Assignment 9

!"# P A R T V I Options, Futures, and Other Derivatives

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The open interest on the contract is the number of contracts outstanding. (Long and short positions are not counted separately, meaning that open interest can be defined either as the number of long or short contracts outstanding. The clearinghouse’s position nets out to zero, and so is not counted in the computation of open interest.) When contracts begin trading, open interest is zero. As time passes, open interest increases as progressively more contracts are entered.

There are many apocryphal stories about futures traders who wake up to discover a small mountain of wheat or corn on their front lawn. But the truth is that futures contracts rarely result in actual delivery of the underlying asset. Traders establish long or short positions in contracts that will benefit from a rise or fall in the futures price and almost always close out, or reverse, those positions before the contract expires. The fraction of contracts that result in actual delivery is estimated to range from less than 1% to 3%, depending on the commodity and activity in the contract. In the unusual case of actual deliveries of commodities, they occur via regular channels of supply, most often ware- house receipts.

You can see the typical pattern of open interest in Figure 22.1. In the gold contract, for example, the July contract is approaching maturity, and open interest is small; most contracts have been reversed already. The greatest open interest is in the August contract. For other contracts, for example, crude oil, for which the nearest maturity date isn’t until August, open interest is highest in the nearest contract.

The Margin Account and Marking to Market The total profit or loss realized by the long trader who buys a contract at time 0 and closes, or reverses, it at time t is just the change in the futures price over the period, Ft ! F0. Sym- metrically, the short trader earns F0 ! Ft.

The process by which profits or losses accrue to traders is called marking to market. At initial execution of a trade, each trader establishes a margin account. The margin is a secu- rity account consisting of cash or near-cash securities, such as Treasury bills, that ensures the trader is able to satisfy the obligations of the futures contract. Because both parties to a contract are exposed to losses, both must post margin. To illustrate, return to the first corn contract listed in Figure 22.1. If the initial required margin on corn, for example, is 10%, then each trader must put up $1,720 per contract. This is 10% of the value of the contract, $3.44 per bushel " 5,000 bushels per contract.

Because the initial margin may be satisfied by posting interest-earning securities, the requirement does not impose a significant opportunity cost of funds on the trader. The ini- tial margin is usually set between 5% and 15% of the total value of the contract. Contracts written on assets with more volatile prices require higher margins.

On any day that futures contracts trade, futures prices may rise or may fall. Instead of waiting until the maturity date for traders to realize all gains and losses, the clearinghouse requires all positions to recognize profits as they accrue daily. If the futures price of corn rises from 344 to 346 cents per bushel, the clearinghouse credits the margin account of the long position for 5,000 bushels times 2 cents per bushel, or $100 per contract. Conversely, the clearinghouse takes this amount from the margin account of the short position for each contract held.

This daily settling is called marking to market. It means the maturity date of the con- tract does not govern realization of profit or loss. Instead, as futures prices change, the pro- ceeds accrue to the trader’s margin account immediately. We will provide a more detailed example of this process shortly.

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Assume the !-day maturity futures price for silver is currently $"#.$# per ounce. Suppose that over the next ! days, the futures price evolves as follows:

The daily mark-to-market settlements for each contract held by the long position will be as follows:

The profit on Day $ is the increase in the futures price from the previous day, or ($"#."# ! $"#.$#) per ounce. Because each silver contract on the Commodity Exchange (CMX) calls for purchase and delivery of !,### ounces, the total profit per contract is !,### times $.$#, or $!##. On Day %, when the futures price falls, the long position’s margin account will be debited by $%!#. By Day !, the sum of all daily proceeds is $!!#. This is exactly equal to !,### times the difference between the final futures price of $"#."$ and the original futures price of $"#.$#. Because the final futures price equals the spot price on that date, the sum of all the daily proceeds (per ounce of silver held long) also equals PT ! F#.

Example 22.2 Marking to Market

Day Profit (Loss) per Ounce ! !,""" Ounces/Contract = Daily Proceeds

$ "#."# ! "#.$# = &&&#.$# $!## " "#."! ! "#."# = & &#.#! &"!# % "#.$' ! "#."! = !#.#( &!%!# ) "#.$' ! "#.$' = & &# & && &# ! "#."$ ! "#.$' = & &#.#% & $!# & & Sum = $!!#

Day Futures Price

# (today) $"#.$#& $ &"#."# " &"#."! % &"#.$' ) &"#.$' ! (maturity) &"#."$

Cash versus Actual Delivery Most futures contracts call for delivery of an actual commodity such as a particular grade of wheat or a specified amount of foreign currency if the contract is not reversed before maturity. For agricultural commodities, where quality of the delivered good may vary, the exchange sets quality standards as part of the futures contract. In some cases, contracts may be settled with higher- or lower-grade commodities. In these cases, a premium or dis- count is applied to the delivered commodity to adjust for the quality difference.

