Futures

A futures contract is a type of forward contract with highly standardized and precisely specified contract terms. As in all forward contracts, a futures contract calls for the exchange of some good at a future date for cash, with the payment for the good to occur at that future date. The purchaser of a futures contract undertakes to receive delivery of the good and pay for it, while the seller of a futures contract promises to deliver the goods and receive payment. The price of the good is determined at the initial time of contracting.

Spot contact = is an agreement to buy or sell an asset today (traded OTC).

Forward Contract = is an agreement to buy or sell an asset at a certain time for a certain price. One of the parties to a forward contract assumes a "long position" and agrees to buy the underlying asset on certain date in the future for a certain specified price. The other party assumes a "short position" and agrees to sell the asset on the same date for the same price (traded OTC).

Futures Contract = is an agreement to buy or sell an asset at a certain time for a certain price. One of the parties to a forward contract assumes a "long position" and agrees to buy the underlying asset on certain date in the future for a certain specified price. The other party assumes a "short position" and agrees to sell the asset on the same date for the same price (traded on an Exchange).

Forwards vs. Futures Contracts
Forward	Futures
Private contract between two parties	Traded on an exchange
Not standardized	Standardized contract
Usually one specified delivery date	Range of delivery dates
Settled at end of contract	Settled daily
Delivery or final cash settlement usually takes place	Contract is usually closed out prior to maturity
Some credit risk	Virtually no credit risk

Basis

The basis in a hedging situation is as follows:

Basis = Spot price of asset to be hedged - Futures price of contract used

If the asset to be hedged and the asset underlying the futures contract are the same, the basis should be zero at the expiration of the futures contract. Prior to expiration, the basis may be positive or negative.

As time passes, the spot price and the futures price do not necessarily change by the same amount. As a result, the basis changes. An increase in the basis is referred to as a strengthening of the basis; a decrease in the basis is referred to as a weakening of the basis.

To examine the nature of basis risk we will use the following notation:
S_t: Spot price at time t₁
S₂: Spot price at time t₂
F₁: Futures price at time t₁
F₂: Futures price at time t₂
b₁: Basis at time t₁
b₂: Basis price at time t₂

We will assume that a hedge is put in place at time t₁ and closed out at time t₂. As an example, we will consider the case where the spot and futures price at the time the hedge is initiated are $2.50 and $2.20, respectively, and that at the time the hedge is closed out they are $2.00 and $1.90, respectively. This means that S₁ = 2.50, F₁ = 2.20, S₂ = 2.00, and F₂ = 1.90.

From the definition of the basis, we have

b_{1 = S₁ - F₁ and b₂ = S₂ - F₂}

b₁ = 2.50 -2.20 =.30 and b₂ = 2.00 - 1.90 =.10

Consider first the situation of a hedger who knows that the asset will be sold at times t₂ and takes a short futures position at time t₁. The price realized for the asset is S₂ and the profit on the futures position is F₁ - F₂. The effective price that is obtained for the asset with hedging is therefore:

S₂ + F₁ - F₂ = F₁ + b₂

=2+2.2-1.9 = 2.20

In our example, this is $2.30. The value of F₁ is known at time t₁. If b₂ were also known at this time, a perfect hedge would result. The hedging risk is the uncertainty associated with b₂ and is known as basis risk. Consider next a situation where a company knows it will buy the asset at time t₂ and initiates a long hedge at time t₁. The price paid for the asset is S₂ and the loss on the hedge is F₁ - F₂. The effective price that is paid with hedging is therefore:

S₂ + F₁ - F₁ = F₁ + b₂

This is the same example expression as before and is $2.30 in the example. The value of F₁ is known at time t₁, and the term b₂ represents basis risk.

Note the basis risk can lead to an improvement or a worsening of a hedger's position. Consider a short hedge. If the basis strengthens (i.e., increases) unexpectedly, the hedger's position improves; if the basis weakens (i.e., decreases) unexpectedly, the hedger's position worsens. For a long hedge, the reverse holds. If the basis strengthens unexpectedly, the hedger's position worsens; if the basis weakens unexpectedly, the hedger's position improves.

Example I: Basis Risk on Currency Futures

It is March 1. A US company expect to receive 50 million Japanese yen at the end of July. Yen futures contracts on the CME have delivery months of March, June, September and December. One contract is for the delivery of 12.5 million yen. The company therefore shorts four September yen futures contracts on March 1. When the yen are received at the end of July, the company closes its futures position. We suppose that the futures price on March 1 in cents per yen is 0.780 and that the spot and future prices when the contact is closed out are 0.720 and 0.725 yen respectively.

The gain on the futures is 0.780 - 0.725 = -.055 cents per yen. The basis is 0.720-.0725 = -0.005 cents per yen when the contract is closed out. The effective price obtained in cents per yen is the final spot price plus the gain on the futures:

0.72 + 0.055 = 0.775

This can be written as the initial futures price plus the final basis:

0.780 + (-0.0050) = .7750

Example II: Basis Risk Oil Futures

It is June 8 and a company knows that it will need to purchase 20,000 barrels of crude oil at some time in October or November. Oil futures contracts are currently traded for delivery every month on NYMEX and the contract size is 1,000 barrels. The company therefore decides to use the December contract for hedging and takes a long position in 20 December contracts. The futures price on June 8 is $68.00 per barrel. The company finds that it is ready to purchase the crude oil on November 10. It therefore closes out it futures contract on that date. This spot price and futures price on November 10 are $70.00 per barrel and $69.10 per barrel.

