of chapter 4

Derivatives Markets Leonidas Rompolis

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Chapter 4: Hedging Strategies Using Futures Many of the participants in futures markets are hedgers. Their aim is to use futures markets to reduce a particular risk that they face. This risk might relate to fluctuations in the price of oil, a foreign exchange rate, the level of the stock market, or some other variable. A perfect hedge is one that completely eliminates the risk. Perfect hedges are rare. For the most part, therefore, a study of hedging using futures contracts is a study of the ways in which hedges can be constructed so that they perform as close to perfect as possible. In this chapter we consider a number of general issues associated with the way hedges are set up. These issues concerns the choice of long or short position, the kind of the futures contract used and the optimal size of the futures position. We initially treat futures contracts as forward contracts. Later we explain an adjustment known as tailing that takes account of the difference between futures and forwards. 4.1. Basic strategies A short hedge is a hedge that involves a short position in futures contracts. A short hedge is appropriate when the hedger already owns as asset and expects to sell it at some future time. A short hedge can also be used when an asset is not owned right now but will be owned at some time in the future. Example: Assume that today an oil producer has just negotiated a contract to sell 1 million barrels of crude oil three months later. It has been agreed that the price that will apply at the contract is the market price at the delivery day. Suppose that the current spot price is $60 per barrel and the crude oil futures price on NYMEX for delivery in 3 months is $59 per barrel. Because each futures contract on NYMEX is for delivery of 1,000 barrels, the company hedges its exposure by shorting 1,000 futures contract. Suppose that the spot price three months later proves to be $55 per barrel. The company realizes $55 million for the oil under its sale contract. The company also realizes a gain of $59 - $55 = $4 per barrel from the short position in the futures contract, or $4 million in total. The total amount realized from both the futures position and the sales contract is therefore $59 per barrel, or $59 million in total. Suppose that the spot price three months later proves to be $65 per barrel. The company realizes $65 million for the oil and loses $65 - $59 = $6 per barrel on the short futures position. Again, the total amount realized is $59 million. It is easy to see that the company ends up with $59 million. Hedges that involve taking a long position in a futures contract are known as long hedges. A long hedge is appropriate when a company knows it will have to purchase a certain asset in the future and wants to lock in a price now. Example: Suppose a copper fabricator knows it will require 100,000 pounds of copper three months later to meet a certain contract. The spot price of the copper is $3.40 per pound, and the futures price for delivery three months later is $3.20 per


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pound. The fabricator hedges it position by taking a long position in four futures contracts on the COMEX division of NYMEX. Each contract is for a delivery of 25,000 pounds of copper. The strategy has the effect of locking in the price of the required copper at $3.20 per pound. Suppose that the spot price of copper three months later proves to be $3.25 per pound. The fabricator gain from the long position on the futures contract is 100,000 (3.25 3.20) = $5,000. It pays 100,000 3.25 = $325,000 for the copper, making the net cost $325,000 - $5,000 = $320,000. For an alternative outcome, suppose that the spot price is $3.05 per pound. The fabricator loses 100,000 (3.20 3.05) = $15,000 on the futures contract and pays 100,000 3.05 = $305,000 for the copper. Again, the net cost is $320,000. It is easy to see that in all cases the company ends up to pay $320,000. Note that it is better for the company to use futures contracts than to buy the copper today in the spot market. If it does the latter, it will pay $3.40 per pound instead of $3.20 per pound and will incur both interest costs and storage costs. For a company using copper on a regular basis, the disadvantage would be offset by the convenience of having the copper on hand. However, for a company that knows it will not require the copper for the next three months, the futures contract alternative is likely to be preferred. Long hedges can be used to manage an existing short position. Consider an investor who has shorted a certain stock. Part of the risk faced by the investor is related to the performance of the whole stock market. The investor can neutralize the risk with a long position in index futures contracts. This type of hedging strategy will be discussed later. The examples we have looked at assume that the futures position is closed out in the delivery month. However, making or taking delivery can be costly and inconvenient. For this reason, delivery is not usually made even when the hedger keeps the futures contracts until the delivery month. Hedgers with long positions usually avoid any possibility of having to take delivery closing out their positions before the delivery month. The arguments in favor of hedging are obvious. Companies, which do not have particular skills or expertise in predicting variables such as interest rates, exchange rates and commodity prices, can avoid the random fluctuations of these variables by hedging. However, there are some issues that we should take into account when we decide to hedge. 1. Hedging and competitors. If hedging is not the norm in a certain industry, it may

