asset underlying the futures contract2. The choice of the delivery month
If the asset being hedged exactly matches an asset underlying a futures contract, the first choice isgenerally fairly easy. In other circumstances, it is necessary to carry out a careful analysis todetermine which of the available futures contracts has futures prices that are most closelycorrelated with the price of the asset being hedged.
The choice of the delivery month is likely to be influenced by several factors. In the examplesgiven earlier in this chapter, we assumed that when the expiration of the hedge corresponds to adelivery month, the contract with that delivery month is chosen. In fact, a contract with a laterdelivery month is usually chosen in these circumstances. The reason is that futures prices are insome instances quite erratic during the delivery month. Also, a long hedger runs the risk of havingto take delivery of the physical asset if the contract is held during the delivery month. Takingdelivery can be expensive and inconvenient.
In general, basis risk increases as the time difference between the hedge expiration and thedelivery month increases. A good rule of thumb is therefore to choose a delivery month that is asclose as possible to, but later than, the expiration of the hedge. Suppose delivery months areMarch, June, September, and December for a particular contract. For hedge expirations inDecember, January, and February, the March contract will be chosen; for hedge expirations inMarch, April, and May, the June contract will be chosen; and so on. This rule of thumb assumes78 CHAPTER 4
that there is sufficient liquidity in all contracts to meet the hedger’s requirements. In practice,Vineet Vides tae do te be greatest in choet mastourity Situres contracts The bhedAoer may the nen Gee lee eeeUyuidicy Loss to De Bltatest dn SHOTE MMA LULity JULUIES CUMUIALLS. Lt meager ilay Liererure, If sumesituations, be inclined to use short maturity contracts and roll them forward. This strategy isdiscussed later in the chapter.
Example 4.1 It is March 1. A U.S. company expects to receive 50 million Japanese yen at the endof July. Yen futures contracts on the Chicago Mercantile Exchange have delivery months of March,June, September, and December. One contract is for the delivery of 12.5 million yen. The companytherefore shoris four September yen futures contracts on March 1. When the yen are received at theend of July, the company closes out its position. We suppose that the futures price on March 1 incents per yen is 0.7800 and that the spot and futures prices when the contract is closed out are 0.7200and 0.7250, respectively.
The gain on the futures contract is 0.7800 — 0.7250 = 0.0550 cents per yen. The basis is0.7200 — 0.7250 = —0.0050 cents per yen when the contract is closed out. The effective price obtainedin cents per yen is the final spot price plus the gain on the futures:
0.7200 + 0.0550 = 0.7750This can also be written as the initial futures price plus the final basis:0.7800 — 0.0050 = 0.7750
The total amount received by the company for the 50 million yen is 50 x 0.00775 million dollars, or$387,500.
Example 4.2. It is June 8 and a company knows that it will need to purchase 20,000 barrels ofcrude oil at some time in October or November. Oil futures contracts are currently traded for deliveryevery month on NYMEX and the contract size is 1,000 barrels. The company therefore decides touse the December contract for hedging and takes a long position in 20 December contracts. Thefutures price on June 8 is $18.00 per barrel. The company finds that it is ready to purchase the crudeoil on November 10. It therefore closes out its futures contract on that date. The spot price andfutures price on November 10 are $20.00 per barrel and $19.10 per barrel.
The gain on the futures contract is 19.10 — 18.00 = $1.10 per barrel. The basis when the contract isclosed out is 20.00 — 19.10 = $0.90 per barrel. The effective price paid (in dollars per barrel) is the finalspot price less the gain on the futures, or
20.00 — 1.10 = 18.90
This can also be calculated as the initial futures price plus the final basis:18.00 + 0.90 = 18.90
The total price received is 18.90 x 20,000 = $378,000.
4.4 MINIMUM VARIANCE HEDGE RATIO
 
