Futures Prices TERMINOLOGY To understand how to read the Wall Street Journal futures price quotations, we need to first understand some terminology. Spot Price Spot price is the price of a good for immediate delivery. Nearby Contract Nearby contracts are the next contract to mature. Distant Contract Distant contracts are contracts that mature sometime after the nearby contracts. Chapter 3 2 Reading Futures Prices TERMINOLOGY Settlement Price Settlement price is the price that contracts are traded at the end of the trading day. Trading Session Settlement Price New term used to reflect round-the-clock trading. Open Interest Open interest is the number of futures contracts for which delivery is currently obligated. Chapter 3 3 Reading Futures Prices Insert figure 3.1 Here Chapter 3 4 How Trading Affects Open Interest The last column in Figure 3.1 shows the open interest or total number of contracts outstanding for each maturity month. Assume that today, Dec 1997, widget contract has just been listed for trading, but that the contract has not traded yet. Table 3.1 shows how trading affects open interest at different times (t). Table 3.1 How Trading Affects Open Interest Time Action Open Interest t=0 Trading opens for the popular widget contract. 0 t=1 Trader A buys and Trader B sells 1 widget contract. 1 t=2 Trader C buys and Trader D sells 3 widget contracts. 4 t=3 Trader A sells and Trader D buys 1 widget contract. (Trader A has offset 1 contract and is out of the market. Trader D has offset 1 contract and is now short 2 contracts.) 3 t=4 Trader C sells and Trader E buys 1 widget contract. 3 Ending Positions Trader Long Position B C D E All Traders Short Position 1 2 2 1 3 Chapter 3 3 5 Open Interest &Trading Volume Patterns Insert Figure 3.2 Here Insert Figure 3.3 Here Chapter 3 6 The Basis The Basis The basis is the difference between the current cash price of a commodity and the futures price for the same commodity. Basis S 0 F 0, t S0 = current spot price F0,t = current futures price for delivery of the product at time t. The basis can be positive or negative at any given time. Normal Market Price for more distant futures are higher than for nearby futures. Inverted Market Distant futures prices are lower than the price for contracts nearer to expiration. Chapter 3 7 The Basis Table 3.2 Gold Prices and the Basis (July 11) Contract Prices CASH JUL (this year) AUG OCT DEC FEB (next year) APR JUN AUG OCT DEC 353.70 354.10 355.60 359.80 364.20 368.70 373.00 377.50 381.90 386.70 391.50 The Basis -.40 -1.90 -6.10 -10.50 -15.00 -19.30 -23.80 -28.20 -33.00 -37.80 Example: if the current price of gold in the cash market is $353.70 (July 11) and a futures contract with delivery in December is $364.20. How much is the basis? Basis S 0 F 0, t Basis $353 .7 364 .20 $10.50 Basis $10.50 Chapter 3 8 The Basis Convergence As the time to delivery passes, the futures price will change to approach the spot price. When the futures contract matures, the futures price and the spot price must be the same. That is, the basis must be equal to zero, except for minor discrepancies due to transportation and other transactions costs. The relatively low variability of the basis is very important for hedging. Insert Figure 3.4 here Insert Figure 3.5 here Chapter 3 9 Spreads Spread A spread is the difference in price between two futures contracts on the same commodity for two different maturity dates: Spread F 0, t k F 0, t Where F0,t = The current futures price for delivery of the product at time t. This might be the price of a futures contract on wheat for delivery in 3 months. F0,t+k = The current futures price for delivery of the product at time t +k. This might be the price of a futures contract for wheat for delivery in 6 months. Spread relationships are important to speculators. Chapter 3 10 Spreads Suppose that the price of a futures contract on wheat for delivery in 3 months is $3.25 per bushel. Suppose further that the price of a futures contract on wheat for delivery in 6 months is $3.30/bushel. What is the spread? Spread F 0, t k F 0, t Spread $3.30 $3.25 $0.05 Insert Figure 3.7 Here Chapter 3 11 Repo Rate Repo Rate The repo rate is the finance charges faced by traders. The repo rate is the interest rate on repurchase agreements. A Repurchase Agreement An agreement where a person sells securities