Some futures contracts call for cash settlement. An example is a stock index futures contract where the underlying asset is an index such as the Standard & Poor’s 500. Deliv- ery of every stock in the index clearly would be impractical. Hence the contract calls for “delivery” of a cash amount equal to the value that the index attains on the maturity date of the contract. The sum of all the daily settlements from marking to market results in the long position realizing total profits or losses of ST ! F0, where ST is the value of the stock index on the maturity date T and F0 is the original futures price. Cash settlement closely mimics actual delivery, except the cash value of the asset rather than the asset itself is delivered.

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More concretely, the widely traded E-mini S&P 500 index contract calls for delivery of $50 times the value of the index.2 Suppose that at maturity, the S&P 500 is at a level of 2,000. Instead of delivering shares of all 500 stocks included in the index, cash settlement would require the short trader to deliver $50 ! 2,000, or $100,000. This provides the trader exactly the same profit as would result from directly purchasing 50 units of the index for $100,000 and then delivering it for $50 times the original futures price.

Regulations Futures markets are regulated by the federal Commodities Futures Trading Commission. The CFTC sets capital requirements for member firms of the futures exchanges, authorizes trading in new contracts, and oversees maintenance of daily trading records.

The futures exchange may set limits on the amount by which futures prices may change from one day to the next. For example, if the price limit on silver contracts were set at $1 and silver futures close today at $22.10 per ounce, then trades in silver tomorrow may vary only between $21.10 and $23.10 per ounce. The exchanges may increase or reduce price limits in response to perceived changes in the price volatility of the underlying asset. Price limits are often eliminated as contracts approach maturity, usually in the last month of trading.

Price limits traditionally are viewed as a means to limit violent price fluctuations. This reasoning seems dubious. Suppose an international monetary crisis overnight drives up the spot price of silver to $30. No one would sell silver futures at prices for future delivery as low as $22.10. Instead, the futures price would rise each day by the $1 limit, although the quoted price would represent only an unfilled bid order—no contracts would trade at the low quoted price. After several days of limit moves of $1 per day, the futures price would finally reach its equilibrium level, and trading would occur again. This process means no one could unload a position until the price reached its equilibrium level. We conclude that price limits offer no real protection against fluctuations in equilibrium prices.

Taxation Because of the mark-to-market procedure, investors do not have control over the tax year in which they realize gains or losses. Instead, price changes are realized gradually, with each daily settlement. Therefore, taxes are paid at year-end on cumulated profits or losses, regardless of whether the position has been closed out. As a general rule, 60% of futures gains or losses are treated as long term, and 40% are treated as short term.

2The original S&P 500 contract had a multiplier of $500, later reduced to $250. However, the all-electronically traded version of the contract, usually referred to as the E-mini, has a multiplier of $50. The great majority of futures trading on the S&P 500 is now conducted on the E-mini rather than the original “big” contract.

22.3 Futures Markets Strategies Hedging and Speculation Hedging and speculating are two polar uses of futures markets. A speculator uses a futures contract to profit from movements in futures prices, a hedger to protect against price movement.

If speculators believe prices will increase, they will take a long position for expected profits. Conversely, they exploit expected price declines by taking a short position.

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Suppose you believe that crude oil prices are going to increase and therefore decide to pur- chase crude oil futures. Each contract calls for delivery of !,""" barrels of oil, so for every $! increase in the futures price of crude, the long position gains $!,""" and the short position loses that amount.

Conversely, suppose you think that prices are heading lower and therefore sell a con- tract. If crude oil prices do in fact fall, then you will gain $!,""" per contract for every $! that prices decline.

If the futures price for delivery in February is $#$ and crude oil is selling at the contract maturity date for $#%, the long side will profit by $!,""" per contract purchased. The short side will lose an identical amount on each contract sold. On the other hand, if oil has fallen to $#!, the long side will lose, and the short side will gain, $!,""" per contract.

Example 22.3 Speculating with Oil Futures

Suppose the initial margin requirement for the oil contract is !"%. At a current futures price of $#$, and contract size of !,""" barrels, this would require margin of .!" ! #$ ! !,""" = $#,$"". A $! increase in oil prices represents an increase of !.&$%, and results in a $!,""" gain on the contract for the long position. This is a percentage gain of !&.$% on the $#,$"" posted as margin, precisely !" times the percentage increase in the oil price. The !"-to-! ratio of percentage changes reflects the leverage inherent in the futures position, because the contract was established with an initial margin of one-tenth the value of the underlying asset.