 
The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of theexposure. Up to now we have always used a hedge ratio of 1.0. For instance, in Example 4.2, thehedger’s exposure was to 20,000 barrels of oil and futures contracts were entered into for thedelivery of exactly this amount of oil. If the objective of the hedger is to minimize risk, setting thehedge ratio equal to 1.0 is not necessarily optimal.Hedging Strategies Using Futures 79
We will use the following notation:
6S: Change in spot price, 5, during a period of time equal to the life of the hedge5F: Change in futures price, F, during a period of time equal to the life of the hedgeos: Standard deviation of 5Sor: Standard deviation of 6F
p: Coefficient of correlation between 5S and $F
h*: Hedge ratio that minimizes the variance of the hedger’s position
In Appendix 4A, we show thath* = p= (4.1)
The optimal hedge ratio is the product of the coefficient of correlation between 6S and $F and theratio of the standard deviation of 8S to the standard deviation of 5F. Figure 4.2 shows how thevariance of the value of the hedger’s position depends on the hedge ratio chosen.
If p= and op = 0g, the hedge ratio, #*, is 1.0. This result is to be expected, because im this casethe futures price mirrors the spot price perfectly. If p = 1 and of = 20¢, the hedge ratio A* is 0.5.This result is also as expected, hecause in this case the futures price always changes by twice asmuch as the spot price.
The optimal hedge ratio, h*, is the slope of the best fit line when 8S is regressed against 5F, asindicated in Figure 4.3. This is intuitively reasonable, because we require h* to correspond to theratio of changes in 5S to changes in 6F. The hedge effectiveness can be defined as the proportion of
the variance that is eliminated hv hedoing Thisis 4° ortie variance that is ciiminatca oy neaging. 1niS iS p", OT
ne oF2
oSThe parameters p, of, and oy in equation (4.1) are usually estimated from historical data on 6S and6F. (The implicit assumption is that the future will in some sense be like the past.) A number ofequal nonoverlapping time intervals are chosen, and the values of 5S and 45F for each of the
Variance ofposition
Hedge ratio, A1 >
h*
Figure 4.2 Dependence of variance of hedger’s position on hedge ratio80 CHAPTER 4
‘s
 
 
Figure 4.3 Regression of change in spot price against change in futures price
intervals are observed. Ideally, the length of each time interval is the same as the length of the timeinterval for which the hedge is in effect. In practice, this sometimes severely limits the number ofobservations that are available, and a shorter time interval is used.
Optimal Number of Contracts
Define variables as follows:
N,: Size of position being hedged (units)Q;: Size of one futures contract (units)
N*: Optimal number of futures contracts for hedging
The futures contracts used should have a face value of h*N,. The number of futures contractsrequired is therefore given byh'N«g
N*=Or
 
(4.2)
Example 4.3. An airline expects to purchase two million gallons of jet fuel in one month anddecides to use heating oil futures for hedging. (The article by Nikkhah referenced at the end of thechapter discusses this type of strategy.) We suppose that Table 4.2 gives, for 15 successive months,data on the change, 3S, in the jet fuel price per gallon and the corresponding change, SF, in thefutures price for the contract on heating oil that would be used for hedging price changes duringthe month. The number of observations, which we will denote by n, is 15. We will denote the ithHedging Strategies Using Futures
81
Table 4.2 Data to calculate minimum variance hedgeratio when heating oil futures contract is used to hedge
purchase of jet fuel
Month Change in futures price Change in fuel price
L
COraynuwn sk wn
Bee He eeUM bw he — oS
 
per gallon (= x;) per gallon (= y;)0.021 0.0290.035 0.020
—0.046 —0.0440.001 0.0080.044 0.026
—0.029 —0.019
—0.026 —0.010
—9.029 —0.0070.048 0.043
—0.006 0.011
—0.036 —0.036
—0.011 —0.0180.019 0.009
—0.027 —0.0320.029 0.023
observations on éF and 4S by x; and y,, respectively. From Table 4.2,
yx, =-0.013, Px? = 0.0138
Yiy, =0.003, oy? = 0.0097
¥ x;y, = 0.0107
Standard formulas from statistics give the estimate of of as
The estimate of oy is
The estimate of p is
Ee Oxy
n-1 ~ na —T
 
