CHAPTER 11: FORWARD AND FUTURES HEDGING, SPREAD, AND TARGET STRATEGIES

END-OF-CHAPTER QUESTIONS AND PROBLEMS

1. (Short hedge and long hedge) Another type of hedge situation is faced when a party plans to purchase an
asset at a later date, such as a bread maker. Fearing an increase in wheat prices, the bread maker would
buy futures contracts. Then, if the price of wheat increases, the wheat futures price also will increase and
produce a profit on the futures position. That profit will at least partially offset the higher cost of
purchasing wheat. This is a long hedge, because the hedger, the bread maker here, is long in the futures
market. Because it involves an anticipated transaction, it is sometimes called an anticipatory hedge.

2. (Spread Strategies) The implied repo on a cash-and-carry transaction with the nearby futures is the return
from buying an asset and selling it at the futures price at the expiration date of the futures. Thus, it is a spot
rate. The implied repo rate from a spread is the return from a transaction involving the purchase of the asset
at a given futures price at the expiration of the nearby futures contract and the sale of the asset at another
futures price at the expiration of the deferred futures contract. Thus, it is a forward rate. The two rates are
connected by the normal relationship between spot and forward rates but they certainly need not be the
same.

3. (The Basis) a. The dealer is long sugar in the spot market and should sell sugar futures to set up a hedge

S0 = 0.0479 f0 = 0.0550
b0 = S0 – f0 = 0.0479 – 0.0550 = –0.0071
π = ST – S0 – (fT – f0)

We are not given ST but it will not matter since ST and fT will cancel. So make up a value of ST, say 0.0465.

π = 0.0465 – 0.0479 –(0.0465 – 0.0550) = 0.0071

In terms of the basis,

π = – b0 + bt = – (–0.0071) + 0 = 0.0071

In dollars,

π = 112,000($0.0071) = $795.20

Thus, the profit on the hedge is –1 times the original basis times the number of pounds.

b. bt = St – ft = 0.0574 – 0.0590 = –0.0016

π = St – S0 – (ft – f0) = 0.0574 – 0.0479 – (0.0590 – 0.0550) = 0.0055

In terms of the basis,

π = – b0 + bt = – (–0.0071) + (–0.0016) = 0.0055

The basis went from –0.0071 to –0.0016, a profit of 0.0055. In dollars,

π = 112,000($0.0055) = $616

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 Thus, the basis strengthened so the hedger gained, though not as much as if the hedge had been held to
expiration.

4. (Minimum Variance Hedge Ratio) a. The minimum variance hedge ratio is defined by specifying the
equation for the profit from a hedge consisting of one unit of the spot commodity and N f futures contracts.
Nf is the number of futures contracts that minimizes the variance of the profit on the hedge. The measure of
hedging effectiveness is the amount of risk reduced divided by the original risk. This measures the
percentage of the risk in the spot position that is eliminated by the hedge.

b. (Price Sensitivity Hedge Ratio) The price sensitivity formula gives a value of N f that assures that the
value of the overall position does not change as interest rates change. The price sensitivity formula and the
minimum variance hedge ratio are both risk minimizing hedge ratios. The latter incorporates past
information on the covariance between the spot and futures price changes, while the former utilizes more
current information on the sensitivity of the spot and futures prices to changes in interest rates. If the past
relationship between spot and futures prices holds in the future, the two formulas would produce identical
hedge results.

5. (Contract Choice) The decision of whether to buy or sell futures when hedging is extremely important.
There are three easy approaches:

The first is to identify the worst outcome for an unhedged position and to assume that it will occur.
Then select a futures transaction that will profit if this worst outcome does happen.

The second approach is to identify if the current spot position is short or long. Then take the
opposite position in futures.

The third approach is to identify the spot transaction that you will undertake at the end of the
hedge. The futures transaction at the end of the hedge will then be the opposite of this spot
transaction. Given this futures transaction, do the opposite futures transaction today.

6. (Short Hedge and Long Hedge) a. The firm is exposed to the risk of a falling stock market. Thus, to
profit it would need to execute a short hedge.

b. The investor is exposed to the risk of the bond price rising. Thus, to profit he would need to
execute a long hedge.

c. The firm is long the currency and is exposed to the risk of a fall in the value of the currency. Thus,
to profit it would need to execute a short hedge.

