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CHAPTER 11: FORWARD AND FUTURES HEDGING, SPREAD, AND TARGET STRATEGIES
END-OF-CHAPTER QUESTIONS AND PROBLEMS
- ( Short Hedge and Long Hedge ) The terms short and long refer to the position taken in the futures contract.
A short (long) hedge means that you are short (long) futures. Since a hedge implies opposite positions in
the spot and futures markets, a short (long) hedge means that you are long (short) in the spot market.
- ( The Basis ) a. The basis is defined as the difference between the spot price and the futures price.
b. At expiration, the spot price must equal the futures price, give or take a small differential for transaction
costs. Therefore, over the life of the contract, the spot and futures prices will converge and the basis
will go to zero at expiration.
c. The basis is the difference between the spot price and the futures price. If the basis is positive and
strengthens, the spot price increases more or decreases less than the futures price (or the spot price goes
up and the futures price goes down). Since a short hedge is long the spot and short the futures, this is
beneficial. Since the long hedge is long the futures and short the spot, this hurts the long hedge.
- ( The Basis ) a. The dealer is long sugar in the spot market and should sell sugar futures to set up a hedge
S 0 = 0.0479 f 0 = 0.
b 0 = S 0 โ f 0 = 0.0479 โ 0.0550 = โ0.
ฯ = ST โ S 0 โ (fT โ f 0 )
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.
ฯ = 0.0465 โ 0.0479 โ(0.0465 โ 0.0550) = 0.
In terms of the basis,
ฯ = โ b 0 + bt = โ (โ0.0071) + 0 = 0.
In dollars,
ฯ = 112,000($0.0071) = $795.
Thus, the profit on the hedge is โ1 times the original basis times the number of pounds.
b. bt = St โ ft = 0.0574 โ 0.0590 = โ0.
ฯ = St โ S 0 โ (ft โ f 0 ) = 0.0574 โ 0.0479 โ (0.0590 โ 0.0550) = 0.
In terms of the basis,
ฯ = โ b 0 + bt = โ (โ0.0071) + (โ0.0016) = 0.
The basis went from โ0.0071 to โ0.0016, a profit of 0.0055. In dollars,
ฯ = 112,000($0.0055) = $
Thus, the basis strengthened so the hedger gained, though not as much as if the hedge had been held to
expiration.
- ( Contract Choice ) The most important factor is to have a strong correlation between the spot and futures
prices. It is also important that the futures contract have sufficient liquidity. If the contract is not very
liquid, then the hedger may be unable to close the position at the appropriate time without making a
significant price concession. This weakens the effectiveness of the hedge by making the futures price less
dependent on the spot market and the normal cost-of-carry relationship between the two markets. In
addition, the contract should be correctly priced or at least priced in favor of the hedger. For example, a
short (long) hedger would not want to sell (buy) a futures contract that was underpriced (overpriced) as this
would reduce the hedging effectiveness.
- ( Contract Choice ) The rule of thumb is that the contract chosen should expire as soon as possible after the
hedge termination date but not during the month of the hedge termination date. This is because there is
sometimes unusual price behavior in the expiration month resulting from a possible shortage of the
deliverable good. If the contract expires before the hedge is terminated, the hedger will have to roll the
contract into the next expiration. This would incur additional transaction costs. The appropriate expirations
are
a. September
b. March of the next year
c. March of the current year
d. September
- ( Why Hedge? ) One reason firms hedge is because they can do it more effectively than their shareholders.
They are better able to assess the risks, and they have lower transaction costs. Of course, this does not
address the question of why the shareholders would want to hedge in the first place, but this may be because
they want to find a more acceptable combination of risk and return. Firms also hedge for tax advantages, to
reduce the probability of bankruptcy (which has many costs associated with it), and also because the
managers are hedging to protect their own wealth, which is tied so closely to that of the firm.
- ( 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 Nf 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 Nf 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.
- ( 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.
b. The spot bonds are worth
This is an increase of $193,750.
The futures price is 76.4375. The profit from the futures transaction is
The net profit on the hedge is
- ( 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
The profit is
The futures price is 77.15625. The profit from the futures transaction is
The overall loss is
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.