 
= 0.0313
 
 
nx — DY
 
[no - (Sa fa E72 - (oy)
= 0.92882 CHAPTER 4
 
From equation (4.1), the minimum variance hedge ratio, A”, is therefore
0.02630.928 x 00313 = 0.78
Each heating oil contract traded on NYMEX is on 42,000 gallons of heating oil. From equation(4.2), the optimal number of contracts is
0.78 x 2,000,000
#000. 14
or, rounding to the nearest whole number, 37.
4.5 STOCK INDEX FUTURES
 
 
Stock index futures can be used to hedge an equity portfolio. Define:
P: Current value of the portfolioA: Current value of the stocks underlying one futures contract
If the portfolio mirrors the index, a hedge ratio of 1.0 is clearly appropriate, and equation (4.2)shows that the number of futures contracts that should be shorted is
N*= A (4.3)Suppose, for example, that a portfolio worth $1 million mirrors the S&P 500. The current value ofthe index is 1,000, and each futures contract is on $250 times the index. In this case P = 1,000,000and A = 250,000, so that four contracts should be shorted to hedge the portfolio.
When the portfolio does not exactly mirror the index, we can use the parameter beta (8) fromthe capital asset pricing model to determine the appropriate hedge ratio. Beta is the slope of thebest fit line obtained when excess return on the portfolio over the risk-free rate is regressedagainst the excess return of the market over the risk-free rate. When 6 = 1.0, the return on theportfolio tends to mirror the return on the market; when B= 2.0, the excess return on theportfolio tends to be twice as great as the excess return on the market; when f = 0.5, it tends tobe half as great; and so on.
Assuming that the index underlying the futures contract is a proxy for the mtarket, it can beshown that the appropriate hedge ratio is the beta of the portfolio. From equation (4.2) thismeans that
Nts pa (4.4)
This formula assumes that the maturity of the futures contract is close to the maturity of the hedgeand ignores the daily settlement
> A small adjustment known as tailing the hedge can be used to take account of the daily settlement when a futurescontract is used for hedging. For a discussion of this, see D. Duffie, Futures Markets, Prentice Hall, Upper SaddleRiver, NJ, 1989; R. Rendleman, “A Reconciliation of Potentially Conflicting Approaches to Hedging with Futures,”Advances in Futures and Options Research, 6 (1993), 81-92. Problem 4.20 deals with this issue.Hedging Strategies Using Futures 83
We illustrate that this formula gives good results with an example. Suppose thatValue of S&P 500 index = 1,000Value of portfolio = $5,000,000Risk-free interest rate = 10% per annumDividend yield on index = 4% per annum
Beta of portfolio = 1.5
We assume that a futures contract on the S&P 500 with four months to maturity is used to hedgethe value of the portfolio over the next three months. One futures contract is for delivery of $250times the index. From equation (3.12), the current futures price should be
1,000¢10-0.08%4/12 _ 1,020.20
From equation (4.4), the number of futures contracts that should be shorted to hedge theportfolio is
Suppose the index turns out to be 900 in three months. The futures price will be900e 0:10.04) 1/12 _ 994.5]The gain from the short futures position is therefore30 x (1,020.20 — 904.51) x 250 = $867,676
The loss on the index is 10%. The index pays a dividend of 4% per annum, or 1% per threemonths. When dividends are taken into account, an investor in the index would therefore earn-9% in the three-month period. The risk-free interest rate is approximately 2.5% per threemonths.* Because the portfolio has a 8 of 1.5,
Expected return on portfolio — Risk-free interest rate= 1.5 x (Return on index — Risk-free interest rate)It follows that the expected return (%) on the portfolio is2.54 [1.5 x (~9.0 ~ 2.5)] = -14.75
The expected value of the portfolio (inclusive of dividends) at the end of the three months istherefore$5,000,000 x (1 — 0.1475) = $4,262,500
It follows that the expected value of the hedger’s position, including the gain on the hedge, is
$4,262,500 + $867,676 = $5,130,176
3 For ease of presentation, the fact that the interest rate and dividend yield are continuously compounded has beenignored. This makes very little difference.84 CHAPTER 4
Table 4.3 Performance of stock index hedge
Value of index
in three months 900.00 950.00 1,000.00 1,050.00 1,100.00
Futures price of index
in three months 904.51 954.76 1,005.01 1,055.26 1,105.51
Gain (loss) on futures
position ($000) 867,676 490,796 113,916 (262,964) (639,843)
Value of portfolio(including dividends)in three months ($000) 4,262,500 4,637,500 5,012,500 5,387,500 5,762,500
Total value of positionin three months ($000) 5,130,176 5,128,296 5,126,416 5,124,537 5,122,657
Table 4.3 summarizes these calculations, together with similar calculations for other values of theindex at maturity. It can be seen that the total value of the hedger’s position in three months isalmost independent of the value of the index.