at one point in time with the understanding that he/she will repurchase the security at a certain price at a later time. Example: Pawn Shop. Chapter 3 12 Arbitrage An Arbitrageur attempts to exploit any discrepancies in price between the futures and cash markets. An academic arbitrage is a risk-free transaction consisting of purchasing an asset at one price and simultaneously selling it that same asset at a higher price, generating a profit on the difference. Example: riskless arbitrage scenario for IBM stock trading on the NYSE and Pacific Stock Exchange. Assumptions: 1. Perfect futures market 2. No taxes 3. No transactions costs 4. Commodity can be sold short Arbitrageur Buys IBM Arbitrageur Sells IBM Riskless Profit Price Exchange ($105) $110 $ 5 Pacific Stock E. NYSE Chapter 3 13 Models of Futures Prices Cost-of-Carry Model The common way to value a futures contract is by using the Cost-of-Carry Model. The Cost-of-Carry Model says that the futures price should depend upon two things: – The current spot price. – The cost of carrying or storing the underlying good from now until the futures contract matures. Assumptions: – There are no transaction costs or margin requirements. – There are no restrictions on short selling. – Investors can borrow and lend at the same rate of interest. In the next section, we will explore two arbitrage strategies that are associated with the Cost-and-Carry Model: – Cash-and-carry arbitrage – Reserve cash-and-carry arbitrage Chapter 3 14 Cash-and-Carry Arbitrage A cash-and-carry arbitrage occurs when a trader borrows money, buys the goods today for cash and carries the goods to the expiration of the futures contract. Then, delivers the commodity against a futures contract and pays off the loan. Any profit from this strategy would be an arbitrage profit. 0 1. Borrow money 2. Sell futures contract 3. Buy commodity 1 4. Deliver the commodity against the futures contract 5. Recover money & payoff loan Chapter 3 15 Reverse Cash-and-Carry Arbitrage A reverse cash-and-carry arbitrage occurs when a trader sells short a physical asset. The trader purchases a futures contract, which will be used to honor the short sale commitment. Then the trader lends the proceeds at an established rate of interest. In the future, the trader accepts delivery against the futures contract and uses the commodity received to cover the short position. Any profit from this strategy would be an arbitrage profit. 0 1. Sell short the commodity 2. Lend money received from short sale 3. Buy futures contract 1 4. Accept delivery from futures contract 5. Use commodity received to cover the short sale Table 3.5 summarizes the cash-and-carry and the reverse cash-and-carry strategies. Chapter 3 16 Arbitrage Strategies Table 3.5 Transactions for Arbitrage Strategies Market Cash-and-Carry Debt Borrow funds Lend short sale proceeds Physical Buy asset and store; deliver against futures Sell asset short; secure proceeds from short sale Futures Sell futures Buy futures; accept delivery; return physical asset to honor short sale commitment Reverse Cash-and-Carry Chapter 3 17 Cost-of-Carry Model The Cost-of-Carry Model can be expressed as: F 0, t S 0(1 C0, t ) Where: S0 = the current spot price F0,t = the current futures price for delivery of the product at time t. C0,t = the percentage cost required to store (or carry) the commodity from today until time t. The cost of carrying or storing includes: 1. Storage costs 2. Insurance costs 3. Transportation costs 4. Financing costs In the following section, we will examine the cost-of-carry rules. Chapter 3 18 Cost-of-Carry Rule 1 The futures price must be less than or equal to the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. F 0, t S 0(1 C 0, t ) Table 3.3 Cash-and-Carry Gold Arbitrage Transactions Prices for the Analysis: Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction t=0 $400 $450 10% Cash Flow Borrow $400 for one year at 10%. Buy 1 ounce of gold in the spot market for $400. Sell a futures contract for $450 for delivery of one ounce in one year. +$400 - 400 0 Total Cash Flow t=1 Remove the gold from storage. Deliver the ounce of gold against the futures contract. Repay loan, including interest. Chapter 3 $0 $0 +450 -440 Total Cash Flow +$10 19 Cost-of-Carry Rule 1 0 1 1. Borrow $400 2. Buy 1 oz gold 3. Sell futures contract 