7. (Foreign Currency Hedges) The exposure is to 100,000(SF225) = SF22,500,000. The dealer would like to
lock in the cost in dollars. Thus, he could buy 22.5 million Swiss francs in a forward contract at the rate of
$0.3881 with an expiration of August 16. When the contract expires, it does not matter what the spot rate is
as the 22.5 million Swiss francs are purchased at the contract rate of $0.3881. Thus, the total cost is
(22,500,000)($0.3881) = $8,732,250.

8. (Foreign Currency Hedges) The bank currently holds a long position in Canadian dollars, worth

5,000,000($0.7564) = $3,782,000

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 and needs to protect against a decline in the value of the Canadian dollar. Therefore, it needs to sell
Canadian dollar futures, which are priced at

100,000($0.7541) = $75,410

It needs $5,000,000/$100,000 = 50 contracts.

January 2
The bank sells 50 contracts.

February 28
The 5,000,000 Canadian dollars are converted at a rate of $0.7207 for

5,000,000($0.7207) = $3,603,500

a decrease in value of $178,500. The futures contracts are bought for $0.7220 or

100,000($0.7220) = $72,200

This produces a profit on the futures of

50($75,410 – $72,200) = $160,500

This reduces the loss on the currency conversion to only $18,000. The hedge eliminated 90 percent of the
loss.

9. (Intermediate- and Long-Term Interest Rate Hedges) a. The spot bonds are worth

0.78875($5,000,000) = $3,943,750

The futures price is 71.25. The number of futures contracts is

Nf = (7.81/8.32)($3,943,750/$71,250) = 51.9

So buy 52 contracts at a price of $71,250 each.

b. The spot bonds are worth

0.8275($5,000,000) = $4,137,500

This is an increase of $193,750.

The futures price is 76.4375. The profit from the futures transaction is

52($76,437.50 – $71,250) = $269,750

The net profit on the hedge is

$269,750 – $193,750 = $76,000

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10. (Intermediate- and Long-Term Interest Rate Hedges) a. The manger is long the bonds and is exposed to
a fall in the price of the bonds, so the appropriate transaction is to sell 13 contracts.

b. The spot bonds are worth

1.01375($1,000,000) = $1,013,750

The profit is

$1,013,750 – $1,074,375 = –$60,625

The futures price is 77.15625. The profit from the futures transaction is

–13($77,156.25 – $77,468.75) = $4,062.50

The overall loss is

$60,625 – $4,062.50 = $56,562.50

Note that the hedge reduced only a small portion of the loss. This suggests that the relationship between the
spot and futures prices obtained from the price sensitivity hedge ratio did not hold perfectly.

11. (Stock Market Hedges) First find the portfolio beta on October 1.

Stock Shares Price Value Weight

Donnelly 10,000 $19.625 $196,250.00 0.0697
Goodrich 6,200 31.375 194,525.00 0.0691
Raytheon 15,800 49.375 780,125.00 0.2772
Maytag 8,900 55.375 492,837.50 0.1751
Kroger 11,000 42.125 463,375.00 0.1647
Comdisco 14,500 19.375 280,937.50 0.0998
Cessna 9,900 29.75 294,525.00 0.1047
Foxboro 4,500 24.75 111,375.00 0.0396
$2,813,950.00

β = 1.00(0.0697) + 1.05(0.0691) + 1.15(0.2772) + 0.90(0.1751) + 0.85(0.1647)

+ 1.45(0.0998) + 1.20(0.1047) + 0.95(0.0396) = 1.067

The futures price is

376.20($500) = $188,100

Nf = –($2,813,950/$188,100)(1.067) = –15.96

Sell 16 contracts.

