12 Forwards and Futures

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12.1 Introduction

Three questions sit at the heart of the study of forwards and futures. First, what is the precise relationship between the current spot price of an asset and the price agreed upon today for delivery at a future date — and why must that relationship hold by arbitrage? Second, how do futures contracts differ from forward contracts in their institutional structure, and what practical consequences follow from those differences for hedgers? Third, how does the futures price compare to the expected future spot price, and what does that gap reveal about how risk is allocated between hedgers and speculators? These questions are fundamental to financial markets because forwards and futures are among the most widely traded instruments for managing price risk. Commodity producers hedge against price declines; currency traders lock in exchange rates; portfolio managers use equity index futures to adjust market exposure rapidly without trading the underlying securities. The theoretical foundations of these instruments underpin an enormous volume of global financial activity.

The first question is answered by a straightforward arbitrage argument. Owning a forward contract that delivers one unit of an asset in the future is economically equivalent to borrowing money today, buying the asset in the spot market, and holding it until delivery. Because these two strategies produce the same outcome, they must have the same cost, which implies that the forward price must equal the spot price compounded at the risk-free rate over the contract’s life. Any deviation from this cost-of-carry relationship creates a riskless profit opportunity that arbitrageurs will immediately exploit, restoring the equilibrium. This principle is derived formally in the chapter and illustrated with a currency forward example.

The second question concerns the institutional differences between forward and futures contracts and their practical implications. Futures contracts are standardized and traded on organized exchanges, which solves the counterparty credit risk problem inherent in over-the-counter forward contracts but introduces the need for margin accounts and daily marking to market. When a futures position moves against a trader, losses are collected from the margin account each day rather than accumulated until expiration; this daily settlement can create cash flow demands that differ from the economics of the underlying hedge. The chapter works through margin mechanics in detail, using a corn futures example from the CBOT, and explains basis risk — the residual price uncertainty that arises when the hedging contract does not perfectly match the timing or specification of the underlying exposure.

The third question moves from no-arbitrage pricing to an equilibrium view of the futures price. The expectations hypothesis holds that the futures price is simply the market’s best forecast of the future spot price. Normal backwardation and contango offer richer explanations grounded in the asymmetric hedging demands of producers and consumers: if natural sellers dominate, speculators must be compensated with a price discount to take the opposite side, so the futures price lies below the expected spot price. Modern portfolio theory provides a further refinement, linking the gap between the futures price and the expected spot price to the asset’s systematic risk, in a way that is fully consistent with the CAPM.

The chapter opens by defining forward contracts and deriving the spot-forward price relationship through the arbitrage argument. It then turns to futures contracts, explaining exchange-based trading, margin requirements, marking to market, and basis risk through a series of concrete examples. The chapter closes by surveying the three classical theories of futures pricing — the expectations hypothesis, normal backwardation, and the modern portfolio theory perspective — providing both the technical tools and the economic intuition needed to analyze the pricing and strategic use of these instruments.

In the news

Futures prices move on tomorrow’s expected supply, not just today’s. CNBC reported Brent crude swinging around $80 a barrel in mid-2026 as U.S.–Iran talks faltered and traders reassessed the risk of disruption at the Strait of Hormuz — after an earlier conflict had briefly pushed prices above $120. Anyone exposed to those swings, from airlines to producers, manages them with the forward and futures contracts, and the cost-of-carry pricing, developed in this chapter. Read it at CNBC.

12.2 Forward contracts

A forward contract is an agreement to buy or sell a certain asset at a certain future time for a certain price. A spot market is a market where an asset is sold for immediate delivery. Forward contracts are sold in over-the-counter markets. The investor with the long position agrees to buy the asset in the future, while the investor who has the short position agrees to sell the asset in the future. Notice that there are two financial instruments working here. The first is the asset that will be bought or sold in the future. The second is the forward contract, or the agreement to buy or sell that asset in the future. Thus, one can speak of the price of the asset (usually called the spot price if speaking of the current price of the asset) or the price of the forward contract.