The gain on the futures contract is 69.10 - 68.00 = $1.10 per barrel. The basis when the contract is closed out is 70.00 - 69.10 = $0.90 per barrel (Why? Because now you have to buy the oil at the $70 spot price). The effective price paid (in dollars per barrel) is the final spot price less the gain on the futures, or:

70.00 - 1.10 = 68.90

This can also be calculated as the initial futures price plus the final basis, 68.00 + .90 = 68.90.

The total price paid is 68.90 x 20,000 = $1,378,000 (vs. a non-hedged amount of 70 x 20,000 = $1,400,000) a $22,000 difference. If hedged correctly, the hedge should be 70 - 68 x 20,000 = $40,000.

Hedge Ratio [viewable here in Excel]

Optimal Number of Future Contracts

An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating oil futures for "hedging". We suppose that the information below gives, for 15 successive months, data on the change, ΔS, in the jet fuel price per gallon and the corresponding change, ΔF, in the futures prices for the contract on heating oil that would be used for hedging price changes during the month. The number of observations, which we will denote by nis 15. We will denote the ith observation on ΔF and ΔS by x_i and y_i, respectively.

Table 1 Data to calculate minimum variance hedge ratio when heating oil futures contract is used to hedge purchase of jet fuel.

Month	Change in futures price per gallon (=x?)	Change in fuel price per gallon (=y?)	Variance x	Variance y	Covariance
1	0.021	0.029	0.000478151	0.00082944	0.00062976
2	0.035	0.020	0.001286418	0.00039204	0.00071016
3	-0.046	-0.044	0.002037018	0.00195364	0.00199489
4	0.001	0.008	3.48444E-06	0.00006084	0.00001456
5	0.044	0.026	0.002013018	0.00066564	0.00115756
6	-0.029	-0.019	0.000791484	0.00036864	0.00054016
7	-0.026	-0.010	0.000631684	0.00010404	0.00025636
8	-0.029	-0.007	0.000791484	0.00005184	0.00020256
9	0.048	0.043	0.002387951	0.00183184	0.00209149
10	-0.006	0.011	2.63511E-05	0.00011664	(0.00005544)
11	-0.036	-0.036	0.001234351	0.00131044	0.00127183
12	-0.011	-0.018	0.000102684	0.00033124	0.00018443
13	0.019	0.009	0.000394684	0.00007744	0.00017483
14	-0.027	-0.032	0.000682951	0.00103684	0.00084149
15	0.029	0.023	0.000892018	0.00051984	0.00068096
Sum	-0.013000	0.003000	0.013754	0.009650	0.010696	0.00076397
Average	-0.000867	0.000200	0.000917	0.000643	0.00071304	0.928372346
Variance	0.000982	0.00069
Standard Deviation	0.031343	0.026255

Minimum variance hedge = correlation coefficient * standard deviation of change in fuel price per gallon/change in futures prices per gallon:

=0.92837235*(0.026255/0.031343) = .78

Each heating oil contract traded on NYMEX is on 42,000 gallons of heating oil. The optimum number of futures contracts is = correlation coefficient x gallons of fuel need to purchase next month/heating oil contract size (in gallons)

Number of heating oil contracts to purchase =0.78*2000000/42000 = 37

Tailing the Hedge

When futures are used for hedging, a small adjustment, known as tailing the hedge, can be made to allow for the impact of daily settlement where N = h*V_a/V_r where V_a is the dollar value of the position being hedged and V_r is the dollar value of one futures contract (the futures price time Q_r). Suppose that the spot price and the futures price are 1.94 and 1.99 dollars per gallon. Then V_a = 2,000,000 x 1.94 = 3,880,000 while V_r = 42,000 x 1.99 = 83,500, so that the optimal number of contracts is .78 * 3,880,000 / 83,500 = 36.

Heating oil futures contracts needed to hedge fuel purchase: =0.78*(2000000*1.94)/(42000*1.99) = 36

Hedging an Equity Portfolio [viewable here in Excel]

Stock index futures can be used to hedge a well-diversified equity portfolio. Define:

P: Current value of the portfolio
F: Current value of one futures contract (the futures price times the contract size)

If the portfolio mirrors the index, the optimal hedge ratio, h*, equals 1.0 and equation (3.3) from Tailing the Hedge (N* = h*V_a/V_f) shows that the number of futures contracts that should be shorted is:

Equation (3.4) N* = P/F

Suppose, for example, that a portfolio worth $5,050,000 mirrors the S&P 500. The index futures price is 1,010 and each futures contract is on $250 times the index. In this case P = 5,050,000 and F = 1,010 x 250 = 252,500, so that 20 contracts should be shorted to hedge the portfolio.

When the portfolio does not exactly mirror the index, we can use the parameter beta (β) from the capital asset pricing model to determine the appropriate hedge ratio. Beta is the slope of the best-fit line obtained when excess return on the portfolio over the risk-free rate is regressed against the excess return of the market on the portfolio over the risk-free rate. When β = 1, the return on the portfolio tends to mirror the return on the market; when β = 2, the excess return on the portfolio tends to be twice as great as the excess return on the market; when β = 0.5, it tends to be half as great; and so on.

A portfolio with a β of 2.0 is twice as sensitive to market movements as a portfolio with a beta 1.0. It is therefore necessary to use twice as many contracts to hedge the portfolio. Similarly, a portfolio with a beta of 0.5 is half as sensitive to market movements as a portfolio with a beta of 1.0 and we should use half as many contracts to hedge it. In general, h* = β, so that equation (3.3) gives:

Equation (3.5) N* = β(P/F)

This formula assumes that the maturity of the futures contract is close to the maturity of the hedge.

