$646 is less than the maintenance margin $700 so the customer receives a margin
call. He will have to pay the variation margin $946 - $646 = $300. If not, the
customer's position is closed.
Day 3 The price of the corn futures rises to $2.08 per bushel.
Change in futures price: $2.08 - $2.04 = $0.04 per bushel
Investor's gain: 2 x 5, 000 x $0:04 = $400
Margin Account Balance: $946 + $400 = $1, 346
$400 is more than the initial margin so the customer can take out $400.
The effect of the marking to market is that a futures contract is settled daily
rather than at the end of its life.
Hedging with stock index futures
Stock index futures provide no facility for delivery and receipt of stock via exercise of the contract. Stock price movements are matched by compensatory cash flows. Futures contracts are available on many stock indices. Stock indices on which futures are traded include the S&P 100, S&P 500 Nikkei 225, FTSE 100, DAX, CAC 40 and the Hang eng. There are contracts relating to all the major stock markets. Contracts sizes are based on sums of money per index point.
Short Hedge
When using index futures to reduce stock market risk, the anticipation is that any losses arising from movements in stock prices are offset by gains from parallel movements in future prices. An investor could reduce the risk of the reduction in the value of the portfolio by taking a position in the futures market that would provide a gain. By taking this short position, he or she guarantees a notional selling price of a quantity of stock for a specific date in the future.
Short Hedge Example 1: MIDDLEMEN
It’s November. You own a grain elevator and must decide whether to buy and store soybeans until February.
Soybeans cash price = $5.40/bu
Soybeans Mar. futures = $5.96/bu
Expected Feb. basis = -$0.30/bu
Expected Feb. local cash price =
5.96 + (-0.30) = $5.66/bu
Expected gains from storage =
5.66 - 5.40 - 0.16 = $0.10/bu
NOV. Buy $5.40 Sell Mar. $5.96 Expected -$0.30
FEB. Sell $5.28 Buy back $5.58 Actual -$0.30
Cash Price + Futures Gain/Loss = Net Selling Price
$5.28 + $0.38 = $5.66 (as expected)
Long Hedge
The investor could close out also his long position in futures in the same number of contracts, if the excess of the selling price over the buying price is paid to the investor in cash in the form of variation margin. This gain on the futures contracts is received on a daily basis as the future price moves (marking to market)
Had the prices of stocks risen, the investor would have gained from his or her portfolio of equities, but lost on futures dealings. In either case, the investor has succeeded in reducing the extent to which the value of the portfolio fluctuates.
Long Hedge Example: END USER
It’s October. You plan to purchase 160 feeder steers @ 700 lbs each, to place in the feedlot in January
112,000 lbs
Need two feeder cattle futures contracts of 50,000 lbs each
Feeder Cattle Mar. futures =
$75.00/cwt
Expected Jan. basis = -$10.00/cwt
Expected Jan. local cash price =
75.00 + (-10.00) = $65.00/cwt
Scenario 1: Basis WEAKENS
OCT. Expect. $65.00 Buy Mar. $75.00 Expected -$10.00
JAN. Buy $63.30 Sell back $73.50 Actual -$10.20
Cash Price - Futures Gain/Loss = Net Buying Price
$63.30 - (-$1.50) = $64.80 ($0.20 LESS than expected)
Scenario 2: Basis STRENGTHENS
OCT. Expect. $65.00 Buy Mar. $75.00 Expected -$10.00
JAN. Buy $63.60 Sell back $73.50 Actual -$9.90
Cash Price - Futures Gain/Loss = Net Buying Price
$63.60 - (-$1.50) = $65.10 ($0.10 MORE than expected)
Convergence
Carrying costs fall as the futures settlement date approaches because the time period a cash position must be held grows shorter. This causes futures prices to converge to underlying spot market prices as the delivery date draws near. On the final day of trading in a futures contract, a futures transaction is essentially equivalent to a spot transaction, so futures prices should differ little from spot prices. Changes in carrying costs can thus explain the phenomenon of convergence. Because of convergence, basis tends to decline systematically over the life of a hedge
The Hedge Ratio
Portfolios are of different sizes and different risk levels. It is necessary to recognize these differences in order to build a proper hedge. Though there is a high degree of influence with futures. Large portfolios require more contracts than do small ones. Similarly, risky portfolios fluctuate more than the market average; thus they require a larger hedge.
Beta
Beta is also a measure of the relative riskiness of a portfolio compared to a benchmark portfolio like the S&P 500 index. The benchmark has a beta of 1.0.
Suppose a portfolio has a beta of 0.92. This means that for every 1 percent change in the value of the S&P 500 index, the portfolio should change in value by 0.92 percent.
PRINCIPLES OF STOCK INDEX ARBITRAGE
Stock index arbitrage is an investment strategy designed to earn a higher money rate without assuming additional risk or, equivalently, to earn higher returns that the stock index while incurring the same risk. The arbitrage strategy is based on the futures/forward pricing principles that govern the relationship between futures prices and the underlying asset price. The cost of carry model provides a signal for index arbitrage opportunities. The signal indicates whether the actual stock index futures price is out of line with the theoretical price.
The arbitrage activities associated with stock index futures contracts are unique and are made possible by a shift of cash between long and short positions in stocks and money market instruments and corresponding short and long positions in stock index futures contracts. Recall that index arbitrageurs will enter into an arbitrage trade when a long position in the index and an offsetting short position in stock index futures produce a return in excess of the arbitrageur’s financing costs. For cash managers, however, this position is considered an alternative to other money market instruments. Arbitrageurs will also enter into an arbitrage trade of selling, or shorting, the index and buying, or going long, stock index futures whenever the trade can be done below the arbitrageur’s marginal lending rate. This strategy will always perform better than the index itself over the same time period, which will benefit equity index fund managers.