Example.: As of 21 March 2011 the Euro spot rate was $1.4208. The 3 months forward rate for Euros was $1.4187. Forward rates can be used to hedge currency risk. To see this, consider a U.S. bank that knows that in 3 months it will be receiving a payment of EUR 1.5M. The bank faces currency risk because it knows that it will want to exchange the euros that it receives for dollars as soon as it receives them. Thus, if the spot rate is low in three months then the dollar value of the payment will be low and oppositely if the spot rate is high. In order to hedge this risk, the bank could enter into a forward contract today. It would agree to sell EUR 1.5M in three months. The current forward rate indicates that a counterparty would agree to pay $1.4187 \cdot 1,500,000 = \$2,128,050.0$ in three months to receive that EUR 1.5M. The bank is short a forward contract on the euro.

The payoff to a forward contract depends on the spot price of the underlying asset at the date of maturity of the contract. For a trader that is long the forward contract with delivery price $K$, the payoff of the contract at maturity is $S_T - K$ where $S_T$ represents the spot price of the underlying asset at time $T$ (maturity). If the spot price $S_T$ is larger than the delivery price agreed upon in the contract $K$, then at maturity the long trader owns an agreement to purchase the asset for less than its current price. Thus the payoff to owning the contract is positive. If at maturity the spot price $S_T$ is less than the agreed upon delivery price $K$ then the long trader is required to purchase the asset at a price that is higher than she would have to pay if she were purchasing the asset in the spot market. The payoff to being short the contract is $K - S_T$ and analogous statements can be made.

The spot price of an asset and the forward price of the asset are related. To see why, consider what it means to be long a forward contract that matures in 1 year and will deliver $Y$ units of the asset for a price of $F$ per unit. This means that in 1 year the trader will need to pay $FY$ and take receipt of $Y$ units of the asset. An alternative way to have $Y$ units (and pay for them) in one year is for the investor to borrow some amount of money today at interest rate $r$ to be paid back in one year and use it to buy $Y$ units of the asset today. Thus, the investor could borrow $\$SY$ today and pay it back in one year (i.e. pay back $SY(1+r)$). These strategies are equivalent since both give you possession of the asset in one year in exchange for some amount of money. Thus, the payoff to them must be the same. In other words $(1+r)SY = FY$ or $(1+r)S = F$.

To see why, assume that $(1+r)S > F$. Then the investor could sell short the asset today and receive $S$ to be invested at the interest rate $r$. At the same time the investor would go long the forward contract. In one year, the investor now has $\$(1+r)S$ out of which she pays $F$ to receive the asset that is then used to close out the short position. The profit made is $(1+r)S - F$ and it is risk free. On the other hand, if $(1+r)S < F$ then the trader can borrow $S$ to buy one unit of the asset and takes a short position in the forward contract. In one year the investor delivers the unit of the asset to fulfill the short position on the contract and from the proceeds of the short contract ($F$) pays off the loan ($(1+r)S$) that was taken out. The risk-free proceeds are $F - S(1+r)$. Since both of these strategies are risk-free, they should provide the same rate of return in equilibrium. Hence $(1+r)S = F$.

The reason that the forward price is higher than the spot price is that purchasing the asset today requires using money that would otherwise be used for something else. In the forward contract the payment does not have to be made until the contract matures.

Example.

Suppose that the current one-year, continuously compounded interest rate is $r$. The spot price for one Euro is $1.4208 and the three-month forward price is $1.4187. What is the implied interest rate $r$? Using continuous compounding we have that the relationship between the spot and future prices is $Se^{0.25r} = F$ (the horizon is $T = 0.25$ years). Solving, take logs of $e^{0.25r} = F/S$ to get $0.25\,r = \ln(F/S)$, so

\[r = \frac{1}{0.25}\ln\!\left(\frac{F}{S}\right) = 4\ln\!\left(\frac{1.4187}{1.4208}\right) = 4\,(-0.0014791) = -0.0059165.\]

Because the forward price is below the spot price ($F < S$), the log is negative and the implied rate is negative: about $-0.59\%$. (The forward selling at a discount to spot signals a slightly negative annualized dollar interest rate over the quarter.)