Example 22.4 Futures and Leverage

Consider an oil distributor planning to sell !"",""" barrels of oil in February that wishes to hedge against a possible decline in oil prices. Because each contract calls for delivery of !,""" barrels, it would sell !"" contracts that mature in February. Any decrease in prices would then generate a profit on the contracts that would offset the lower sales revenue from the oil.

To illustrate, suppose that the only three possible prices for oil in February are $#!, $#$, and $#% per barrel. The revenue from the oil sale will be !"",""" times the price per barrel.

Example 22.5 Hedging with Oil Futures

Why does a speculator buy a futures contract? Why not buy the underlying asset directly? One reason lies in transaction costs, which are far smaller in futures markets.

Another important reason is the leverage that futures trading provides. Recall that futures contracts require traders to post margin considerably less than the value of the asset underlying the contract. Therefore, they allow speculators to achieve much greater leverage than is available from direct trading in a commodity.

Hedgers, by contrast, use futures to insulate themselves against price movements. A firm planning to sell oil, for example, might anticipate a period of market volatility and wish to protect its revenue against price fluctuations. To hedge the total revenue derived from the sale, the firm enters a short position in oil futures. As the following example illus- trates, this locks in its total proceeds (i.e., revenue from the sale of the oil plus proceeds from its futures position).

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Some speculators try to profit from movements in the basis. Rather than betting on the direction of the futures or spot prices per se, they bet on the changes in the difference between the two. A long spot–short futures position will profit when the basis narrows.

Consider an investor holding !"" ounces of gold, who is short one gold-futures contract. Suppose that gold today sells for $!,#$! an ounce, and the futures price for June delivery is $!,#$% an ounce. Therefore, the basis is currently $&. Tomorrow, the spot price might increase to $!,#$&, while the futures price increases to $!,#$$, so the basis narrows to $'.

The investor’s gains and losses are as follows:

Gain on holdings of gold (per ounce): $!,#$& ! $!,#$! = $' Loss on gold futures position (per ounce): $!,#$$ ! $!,#$% = $#

The net gain is the decrease in the basis, or $! per ounce.

Example 22.6 Speculating on the Basis

Consider an investor who holds a September maturity contract long and a June contract short. If the September futures price increases by & cents while the June futures price increases by ' cents, the net gain will be & cents ! ' cents, or ! cent. Like basis strategies, spread positions aim to exploit movements in relative price structures rather than profit from movements in the general level of prices.

Example 22.7 Speculating on the Spread

A related strategy is a calendar spread position, where the investor takes a long posi- tion in a futures contract of one maturity and a short position in a contract on the same commodity, but with a different maturity.4 Profits accrue if the difference in futures prices between the two contracts changes in the hoped-for direction, that is, if the futures price on the contract held long increases by more (or decreases by less) than the futures price on the contract held short.

4Yet another strategy is an intercommodity spread, in which the investor buys a contract on one commodity and sells a contract on a different commodity.

22.4 Futures Prices The Spot-Futures Parity Theorem We have seen that a futures contract can be used to hedge changes in the value of the underlying asset. If the hedge is perfect, meaning that the asset-plus-futures portfolio has no risk, then the hedged position must provide a rate of return equal to the rate on other risk-free investments. Otherwise, there will be arbitrage opportunities that investors will exploit until prices are brought back into line. This insight can be used to derive the theo- retical relationship between a futures price and the price of its underlying asset.

Suppose for simplicity that a stock market index such as the S&P 500 currently is at 1,000 and an investor who holds $1,000 in an indexed mutual fund wishes to tempo- rarily hedge her exposure to market risk. Assume that the indexed portfolio pays year- end dividends of $20. Finally, assume that the futures price for year-end delivery of the

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contract is 1,010.5 Let’s examine the end-of-year proceeds for various values of the stock index if the investor hedges her portfolio by entering the short side of the futures contract.

5Actually, the E-mini futures contract on the S&P 500 calls for delivery of $50 times the value of the index, so that each contract would be settled for $50 times the index. With the index at the assumed value of 1,000, each contract would hedge about $50 ! 1,000 = $50,000 worth of stock. Of course, institutional investors would consider a stock portfolio of this size to be quite small. We will simplify by assuming that you can buy a contract for one unit rather than 50 units of the index.

Final value of stock portfolio, ST $ ! "#$ $ ! ""$ $%,$%$ $%,$&$ $%,$'$ $%,$#$ Payoff from short futures position (equals F$ " FT = $%,$%$ " ST)

! ! ! !($ ! ! ! !)$ ! ! ! ! !$ ! ! ")$ ! ! "($ ! ! "*$

Dividend income ! !++ ! !)$ ! ! !++ !)$ ! !++ ! )$ ! !++ ! !)$ ! ++! ! !)$ ! ++! ! !)$ Total $%,$&$ $%,$&$ $%,$&$ $%,$&$ $%,$&$ $%,$&$

The payoff from the short futures position equals the difference between the original futures price, $1,010, and the year-end stock price. This is because of convergence: The futures price at contract maturity will equal the stock price at that time.