We illustrate that this formula gives good results by extending our earlier example. Suppose that a futures contract with 4 months to maturity is used to hedge the value of a portfolio over the next 3 months in the following situation:

Value of S&P index = 1,000
S&P 500 futures price = 1,010
Value of portfolio = $5,050,000
Risk-free interest rate = 4% per annum
Dividend yield on index = 1% per annum
Beta of portfolio = 1.5

Our futures contract is for delivery of $250 times the index. If follows that F = 250 x 1,010 = 252,500 and from Equation (3.5), the number of futures contracts that should be shorted to hedge the portfolio is:

1.5 x 5,050,000/252,500 = 30

Suppose the index turns out to be 900 in 3 months and the futures price is 902. The gain from the short futures position is then:

30 x (1010 - 902) x 250 = $810,000

The loss on the index is 10%. The index pays a dividend of 1% per annum, or 0.25% per 3 months. When dividends are taken into account, an investor in the index would therefore earn -9.75% in the 3-month period. Because the portfolio has a ? of 1.50, the capital asset pricing model gives:

Expected return on portfolio - Risk-free interest rate = 1.5 x (Return on index - Risk-free interest rate)

The risk-free interest rate is approximately 1% for 3 months. It follows that the expected return (%) on the portfolio during the 3 months when the 3-month return on the index is -9.75% is:

1.0 + (1.5 x (-9.75 - 1.0) = -15.125

The expected value of the portfolio (inclusive of dividends) at the end of the 3 months is therefore:

$5,050,00 x (1 - 0.15125) = $4,286,187

If follows that the expected value of the hedge's position, including the gain on the hedge is:

$4,286,187 + $810,000 = $5,096,187

Table 3.4 summarizes these calculations together with similar calculations for other values of the index at maturity. It can be seen that the total expected value of the hedger's position in 3 months is almost independent of the value of the index.

Table 3.4 Performance of stock index hedge

Value of index in three months:	900	950	1000	1050	1100
Futures price of index today:	1,010	1,010	1,010	1,010	1,010
Futures price of index in three months:	902	952	1003	1053	1103
Gain in futures position ($):	810,000	435,000	52,500	(322,500)	(697,500)
Return on market:	-9.75%	-4.75%	0.25%	5.25%	10.25%
Expected return on portfolio:	-15.125%	-7.625%	-0.125%	7.375%	14.88%
Expected portfolio value in three months including dividends ($):	4,286,188	4,664,938	5,043,688	5,422,438	5,801,188
Total value of position in three months ($):	5,096,188	5,099,938	5,096,188	5,099,938	5,103,688

Changing the Beta of a Portfolio

In the examples n Table 3.4 "Hedging an Equity Portfolio", the beta of the hedger's portfolio is reduced to zero. (The hedger's expected return is independent of the performance of the index.) Sometimes futures contracts are used to change the beta of a portfolio to some value other than zero. Continuing with our earlier example:

Value of S&P index = 1,000
S&P 500 futures price = 1,010
Value of portfolio = $5,050,000
Beta of portfolio = 1.5

As before F = 250 x 1,000 = 252,000 and a complete hedge requires:

1.5 x 5,050,000/252,000 = 30

contracts to be shorted. To reduce the beta of the portfolio from 1.5 to 0.75, the number of contracts shorted should be 1.5 rather than 30; to increase the beta of the portfolio, a long position in 10 contracts should be taken; and so on. In general, to change the beta of the portfolio from β to β*, where β > β* , a short position in:

(β - β*)P/F

contracts is required. When β < β*, a long position in:

(β* - β )P/F

contracts is required.

Exchange Rates and Triangular Arbitrage [viewable here in Excel]

The Law of One Price applies to the foreign exchange market as well as to other financial markets. Arbitrage ensures that for any three currencies that are freely convertible in competitive markets, it is enough to know the exchange rates between any two in order to determine the third. Thus, as we show, if you know that the yen price of the U.S. dollar is ¥100 and the yen price of the U.K pound is ¥200, if follows by the Law of One Price that the dollar price of the pound is $2.

To understand how arbitrage works in the foreign exchange market, it is helpful to start by considering the price of gold in different currencies. Suppose you know that the current dollar price of gold is $100 per ounce and its price in yen is ¥10,000 per ounce. What would you expect the exchange rate to be between the dollar and yen?

The Law of One Price implies that it should not matter which currency you use to pay for gold. Thus, the ¥10,000 price should be equivalent to $100, which implies that the dollar price of the yen must be $0.01 or 1 cent per yen.

Suppose that the Law of One Price was violated, and the dollar price of the yen $.009 rather than $0.01. Suppose you currently have $10,000 in cash in the bank. Since, by assumption, you can buy or sell gold either for ¥10,000 per ounce or for $100 per ounce, you would convert your $10,000 into $10,000/$0.009 = ¥1,111,111. You would use the yen to buy 111.111 ounces of gold (¥1,111,111/¥10,000 per ounce) and sell the gold for dollars to receive $11,111 (111.11 x $100 per ounce). You would now have $11,111 less the transaction costs of buying and selling the gold and the yen. As long as these transaction costs are less than $1,111, it would pay you to engage in arbitrage.

Yen vs. Gold price Mismatch
Gold per oz.	US	Yen (1 cent per yen)		Yen - Gold by oz
$100	$100	10,000	0.01	1.00
Yen gold mispriced	$10,000	1,111,111.11	0.009	111
	$11,111
Gain	$1,111

This type of transaction is called triangular arbitrage because it involves three assets: gold, dollars and yen.

Now let's look at the relations among the prices of three different currencies: yen, dollars, and pounds. Suppose the U.S. dollar price of the yen is $0.01 per yen (or equivalent ¥100 yen to the dollar), and the price of the yen in terms of the British pound is a half pence (£0.005) to the yen (or equivalent, ¥200 to the pound). From these two exchange rates, we can determine that the U.S. dollar price of the pound is $2.