In the case of stock index futures, the arbitrager might buy the portfolio of stocks on which the index is based and hold it until the maturity date of the futures contract. In true arbitrage this portfolio would be financed by borrowing money. This purchase of stock via borrowing money with a view to holding the stock until a time in the future constitutes a financially engineered (or synthetic) futures position. If the cost of this synthetic futures position differs from the price of traded futures, an arbitrage opportunity may be available. If the actual futures price exceeds that of the synthetic, then a profit might arise from long cash-and-carry arbitrage that involves borrowing money and using it to buy the stock index portfolio while simultaneously selling futures. The proceeds from selling the stock via the futures would exceed the sum of money to be repaid. The excess would be the arbitrage profit.
If stock index futures are trading at a price below that of the synthetic, they would be bought and a synthetic short position taken. So in this case the cash-and-carry arbitrage would involve buying futures while short selling stock (selling borrowed stock) and putting the proceeds on deposit (or into other short-term risk-free assets) the stock acquired when the futures contract matures is used to repay the stock borrowing. In this case the proceeds from depositing money (net of dividend obligations) exceed the amount required to buy the stock at the price guaranteed by the futures contracts, and this excess constitutes the arbitrage profit. It is to be noted that the stock transactions, the futures trades and the borrowing/depositing are all carried out simultaneously.
There are three kinds of arbitrage that determines the prices of future contracts.
- Covered interest arbitrage, to establish interest rate parity in the case of currency futures
- Arbitrage based on forward rates, in the case of short- term interest rate futures
- Cash-and-carry in the case of long-term interest rate futures.
Cash and Carry Arbitrage
To see why futures prices should conform to the cost of carry model, consider the arbitrage opportunities that would exist if they did not. Suppose the futures price exceeds the cost of the underlying item plus carrying costs; that is,
F(0,T) > S(0) + c(0,T).
In this case, an arbitrageur could earn a positive profit of F(0,T) - S(0) - c(0,T) dollars by selling the overpriced futures contract while buying the underlying item, storing it until the futures delivery date, and using it to satisfy delivery requirements.
This type of transaction is known as cash-and-carry arbitrage because it involves buying the underlying item in the cash market and carrying it until the futures delivery date. Ultimately, the market forces created by arbitrageurs selling the overpriced futures contract and buying the underlying item should force the spread between futures and spot prices down to a level just equal to the cost of carry, where arbitrage is no longer profitable. In practice, arbitrageurs rarely find it necessary to hold their positions to contract maturity; instead, they undertake offsetting transactions when market forces bring the spot-futures price relationship back into alignment
The Cash and Carry Arbitrage
• Example: Spot gold price: $400
Futures gold price: $450
Interest rate: 10%
Transaction Cash Flow
Borrow $400 for one year at 10% +$400
Buy one oz. of gold at S = $400 -$400
Sell 1 year futures @ $450 $0
Total Cash Flow $0
Deliver gold against futures next year. $450
Repay loan with interest next year $440
Total Cash Flow $10
Fair future prices and no-arbitrage bands
The fair futures price is upon arbitrage. In the case of stock index futures, this would be a cash-and-carry arbitrage. The futures price should be such that there is no arbitrage profit from buying stock and selling futures. The future price should provide a guaranteed capital gain that exactly compensates for the excess of the interest payments over the (expected) dividend receipts.
When the future price is within the no-arbitrage band there will be no further buying or selling by arbitragers to move the future price towards the fair futures price.
If the future price falls below the bottom of the no-arbitrage, arbitragers would buy futures until the future price reaches the bottom of the band, at which point arbitrage will stop. A futures price above the top of the no-arbitrage band would induce long cash and carry arbitrage, which involves selling futures.
Short cash-and-carry involves selling stock and buying futures. In this case the excess of interest over dividends is a net inflow and this gain should be matched by having a guaranteed future purchase price that exceeds the spot sale price by the amount of this net inflow.
The formula for fair value premium is:
FP = I x [(r-y) x (d/365)]
Where:
FP = Fair futures premium
I = Spot FTSE 100 Index
r = Interest rate
y = Dividend yield
d = days to future maturity
Conclusion
Stock index futures can be used to reduce systematic risk associated with a well-diversified stock portfolio. The price of stock index futures contract is determined by adjusting the current value of the index for the differential between the dividend yield on the index portfolio and the prevailing treasury bill rate.
With stock index futures the hedge ratio depends on the size of the portfolio, its beta, and the value of the chosen futures contract. Furthermore, we have short and long cash-and-carry and hedge.
BIBLIOGRAPHY
- Keith Redhead, Introduction to financial futures and options, New York : London : Woodhead-Faulkner, 1990
- Keith Redhead, Financial Derivatives, An introduction to Futures, Forwards, Options and Swaps. London: Prentice Hall, c1997, (i.e. 1996)
- Robert A. Strong, Speculative Markets, 2nd ed, New York: HarperCollins, 1994
- Don M. Chance, An introduction to derivatives, 3rd ed, Fort Worth, Tex.: London: Dryden Press, 1995
- Robert T. Daigler, Financial futures markets: concepts, evidence and application, New York: HarperCollins, 1993
- Robert W. Kolb, Understanding futures markets, 5th ed, Oxford: Blackwell Business, 1997
- Todd Lofton, Getting started in futures, 3rd ed, New York: Chichester: Wiley, c1997
- Bruce M. Collins and Frank J. Fabozzi, Derivatives and equity portfolio management, New Hope, PA: Frank J. Fabozzi Associates, 1999
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