Notice that the overall position is perfectly hedged. Any increase in the value of the indexed stock portfolio is offset by an equal decrease in the payoff of the short futures posi- tion, resulting in a final value independent of the stock price. The $1,030 total payoff is the sum of the current futures price, F0 = $1,010, and the $20 dividend. It is as though the investor arranged to sell the stock at year-end for the current futures price, thereby elimi- nating price risk and locking in total proceeds equal to the futures price plus dividends paid before the sale.

What rate of return is earned on this riskless position? The stock investment requires an initial outlay of $1,000, whereas the futures position is established without an initial cash outflow. Therefore, the $1,000 portfolio grows to a year-end value of $1,030, providing a rate of return of 3%. More generally, a total investment of S0, the current stock price, grows to a final value of F0 + D, where D is the dividend payout on the portfolio. The rate of return is therefore

Rate#of#return#on#hedged#stock#portfolio = ( F 0 + D) " S 0 __________ S 0

This return is essentially riskless. We observe F0 at the beginning of the period when we enter the futures contract. While dividend payouts are not perfectly riskless, they are highly predictable over short periods, especially for diversified portfolios. Any uncertainty is extremely small compared to the uncertainty in stock prices.

Presumably, 3% must be the rate of return available on other riskless investments. If not, then investors would face two competing risk-free strategies with different rates of return, a situation that could not last. Therefore, we conclude that

( F 0 + D) " S 0 __________ S 0

= r f

Rearranging, we find that the futures price must be F 0 = S 0 (1 + r f ) " D = S 0 (1 + r f " d ) (22.1) where d is the dividend yield on the indexed stock portfolio, defined as D/S0. This result is called the spot-futures parity theorem. It gives the normal or theoretically correct relationship between spot and futures prices. Any deviation from parity would give rise to risk-free arbitrage opportunities.

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on the arbitrage opportunity. Ultimately prices would change until the year-end cash flow is reduced to zero, at which point F0 would equal S0(1 + rf ) ! D.

The parity relationship also is called the cost-of-carry relationship because it asserts that the futures price is determined by the relative costs of buying a stock with deferred delivery in the futures market versus buying it in the spot market with immediate delivery and “carrying” it in inventory. If you buy stock now, you tie up your funds and incur a time-value-of-money cost of rf per period. On the other hand, you receive dividend pay- ments with a current yield of d. The net carrying cost advantage of deferring delivery of the stock is therefore rf ! d per period. This advantage must be offset by a differential between the futures price and the spot price. The price differential just offsets the cost-of- carry advantage when F0 = S0(1 + rf ! d).

The parity relationship is easily generalized to multiperiod applications. We simply rec- ognize that the difference between the futures and spot price will be larger as the maturity of the contract is longer. This reflects the longer period to which we apply the net cost of carry. For contract maturity of T periods, the parity relationship is F 0 = S 0 (1 + r f ! d ) T (22.2) Notice that when the dividend yield is less than the risk-free rate, Equation 22.2 implies that futures prices will exceed spot prices, and by greater amounts for longer times to contract maturity. But when d > rf , as is the case today, the income yield on the stock exceeds the for- gone (risk-free) interest that could be earned on the money invested; in this event, the futures price will be less than the current stock price, again by greater amounts for longer maturities. You can confirm that this is so by examining the S&P 500 contract listings in Figure 22.1.

Although dividends of individual securities may fluctuate unpredictably, the annual- ized dividend yield of a broad-based index such as the S&P 500 is fairly stable, recently in the neighborhood of a bit more than 2% per year. The yield is seasonal, however, with regular peaks and troughs, so the dividend yield for the relevant months must be the one used. Figure 22.5 illustrates the yield pattern for the S&P 500. Some months, such as January or April, have consistently low yields, while others, such as May, have consis- tently high ones.6

We have described parity in terms of stocks and stock index futures, but it should be clear that the logic applies as well to any financial futures contract. For gold futures, for example, we would simply set the dividend yield to zero. For bond contracts, we would let the coupon income on the bond play the role of dividend payments. In both cases, the par- ity relationship would be essentially the same as Equation 22.2.

The arbitrage strategy described above should convince you that these parity relation- ships are more than just theoretical results. Any violations of the parity relationship give rise to arbitrage opportunities that can provide large profits to traders. We will see in Chapter 23 that index arbitrage in the stock market is a tool to exploit violations of the parity relationship for stock index futures contracts.