12.3 Types of traders

Traders can generally (although not perfectly) be categorized as hedgers or speculators. Hedgers wish to reduce the risk inherent in their portfolios by taking forward and futures positions that offset their asset positions. Speculators generally have no asset position and wish to take a view (i.e. speculate) on changes in either the future/forward price, or the relative price of the futures and spot markets.

12.4 Futures contracts

Futures contracts are fundamentally similar to forward contracts. A futues contract is an agreement to buy or sell an asset at a future date for a specified price. There are some important differences between futures and forwards however. The first is that futures contracts are generally standardized to the point that they can be traded on an exchange.

Example.: The futures contract for corn trades on the Chicago Board of Trade and is a contract for 5000 bushels of number 2 yellow corn. The party that is short the futures contract (that is has agreed to deliver the corn) may also substitute number 1 yellow corn for an increased price to the long position or number 3 yellow corn for a discount. The CBOT establishes contracts that expire in December, March, May, July and September on the 15th of the month. The last day of trading is the last business day before the 15th and delivery must be made by the second business day following the last trading date of the delivery month. The exchange also stipulates that for each contract, those in a hedging position must have $1500 in margin on reserve in their margin account while those in a speculative position must have $2025 in their margin account. Currently (as of 13 November 2008) the futures price for March 09 delivery is $388 \frac{6}{32}$ cents per bushel or $19,409.375. The contract also must specify where delivery will occur.

12.4.1 Convergence of futures and spot prices

The first thing to note about futures prices is that they must converge to the spot price on the date of delivery. To see why this must be the case, suppose that the price of delivery of March delivery of on 13 March 2009 (the last day of trading of the contract) is something other than the spot price for the purchase of corn. If the futures price is higher, then an investor would sell the futures contract agreeing to deliver corn and buy the corn to be delivered in the spot market for less than the futures price. This profit would be risk free and nearly immediate. Similarly if the futures price is lower than the spot price on the last day of trading then a trader could go long the futures contract agreeing to take delivery of the corn and sell the corn short in the spot market. The delivery from the futures contract would fulfill the obligation of the short selling in the spot market and the difference in prices would be risk-free profit for the trader.

12.4.2 Closing out positions

Most futures contracts are closed out before the end of trading. That is, most of the corn traded on the CBOT is never delivered. This is done by entering into the opposite trade with the market before the close of trading. If a trader is short 15 futures contracts then she must purchase 15 contracts to close her position. Notice that since a futures contract is an agreement to sell or buy in the future, going short in the futures markets is easy and does not require finding some of the asset in the market to borrow before shorting. We will see that the fact that the futures price converges to the spot price means that the futures contract still provides a hedge against price movements even if delivery is not taken.

Looking at futures price data¹ notice the column OpenInt is the measure of one side of the market. That is, it is the number of short positions, or equivalently long positions. If you look at rice today (13 March 2008) it can be noticed that the open interest is rather small since the November contract is about to stop trading.

12.4.3 Settlement and Margins

Futures contracts are settled daily based on the market value of that futures contract. This process is called marking to market.

Example.: Consider the March 2009 corn contract discussed earlier. The current value of one contract is $19,409.375. The margin required for a hedger is $1,500 per contract. Consider a hedger that is long one contract and whose margin account has $2000 in it. Suppose that tomorrow the price per bushel of corn for March delivery goes up to 390 cents per bushel. Now being long a March corn contract has value $3.90 \cdot 5000 = 19,500$. Thus, the value of one contract has increased by 19,500 - 19,409.375 = 90.625. At the end of the day then, the clearinghouse/exchange deducts $90.625 from the margin account of each trader that is short the March contract and adds $90.625 to the margin account of each trader that is long the March contract. So the balance of our trader's margin account is now $2090.625. If the maintenance margin is $1500, then when the price/bushel of the futures contract solves $19409.375 - 5000 \cdot \frac{p}{100} = 500$ then the investor will have to meet a margin call. This occurs when the price is $378 \frac{6}{32}$ cents.