4. Deliver gold against futures contract 5. Repay loan Chapter 3 20 The Cost-of-Carry Rule 2 The futures price must be equal to or greater than the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. F 0, t S 0(1 C 0, t ) Table 3.4 Reverse Cash-and-Carry Gold Arbitrage Transactions Prices for the Analysis Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction t=0 $420 $450 10% Cash Flow Sell 1 ounce of gold short. Lend the $420 for one year at 10%. Buy 1 ounce of gold futures for delivery in 1 year. +$420 - 420 0 Total Cash Flow t=1 Collect proceeds from the loan ($420 x 1.1). Accept delivery on the futures contract. Use gold from futures delivery to repay short sale. $0 +$462 -450 0 Total Cash Flow +$12 Chapter 3 21 The Cost-of-Carry Rule 2 0 1. Sell short 1 oz. gold 2. Lend $420 at 10% interest 3. Buy a futures contract 1 4. Collect proceeds from loan 5. Accept delivery on futures contract 6. Use gold from futures contract to repay the short sale Chapter 3 22 The Cost-of-Carry Rule 3 Since the futures price must be either less than or equal to the spot price plus the cost of carrying the commodity forward by rule #1. And the futures price must be greater than or equal to the spot price plus the cost of carrying the commodity forward by rule #2. The only way that these two rules can reconciled so there is no arbitrage opportunity is by the cost of carry rule #3. Rule #3: the futures price must be equal to the spot price plus the cost of carrying the commodity forward to the delivery date of the futures contract. F 0, t S 0(1 C 0, t ) If prices were not to conform to cost of carry rule #3, a cash-and carry arbitrage profit could be earned. Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling. Chapter 3 23 Spreads and The Cost-of-Carry As we have just seen, there must be a relationship between the futures price and the spot price on the same commodity. Similarly, there must be a relationship between the futures prices on the same commodity with differing times to maturity. The following rules address these relationships: Cost-of-Carry Rule 4 Cost-of-Carry Rule 5 Cost-of-Carry Rule 6 Chapter 3 24 The Cost-of-Carry Rule 4 The distant futures price must be less than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date. F 0, d F 0, n(1 Cn, d ) where d > n F0,d = the futures price at t=0 for the distant delivery contract maturing at t=d. Fo,n= the futures price at t=0 for the nearby delivery contract maturing at t=n. Cn,d= the percentage cost of carrying the good from t=n to t=d. If prices were not to conform to cost of carry rule # 4, a cash-and-carry arbitrage profit could be earned. Chapter 3 25 Spreads and the Cost-of-Carry Table 3.6 shows that the spread between two futures contracts can not exceed the cost of carrying the good from one delivery date forward to the next, as required by the cost-of-carry rule #4. Table 3.6 Gold Forward Cash-and-Carry Arbitrage Prices for the Analysis Futures price for gold expiring in 1 year Futures price for gold expiring in 2 years Interest rate (to cover from year 1 to year 2) Transaction t=0 $400 $450 10% Cash Flow Buy the futures expiring in 1 year. Sell the futures expiring in 2 years. Contract to borrow $400 at 10% for year 1 to year 2. +$0 0 0 Total Cash Flow t=1 t=2 Borrow $400 for 1 year at 10% as contracted at t = 0. Take delivery on the futures contract. Begin to store gold for one year. Deliver gold to honor futures contract. Repay loan ($400 x 1.1) $0 +$400 - 400 0 Total Cash Flow $0 +$450 - 440 Total Cash Flow + $10 Chapter 3 26 The Cost-of-Carry Rule 4 0 1 1. Buy futures contract w/exp in 1 yrs. 2. Sell futures contract w/exp in 2 years 3. Contract to borrow $400 from yr 1-2 4. Borrow $400 5. Take delivery on 1 yr to exp futures contract. 