Table 4.3 assumes that the dividend yield on the index is predictable, the risk-free interest rateremains constant, and the return on the index over the three-month period is perfectly correlatedwith the return on the portfolio. In practice, these assumptions do not hold perfectly, and thehedge works rather less well than is indicated by Table 4.3.
Reasons for Hedging an Equity Portfolio
Table 4.3 shows that the hedging scheme results in a value for the hedger’s position close to$5,125,000 at the end of three months. This is greater than the $5,000,000 initial value of theposition by about 2.5%. There is no surprise here. The risk-free interest rate is 10% per annum, orabout 2.5% per quarter. The hedge results in the hedger’s position growing at the risk-free interestfate,
It is natural to ask why the hedger should go to the trouble of using futures contracts. To earnthe risk-free interest rate, the hedger can simply sell the portfolio and invest the proceeds inTreasury bills.
One answer to this question is that hedging can be justified if the hedger feels that the stocks inthe portfolio have been chosen well. In these circumstances, the hedger might be very uncertainabout the performance of the market as a whole, but confident that the stocks in the portfolio willoutperform the market (aftcr appropriate adjustments have been made for the beta of theportfolio). A hedge using index futures removes the risk arising from market moves and leavesthe hedger exposed only to the performance of the portfolio relative to the market. Another reasonfor hedging may be that the hedger is planning to hold a portfolio for a long period of time andrequires short-term protection in an uncertain market situation. The alternative strategy of sellingthe portfolio and buying it back later might involve unacceptably high transaction costs.
Changing Beta
In the example in Table 4.3, the beta of the hedger’s portfolio is reduced to zero. Sometimesfutures contracts are used to change the beta of a portfolio to some value other than zero. In theHedging Strategies Using Futures $5
example, to reduce the beta of the portfolio from 1.5 to 0.75, the number of contracts shortedshould be 15 rather than 30; to increase the beta of the portfolio to 2.0, a long position in 10contracts should be taken; and so on. In general, to change the beta of the portfolio from £ to f*,where 8 > 6*, a short position in
a P(B~ BY
contracts is required. When f < §*, a long position in
P(B* — BZ
contracts is required.
Exposure to the Price of an Individual Stock
Some exchanges do trade futures contracts on selected individual stocks, but in most cases aposition in an individual stock can only be hedged using a stock index futures contact.
Hedging an exposure to the price of an individual stock using index futures contracts is similarto hedging a stock portfolio. The number of index futures contracts that the hedger should shortinto is given by BP/A, where f is the beta of the stock, P is the total value of the shares owned,and A is the current value of the stocks underlying one index futures contract. Note thatalthough the number of contracts entered into is calculated in the same way as it is when aportfolio of stocks is being hedged, the performance of the hedge is considerably worse. Thehedge provides protection only against the risk arising from market movements, and this risk is arelatively small proportion of the total risk in the price movements of individual stocks. Thehedge is appropriate when an investor feels that the stock will outperform the market but isunsure about the performance of the market. It can also be used by an investment bank that has
underwritten a new issue of the stock and wants protection against moves in the market as awhole.
Consider an investor who in June holds 20,000 IBM shares, each worth $100. The investor feelsthat the market will be very volatile over the next month but that IBM has a good chance ofoutperforming the market. The investor decides to use the August futures contract on the S&P 500to hedge the posilion during the one-month period. The # of IBM is estimated at 1.1. The currentlevel of the index is 900, and the current futures price for the August contract on the S&P 500 is 908.Each contract is for delivery of $250 times the index. In this case P = 20,000 x 100 = 2,000,000 andA= 900 x 250 = 225,000. The number of contracts that should be shorted is therefore
2,000,000 _Lx 35,000 7 9.78
Rounding to the nearest integer, the hedger shorts 10 contracts, closing out the position onemonth later. Suppose IBM rises to $125 during the month, and the futures price of the S&P 500rises to 1080. The investor gains 20,00U x ($125 — $100) = $500,000 on 1BM while losing10 x 250 x (1080 — 908) = $430,000 on the futures contracts.
In this example, the hedge offsets a gain on the underlying asset with a loss on the futurescontracts. The offset might seem to be counterproductive. However, it cannot be emphasized oftenenough that the purpose of a hedge is to reduce risk. A hedge tends to make unfavorable outcomesless unfavorable and favorable outcomes less favorable.86 CHAPTER 4
4.6 ROLLING THE HEDGE FORWARD
 