On December 31, the market value of the portfolio is

10,000($27.375) + 6,200($32.875) + 15,800($53.625) + 8,900($77.875)

+ 11,000($47.875) + 14,500($28.625) + 9,900($30.125) + $4,500($26)
= $3,374,862.50

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 The gain on the portfolio is

$3,374,862.50 – $2,813,950 = $560,912.50

The futures price is

424.90($500) = $212,450

The profit on the futures transaction is

–16($212,450 – $188,100) = –$389,600

The overall gain is

$560,912.50 – $389,600 = $171,312.50

The portfolio gained in value, but some of the gain was offset by the loss on the futures. After the fact, the
firm should not have hedged, but, of course, it did not know that the market would have increased.

12. (Stock Market Hedges) The stock is worth

20,000($32.875) = $657,500

The futures price is

375.30($500) = $187,650

The number of futures contracts required is

Nf = ($657,500/$187,650)(1.10) = 3.85

So buy 4 contracts.

On June 1 the stock is bought for

20,000($38.625) = $772,500

The increased cost of the stock is

$772,500 – $657,500 = $115,000

The futures price is

387.30($500) = $193,650

The profit on the futures transaction is

4($193,650 – $187,650) = $24,000

The net additional cost of the stock is

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 $115,000 – $24,000 = $91,000

The hedge eliminated about 21 percent of the increase in the cost of the stock.

13. (Intramarket Spreads) Because you are bearish, you sell the more volatile contract–the nearby–which is
the December, and buy the March contract. The outcome is

July 15

Sell December at 76-9/32 = 76.28125

Buy March at 75-9/32 = 75.28125
Net price = 1

November 15

Buy December at 79-13/32 = 79.40625

Sell March at 78-9/32 = 78.28125
Net price = –1.125

The investor sold the spread at 1 and bought it back at 1.125. The result is a loss of 0.125 per $100 of face
value, or $125 per $100,000 of face value.

14. (Alpha Capture)

S = 100,000($27.625) = $2,762,500
f = $500(393) = $196,500
Nf = ($2,762,500/$196,500)(.95) = 13.36
So sell 13 contracts.

On December 31, the stock is worth

100,000($28.875) = $2,887,500

The futures price is

$500(432.30) = $216,150

The profit on the futures contract is

–13($216,150 – $196,500) = –$255,450

The net value of the stock on December 31 is

$2,887,500 – $255,450 = $2,632,050

Consequently, the result is about a 4.7 percent loss.

This example illustrates the speculative nature of this type of transaction. The market increased by about 10
percent; however, the stock increased by only about 4.5 percent. Thus, its beta was really only about 0.45
instead of 0.95. Because the beta was not accurately predicted, the transaction was unable to do what it was

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 supposed to do–remove the systematic risk, leaving the alpha. However, even if the beta were accurately
measured, the actual alpha might not be the expected 10 percent.

15. (Target Duration with Bond Futures) a. To lower duration you must sell futures.

(5 - 8.33) (9,448,456) (1.12)

Nf = = - 51.65
(8.43) (72,094) (1.1225)

or 52 contracts.

b. Profit on spot = 8,952,597 – 9,448,456 = –495,859

Profit on futures = –52(68,500 – 72,094) = 186,888
–308,971

16. (Target Beta with Stock Index Futures) a. To lower the beta, you must sell futures.

Nf = (β T – β S)(S/f)

= (1 – 1.15)(10,500,000/(425.75(500))) = –7.40

so sell 7 contracts.

b. Profit on spot = 9,870,000 – 10,500,000 = –630,000

Profit on futures = –7(402.35 – 425.75)(500) = 81,900
–548,100

The portfolio return was –548,100/10,500,000 = –0.0522. The market fell (402.35 –
425.75)/425.75 = –0.0550. This was an effective beta of 0.95.

17. (Tactical Asset Allocation) a. To synthetically sell $5 million of domestic stock with a beta of 1.10
would require NDf futures as follows:

 $5,000,000 
N Df   (0  1.10)  22
 $250,000 
In other words, sell 22 contracts to reduce the beta on $5 million to zero. To synthetically buy $5 million of
foreign stock with a beta of 1.05 would require NFf futures as follows:

 $5,000,000 
N Ff   (1.05  0)  35
 $150,000 
In other words, buy 35 contracts to increase the beta on $5 million to 1.05.

b. The domestic stock futures price goes to $250,000(1.018) = $254,500. The profit is

–22($254,500 – $250,000) = –$99,000

The foreign stock futures price goes to $150,000(1.014) = $152,100. The profit is