As mentioned previously, even if the position is closed before the close of trading, the futures contract provides a hedge for a hedging investor. Consider the following example.

Example.: Consider an oil producer that has just agreed to sell 1 million barrels of crude oil. It is now 15 May. The price of the sale will be the market price on 15 August. Thus, if the price of oil increases by $0.01, then the increase in the proceeds of this sale increase by $1,000,000\cdot 0.01 = 10,000$. Likewise, for every penny that the price of oil decreases the revenue from the sale decreases by $10,000. Suppose that the spot price of crude oil on 15 May is $60 per barrel and the crude oil futures price for August delivery (NYMEX) is $59/barrel. Each oil futures contract on NYMEX is for 1000 barrels, so to hedge this position the producer will go short 1000 oil contracts. Closing out the contract near 15 August will imply that the producer locks in an approximate price of $59/barrel. To see this, suppose that the spot price on 15 August is $56. Then the sale of oil under the original agreement brings in $56M. Since the futures contract expires near 15 August, the price of the futures contract will be approximately $56/barrel. Therefore, for each contract the company has gained $3\cdot1000 = \$3000$. Multiplying this by the number of contracts shows that the gains from the futures position are $3000*1000 = 3,000,000$. So the net proceeds from the futures position and the sale of oil is $\$56M + \$3M = \$59M$. On the other hand, suppose that the spot price of oil on 15 August is $63. In this case, the sale of the oil brings revenue of $63M while the futures contracts show a loss of $4\cdot 1000 \cdot 1000 = 4,000,000$. Thus, the total proceeds from the oil sale and the futures contracts is $\$63M - \$4M = \$59M$.

12.4.4 Basis risk

Many times when using futures to hedge there does not exist a contract that expires at exactly the same time as the asset that will be delivered. As such, one may have to hedge using an imperfect instrument. This leads to basis risk. For example, if a farmer is going to deliver wheat in April 2009, then she may use the May 2009 contract to lock in the price of the wheat for delivery. However, since May comes after April, it is likely that the spot price of wheat in April will not be equal to the futures price of wheat for May delivery. The difference between these is the basis. In particular, if $T$ is the time at which the farmer will make delivery and $S(t)$ is the spot price with $F_{\tau}(t)$ the futures price for delivery at time $\tau$ when the time is $t$ then

\[Basis(T) = S(T) - F_{\tau}(T)\]

12.5 Pricing futures contracts

Consider first the forward price. By a cost-of-carry (no-arbitrage) argument, buying the asset today at the spot price $S$ with money borrowed at the continuously compounded rate $r$ and holding it to delivery costs $Se^{rt}$ at time $t$; since this replicates the forward’s delivery, the forward price must be $\tilde F = Se^{rt}$, for otherwise one could borrow-and-carry (or short-and-lend) for a riskless profit.