not make sense for one particular company to choose to be different from all others. Competitive pressures within the industry may be such that the prices of the goods and services produced by the industry fluctuate to reflect raw materials, interest rates, exchange rates, and so on. A company that does not hedge can expect its profits margins to be roughly constant. However, a company that does hedge can expect its profits margins to fluctuate.

2. Hedging can lead to a worse outcome. It is important to realize that a hedge using futures contracts can result in a decrease or increase in a companys profits relative to the position it would be in with no hedging. In the example involving the oil producer considered earlier, if the price of oil goes down, the company loses money and the futures position leads to an offsetting gain. If the price goes up, the company gains from the sale of the oil, and the futures position leads to an


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offsetting loss. The company is in worse position than it would be with no hedging.

4.2. Basis risk In the examples considered so far the hedger was able to identify the precise date in the future when an asset would be bought or sold. The hedger was then able to use futures contracts to remove almost all the risk arising from the price of the asset on that date. In practice, hedging is often not quite as straightforward. Some of the reasons are as follows: 1. The asset whose price is to be hedged may not be exactly the same as the asset

underlying the futures contract. 2. The hedger may be uncertain as to the exact date when the asset will be bought or

sold. 3. The hedge may require the futures contract to be closed out before the delivery

month. These problems give rise to what is termed basis risk, which will now be explained. 4.2.1. The basis The basis, denoted as bt, in a hedging situation is defined as follows:

bt = St Ft where St is the spot price of the asset to be hedged at time t, and Ft is the futures price of the contract used at time t. 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. Assume that a hedge is put in place at t1 and closed out at time t2. From the definition of the basis we have

1 1 1 2 2 2t t t t t tb S F and b S F= =

Consider first the situation of a hedger who knows that the asset will be sold at time t2 and takes a short futures position at time t1. The price realized for the asset is

2tS and

the profit on the futures position is 1 2t t

F F . The effective price that is obtained for the asset with hedging is therefore

2 1 2 1 2t t t t tS F F F b+ = +

The value of 1t

F is known at time t1. If 2t

b were also known at time t1, a perfect hedge would result. The hedging risk is the uncertainty associated with

2tb and is known as

basis risk. Consider next the situation where the 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

2tS and

the loss on the hedge is 1 2t t

F F . The effective price that is paid with hedging is therefore

2 1 2 1 2t t t t tS F F F b+ = +


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This is the same expression as before. The value of 1t

F is known at time t1, and the term

2tb represents basis risk.

The asset that gives rise to the hedgers exposure is sometimes different from the asset underlying the futures contract that is used for hedging. This increases the basis risk. Define

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*tS as the price of the asset underlying the futures contract at time t2. By

hedging, a company ensures that the price that will be paid (or received) for the asset is:

( )2 1 2 1 2 2 2*t t t t t t tS F F F b S S+ = + + The terms

2tb and

2 2

*t tS S represent the two components of the basis. The first term

is the basis that quantifies the uncertainty of the futures position when it is closed out. The second term is the basis arising from the difference between the two assets. 4.2.2. Choice of contract One key factor affecting basis risk is the choice of the futures contract to be used for hedging. This choice has two components: 1. The choice of the asset underlying the future contract 2. The choice of delivery month If the asset being hedged exactly matches an asset underlying a futures contract, the first choice is generally fairly easy. In other circumstances, we must determine which of the available futures contracts has futures prices that are most closely correlated with the price of the asset being hedged. The choice of a delivery month is influenced by several factors. In fact, a contract with a later delivery month is usually chosen. The reason is that futures prices are quite erratic during the delivery month. Moreover, a long hedger runs the risk of having to take delivery of the physical asset if the contract is held during the delivery month. This could be expensive and inconvenient. However, basis risk increases as the time difference between the hedge expiration and the delivery month increases. A good rule of thumb is therefore to choose a delivery month that is as close as possible to, but later than, the expiration of the hedge. This rule of thumb assumes that there is sufficient liquidity in all contracts to meet the hedgers requirements. In practice, liquidity tends to be greatest in short-maturity futures contracts. Therefore, in some situations, the hedger may be inclined to use short-maturity contracts and roll them forward. This is done by closing out one futures contract and taking the same position in a futures contract with a later maturity. Example: It is March 1. A US company expects 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 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 out its position. We suppose that the futures price on March 1 in cents per yen is 0.78 and that the spot and futures prices when the contract is closed out are 0.72 and 0.725, respectively. The gain on the futures contract is 0.78 0.725 = 0.055 cents per yen. The basis is 0.72 0.725 = -0.005 cents per yen when the contract is closed out. The effective


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price obtained is the final spot price plus the gain on the futures (which can also be written as the initial futures price plus the final basis): 0.72 + 0.055 = 0.775. The total amount received by the company for the 50 million yen is 50 0.775 = $387,000. 4.3. Cross hedging In the examples considered up to now, the asset underlying the futures contract has been the same as the asset whose price is being hedged. Cross hedging occurs when the two assets are different. Consider, for example, an airline that is concerned about the future price of jet fuel. Because there is no futures contract on jet fuel, it might choose to use heating or crude oil futures contracts to hedge its exposure. The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure. The hedge ratio we have used so far was equal to 1. When cross hedging is used, setting the hedge ratio equal to one is not always optimal. The hedger should choose a value for the hedge ratio that minimizes the variance of the value of the hedged position. Suppose that we expect to sell NA units of an asset at time t2 and choose to hedge at time t1 by shorting futures contract on NF units of another asset. Using the same notations as before we can write the total amount realized for the asset when the profit or loss on the hedge is taken into account by:

( )2 2 1A t F t tY N S N F F= or

( ) ( )1 2 1 2 1A t A t t F t tY N S N S S N F F= + Because

1tS is known at time t1 the variance of Y can be written as:

2 2 2 2 2Y A S F F A F S FN N 2N N = +

where 2S and 2

F is the variance of 2 1t tS S S = and 2 1t tF F F = , respectively. denotes the correlation between S and F . The minimum variance with respect to NF is achieved when

22 SY

F F A S F FF F

2N 2N 0 NN

= = =

Thus the optimal hedge ratio is:1

* SF

F

NH

= =

(1)

If = 1 and F S = , the hedge ratio is 1. This result is to be expected, because in this case the futures price mirrors the spot price perfectly (see previous example). If = 1 and F S2 = , the hedge ratio is 0.5. This result is also as expected, because in this case the futures price always changes by twice as much as the spot price. The optimal hedge ratio H* is the slope of the best-fit line when S is regressed against F . This is intuitively reasonable, because we require H* to correspond to the ratio of changes in S to changes in F . The hedge effectiveness can be defined as

1 The solution is the same if we go long futures contracts.


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the proportion of the variance that is eliminated by hedging. This the R2 from the regression of S against F and equals 2 or,

2*2 F

2S

H

The parameters , F and S in equation (1) are usually estimated from the historical data on S and F (the implicit assumption is that the future will in some sense be like the past). If we define as QF the size of one futures contract (units) then the optimal number of futures contract for hedging is:

** A

F

H NNQ

= (2)

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. In practice this means that equation (2) becomes

** A

F

H VNV

= (3)

where VA is the dollar value of the position being hedged and VF is the dollar value of one futures contract (the futures price times QF). Example: Jet fuel futures do not exist in the US, but firms sometimes hedge jet fuel with crude oil futures along with futures for related petroleum products (for example heating oil futures). If we own a quantity of jet fuel and hedge by holding H crude oil futures contracts, our mark-to-market profit depends on the change in the jet fuel price and the change in the futures price:

( ) ( )t t 1 t t 1S S H F F + We can estimate H by regressing the change in the jet fuel price (denominated in cents per gallon) on the change in the crude oil (denominated in dollars per barrel). Doing so using daily data for January 2000 January 2004 gives

( ) 2t t 1 t t 1(0.069) (0.094)S S 0.009 2.037 F F R 0.287 = + = The coefficient on the futures price change tells us that, on average, when the crude oil futures price increase by $1, a gallon of jet fuel increases by $0.02. An airline expects to purchase 2 millions gallons of jet fuel in 1 month and decides to use crude oil futures for hedging. From the previous regression the optimal hedge ratio is 2.037. Each crude oil contract traded on NYMEX is on 42,000 gallons (1000 barrels) of crude oil. Thus the optimal number of contracts, given by equation (2), is

2.037 2,000,000 9742,000

=

Suppose that the spot price and the futures price are 1.94 and 1.99 dollars per gallon. Then VA = 2,000,000 1.94 = 3,880,000 while VF = 42,000 1.99 = 83,580, so that when tailing the hedge the optimal number of contracts is

2.037 3,880,000 94.5983,550

=

If we round this to the nearest whole number, the optimal number of contracts is now 95 rather than 97.


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4.4. Stock index futures Stock index futures can be used to hedge a well-diversified portfolio. If the portfolio mirrors the index, the optimal hedge ratio equals 1. When the portfolio does not exactly mirrors the index, that is the two assets are not perfectly correlated, we can use the parameter beta () from the CAPM to determine the appropriate hedge ratio. The CAPM implies that the return of our portfolio, denoted as rp, is related to the return of the index (which duplicates the market portfolio), denoted as rI, through its beta p, by

rp = r + p(rI r) where r is the risk-free rate. When p = 1, the return on the portfolio tends to mirror the return on the market; when p = 2, the excess return on the portfolio tends to be twice as great as the excess return on the market; when p = 0.5, it tends to be half as great; and so on. The CAPM implies that the beta coefficient is the slope of the best-fit line obtained when excess return on the portfolio is regressed against the excess return of the index. Therefore, following the theory elaborated in Section 4.3 the optimal hedge ratio is equal to the beta, i.e.

H* = p It Vp is the current value of the portfolio and VF is the current value of one futures contract (the futures price times the contract size) then equation (3) gives that the optimal number of futures contract is:

p*p

F

VN

V= (4)

This formula assumes that the maturity of the futures contract is close to the maturity of hedge. Example: Suppose that the S&P 500 futures contract with 4 months to maturity is used to hedge the value of a portfolio over the next 3 months. We obtain the variance-minimizing position in the S&P 500 by using equation (4). A 5-year regression (from June 1999 to June 2004) of the daily portfolio return on the S&P 500 return gives

( ) 2p S&P500(0.0003) (0.0262)r r 0.0001 1.5 r r R 0.7188 = + = Suppose that the value of the portfolio is $5,050,000 the value of the S&P 500 index is 1,000 and the S&P 500 futures price is 1,010. One futures contract is for delivery of $250 times the index. It follows that VF = 250 1,010 = 252,500 and from equation (4), the number of futures contracts that should be shorted to hedge the portfolio is

5,050,0001.5 30252,500

=

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

30 (1,010 902) 250 = $810,000 The loss of 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 lose 9.75% in the 3-months period. The 3-month risk-free interest rate is 1%. The CAPM gives that the expected return on the portfolio is:

( )pr 0.01 1.5 0.0975 0.01 0.1512= + = The expected value of the portfolio at the end of the 3 months is therefore


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5,050,000 (1 0.1512) = $4,286,187 It follows that the expected value of the hedgers position, including the gain on the hedge is

4,286,187 + 810,000 = $5,096,187 Table 1 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 hedgers position is almost independent of the value of the index.

Table 1: Performance of stock index hedge.