 
Sometimes the expiration date of the hedge is later than the delivery dates of all the futurescontracts that can be used. The hedger must then roll the hedge forward by closing out onefutures contract and taking the same position in a futures contract with a later delivery date.Hedges can be rolled forward many times. Consider a company that wishes to use a short hedgeto reduce the risk associated with the price to be received for an asset at time 7. If there arefutures contracts 1, 2, 3,..., a (not all necessarily in existence at the present time) withprogressively later delivery dates, the company can use the following strategy:
Time t,: Short futures contract 1.
Time t,: Close out futures contract 1.Short futures contract 2.
Time t3: Close out futures contract 2.Short futures contract 3.
Time ¢,: Close out futures contract n — 1.Short futures contract n.
Time T: Close out futures contract n.
In this strategy there are n basis risks or sources of uncertainty. At time 7 there is uncertaintyabout the difference between the futures price for contract n and the spot price of the asset beinghedged. In addition, on each of the n — 1 occasions when the hedge is rolled forward, there isuncertainty about the difference between the futures price for the contract being closed out and thefutures price for the new contract being entered into. (We will refer to the latter as the rolloverbasis.) In many situations the hedger has some flexibility on the exact time when a switch is made
from one contract to the next, This can be used to reduce the rollover basis risk. For example, if the
rollover basis is unattractive at the beginning of the period during which the rollover must be made,the hedger can delay the rollover in the hope that the rollover basis will improve.
Example4.4 In April 2002, a company realizes that it will have 100,000 barrels of oil,to sell in June2003 and decides to hedge its risk with a hedge ratio of 1.0. The current spot price is $19. Althoughcrude oil futures are traded on the New York Mercantile Exchange with maturities up to six years, wesuppose that only the first six delivery months have sufficient liquidity to meet the company’s needs.The company therefore shorts 100 October 2002 contracts. In September 2002 it rofls the hedgeforward into the March 2003 contract. In February 2003 it rolls the hedge forward again into theJuly 2003 contract.
One possible outcome is that the price of oil drops from $19 to $16 per barrel between April 2002and June 2003. Suppose that the October 2002 futures contract is shorted at $18.20 per barrel andclosed out at $17.40 per barrel for a profit of $0.80 per barrel; the March 2003 contract is shorted at$17.00 per barrel and closed out at $16.50 per barrel for a profit of $0.50 per barrel; the July 2003contract is shorted at $16.30 per barrel and closed out at $15.90 per barrel for a profit of $0.40 perbarrel. In this case the futures contracts provide a total of $1.70 per barrel compensation for the $3per barrel oil price decline. .Hedging Strategies Using Futures 87
Metallgesellschaft
Sometimes rolling the hedge forward can lead to cash flow pressures. The problem wasillustrated dramatically by the activities of a German company, Metallgesellschaft (MG), inthe early 1990s.
MG sold a huge volume of 5- to 10-year heating oil and gasoline fixed-price supply contracts toits customers at 6 to 8 cents above market prices. It hedged its exposure with long positions inshort-dated futures contracts that were rolled over. As it turned out, the price of oil fell and therewere margin calls on the futures position. Considerable short-term cash flow pressures were placedon MG. The members of MG who devised the hedging strategy argued that these short-term cashoutflows were offset by positive cash flows that would ultimately be realized on the long-term fixed-price contracts. However, the company’s senior management and its bankers became concernedabout the huge cash drain. As a result, the company closed out all the hedge positions and agreedwith its customers that the fixed-price contracts would be abandoned. The result was a loss to MGof $1.33 billion.*
SUMMARY
 