Spreads Just as we can predict the relationship between spot and futures prices, there are similar ways to determine the proper relationships among futures prices for contracts of different maturity dates. Equation 22.2 shows that the futures price is in part determined by time to maturity. If the risk-free rate is greater than the dividend yield (i.e., rf > d), then the

6The high value for the dividend yield in 2009 reflects the financial crisis. You learned in your corporate finance class that firms are reluctant to reduce dividends. When the economy entered the crisis and stock prices fell dra- matically, dividend payouts did not fall as precipitously. Therefore, the ratio of dividends to stock prices increased.

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level below the expected spot price and will rise over the life of the contract until the matu- rity date, at which point FT = PT.

Although this theory recognizes the important role of risk premiums in futures markets, it is based on total variability rather than on systematic risk. (This is not surprising, as Keynes wrote almost 40 years before the development of modern portfolio theory.) The modern view refines the measure of risk used to determine appropriate risk premiums.

Contango The polar hypothesis to backwardation holds that the natural hedgers are the purchasers of a commodity, rather than the suppliers. In the case of wheat, for example, we would view grain processors as willing to pay a premium to lock in the price that they must pay for wheat. These processors hedge by taking a long position in the futures market; therefore, they are called long hedgers, whereas farmers are short hedgers. Because long hedgers will agree to pay high futures prices to shed risk, and because speculators must be paid a premium to enter the short position, the contango theory holds that F0 must exceed E(PT).

It is clear that any commodity will have both natural long hedgers and short hedgers. The compromise traditional view, called the “net hedging hypothesis,” is that F0 will be less than E(PT) when short hedgers outnumber long hedgers and vice versa. The strong side of the market will be the side (short or long) that has more natural hedgers. The strong side must pay a premium to induce speculators to enter into enough contracts to balance the “natural” supply of long and short hedgers.

Modern Portfolio Theory The three traditional hypotheses all envision a mass of speculators willing to enter either side of the futures market if they are sufficiently compensated for the risk they incur. Mod- ern portfolio theory fine-tunes this approach by refining the notion of risk used in the determination of risk premiums. Simply put, if commodity prices pose positive systematic risk, futures prices must be lower than expected spot prices.

To illustrate this approach, consider once again a stock paying no dividends. If E(PT) denotes the expected time-T stock price and k denotes the required rate of return on the stock, then the price of the stock today must equal the present value of its expected future payoff as follows:

P 0 = E( P T ) ______

(1 + k) T (22.4)

We also know from the spot-futures parity relationship that

P 0 = F 0 _______

(1 + r f ) T (22.5)

Therefore, the right-hand sides of Equations 22.4 and 22.5 must be equal. Equating these terms allows us to solve for F0!in terms of the expected spot price:

F 0 = E( P T ) ( 1 + r f _____ 1 + k ) T

(22.6)

You can see immediately from Equation 22.6 that F0 will be less than the expectation of PT whenever k is greater than rf , which will be the case for any positive-beta asset. This means that the long side of the contract will make an expected profit [F0 will be lower than E(PT)] when the commodity exhibits positive systematic risk (k is greater than rf ).

Why should this be? A long futures position will provide a profit (or loss) of PT " F0. If the ultimate value of PT entails positive systematic risk, so will the profit to the long

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forward contract futures price long position short position single-stock futures clearinghouse

KEY TERMSopen interest marking to market maintenance margin convergence property cash settlement basis

basis risk calendar spread spot-futures parity theorem cost-of-carry relationship

Spot-futures parity:! F 0 ( T ) = S 0 !(1 + r " d ) T Futures spread parity:! F 0 ( T 2 ) = F 0 ( T 1 )!(1 + r " d ) ( T 2 " T 1 )

Futures vs. expected spot prices: F 0 = E( P T ) ( 1 + r f _____ 1 + k ) T

KEY EQUATIONS

1. Why is there no futures market in cement? 2. Why might individuals purchase futures contracts rather than the underlying asset? 3. What is the difference in cash flow between short-selling an asset and entering a short futures

position? 4. Are the following statements true or false? Why?

a. All else equal, the futures price on a stock index with a high dividend yield should be higher than the futures price on an index with a low dividend yield.

b. All else equal, the futures price on a high-beta stock should be higher than the futures price on a low-beta stock.

c. The beta of a short position in the S&P 500 futures contract is negative. 5. What is the difference between the futures price and the value of the futures contract? 6. Evaluate the criticism that futures markets siphon off capital from more productive uses. 7. a. Turn to the Mini-S&P 500 contract in Figure 22.1. If the margin requirement is 10% of the

futures price times the contract multiplier of $50, how much must you deposit with your broker to trade the September maturity contract?

b. If the September futures price were to increase to 2,090, what percentage return would you earn on your net investment if you entered the long side of the contract at the price shown in the figure?

c. If the September futures price falls by 1%, what is your percentage return? 8. a. A single-stock futures contract on a non-dividend-paying stock with current price $150 has

a maturity of 1 year. If the T-bill rate is 3%, what should the futures price be? b. What should the futures price be if the maturity of the contract is 3 years? c. What if the interest rate is 6% and the maturity of the contract is 3 years?