Although it may not be immediately obvious, there are two ways to buy pounds for dollar. One way is indirectly through the yen market - by first buying yen for dollars and then using the yen to buy pounds. Since, by assumption, one pound costs ¥200, and ¥200 costs $2.00, this indirect way costs $2.00 per pound. Another way to buy pounds for dollars is to just do it directly.

The direct purchase of pounds for dollars must cost the same as the indirect purchase of pounds for dollars because of the Law of One Price. If it is violated, there will be an arbitrage opportunity that cannot persist for very long.

To see how the force of arbitrage works to uphold the Law of One Price in this example, let's look at what would happen if the price of the pound were $2.10 rather than $2. Suppose you walk into a bank in New York City, and you observe the following three exchange rates - $0.01 per yen, ¥200 per pound, and $2.10 per pound. Suppose that there is a window for exchanging dollars and yen, another for exchanging yen and pounds and a third window for exchanging dollars and pounds.

US dollar to the Yen is $0.01 per yen (or equivalently 100 Yen to US dollar). Japanese yen to British pound is a half pence (Pound 0.005) or Yen 200 to the pound. With these two exchange rates, the computed US dollar to pound is $2.00 (.01/.005 = 2).

You can possibly arbitrage currencies if a mismatch occurs. You can buy yen for US dollars and US dollars to buy pounds. Since one pound costs Yen 200, and Yen 200 costs $2.00, pound cost is $2.00.

If this method is mismatched an arbitrage opportunity exists. Let's say the pound is selling for $2.10 rather than $2.00.

Here is how you could make an immediate risk free $10 profit:

Exchange Rates
US Dollar	Yen:USD	Yen: Pound	USD:Pound
	$0.01	$2.00	$2.00
$100.00	$10,000.00	$200.00	$50.00
		$0.00500
What if pound is $2.10 to USD?
1. Convert )Dollar/Yen exchange) $200 USD into Y20,000		$20,000	$100	$210
2. Convert (Yen/Pound exchange) Y$20,000 to $100 Pound
3. Convert (Dollar/Pound exchange) $100 Pound into $2.10 USD for an immediate risk-free trade of $10.

Congratulations, you have converted $200 into $210!

Three exchange rates
Yen	0.01
Pound	0.005
Dollar	2

(Dollar/Yen) / (Pound/Yen) = Dollar/Pound	2
(Dollar/Yen) = Dollar/Pound x (Pound/Yen)	0.01
(Pound/Yen) = (Dollar Yen) / (Dollar/Pound)	0.005

Yen/Pound = US Dollar	2
Yen = Dollar x Pound	0.01
Pound = Yen/Dollar	0.005

Currency One-Year Future Price
One-year interest rate in the US	5%
One-year interest rate in Germany	4%
Spot rate (deutsche mark)	0.65

Forward Contract [viewable here in Excel]

Any time two parties agree to exchange some item in the future at a prearranged price, they are entering into a forward contract. Often people enter into forward contracts without knowing that is what they are called.

For example, you might be planning a trip from Boston to Tokyo a year from now. You make your flight reservations now, and the airline reservation clerk tells you that you can either lock in a price of $1,000 by committing now or you can wait and pay whatever the price may be on the day of your flight. In either case, payment will not take place until the day of your flight. If you decide to lock in the $1,000 price, you have entered into a forward contract with the airline.

In entering the forward contract you eliminate the risk of the cost of your airfare going above $1,000. If the price of a ticket turns out to be $1,500 a year from now, you will be happy that you had the good sense to lock in a forward price of $1,000. On the other hand, if the price turns out to be $500 on the day of your flight, you will still have to pay the $1,000 forward price you agreed to. In that case, you will regret your decision.

The main features of forward contracts and the terms used to describe them are as follows:

Two parties agree to exchange some item in the future at a price specified now - forward price.
The price for immediate delivery of the item is called the spot price.
No money is paid in the present by either party to the other.
The face value of the contract is the quantity of the item specified in the contract multiplied by the forward price.
The party who commits to buy the specified item is said to take a long position, and the party who commits to sell the item is said to take a short position.

Forward contract = an agreement between two parties that call for the delivery of an item on a specified date for an agreed-upon price that is paid in the future.

Long position = buyer
Short position = seller

If the spot price on the contract maturity date is higher than the forward price, the party who is long makes money. But if the spot price on the contract maturity date is lower than the forward price, the party who is short makes money.

Forward contracts are somewhat "customizable"
Future contracts are "standardized" and no changes are permitable

Future Contract Example

Suppose you are a wheat speculator. You gather information on all the supply and demand factors that determine the price of wheat, such as total acreage planted, rainfall, production plans of major baked goods producers, and so on, and come up with a forecast of next month's spot price of wheat. Say it is $2 per bushel. If the current futures price for delivery a month from now is less than $2 per bushel, you buy the futures contract (take a long position), because you expect to make a profit from it.

To see this, suppose the current futures price for wheat to be delivered a month from now is $1.50 per bushel. By taking a long position in this futures contract, you lock in a buying price of $1.50 per bushel for wheat to be delivered a month from now. Since you expect the spot price to be $2 at that time, your expected gain is $.50 per bushel.

On the other hand, suppose that the current futures price for delivery a month from now is greater than $2 per bushel (your forecast); say it is $2.50 per bushel. Then to earn an expected profit, you sell the futures contract (take a short position). By taking a short position in this futures contract, you lock in a selling price of $2.50 per bushel for wheat to be delivered a month from now. You expect to be able to buy wheat at a spot price of $2 per bushel at that time. You therefore expect a gain of $.50 per bushel.