6. Place the gold in storage for one yr. Chapter 3 2 7. Remove gold from storage 8. Deliver gold against 2 yr. futures contract 9. Pay back loan 27 The Cost-of-Carry Rule 5 The nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date cannot exceed the distant futures price. Or alternatively, the distant futures price must be greater than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby futures date to the distant futures date. F0,d F0,n 1 Cn,d If prices were not to conform to cost of carry rule # 5, a reverse cash-and-carry arbitrage profit could be earned. Chapter 3 28 The Cost-of-Carry Rule 5 Table 3.7 illustrates what happens if the nearby futures price is too high relative to the distant futures price. When this is the case, a forward reverse cash-and-carry arbitrage is possible. Table 3.7 Gold Forward Reverse Cash-and-Carry Arbitrage Prices for the Analysis: Futures price for gold expiring in 1 year Futures price for gold expiring in 2 years Interest rate (to cover from year 1 to year 2) Transaction t=0 $440 $450 10% Cash Flow Sell the futures expiring in one year. Buy the futures expiring in two years. Contract to lend $440 at 10% from year 1 to year 2. +$0 0 0 Total Cash Flow t=1 Borrow 1 ounce of gold for one year. Deliver gold against the expiring futures. Invest proceeds from delivery for one year. $0 + 440 - 440 Total Cash Flow t=2 Accept delivery on expiring futures. Repay 1 ounce of borrowed gold. Collect on loan of $440 made at t = 1. $0 $0 - $450 0 + 484 Total Cash Flow + $34 Chapter 3 29 The Cost-of-Carry Rule 5 0 1 1. Sell futures contract w/exp in 1 yrs. 2. Buy futures contract w/exp in 2 years 3. Contract to lend $400 from yr 1-2 4. Borrow 1 oz. gold 5. Deliver gold on 1 yr to exp futures contract. 6. Invest proceeds from delivery for one yr. Chapter 3 2 7. Accept delivery on exp 2 yr futures contract 8. Repay 1 oz. borrowed gold. 9. Collect $400 loan 30 Cost-of-Carry Rule 6 Since the distant futures price must be either less than or equal to the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date by rule #4. And the nearby futures price plus the cost of carrying the commodity from the nearby delivery date to the distant delivery date can not exceed the distant futures price by rule #5. The only way that rules 4 and 5 can be reconciled so there is no arbitrage opportunity is by cost of carry rule #6. Chapter 3 31 Cost-of-Carry Rule 6 The distant futures price must equal the nearby futures price plus the cost of carrying the commodity from the nearby to the distant delivery date. F 0, d F 0, n(1 Cn, d ) If prices were not to conform to cost of carry rule #6, a cash-and-carry arbitrage profit or reverse cash-and-carry arbitrage profit could be earned. Recall that we have assumed away transaction costs, margin requirements, and restrictions against short selling. Chapter 3 32 Implied Repo Rates If we solve for C0,t in the above equation, and assume that financing costs are the only costs associated with holding an asset, the implied cost of carrying the asset from one time point to another can be estimated. This rate is called the implied repo rate. The Cost-of-Carry model gives us: F 0, t S 0(1 C 0, t ) Solving for C 0, t F 0, t (1 C 0, t ) S0 And F 0, t 1 C 0, t S0 Chapter 3 33 Implied Repo Rates Example: cash price is $3.45 and the futures price is $3.75. The implied repo rate is? F 0, t 1 C 0, t S0 $3.75 1 0.086956 $3.45 That is, the cost of carrying the asset from today until the expiration of the futures contract is 8.6956%. Chapter 3 34 The Cost-of-Carry Model in Imperfect Markets In real markets, no less than four factors complicate the Cost-of-Carry Model: 1. Direct transactions costs 2. Unequal borrowing and lending rates 3. Margin and restrictions on short selling 4. Limitations to storage Chapter 3 35 Transaction Costs Transaction Costs Traders generally are faced with transaction costs when they trade. In this case, the profit on arbitrage transactions might be reduced or disappear altogether. Types of Transaction Costs: – Brokerage fees to have their orders executed – A bid ask spread A market maker on the floor of the exchange needs to make a profit. He/She does so by paying one price (the bid price) for a product and selling it for a higher price (the ask price). Chapter 3 36 Cost-of-Carry Rule 1 with Transaction Costs Recall that the futures price must be less than or equal to the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. F 0, t S 0(1 C 0, t ) We can modify this rule to account for transaction costs: F 0, t S 0(1 T )(1 C 0, t ) Where T is the percentage transaction cost. Chapter 3 37 Cost-of-Carry Rule 1 with Transaction Costs To show how transaction costs can frustrate an arbitrage consider Table 3.8. Table 3.8 Attempted Cash-and-Carry Gold Arbitrage Transactions Prices for the Analysis: Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction cost (T ) Transaction t=0 t=1 $400 $450 10% 3% Cash Flow Borrow $412 for one year at 10%. Buy 1 ounce of gold in the spot market for $400 and pay 3% transaction costs, to total $412. Sell a futures contract for $450 for delivery of one ounce in one year. Remove the gold from storage. Deliver the ounce of gold to close futures contract. Repay loan, including interest. Chapter 3 +$412 - 412 0 Total Cash Flow $0 $0 +450.00 -453.20 Total Cash Flow -$3.20 38 Cost-of-Carry Rule 2 with Transaction Costs Recall from Cost-of-Carry Rule 2 that the futures price must be equal to or greater than the spot price of the commodity plus the carrying charges necessary to carry the spot commodity forward to delivery. F 0, t S 0(1 C 0, t ) We can modify this rule to allow for transaction costs as follows: F 0, t S 0(1 T )(1 C 0, t ) Chapter 3 39 Cost-of-Carry Rule 2 with Transaction Costs To show how transaction costs can frustrate an attempt to reserve cash-and-carry arbitrage. Consider Table 3.9. Table 3.9 Attempted Reverse Cash-and-Carry Gold Arbitrage Prices for the Analysis: Spot price of gold Future price of gold (for delivery in one year) Interest rate Transaction costs (T ) Transaction t=0 t=1 $420 $450 10% 3% Cash Flow Sell 1 ounce of gold short, paying 3% transaction costs. Receive $420(.97) = $407.40. Lend the $407.40 for one year at 10%. Buy 1 ounce of gold futures for delivery in 1 year. Collect loan proceeds ($407.40 x 1.1). Accept gold delivery on the futures contract. Use gold from futures delivery to repay short sale. +$407.40 - 407.40 0 Total Cash Flow $0 +$448.14 -450.00 0 Total Cash Flow -$1.86 Chapter 3 40 No-Arbitrage Bounds Incorporating transaction costs and combining cost-ofcarry rules 1 and 2, we have the following. S 0(1 T )(1 C 0, t ) F 0, t S 0(1 T )(1 C 0, t ) This equation defines the “No Arbitrage Bounds”. That is, as long as the futures price trades within this range, no cash-and-carry or reverse cash-and-carry arbitrage transactions will be profitable. Table 3.10 illustrates this equation. Chapter 3 41 No-Arbitrage Bounds Table 3.10 Illustration of No-Arbitrage Bounds Prices for the Analysis: Spot price of gold $400 Interest rate Transaction costs (T ) 10% 3% No-Arbitrage Futures Price in Perfect Markets F0,t = S0(1 + C ) = $400(1.1) = $440 Upper No-Arbitrage Bound with Transaction Costs F0,t < S0(1 + T )(1 + C ) = $400(1.03)(1.1) = $453.20 Lower No-Arbitrage Bound with Transaction Costs F0,t > S0(1 B T )(1 + C ) = $400(.97)(1.1) = $426.80 In this case, as long as the futures price is between $426.80 and $453.20, arbitrage transactions will not be profitable. Chapter 3 42 No-Arbitrage Bounds $453.20 Futures Price $426.80 Time Chapter 3 43 Differential Transaction Costs Situations occur where all traders do not have equal transaction costs. For example, a floor trader, trading on his own behalf would have a lower transaction cost than others. So while he/she might be able to earn an arbitrage profit, others could not. Such a transaction is called a quasi-arbitrage. Chapter 3 44 Unequal Borrowing & Lending Rates Thus far we have assumed that investors can borrow and lend at the same rate of interest. Anyone going to a bank knows that this possibility generally does not exist. Incorporating differential borrowing and lending rates into the Cost-of-Carry Model gives us: S0 (1 T )(1 CL ) F 0, t S 0(1 T )(1 CB ) Where: CL = lending rate CB = borrowing rate Chapter 3 45 Unequal Borrowing & Lending Rates Table 3.11 Illustration of No-Arbitrage Bounds with Differential Borrowing and Lending Rates Prices for the Analysis: Spot price of gold $400 Interest rate (borrowing) 12% Interest rate (lending) 8% Transaction costs (T ) 3% Upper No-Arbitrage Bound with Transaction Costs and a Borrowing Rate F0,t < S0(1 + T )(1 + CB ) = $400(1.03)(1.12) = $461.44 Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate F0,t > S0(1 B T )(1 + CL ) = $400(.97)(1.08) = $419.04 Chapter 3 46 Restrictions on Short Selling Thus far we have assumed that arbitrageurs can sell short commodities and have unlimited use of the proceeds. There are two limitations to this in the real world: – It is difficult to sell some commodities short. – Investors are generally not allowed to use all proceeds from the short sale. How do limitations on the use of funds from a short sale affect the Cost-of-Carry Model? We can examine this by editing our transaction cost and