9. Determine how a portfolio manager might use financial futures to hedge risk in each of the fol- lowing circumstances: a. You own a large position in a relatively illiquid bond that you want to sell. b. You have a large gain on one of your Treasuries and want to sell it, but you would like to

defer the gain until the next tax year. c. You will receive your annual bonus next month that you hope to invest in long-term corpo-

rate bonds. You believe that bonds today are selling at quite attractive yields, and you are concerned that bond prices will rise over the next few weeks.

10. Suppose the value of the S&P 500 stock index is currently 2,000. a. If the 1-year T-bill rate is 3% and the expected dividend yield on the S&P 500 is 2%, what

should the 1-year maturity futures price be? b. What if the T-bill rate is less than the dividend yield, for example, 1%?

11. Consider a stock that pays no dividends on which a futures contract, a call option, and a put option trade. The maturity date for all three contracts is T, the exercise price of both the put and the call is X, and the futures price is F. Show that if X = F, then the call price equals the put price. Use parity conditions to guide your demonstration.

12. It is now January. The current interest rate is 2%. The June futures price for gold is $1,500, whereas the December futures price is $1,510. Is there an arbitrage opportunity here? If so, how would you exploit it?

PROBLEM SETS

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13. OneChicago has just introduced a single-stock futures contract on Brandex stock, a company that currently pays no dividends. Each contract calls for delivery of 1,000 shares of stock in 1 year. The T-bill rate is 6% per year. a. If Brandex stock now sells at $120 per share, what should the futures price be? b. If the Brandex price drops by 3%, what will be the change in the futures price and the change

in the investor’s margin account? c. If the margin on the contract is $12,000, what is the percentage return on the investor’s position?

14. The multiplier for a futures contract on a stock market index is $50. The maturity of the contract is 1 year, the current level of the index is 1,800, and the risk-free interest rate is .5% per month. The dividend yield on the index is .2% per month. Suppose that after 1 month, the stock index is at 1,820. a. Find the cash flow from the mark-to-market proceeds on the contract. Assume that the parity

condition always holds exactly. b. Find the holding-period return if the initial margin on the contract is $5,000.

15. You are a corporate treasurer who will purchase $1 million of bonds for the sinking fund in 3 months. You believe rates will soon fall, and you would like to repurchase the company’s sinking fund bonds (which currently are selling below par) in advance of requirements. Unfor- tunately, you must obtain approval from the board of directors for such a purchase, and this can take up to 2 months. What action can you take in the futures market to hedge any adverse move- ments in bond yields and prices until you can actually buy the bonds? Will you be long or short? Why? A qualitative answer is fine.

16. The S&P portfolio pays a dividend yield of 1% annually. Its current value is 2,000. The T-bill rate is 4%. Suppose the S&P futures price for delivery in 1 year is 2,050. Construct an arbitrage strategy to exploit the mispricing and show that your profits 1 year hence will equal the mispric- ing in the futures market.

17. The Excel Application box in the chapter (available in Connect; link to Chapter 22 material) shows how to use the spot-futures parity relationship to find a “term structure of futures prices,” that is, futures prices for various maturity dates. a. Suppose that today is January 1, 2016. Assume the interest rate is 3% per year and a stock

index currently at 2,000 pays a dividend yield of 2.0%. Find the futures price for contract maturity dates of: (i) February 14, 2016, (ii) May 21, 2016, and (iii) November 18, 2016.

b. What happens to the term structure of futures prices if the dividend yield is higher than the risk-free rate? For example, what if the dividend yield is 4%?

18. a. How should the parity condition (Equation 22.2) for stocks be modified for futures contracts on Treasury bonds? What should play the role of the dividend yield in that equation?

b. In an environment with an upward-sloping yield curve, should T-bond futures prices on more-distant contracts be higher or lower than those on near-term contracts?

c. Confirm your intuition by examining Figure 22.1. 19. Consider this arbitrage strategy to derive the parity relationship for spreads: (1) enter a long

futures position with maturity date T1 and futures price F(T1); (2) enter a short position with maturity T2 and futures price F(T2); (3) at T1, when the first contract expires, buy the asset and borrow F(T1) dollars at rate rf ; (4) pay back the loan with interest at time T2. a. What are the total cash flows to this strategy at times 0, T1, and T2? b. Why must profits at time T2 be zero if no arbitrage opportunities are present? c. What must the relationship between F(T1) and F(T2) be for the profits at T2 to be equal to

zero? This relationship is the parity relationship for spreads.