4-Aug-21	You take a long position in a September wheat futures contract.
Margin Deposit	$1,500	8.11%
Bushels	5000
Price on Aug. 4, 1991	$3.6975
Contract Value	$18,488
Typical Margin - 5%	$924.38
Price on Aug. 5, 1991	$3.62500
Difference	$(0.072500)
Margin reduction	$(362.50)
Current Margin Balance	$1,137.50

Hedging [viewable here in Excel]

A wheat farmer wants to protect the price of his wheat crop (30-days before harvest).

Bushels	10,000
Spot price (today's price)	$2.00
Future price (30 days from today)	?

The farmer can sell the crop today at the spot price for delivery in 30 days. Or,

The farmer could short two wheat contracts for delivery in 30 days.

What is more profitable, the spot or the short?

The answer depends on the future price and the cost to carry. If the cost of carry is greater then futures price + storage + insurance + any finance cost then the spot price is better.

Spot price	$2.00
Cost to carry per month	$0.10
Futures price	$2.12

Speculators
Motivated to profit on future price moves
Wheat price today:	$1.50
You believe the price should be:	$2.00
You would then take a long position (30 days delivery):	$1.50
If price moves to say 2.00 w/in 30 days, sell the long position and profit .50 x 5,000 =	$2,500.00
Vice-versa if you think prices will drop, reverse the trade - short wheat 30-days.	$2.00
Profit if wheat drops below $2 w/in 30 days.

Futures cannot be higher then forward price + cost of carry, see gold example below (otherwise it creates an arbitrage position).

1. If future prices are lower than current spot price, then we can take it to be an indicator of the expected spot price (there are some exceptions - risk premium or discount to holding the commodity).

2. Future price is higher than spot price leads to no inference of miss pricing (providing the price is not greater than futures + cost of carry).

F = (1 + r + s)S = 1.10 300 = 330	Buy gold & store it	Buy synthetic gold using forward contract
Gold = $300/oz	Return = S_i - 300/300 - .02	Return = S_i - F/300 -.08
Risk free rate 8%	You lose the 8% risk free rate	Here you buy 1 gold forward contract for a year
Storage cost = 2%		Invest the $300 in a 1-yr 8% risk free treasury bill

	Gold	Forward/Cost of Carry	What if Forward Price was	Dealer
If spot price was higher than futures price for gold (higher than $330) arbitragers would:	$300	$330	$340	Borrow, use funds to buy gold for $300 oz, simultaneously sell gold forward at $340 oz. After paying off the loan & storage cost a yr from now, dealer makes $10/oz.
Buy gold at the spot price and simultaneously sell it for future delivery at the forward price.	$300	$330	$320	Sell gold short in spot at $330 oz, invest the funds in risk-free asset, simultaneously sell gold forward at $320 oz. After paying off loan and collecting storage cost for a yr, dealer makes $10/oz.

If spot price was lower than futures price for gold (higher than $330) arbitragers would: sell short gold at the spot price (i.e., borrow it and sell it immediately), invest the proceeds of the short sale in the risk-free asset, and go long the forward contract.

Hedging Basics

A wheat farmer wants to protect the pricing of his wheat crop and a baker who needs wheat and wants to cap the price of future wheat pricing. Hedging or risk reduction is available for parties through futures contracts. A futures contract is a standardized forward contract traded on an open exchange.

A forward contract is typical bilateral agreement between two or more parties for the actual delivery and payment of goods, services or exchange.

A farmer is expecting 100,000 bushels from this year's crop available in one month. The forward price for delivery next month is $2.00 per bushel. A baker needs 100,000 bushel of wheat for delivery next month. The farmer agrees to sell to the baker for delivery next month 100,000 bushel of wheat for $2.00 per bushel. The farmer is responsible for the cost of delivering wheat on a specific date, but otherwise the farmer and baker have hedged their wheat requirements; the farmer has insured against lower wheat prices and the baker has insured against rising wheat prices. However, unless farmer and baker know each other, arranging such a transaction would be cumbersome. This is where futures contracts come into play. Typically the farmer will sell his wheat to a local coop or supplier and a baker will typically buy wheat from a local distributor. By using wheat futures contracts (and saving on shipping costs) the farmer can sell his wheat through a futures exchange (or at least hedge his future crop with future contracts) while the baker can buy (or hedge future wheat price increase) using future contracts.

Hedging

A hedger is a trader who enters the futures market in order to reduce preexisting risk. If a trader trades futures contracts on commodities in which he or she has no initial position, and in which he or she does not contemplate taking a cash position, then the trader cannot be a hedger. The futures transaction cannot serve as a substitute for a cash market transaction Having a position, in this case, does not mean that the trader must actually own a commodity. An individual or firm that anticipates the need for a certain commodity in the future or a person who plans to acquire a certain commodity later also has a position in that commodity. In many cases, a hedger has a certain hedging horizon - the future date on which the hedge will terminate. For example, a farmer can anticipate that he or she will want to hedge from planting to the harvest. In other cases, there will be no specific horizon. We begin with two examples in which hedgers have definite hedging horizons.

Because the farmer owns the wheat and wants to protect against lower wheat prices, the farmer will short wheat futures contracts. One wheat futures contract equals 5,000 bushels so the farmer will short 20 contacts at $2.00 per bushel x 5,000 x 20 = $200,000 for delivery in one month. If wheat prices move higher the farmer could sell his wheat to a local distributor but will have to pay to cover the 20 short wheat contracts. Let's say wheat prices increase to $2.20 per bushel. The farmer will receive $220,000 from the local distributor and pay $20,000 to cover the short position, netting $200,000. If wheat prices drop to $1.80 per bushel, the farmer will lose $20,000 on his wheat but will make $20,000 on the short position to net $200,000. The same hold true with the baker. If wheat prices move high or lower the baker will not pay more then $200,000 for the100,000 bushels of wheat.