When the interest rate is constant from the time that the futures contract is purchased until it expires, the futures price equals this forward price, $F = \tilde F = Se^{rt}$. The reason is a rollover argument. A futures position is marked to market daily, so it throws off (or absorbs) cash each day rather than only at maturity. But with a known, constant rate $r$ those daily settlement flows can be borrowed or invested at exactly that same rate, and by holding a slightly scaled (“tailed”) number of contracts — $e^{-r(t-s)}$ contracts on day $s$ — the reinvested daily gains accumulate to precisely the single terminal payoff of the forward. The two contracts therefore deliver identical cash at time $t$ and must carry the same price, $Se^{rt}$. However, there are differences between these in the case when interest rates are not constant. This arises because futures contracts are marked to market, so the proceeds from movements in the value of the contract accrue every day, as opposed to when the contract matures. Thus, the present value of the contract must discount immediate payments less than the payments at maturity are discounted. For example, consider a contract for which the value of a long position in the contract increases when the interest rate increases. Thus, the futures contract provides income to the long position exactly when this investor can invest those payments for a high rate. As such, the long position becomes relatively more desirable than the short position and the price of the futures contract will increase above the equivalent forward contract. On the other hand, if the value of being long the contract tends to decrease when the interest rate increases then the futures contract will be less desirable than the forward contract and the price will go down.

12.6 Futures prices and expected spot prices

There are three traditional hypotheses of the relationship between the futures price and the expected future spot price. These are:

Expectations hypothesis.

That $ES_T = F$.

Normal Backwardation.

For many contracts there are natural hedgers who want to reduce their risk. As such, speculators have to be rewarded to hold the risk that the hedgers are shedding. Thus it should be that $ES_T > F$.

Contango.

The opposite of normal backwardation. The natural hedgers want to take a long position so the short side of the market must be rewarded to hold that risk. Thus $ES_T < F$.

Modern portfolio theory.

Investors will enter the market if the rate of return is commensurate with the risk involved. Thus, if $k$ is the discount rate required in order to hold the asset we have that

\[S = e^{-kT}ES_T\]

from the spot-futures relationship (assuming no correlation between interest rates and mark-to-market)

\[S = e^{-rT}F.\]

Both expressions equal the same spot price $S$, so we can eliminate $S$ by setting them equal:

\[e^{-rT}F = e^{-kT}ES_T.\]

Multiplying both sides by $e^{rT}$ isolates $F$,

\[F = e^{rT}e^{-kT}ES_T = e^{(r-k)T}ES_T.\]

It can be seen from this that if the variability of the spot price $S_T$ is really high (which implies that $k$ is high) then the price of the futures contract will be less than the expected spot price. In general this will happen for all positive $\beta$ stocks. Thus, in general the long position in the futures contract will cost less than the expected future spot price.

12.7 Application: An airline hedging fuel

The treasurer of an airline knows that a sudden jump in the price of crude oil can erase a quarter’s profit, and must decide how to protect the company against a spike it cannot control. Forward and futures contracts are the answer, and the cost-of-carry relationship developed in this chapter is what lets the treasurer use them intelligently. By locking in a price for fuel months ahead, the airline converts an uncertain future cost into a known one; understanding how futures prices relate to the spot price through storage, financing, and convenience yields tells the treasurer whether the hedge is fairly priced and how the futures curve will behave as delivery approaches. When geopolitical news sends oil swinging, the difference between a hedged and an unhedged airline is the difference this chapter’s tools are built to manage.

Further listening

Planet Money (NPR) — “Oil #2: The Price of Oil”: how the price of oil is really set in the futures market, where traders buy and sell promises to deliver months or years ahead.

12.8 Homework problems

12.8.1 Conceptual

FF-C1. A gold-mining company will produce and sell 10,000 ounces of gold in six months and worries that the price could fall before then. A trader at a hedge fund, holding no gold, believes prices will rise and wants to profit from that view. Explain how each party would use a gold forward or futures contract, identifying which side (long or short) each takes and why. Which party is the hedger and which is the speculator?

FF-C2. A wheat farmer expecting to harvest and sell wheat in three months hedges by going short wheat futures. Suppose that, contrary to fears, the spot price of wheat rises sharply by delivery. Explain what happens to the farmer’s revenue from selling the physical wheat versus the payoff on the short futures position, and why the farmer is not made worse off by having hedged even though prices rose. What has the farmer actually “locked in,” and what has the farmer given up?