differential borrowing Cost-of-Carry Model as follows: Chapter 3 47 Restrictions on Short Selling The transaction cost and differential cost of borrowing model is as follows: S0 (1 T )(1 CL ) F 0, t S 0(1 T )(1 CB ) We modify this by recognizing that you will not get all of the proceeds from the short sale. You will get some portion of the proceeds. S 0(1 T )(1 fCL ) F 0, t S 0(1 T )(1 CB ) Where: ƒ = the proportion of funds received Chapter 3 48 Restrictions on Short Selling Table 3.12 illustrates the effect of limitations on the use of short sale proceeds. Table 3.12 Illustration of No-Arbitrage Bounds with Various Short Selling Restrictions Prices for the Analysis: Spot price of gold $400 Interest rate (borrowing) 12% Interest rate (lending) 8% Transaction costs (T ) 3% Upper No-Arbitrage Bound with Transaction Costs and a Borrowing Rate F0,t < S0(1 + T )(1 + CB ) = $400(1.03)(1.12) = $461.44 Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 1.0 F0,t > S0(1 B T )(1 + fCL ) = $400(.97)[1 + (1.0)(.08)] = $419.04 Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 0.75 F0,t > S0(1 B T)(1 + fCL ) = $400(.97)[1 + (.75)(.08)] = $411.28 Lower No-Arbitrage Bound with Transaction Costs and a Lending Rate, f = 0.5 F0,t > S0(1 B T )(1 + f CL ) = $400(.97)[1 + (0.5)(.08)] = $403.52 Chapter 3 49 Restrictions on Short Selling The effect of the proceed use limitation is to widen the noarbitrage trading bands. $461.44 Futures Price $403.52 Time Chapter 3 50 Limitations on Storage The ability to undertake certain arbitrage transactions requires storing the product. Some items are easier to store than others. Gold is very easy to store. You simply rent a safe deposit box at the bank and place your gold there for safekeeping. Wheat is moderately easy to store. How about milk or eggs? They can be stored, but not for long periods of time. To the extent that a commodity can not be stored, or has limited storage life, the Cost-of-Carry Model may not hold. Chapter 3 51 How Traders Deal with Market Imperfections The costs associated with carrying commodities forward vary widely among traders. If you are a floor trader, your transaction costs will be very low. If you are a farmer with unused grain storage on your farm, your cost of storage will be very low. Individuals with lowest trading costs (storage costs, and cost of borrowing) will have the most profitable arbitrage opportunities. The ability to sell short varies between traders. Chapter 3 52 The Concept of Full Carry Market To the extent that markets adhere to the following equations markets are said to be at “full carry”: F 0, t S 0(1 C 0, t ) F 0, d F 0, n(1 Cn, d ) If the futures price is higher than that specified by above equations, the market is said to be above full carry. If the futures price is below that specified by the above equations, the market is said to be below full carry. To determine if a market is at full carry, consider the following example: Suppose that: September Gold December Gold Bankers Acceptance Rate Chapter 3 $410.20 $417.90 7.8% 53 The Concept of Full Carry Market Step 1: compute the annualized percentage difference between two futures contracts. AD ( FF ) 0, d M 12 1 0. N Where AD = Annualized percentage difference M = Number of months between the maturity of the futures contracts. AD ( 3 $417.90 12 1 $410.20 ) AD 1.0772 1 AD 0.0772 Step 2: compare the annualized difference to the interest rate in the market. The gold market is almost always at full carry. Other markets can diverge substantially from full carry. Chapter 3 54 The Concept of Full Carry Market Insert Figure 3.1 here Futures Price Quotations Chapter 3 55 Market Features That Promote Full Carry Ease of Short Selling To the extent that it is easy to short sell a commodity, the market will become closer to full carry. Difficulties in short selling will move a market away from full carry. Selling short of physical goods like wheat is more difficult, while selling short of financial assets like Eurodollars is much easier. For this reason, markets for financial assets tend to be closer to full carry than markets for physical assets. Large Supply If the supply of an asset is large relative to its consumption, the market will tend to be closer to full carry. If the supply of an asset is low relative to its consumption, the market will tend to be further