A Long Hedge

The idea that you may be at risk in a certain commodity without actually owning it may be confusing to some, but consider the following example. Silver is an essential input for the production of most types of photographic films and papers, and the price of silver is quite volatile. For a manufacturer of film, there is a considerable risk that profits could be dramatically affected by fluctuations in the price of silver. If production schedules are to be maintained, it is absolutely essential that silver be acquired on a regular basis in large quantities. Assume that the film manufacturer needs 50,000 troy ounces of silver in two months and confronts the silver prices show in table 4.7 on May 10. The current spot price is 1,052.5 cents per ounce, and the price of the JUL futures contract lies above that at 1,068.0, with the SEP futures contract trading at 1,084.0.

Table 4.7 Silver futures prices on May 10

Contract	Prices (cents per troy ounce)
Spot	1,052.50
JUL	1,068.00
SEP	1,084.00

Note: The COMEX trades a silver contract for 5,000 troy ounces.

Fearing that silver prices may rise unexpectedly, the film manufacturer decides that the price of 1,068.0 is acceptable for the silver that he will need in July. He realizes that it is hopeless to buy the silver on the spot market at 1,052.5 and to store the silver for two months. The price differential of 15.5 cents per ounce would not cover his storage costs. Also, the manufacturer will receive an acceptable level of profits even if he pays 1,068.0 for the silver to be delivered in July. To pay a price higher that 1,068.0, however, could jeopardize profitability seriously. With these reasons in mind, he decides to enter the futures market to hedge against the possibility of future unexpected increases in prices, and accordingly, he enters the trades shown in table 4.8

Table 4.8 A long hedge in silver

Date	Cash Market	Futures Market
May 10	Anticipates the need for 50,000 troy ounces in two months and expects to pay 1,068.0 cent per ounce, or a total of $534,000	Buys ten 5,000 troy ounce JUL futures contract at 1,068.0 cents per ounce
July 10	The spot price of silver is now 1,071.0 cents per ounce; the manufacturer buys 50,000 ounces, paying $535,500	Since the futures contract is at maturity, the futures and spot prices are equal, and the ten contracts are sold at 1,071.0 cents per ounce.
	Loss = -$1,500	Profit = $1,500
	Net wealth change = 0

Taking the futures price as the best estimate if the future spot price, the manufacturer expects to pay 1,068.0 cents per ounce for silver in the spot market two months from now, in July. At the same time, he buys ten 5,000-ounce JUL futures contract at 1,068.0 cents per ounce. Since he buys a futures contract in order to hedge, this transaction is known as a long hedge. The trader is also purchasing a futures contract in anticipation of needing the silver at a future date, so these transaction also represent an anticipatory hedge. Time passes, and by July the spot price of silver has risen to 1,071.0 cents per ounce, 3 cents higher than expected. Needing the silver, the manufacturer purchases the silver on the spot market, paying a total of $535,000. This is $1,500 more than expected. Since the futures contract is about to mature, the futures price must equal the spot price, so the film manufacturer is able to sell his ten futures contracts at the same price of 1,071.0 cents per ounce, making a 3 cent profit on each ounce, and a total profit of $1,500 on the futures position. The cash and futures results net to zero. In the cash market, the price was $1,500 more than expected, but there was an offsetting futures profit of $1,500, which generated a net wealth change of zero.

The Reversing Trade and Hedging

One particular feature of these transactions is that the manufacturer did not accept delivery on the futures contract, but offset the contract instead. Rather than accepting delivery on a contract, it usually is better to reverse the trade, because offsetting saves on transaction costs and administrative difficulties. The short trader has the right to choose the delivery destination and the long trader must fear that the short trader will select an unpalatable destination. Instead of taking delivery, the long trader can acquire the physical commodity from normal suppliers. The hedger in this example could have achieved the same results by accepting delivery. If delivery were accepted on the futures contract, the silver would have been secured at a price of 1,068.0 cents per ounce, which is what happened when the reversing trade was used.

A Short Hedge

Although the long silver hedge involved the purchase of a futures contract, hedges do not necessarily involve long futures positions. A short hedge is a hedge in which the hedger sells a futures contract. As an example, we assume the same silver prices and a date of May 10, as shown in Table 4.7. A Nevada silver mine owner is concerned about the price of silver, since she wants to be able to plan for the profitability of her firm. If silver prices fall, she may be forced to suspend production. Given the current level of production, she expects to have about 50,000 ounces of silver ready for shipment in two months. Considering the silver prices shown in Table 4.7, she decides that she would be satisfied to receive 1,068.0 cents per ounce for her silver.

To establish the price of 1,068.0 cents per ounce, the mine owner decides to enter the silver futures market. By hedging, she can avoid the risk that silver prices might fall in the next two months. Table 4.9 shows the mine owner's transaction Notice that these are exactly the mirror image of the film manufacturer's transactions. Anticipating the need to sell 50,000 ounces of silver in two months, the mine owner sells ten 5,000 ounce futures contracts for July delivery at 1,068.0 cents per ounce. On July 10, with silver prices at 1,071.0 cents per ounce, the mine owner sells the silver and receives $535,000. This is $1,500 more than she originally expected. In the futures market, however, the mine owner suffers an offsetting loss. The futures contracts she sold at 1,068.0 cents per ounce, she offsets in July at 1,071.0 cents per ounce. Once again, the profits and losses in the two markets offset each other, and produce a net wealth change of zero.