FF-C3. Explain the no-arbitrage (cost-of-carry) argument that pins down the forward price. Describe the two portfolios that must have the same payoff — a long forward, versus borrowing to buy and carry the asset — and show how their equivalence forces $F = (1+r)S$. Then explain what riskless trade an arbitrageur would execute if instead $(1+r)S > F$.

FF-C4. The chapter states that the forward price exceeds the spot price for a non-dividend asset because “purchasing the asset today requires using money that would otherwise be used for something else.” Explain this financing (cost-of-carry) intuition, and describe how the relationship would change if holding the asset instead threw off income or a convenience yield $k$, giving $F = Se^{(r-k)T}$.

FF-C5. Compare a forward contract negotiated over the counter with an exchange-traded futures contract on the same underlying asset with the same maturity. Explain how the futures contract addresses counterparty credit risk that the forward does not, and describe the mechanism (standardization, margin accounts, and daily marking to market) the exchange uses to do so.

FF-C6. Two traders each hold a long futures position on corn. Over a week the corn futures price falls steadily. Explain, using the marking-to-market mechanism, what happens in each trader’s margin account each day, when a margin call is triggered, and why this daily cash-flow pattern differs from the economics of an otherwise-identical forward contract that settles only at maturity.

FF-C7. A jet-fuel-consuming airline cannot find a jet-fuel futures contract with enough liquidity, so it hedges using crude-oil futures whose delivery date is one month after it will actually buy fuel. Explain what basis risk is in this setting and identify the two distinct sources of basis risk in the airline’s hedge (the mismatch in asset and the mismatch in timing). Why does basis risk mean the hedge is imperfect even though the position is closed out?

FF-C8. A farmer hedging a June wheat delivery uses the July futures contract. Explain, using the definition $Basis(T) = S(T) - F_\tau(T)$, why the basis is generally not zero at the time the farmer closes the position, but why it is nonetheless smaller and more predictable than the outright price risk the farmer would face with no hedge at all.

FF-C9. Consider a commodity for which producers (natural sellers) overwhelmingly dominate the demand for hedging, so they crowd the short side of the futures market. Using the theory of normal backwardation, explain why speculators must be offered a futures price below the expected future spot price ($ES_T > F$) to induce them to take the long side, and what this price discount represents economically.

FF-C10. Contrast the expectations hypothesis ($ES_T = F$) with contango. Describe a market — in terms of which side (producers or consumers) dominates the natural demand for hedging — in which contango arises, and explain why in that case the short side of the market must be rewarded so that $ES_T < F$.

FF-C11. Under the modern portfolio theory view, the relationship $F = e^{(r-k)T}ES_T$ ties the gap between the futures price and the expected spot price to the discount rate $k$ required to hold the asset. Explain what happens to the futures price relative to the expected spot price for a high-$\beta$ (positive systematic risk) commodity, and contrast this with what the pure expectations hypothesis ($ES_T = F$) would predict. Why are the two views inconsistent for a positive-$\beta$ asset?

FF-C12. Derive the modern-portfolio-theory relation $F = e^{(r-k)T}ES_T$ from the two pricing statements $S = e^{-kT}ES_T$ and $S = e^{-rT}F$. Explain the economic content of each of the two starting equations, and interpret why a negative-$\beta$ asset (with $k < r$) ends up with a futures price above its expected spot price.

12.8.2 Quantitative

FF-Q1. The spot price of one ounce of silver is $S = \$30$ and the annual simple risk-free interest rate is $r = 0.04$. Using the cost-of-carry relationship $F = (1+r)S$, compute the fair one-year forward price $F$ per ounce. Then suppose the one-year forward is instead quoted in the market at $F = \$32$. Describe the arbitrage trade and compute the risk-free profit per ounce.

FF-Q2. A stock index has spot value $S = \$2{,}000$ and the continuously compounded risk-free rate is $r = 0.05$. Using $F = Se^{rt}$, find the fair forward price for delivery in $t = 0.5$ years. If the actual six-month forward price is $\$2{,}080$, is the forward overpriced or underpriced relative to cost-of-carry, and by how much?