away from full carry. Chapter 3 56 Market Features that Promote Full Carry Non-Seasonal Production To the extent that production of a crop is seasonal, temporary imbalances between supply and demand can occur. In this case, prices can vary widely. – Example: in North America, wheat harvest occurs between May and September. Non-Seasonal Consumption To the extent that consumption of commodity is seasonal, temporary imbalances between supply and demand can occur. – Example: propane gas during winter Turkeys during thanksgiving High Storability A market moves closer to full carry if its underline commodity can be stored easily. The Cost-of-Carry Model is not likely to apply to commodities that have poor storage characteristics. – Example: eggs Chapter 3 57 Convenience Yield When there is a return for holding a physical asset, we say there is a convenience yield. A convenience yield can cause futures prices to be below full carry. In extreme cases, the cash price can exceed the futures price. When the cash price exceeds the futures price, the market is said to be in “backwardation.” Chapter 3 58 Futures Prices and Expectations If futures contracts are priced appropriately, the current futures price should tell us something about what the spot price will be at some point in the future. There are four theories about futures prices and future spot prices: – Expectations or Risk Neutral Theory – Normal Backwardation – Contango – Capital Asset Pricing Model (CAPM) Speculators play an important role in the futures market, they ensure that futures prices approximately equal the expected future spot price. Chapter 3 59 Expectations or Risk Neutral Theory The Expectations Theory says that the futures price equals the expected future spot price. F 0, t E(S 0) Where E(S 0) = the expected future spot price Chapter 3 60 Normal Backwardation The Normal Backwardation Theory says that futures markets are primarily driven by hedgers who hold short positions. For example, farmers who have sold futures contracts to reduce their price risk. The hedgers must pay speculators a premium in order to assume the price risk that the farmer wishes to get rid of. So speculators take long positions to assume this price risk. They are rewarded for assuming this price risk when the futures price increases to match the spot price at maturity. So this theory implies that the futures price is less than the expected future spot price. F 0, t E(S 0) Chapter 3 61 Normal Backwardation Figure 3.9 depicts a situation that might prevail in the futures market for a commodity. Insert figure 3.9 here Chapter 3 62 Contango The Contango Theory says that futures markets are primarily driven by hedgers who hold long positions. For example, grain millers who have purchased futures contracts to reduce their price risk. The hedgers must pay speculators a premium in order to assume the price risk that the grain miller wishes to get rid of. So speculators take short positions to assume this price risk. They are rewarded for assuming this price risk when the futures price declines to match the spot price at maturity. So this theory implies that the futures price is greater than the expected future spot price. F 0, t E(S 0) Chapter 3 63 Contango Figure 3.10 illustrates the price patterns for futures under different scenarios. Insert Figure 3.10 here Chapter 3 64 Capital Asset Pricing Model (CAPM) The CAPM Theory is consistent with the Normal Backwardation Theory, the Contango Theory, and the Expectations Theories. However, the CAPM Theory suggests that speculators will be rewarded only for the systematic portion of the risk that they are assuming. Chapter 3 65 Statistical Characteristics of Futures Prices Futures prices exhibit statistically significant first-order autocorrelation. However, not strong enough to allow profitable trading strategies. Autocorrelation – A time series is correlated when one observation in the series is statistically related to another. – first-order autocorrelation occurs when one observation is related to the immediately preceding observations. The Volatility of Futures Prices – Evidence suggests that futures trading does not increase the volatility of the cash market. Time to Expiration and Futures Price Volatility – Price changes are large when more information is known about the future spot price. Chapter 3 66