This chapter has discussed various ways in which a company can take a position in futurescontracts to offset an exposure to the price of an asset. If the exposure is such that the companygains when the price of the asset increases and loses when the price of the asset decreases, a shorthedge is appropriate. If the exposure is the other way round (i.e., the company gains when the priceof the asset decreases and loses when the price of the asset increases), a long hedge is appropriate.
Hedging is a way of reducing risk. As such, it should be welcomed by most executives. In reality,there are a number of theoretical and practical reasons that companies do not hedge. On atheoretical level, we can argue that shareholders, by holding well-diversified portfolios, caneliminate many of the risks faced by a company. They do not require the company to hedge theserisks. On a practical level, a company may find that it is increasing rather than decreasing risk byhedging if none of its competitors does so. Also, a treasurer may fear criticism from otherexecutives if the company makes a gain from movements in the price of the underlying assetand a loss on the hedge.
An important concept in hedging is basis risk. The basis is the difference between the spot priceof an asset and its futures price. Basis risk is created by a hedger’s uncertainty as to what the basiswill be at maturity of the hedge. Basis risk is generally greater for consumption assets than for
investment assets. yThe hedge ratio is the ratio of the size of the position taken m futures contracts to the size of theexposure. It is not always optimal to use a hedge ratio of 1.0. If the hedger wishes to minimize the
variance of a position, a hedge ratio different from 10 may be appropriate. The optimal hedgeratio is the slope of the best fit line obtained when changes in the spot price are regressed againstchanges in the futures price.
Stock index futures can be used to hedge the systematic risk in an equity portfolio. The numberof futures contracts required is the beta of the portfolio multiplied by the ratio of the value of the
‘ For a discussion of MG, see “MG’s Trial by Essay,” RISK, October 1994, pp. 228-34; M. Miller and C. Culp,“Risk Management Lessons from Metallgesellschaft,” Journal of Applied Corporate Finance, 7, no. 4 (Winter 1995),62-76.
| denond/eupely
” covtongo> wormal88 CHAPTER 4
portfolio to the value of one futures contract. Stock index futures can also be used to change thebeta of a portfolio without changing the stocks comprising the portfolio.
When there is no liquid futures contract that matures later than the expiration of the hedge, astrategy known as rolling the hedge forward may be appropriate. This involves entering into asequence of futures contracts. When the first futures contract is near expiration, it is closed out andthe hedger enters into a second contract with a later delivery month. When the second contract isclose to expiration, it is closed out and the hedger enters into a third contract with a later deliverymonth; and so on. Rolling the hedge forward works well if there is a close correlation betweenchanges in the futures priccs and changes in the spot prices.
SUGGESTIONS FOR FURTHER READING
 
 
 
Ederington, L.H., ‘‘The Hedging Performance of the New Futures Market,” Journal of Finance,34 (March 1979), 157-70.
Franckle, C.T., “The Hedging Performance of the New Futures Market: Comment,” Journal ofFinance, 35 (December 1980), 1273-79.
Johnson, L. L., “The Theory of Hedging and Speculation in Commodity Futures Markets,” Review ofEconomics Studies