FF-Q3. Gold has spot price $S = \$1{,}800$ per ounce and the continuously compounded risk-free rate is $r = 0.05$. Using $F = Se^{rt}$, compute the fair six-month ($t = 0.5$) forward price. Separately, a currency has spot rate $S = \$1.4208$ and a three-month ($T = 0.25$) forward rate $F = \$1.4187$; using the inverted relationship $r = \tfrac{1}{T}\ln(F/S)$, compute the implied continuously compounded interest rate and state its sign, explaining why $F < S$ produces that sign.

FF-Q4. A trader is long one forward contract on one barrel of oil with delivery price $K = \$59$. Compute the payoff $S_T - K$ to the long position if the spot price at maturity is (a) $S_T = \$56$ and (b) $S_T = \$63$. Then compute the corresponding payoff $K - S_T$ to the short position in each case, and confirm that the long and short payoffs sum to zero.

FF-Q5. An oil producer will sell 1,000,000 barrels at the 15 August spot price and hedges by going short 1,000 NYMEX crude contracts (1,000 barrels each) at a futures price of $\$59$/barrel. Suppose the 15 August spot price turns out to be $S_T = \$52$/barrel and the futures price converges to the spot price. Compute (a) the revenue from selling the physical oil, (b) the total gain or loss on the short futures position using the per-barrel payoff $K - S_T$, and (c) the combined net proceeds, confirming the hedge locks in approximately $\$59$/barrel.

FF-Q6. A farmer will deliver wheat at time $T$ and hedges with a futures contract for a later delivery date $\tau$. At time $T$ the spot price is $S(T) = \$6.20$ per bushel and the futures price for $\tau$ delivery is $F_\tau(T) = \$6.35$ per bushel. Using $Basis(T) = S(T) - F_\tau(T)$, compute the basis. State whether the basis is positive or negative and, if the farmer is short the futures to hedge a sale, whether this basis movement helped or hurt relative to a zero basis.

FF-Q7. On 13 March the spot price of corn is $S(T) = 385$ cents per bushel, while the futures price for the same-month delivery, one day before the contract stops trading, is $F_\tau(T) = 383$ cents per bushel. Compute $Basis(T)$ in cents. Explain, given that convergence forces basis toward zero at expiration, the arbitrage a trader could attempt to exploit this 2-cent gap, and compute the gross profit per 5,000-bushel contract.

FF-Q8. A commodity has expected spot price at maturity $ES_T = \$100$. The continuously compounded risk-free rate is $r = 0.03$, the discount rate required to hold the asset is $k = 0.08$, and the horizon is $T = 1$ year. Using the modern portfolio theory relationship $F = e^{(r-k)T}ES_T$, compute the futures price $F$. Is $F$ above or below $ES_T$, and is this consistent with normal backwardation or contango?

FF-Q9. Suppose instead that a commodity behaves like a negative-$\beta$ asset, so the required discount rate is $k = 0.01$ while $r = 0.04$, with $ES_T = \$50$ and $T = 2$ years. Using $F = e^{(r-k)T}ES_T$, compute the futures price $F$. State whether $F$ exceeds or falls short of $ES_T$ and which of the three classical hypotheses this outcome matches.

FF-Q10. A March corn futures contract (5,000 bushels) is currently valued at $19,409.375, i.e. a futures price of $388\tfrac{6}{32}$ cents per bushel. A hedger is long one contract with $2,000 in a margin account and a maintenance margin of $1,500. Suppose the next day the futures price rises to 390 cents per bushel. Compute (a) the new contract value, (b) the daily marking-to-market gain credited to the long margin account, and (c) the new margin balance. Then compute (d) the futures price (in cents per bushel) at which the account would fall to the $1,500 maintenance level and trigger a margin call.

Data can be found at: http://online.wsj.com/mdc/page/marketsdata.html↩︎