Table 1 shows that the hedging scheme results in a value for the hedgers position at the end of the 3-month period being about 1% higher than at the beginning of the 3-month period. The risk-free rate is 1% per 3 months. The hedge results in an investors position growing at the risk-free rate. Why therefore the hedger should go to the trouble of using futures contract? To earn the risk-free rate, the hedger can simply sell the portfolio and invest the proceeds in risk-free instruments. One answer to this question is that hedging can be justified if the hedger feels that the stocks in the portfolio have been chosen well. In these circumstances, the hedger might be very uncertain about the performance of the market as a whole, but confident that the stocks in the portfolio will outperform the market (after appropriate adjustments have been made for the beta of the portfolio). Algebraically, this means that the return of stock X is given by:

rX = X + r + X(rI r) where X represents the expected abnormal return on X. A hedge using index futures removes the risk arising from the market and leaves the hedger exposed only to the performance of the portfolio relative to the market. The result for the hedged position will be that, on average, we earn X + r. Another reason for hedging may be that the hedger is planning to hold the portfolio for a long period of time and requires short-term protection in an uncertain market situation.


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Some exchanges do trade futures contracts on selected individual stocks, but in most cases a position in an individual stock can only be hedged using stock index futures contracts. Hedging an exposure to the price of an individual stock using index futures contracts is similar to hedging a well-diversified portfolio. The optimal number of index futures contracts that the hedger should short is given by equation (4), where p is the beta of the stock, Vp is the total value of the shares owned, and VF is the current value of one index futures contract. The hedge provides protection only against the risk arising from market movements, and this risk is a relatively small proportion of the total risk in the price movements of individual stocks. The hedge is appropriate when an investor feels that the stock will outperform the market but is unsure about the performance of the market (see also previous paragraph).

Exercises

1. Suppose that the S&P 500 index currently has a level of 1,100. The 6-month risk-free rate is 5% and the dividend yield on the index is 3%. You wish to hedge a $300,000 portfolio that has a beta of 1.2 and a correlation of 1 with the S&P 500. One futures contract is for delivery of $250 times the index. (a) How many S&P 500 futures contracts should you short to hedge your

portfolio? (b) What is the expected return of the hedged portfolio?

2. The standard deviation of monthly changes in the spot price of corn is 1.9. The

standard deviation of monthly changes in the futures price of corn for the closest contract is 2.3. Each contract is for delivery of 5,000 bushels of corn. The correlation between the futures price changes and the spot price changes is 0.75. A food industry needs to buy 2,000,000 bushels of corn in 1 month. What strategy can the industry use to hedge its risk?

3. Discuss the following viewpoints. (a) A corn farmer argues. I dont think that I should use corn futures for

hedging. My real risk is not the price of corn. It is that my whole crop can be destroyed by the weather.

(b) An airline executive argues. There is no point to use oil futures. There is an equal chance that the price of oil in the future will be more or less than the futures price.

4. A fund manager has a portfolio worth $35 million with a beta of 0.71. The

manager is concerned about the performance of the market over the next 2 months and plans to use 3-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 980 and one contract is 250 times the index. The risk-free rate is 5% and the dividend yield on the index is 2%. (a) What position should the fund manager take to hedge all exposure to the

market risk over the next 2 months? (b) Calculate the effect of your strategy on the fund managers returns if the

index in 2 months is 950 and 1,100.

5. The file S&P500 Data.xls contains settlement prices of the S&P 500 futures contract with maturity period December 2012.


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(a) Download to a separate spreadsheet in this file, daily prices for the same period to that of the futures price data for two stocks with a ticker symbol which is close to your family name and the S&P 500 index from finance.yahoo.com or from google.com/finance.

(b) Calculate daily returns for the two stocks and the S&P 500 index. (c) Calculate daily returns of a portfolio with equal investments in each of the

two stocks (If r1 and r2 are the returns of the two stocks, respectively, then the return of the portfolio is equal to rp = 0.5r1 + 0.5r2).

(d) Calculate the beta of the portfolio. Beta is calculated by the Excel function SLOPE where y range refers to the portfolio returns and x range refers to the returns of the S&P 500 index.

(e) Consider that you make a $1,000,000 investment in this portfolio. How many S&P 500 futures contracts should you short to hedge your portfolio at the beginning of the period?

(f) Calculate daily values for the portfolio with and without the position in the futures contract. Present these results in a joint graph. Discuss these results.

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