- ( 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.
Goodrich 6,200 31.375 194,525.00 0.
Raytheon 15,800 49.375 780,125.00 0.
Maytag 8,900 55.375 492,837.50 0.
Kroger 11,000 42.125 463,375.00 0.
Comdisco 14,500 19.375 280,937.50 0.
Cessna 9,900 29.75 294,525.00 0.
Foxboro 4,500 24.75 111,375.00 0.
ฮฒ = 1.00(0.0697) + 1.05(0.0691) + 1.15(0.2772) + 0.90(0.1751) + 0.85(0.1647)
The futures price is
Nf = โ($2,813,950/$188,100)(1.067) = โ15.
Sell 16 contracts.
On December 31, the market value of the portfolio is
The gain on the portfolio is
The futures price is
The profit on the futures transaction is
The overall gain is
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.
- ( Stock Market Hedges ) The stock is worth
The futures price is
The number of futures contracts required is
Nf = ($657,500/$187,650)(1.10) = 3.
So buy 4 contracts.
On June 1 the stock is bought for
The increased cost of the stock is
The futures price is
instead of 0.95. Because the beta was not accurately predicted, the transaction was unable to do what it was
supposed to doโremove the systematic risk, leaving the alpha. However, even if the beta were accurately
measured, the actual alpha might not be the expected 10 percent.
- ( Target Duration with Bond Futures ) a. To lower duration you must sell futures.
N^ =
f
or 52 contracts.
b. Profit on spot = 8,952,597 โ 9,448,456 = โ495,
Profit on futures = โ52(68,500 โ 72,094) = 186,
- ( Target Beta with Stock Index Futures ) a. To lower the beta, you must sell futures.
Nf = (ฮฒ (^) T โ ฮฒ (^) S)(S/f)
so sell 7 contracts.
b. Profit on spot = 9,870,000 โ 10,500,000 = โ630,
Profit on futures = โ7(402.35 โ 425.75)(500) = 81,
The portfolio return was โ548,100/10,500,000 = โ0.0522. The market fell (402.35 โ
425.75)/425.75 = โ0.0550. This was an effective beta of 0.95.
- ( Tactical Asset Allocation ) a. To synthetically sell $5 million of domestic stock with a beta of
1.10 would require NDf futures as follows:
Df
N
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:
Ff
N
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
The foreign stock futures price goes to $150,000(1.014) = $152,100. The profit is
The value of the domestic stock goes up to $20,000,000(1.02) = $20,400,000. The total
value of the portfolio is
- ( Stock Market Hedges ) The transaction costs to sell each group of shares are as follows:
Northrup Grumman 20 + 14,870(0.03) = 466.
H. J. Heinz 20 + 8,755(0.03) = 282.
Washington Post 20 + 1,245(0.03) = 57.
Disney 20 + 8,750(0.03) = 282.
Wang Labs 20 + 33,995(0.03) = 1,039.
Wisconsin Energy 20 + 12,480(0.03) = 394.
General Motors 20 + 14,750(0.03) = 462.
Union Pacific 20 + 12,900(0.03) = 407.
Royal Dutch Shell 20 + 7,500(0.03) = 245.
Illinois Power 20 + 3,550(0.03) = 126.
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.
H. J. Heinz 8,755 (36.125) = 316,274.38 0.
Washington Post 1,245 (264) = 328,680.00 0.
Disney 8,750 (134.5) = 1,176,875.00 0.
Wang Labs 33,995 (4.25) = 144,478.75 0.
Wisconsin Energy 12,480 (29) = 361,920.00 0.
General Motors 14,750 (48.75) = 719,062.50 0.
Union Pacific 12,900 (71.5) = 922,350.00 0.
Royal Dutch Shell 7,500 (78.75) = 590,625.00 0.
Illinois Power 3,550 (15.5) = 55,025.00 0.
The beta is
ฮฒ = 1.10(0.055) + 1.05(0.065) + 1.05(0.067) + 1.25(0.241) + 1.20(0.030) + 0.65(0.74) +
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.
- ( 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
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