Table 4.9 A short hedge in silver

Date	Cash Market	Futures Market
May 10	Anticipates the sale of 50,000 troy ounces in two months and expects to receive 1,068.0 cent per ounce, or a total of $534,000.	Sells ten 5,000 troy ounce JUL
July 10	The spot price of silver is now 1,071.0 cents per ounce; the mine owner sells 50,000 ounces, receiving $535,500.	Buys ten contracts at 1,071.0 cents per ounce.
	Gain = $1,500	Loss = -$1,500
	Net wealth change = 0

Viewing the results from the vantage point of July, it is clear that the mine owner would have been $1,500 richer is she had not hedged. She would have received $1,500 more than originally expected in the physicals market, and she would have incurred no loss in the futures market. However, it does not follow that she was unwise to hedge. The mine owner knew before establishing her hedge that she might have a higher profit margin in certain market conditions if her position was left unhedged. However, she knew that she definitely wanted to avoid a lower profit margin and therefore chose to establish a hedged position with future contracts. In hedging, the mine owner and the film manufacturer both decided that the futures price as an acceptable price at which to complete the transaction in July.

Cross-Hedging

In the example of a long and short hedge in silver, hedgers' needs were perfectly matched with the institutional features of the silver markets. The goods in question were exactly the same goods traded on the futures market, the cash amounts matched the futures contract amounts, and the hedging horizon of the mine owner and film manufacturer matched the delivery date for the futures contract. In actual hedging applications, it will be rare for all factors to match so well. In most cases the hedged and hedging positions will differ in (1) the time span covered, (2) the amount of the commodity, or (3) the particular characteristics of the goods. In such cases, the hedge will be a cross-hedge - a hedge in which the characteristics of the spot and futures positions do not match perfectly.

As an example, consider the problem faced by a film manufacturer that uses silver, a key ingred ient in manufacturing photographic film. Film production is a process industry, with more or less continuous production. However, silver futures, listed at the Commodity Exchange, Inc. COMEX Division of the New York Mercantile Exchange (NYMEX), trade for delivery in January, March, May, July, September, and December. The film manufacturer will also need silver in February, April, and so on. Thus, the futures expiration dates and the hedging horizon for the film manufacturer do not match perfectly. Second, consider the differences in quantity between the futures contract and the film manufacturer's needs. The COMEX contract is for 5,000 troy ounces of silver. The film manufacturer will likely need many thousands of ounces, so it will be fairly easy for the manufacturer to choose and trade a number of contracts that will bring the quantity of silver futures close to the actual need. However, if a hedger needed to hedge 7,500 ounces, he or she might have a problem choosing between one or two contracts. Finally, consider the differences in the physical characteristics of the silver underlying the futures contract and the silver used in manufacturing film. To produce film, silver needs to be in pellet form and it does not to be as pure as silver bullion. Also, the pellets contain other metals besides silver. The COMEX silver contract specifies that deliverable silver must be in 1,000-ounce ingots that are 99.9 percent pure. In other words, the silver in the futures contract is extremely pure and refined, not like the adulterated silver products that are typically used in industry. Thus, the film manufacturer will have to hedge his or her industrial silver with pure silver bullion. Cross-hedging is often particularly problematic in the interest rate futures market. Financial instruments are extremely varied in their characteristics, such as risk level, maturity, and coupon rate. By contrast, really active futures contracts are only traded on a few different types of interest-bearing securities.

When the characteristics of the position to be hedged do not perfectly match the characteristics of the futures contract used for the hedging, the hedger must be sure to trade the right number and kind of futures contract to control the risk in the hedged position as much as possible. In general, we cannot expect a cross-hedge to be as effective in reducing risk as a direct hedge. We consider cross-hedging in more detail in other sections.

Stack Hedges Versus Strip Hedges

Some situations require hedges of cash flow over extended periods. A hedge for this type of long-term risk can be implemented in two different ways. First, futures positions can be established in a series of futures contracts of succesivly longer expirations. This is called a strip hedge. Second, the entire futures position can be stacked in the front month and then rolled forward (less the portion of the hedge that is no longer needed) into the next front month contract. This is called a stack hedge.

Each strategy involves tradeoffs. The strip hedge has a higher correlation with the underlying risks than the stack hedge (i.e., has lower tracking error), but may have higher liquidity costs because the more distant contracts may be very thinly traded and may have high bid-ask spreads accompanied by high trade execution risk. The stack hedge has lower liquidity costs but has higher tracking error.

Figure 4.3 Soybean cash prices

Risk Minimization Hedging

In our first examples, we considered hedges when the hedger had a definite horizon in view. Often, the hedger will not want to hedge for a specific future date. Instead, the hedger may wait to control a continuing risk on an indefinite basis. Consider, for example, a soy dealer who holds an inventory of soybeans. From this inventory, the dealer meets orders from her customers. As her inventory becomes low, she periodically replenishes her own inventory from cash market sources. The inventory that she holds will fluctuate in value with the price of futures contracts, as the following case study shows.

Figure 4.3 shows 300 days of historical soybean cash prices. For this case, we assume that the present is day 60, and soybeans are near their recent high, closing today at 719.5 (719.5 cents per bushel). As figure 4.3 shows, soybeans have been quite volatile in the preceding 60 days. Therefore, the dealer decides to hedge her inventory of 1 million bushels of soybeans by selling soybean futures. After she sells futures, she will be long the physical soybeans in her inventory and short soybean futures. If the hedge works, the risk of the combined cash/futures position should be less than the cash position alone.