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Spot price for commodity $!"# Futures price for commodity expiring in ! year $!"$ Interest rate for ! year %%%%%%%&%

1. Joan Tam, CFA, believes she has identified an arbitrage opportunity for a commodity as indicated by the following information:

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C H A P T E R !! Futures Markets ""#

bod77178_ch22_747-774.indd 773 04/08/17 06:01 PM

a. Describe the transactions necessary to take advantage of this specific arbitrage opportunity. b. Calculate the arbitrage profit.

2. Michelle Industries issued a Swiss franc–denominated 5-year discount note for SFr200 million. The proceeds were converted to U.S. dollars to purchase capital equipment in the United States. The company wants to hedge this currency exposure and is considering the following alternatives: ! At-the-money Swiss franc call options. ! Swiss franc forwards. ! Swiss franc futures. a. Contrast the essential characteristics of each of these three derivative instruments. b. Evaluate the suitability of each in relation to Michelle’s hedging objective, including both

advantages and disadvantages. 3. Identify the fundamental distinction between a futures contract and an option contract, and briefly

explain the difference in the manner that futures and options modify portfolio risk. 4. Maria VanHusen, CFA, suggests that using forward contracts on fixed-income securities can be

used to protect the value of the Star Hospital Pension Plan’s bond portfolio against the possibility of rising interest rates. VanHusen prepares the following example to illustrate how such protec- tion would work: ! A 10-year bond with a face value of $1,000 is issued today at par value. The bond pays an

annual coupon. ! An investor intends to buy this bond today and sell it in 6 months. ! The 6-month risk-free interest rate today is 5% (annualized). ! A 6-month forward contract on this bond is available, with a forward price of $1,024.70. ! In 6 months, the price of the bond, including accrued interest, is forecast to fall to $978.40

as a result of a rise in interest rates. a. Should the investor buy or sell the forward contract to protect the value of the bond against

rising interest rates during the holding period? b. Calculate the value of the forward contract for the investor at the maturity of the forward con-

tract if VanHusen’s bond-price forecast turns out to be accurate. c. Calculate the change in value of the combined portfolio (the underlying bond and the appro-

priate forward contract position) 6 months after contract initiation. 5. Sandra Kapple asks Maria VanHusen about using futures contracts to protect the value of the Star

Hospital Pension Plan’s bond portfolio if interest rates rise. VanHusen states: a. “Selling a bond futures contract will generate positive cash flow in a rising interest rate envi-

ronment prior to the maturity of the futures contract.” b. “The cost of carry causes bond futures contracts to trade for a higher price than the spot price

of the underlying bond prior to the maturity of the futures contract.” Comment on the accuracy of each of VanHusen’s two statements.

E$INVESTMENTS EXERCISES Go to the Chicago Mercantile Exchange site at www.cme.com. From the Trading tab, select the link to Equity Index, and then link to the NASDAQ-%&& E-mini contract. Now find the tab for Con- tract Specifications.

%. What is the contract size for the futures contract?

!. What is the settlement method for the futures contract?

#. For what months are the futures contracts available?

'. Click the link to view Price Limits and then U.S. Equity Price Limits. What is the current value of the "% down limit for the S&P (&& contract?

(. Click on!Calendar. What is the settlement date of the shortest-maturity outstanding contract? The longest-maturity contract?

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!"# P A R T V I Options, Futures, and Other Derivatives

bod77178_ch22_747-774.indd 748 04/08/17 06:01 PM

The miller who must purchase wheat for processing faces a risk management problem that is the mirror image of the farmer’s. He is subject to profit uncertainty because of the unpredictable cost of the wheat.

Both parties can hedge their risk by entering into a forward contract calling for the farmer to deliver the wheat when harvested at a price agreed upon now, regardless of the market price at harvest time. No money need change hands at this time. A forward contract is simply a deferred-delivery sale of some asset with the sales price agreed on now. All that is required is that each party must be willing to lock in the ultimate delivery price. The contract protects each party from future price fluctuations.

Futures markets formalize and standardize forward contracting. Buyers and sellers trade in a centralized futures exchange. The exchange standardizes the types of contracts that may be traded: It establishes contract size, the acceptable grade of commodity, contract delivery dates, and so forth. Although standardization eliminates much of the flexibility available in forward contracting, it offers the offsetting advantage of liquidity because many traders will concentrate on the same small set of contracts. Futures contracts also differ from for- ward contracts in that they call for a daily settling up of any gains or losses on the contract. By contrast, no money changes hands in forward contracts until the delivery date.