35($152,100 – $150,000) = $73,500

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 The value of the domestic stock goes up to $20,000,000(1.02) = $20,400,000. The total value of the
portfolio is

$20,400,000 – $99,000 + $73,500 = $20,374,500

18. (Stock Market Hedges) The transaction costs to sell each group of shares are as follows:

Northrup Grumman 20 + 14,870(0.03) = 466.10

H. J. Heinz 20 + 8,755(0.03) = 282.65
Washington Post 20 + 1,245(0.03) = 57.35
Disney 20 + 8,750(0.03) = 282.50
Wang Labs 20 + 33,995(0.03) = 1,039.85
Wisconsin Energy 20 + 12,480(0.03) = 394.40
General Motors 20 + 14,750(0.03) = 462.50
Union Pacific 20 + 12,900(0.03) = 407.00
Royal Dutch Shell 20 + 7,500(0.03) = 245.00
Illinois Power 20 + 3,550(0.03) = 126.50
3,763.85

To determine the number of futures needed, we need the beta of the portfolio. The market values of the
stocks and their weights are as follows:

Value Weight
Northrup Grumman 14,870 (18.125) = 269,518.75 0.055
H. J. Heinz 8,755 (36.125) = 316,274.38 0.065
Washington Post 1,245 (264) = 328,680.00 0.067
Disney 8,750 (134.5) = 1,176,875.00 0.241
Wang Labs 33,995 (4.25) = 144,478.75 0.030
Wisconsin Energy 12,480 (29) = 361,920.00 0.074
General Motors 14,750 (48.75) = 719,062.50 0.147
Union Pacific 12,900 (71.5) = 922,350.00 0.189
Royal Dutch Shell 7,500 (78.75) = 590,625.00 0.121
Illinois Power 3,550 (15.5) = 55,025.00 0.011
4,884,809.38

The beta is

β = 1.10(0.055) + 1.05(0.065) + 1.05(0.067) + 1.25(0.241) + 1.20(0.030) + 0.65(0.74) +

0.95(0.147) + 1.20(0.189) + 0.75(0.121) + 0.60(0.011) = 1.048

The number of futures is

Nf = (1.048)(4,884,809.38)/(369.45(500))) = 27.7 or 28

Cost of trading futures = 28(27.50) = 770

The stocks would cost $3,763.85 plus $25 for each t-bill the funds were placed in. So the minimum would
be $3,788.85. The futures cost $770 or about 1/5 the cost of selling the stocks.

19. (Hedging Strategies) Current spot position is 1,000(950) = 950,000 in Swiss francs worth
SF950,000($0.7254/SF) = $689,130. You will need to buy Swiss franc futures because you will be hurt on

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 the spot purchase of the stock if the franc rises. You will require 950,000/125,000 = 7.6 contracts. So buy 8
at $0.7250.

At expiration your futures profit is

8($0.7295 – $0.7250)(125,000) = $4,500

The stock will cost

1,000(926.50)($0.7301) = $676,438

which is $12,692 less. Thus your overall gain is $12,692 + $4,500 = $17,192.

The stock's performance was a loss of

1,000(950 – 926.50)($0.7254) = $17,047

The currency strengthened for a gain of

1,000(926.50)($0.7301 – $0.7254) = $4,355

The futures hedge generated a gain of $4,500.

To the hedger the loss on the stock is a gain because the stock will be cheaper to buy. The gain on the
currency is a loss to the hedger because more dollars will be required to acquire the necessary units of
currency. The reason the stock price of 926.50 rather than 950 is used is because that is the number of
Swiss francs that will have to be acquired per share.

Thus, the hedge's performance consists of

+$17,047 (from the stock)

–4,355 (from the currency)
+4,500 (from the futures)
$17,192

20. (Intramarket Spreads) If you are bullish you must think interest rates are going down. Would you believe
the nearby contract or the deferred contract would be more volatile? You would normally expect the nearby
to be more volatile. This is because it must absorb the impact of lower interest rates over a shorter period of
time. The deferred contract has much more time to go during which other news could offset the effect or
reverse the direction of the interest rate move.

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