With an inventory of 1 million bushels, and a soybean contract calling for 5,000 bushels, it might seem wise to sell one bushel in the futures market for each bushel in the cash market. This would call for selling 200 soybean contracts. However, a 1:1 hedge may not be optimal. In our example, the dealer wants to minimize her preexisting risk, which comes from holding her soybean inventory. We assume that she holds a given bean inventory for business reasons, and we treat that inventory decision as fixed. The dealer's problem is to choose the number of futures contracts that will minimize her risk. Thus, we define the hedge ratio (HR) as the number of futures contracts to hold for a given position in the commodity:

HR = -futures position / cash market position

The dealer will trade HR units of the futures to establish the futures market hedge. After establishing the hedge, the trader has a portfolio, P, that consists of the spot position plus the futures position. The profit and losses on the portfolio for one day will be:

P_t+1 - P_t = S_t+1 - S_t + HR(F_t+1 - F_t)

Note that in our initial discussion we considered that the dealer might hedge each bushel in her cash position with one bushel of futures. In that case, the hedge ratio would be -1.0, the negative sign indicates a short position. Generally, if the trader is long the cash commodity, the futures position will be short. Likewise, if the trader is short the cash good, the futures position will be long.

Now, however, the dealer wants to choose the hedge ratio that will minimize the risk of the portfolio of the spot beans and the futures position. The variance of the combined position depends on the variance of the cash price, the variance of the futures price, and the covariance between the two prices. It is a basic statistical rule that the variance of returns on a portfolio, P, of one unit of the spot asset and HR units of a futures contract is given by the following equation:

σ²_P = σ²_S + 2HR σ²_F = 2HRρ_SFσ_Sσ_F

where σ²_P is the variance of the portfolio, P, σ²_S is the variance of S_t, σ²_F is the variance of F_t; and σ_Sσ_F is the correlation between S_t and F_t. The dealer minimizes the variance by choosing a hedge ratio as follows:

HR - ρ_SFσ_Sσ_F / σ²_F = COV_SF / σ²_F

where COV_SF is the covariance between S_t and F_t. As a practical matter, the easiest way to find the risk minimizing hedge ratio is to estimate the following regression:

Equation (4.5) S_t = ∝ + βF_t + ε_t

where ∝ is the constant regression parameter, β is the slope regression parameter, and ε is an error term with zero mean and standard deviation of 1.0. The negative of the estimated beta from this regression is the risk-minimizing hedge ratio, because the estimated beta equals the sample variance between the independent (F_t) and dependent (S_t) variable divided by the minimizing hedge ratio in Equation 4.4.(HR = -ρ_SFσ_Sσ_F / σ²_F = -COV_SF/σ²_F)

From this regression estimation we also obtain a measure of hedging effectiveness. The coefficient of determination, or R², is provided by the regression estimate. Conceptually:

R² = portion of total variance in the cash price changes statistically related to the futures price changes

Thus, the R² will always be a number between 0 and 1. The closer it is to 1.0, the better will be the degree of fit in the regression between the cash and the futures and the better will be the chance for our hedge to work well.

There has been considerable controversy regarding the proper measure. While this controversy is not fully resolved, we recommend using the change in price or the percentage change in price, but not the price level. If the general range of price over the estimation period is fairly stable, the price change measure will be satisfactory. If the price changes dramatically, the use of percentage price changes will given better results.

We now apply this regression approach to the problem of our soybean dealer, using the immediately previous 60 daily data to estimate the following regression equation:

ΔC_t ∝ + βΔF_t + Ε_t

Where ΔCt is the cash price on day t and ΔF_t is the change in the futures price on day t. Estimating the regression gives the following parameter estimates: ∝ = 0.6976; β = 0.8713; R² = 0.56. With the estimated β = 0.8713, the model suggests selling 0.8713 bushels in the futures market for each bushel in inventory. With 1 million bushels in inventory and a futures contract of 5,000 bushels, the model suggests selling 174 contracts because

0.0813 x (1,000,000 / 5,000) = 174.26

From the estimation of the model, we see that the regression accounts for 56 percent of the variance of the cash price change during our sample period. This is important point, because the regression chose an estimate of beta to maximize the R². This provides no certainty that we can expect similar results beyond the estimation period. To this point in our example, we have used the data that would actually be available to a trader on day 60. We assume that our soy-bean dealer estimated her hedge ratio and placed the hedge at the close of the next business day. Next, we want to evaluate the performance of the hedge.

Figure 4.4 shows how soybeans performed from day 61 through day 300, approximately over the next year, when we assume that the dealer offset in the futures market, thereby ending the hedge. The graph in Figure 4.4 shows the wealth change from day 61 onward for one contract of cash soybeans and for a contract (5,000 bushels) of cash soybeans hedged with 0.8713 futures contracts. As the figure shows, soybean prices fell dramatically over the next year (240 trading days). From the time the hedge was placed until about day 150, soybean prices fell about $2.00 per bushel. During the same interval, the hedge position lost about $1.00 per bushel. From about day 150 to day 300, prices drifted somewhat higher.

Figure 4.4 The performance of the hedged and unhedged soybean position

Comparing the unhedged and the hedged strategies, we see that both lost money. However, the hedged strategy avoided about 50 percent of the loss associated with the drop in cash prices. Over the life of the hedge, the unhedged bushel of soybeans lost $1.56 and the hedged bushel lost $0.71. On the inventory of 1 million bushels, this represents a benefit of $850,000 from hedging. From Figure 4.4, we can also see that the hedged position has much less variance than the cash position. For an unhedged bushel, the standard deviation of the price change was $0.0815 per day. For the hedged position the standard deviation was $0.0431.

Several special points need to be made about this particular hedge. First, we see that the hedge made money because the short position in the futures generated profits as soybean prices fell. We must realize that bean prices could just as easily have risen. In that event, the futures position in the hedge would have lost money. This brings us to the second point. Hedging aims at reducing risk, not generating profits. In this case, the goal was to reduce variance, and would have been successful in attaining its goal. Thus, the hedger must expect an equal chance of monetary gains and losses from placing a hedge. However, with a good hedge, the variance can be reduced substantially.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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