The centralized market, standardization of contracts, and depth of trading in each con- tract allows futures positions to be liquidated easily rather than renegotiated with the other party to the contract. Because the exchange guarantees the performance of each party, costly credit checks on other traders are not necessary. Instead, each trader simply posts a good-faith deposit, called the margin, to guarantee contract performance.

The Basics of Futures Contracts The futures contract calls for delivery of a commodity at a specified delivery or maturity date, for an agreed-upon price, called the futures price, to be paid at contract maturity. The contract specifies precise requirements for the commodity. For agricultural commodi- ties, the exchange sets allowable grades (e.g., No. 2 hard winter wheat or No. 1 soft red wheat). The place and means of delivery of the commodity are specified as well. Delivery of agricultural commodities is made by transfer of warehouse receipts issued by approved warehouses. For financial futures, delivery may be made by wire transfer; for index futures, delivery may be accomplished by a cash settlement procedure such as those for index options. Although the futures contract technically calls for delivery of an asset, delivery rarely occurs. Instead, parties to the contract much more commonly close out their posi- tions before contract maturity, taking gains or losses in cash.

Because the futures exchange specifies all the terms of the contract, the traders need bar- gain only over the futures price. The trader taking the long position commits to purchasing the commodity on the delivery date. The trader who takes the short position commits to delivering the commodity at contract maturity. The trader in the long position is said to “buy” a contract; the short-side trader “sells” a contract. The words buy and sell are figura- tive only, because a contract is not really bought or sold like a stock or bond; it is entered into by mutual agreement. At the time the contract is entered into, no money changes hands.

Figure 22.1 shows prices for several futures contracts as they appear in The Wall Street Journal. The boldface heading lists in each case the commodity, the exchange where the futures contract is traded, the contract size, and the pricing unit. The first agricultural contract listed is for corn, traded on the Chicago Board of Trade (CBT). (The CBT merged with the Chicago Mercantile Exchange in 2007 but still maintains a separate identity.) Each contract calls for delivery of 5,000 bushels, and prices in the entry are quoted in cents per bushel.

The next two! rows detail price data for contracts expiring on various dates. The July 2016 maturity corn contract, for example, opened during the day at a futures price of

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!"#

bod77178_ch22_747-774.indd 752 04/08/17 06:01 PM

necessarily incomplete. The nearby box discusses some comparatively fanciful futures markets, sometimes called prediction markets, in which payoffs may be tied to the winner of presidential elections, the box office receipts of a particular movie, or anything else in which participants are willing to take positions.

Prediction Markets W

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If you find S&P "$$ or T-bond contracts a bit dry, perhaps you’d be interested in futures contracts with payoffs that depend on the winner of the next presidential election, or the severity of the next influenza season, or the host city of the #$#% Olym- pics. You can now find “futures markets” in these events and many others.

For example, both Iowa Electronic Markets (www.biz.uiowa .edu/iem) and the Politics page of BetFair (www.betfair.com) maintain presidential futures markets. In September #$&', you could have purchased a contract that would pay off $& in November if Hillary Clinton won the presidential race but noth- ing if she lost. The contract price (expressed as a percentage of face value) therefore may be viewed as the probability of a Clinton victory, at least according to the consensus view of mar- ket participants at the time. If you believed in September that the probability of a Clinton victory was ""%, you would have been prepared to pay up to $."" for the contract. Alternatively, if you had wished to bet against Clinton, you could have(sold the contract. Similarly, you could have bet on (or against) a Donald Trump victory using his contract. (When there are only

two relevant parties, betting on one is equivalent to betting against the other, but in other elections, such as primaries where there are several viable candidates, selling one candi- date’s contract is not the same as buying another’s.)

The accompanying figure shows the price of Democratic and Republican contracts from November #$&) through Election Day. The price clearly tracks each party’s perceived prospects. You can see Clinton’s price rise to above $.*$ in the week just before the election as the polls increasingly suggested she would win. Her price then declined substantially when the FBI announced it was reopening its investigation into her e-mail server. By the day before the election, with the investigation again apparently closed, her price had rebounded to $.%$: Her victory seemed nearly inevitable, at least until the votes were counted.

Interpreting prediction market prices as probabilities actu- ally requires a caveat. Because the contract payoff is risky, the price of the contract may reflect a risk premium. Therefore, to be precise, these probabilities are actually risk-neutral probabilities (see Chapter #&). In practice, however, it seems unlikely that the risk premium associated with these contracts is substantial.

Prediction markets for the !"#$ presidential election. Contract on each party pays $& if the party wins the election. Price is in cents.

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Source: Iowa Electronic Markets, downloaded November &', #$&'.

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