The various indexes use differing computational methods. To understand the trading and pricing of index futures, one must first understand a bit about how the underlying indexes are computed.
The DJIA is computed by adding the share prices of the 30 stocks comprising the index and dividing by the DJIA divisor. The divisor is used to adjust for stock splits, mergers, stock dividends, and changes in the stocks included in the index.
Index Divisor
The index divisor is a computed number that keeps the index unchanged in the event of certain occurrences (e.g., dropping one company from the index and adding another company, mergers and stock splits).
The DJIA can be computed by using the following formula:
Assume that the Dow Jones company decides to delete Boeing from the index and replace it with Dow Chemical. Boeing stock trades at $6.00 and Dow Chemical trades at $47. The current level of the index is 1900.31 with a divisor of .889.
Before the Change
Total 30 stock prices = $1,689.375
After the Change (No New Divisor Is Used) Total new 30 stock price: $1,689.375 - 6+47 = $1,730.375
Thus, to keep the index value unchanged, the new divisor must be 0.9106. If the divisor is not changed the DJIA will be 46 points higher as a result of the component change. Thus, a new divisor must be calculated. A new divisor is computed as follows:
Each of the stocks in these indexes has a different weight in the calculation of the index. The weight is proportional to the total market value of the stock (the price per share times the number of shares outstanding).
The value of the S&P 500 index is reported relative to the average value during the period of 1941-1943, which was assigned an index value of 10.
Assume that the S&P 500 index consists of three stocks ABC, DEF and GHI.
Table 9.2 shows how the value of these 3 firms will be weighted.
Similar to the Market Capitalization Indexes, these indexes reflect the total change in the value of the portfolio from inception to the current date.
Where M t = market capitalization of the index at time t B t = adjusted base date market capitalization of the index at time t base value = the original numerical starting value for the index (e. g.,100 or 1000)
From the above equation, the numerator reflects the total accumulated value of the portfolio and the denominator represents the initial value of the portfolio. As such, both the numerator and denominator are affected by several factors as follows:
Stock index futures trade in a full-carry market. As such, the Cost-of-Carry Model provides a good understanding of index futures pricing.
Recall that the Cost-of-Carry Model for a perfect market with unrestricted short selling is given by:
Applying this model to stock index futures has one complication, dividends. If you purchase the stocks in the index, you will receive dividends. Recall that most indexes ignore dividends in their computation, so the Cost-of-Carry Model must be adjusted to reflect the dividends. The receipt of dividends reduces the cost of carrying the stocks from today until the delivery date on the futures contract.
Today, t 0 , a trader decides to engage in a self-financing cash-and-carry transaction. The trader decides to buy and hold one share of Widget, Inc., currently trading for $100. The trader borrows $100 to buy the stock. The stock will pay a $2 dividend in 6 months and the trader will invest the proceeds for the remaining 6 months at a rate of 10%. Table 9.4 shows the trader's cash flows.
The trader's cash inflow after one year is the future value of the dividend, $2.10, plus the value of the stock in one year, P 1, less the repayment of the loan, $110.
In order to adjust the Cost-of-Carry Model for dividends, the future value of the dividends that will be received is computed at the time the futures contract expires. This amount is then subtracted from the cost of carrying the stocks forward.
Where: S 0 = The current spot price F 0,t = The current futures price for delivery of the product at time t C 0,t = The percentage cost of carrying the stock index from today until time t D i = The i th dividend r i = The interest earned from investing the dividend from the time received until the futures expiration at time t
A stock index futures price has a fair value when the futures price conforms to the Cost-of-Carry Model.
In this section, we use a simplified example to determine the fair value of a stock index futures contract. Assume a futures contract on a price-weighted index, and that there are only two stocks. Table 9.5 provides the information needed to compute the stock index fair value.
The cost of buying the stocks and carrying them to the future is $200.67. Step 6: compute the fair price of the index. To compute the fair value for the index, we must convert the previous answer into index units. Notice that the fair value of the index (111.48) is different than the current level of the index (110.56). This difference suggests that possibility of an arbitrage.
Index arbitrages refer to cash-and-carry strategies in stock index futures. This section examines:
Index arbitrage
Program trading
Recall that deviations from the theoretical price of the Cost-of-Carry Model give rise to arbitrage opportunities.
If the futures price exceeds its fair value, traders will engage in cash-and-carry arbitrage.
A cash-and-carry arbitrage involves purchasing all the stocks in the index and selling the futures contract.
If the futures price falls below its fair value, traders can exploit the pricing discrepancy through a reverse cash-and-carry strategy.
A reserve cash-and-carry arbitrage involves selling the stocks in the index short and buying a futures contract.
We would expect the futures prices to follow those suggested by the Cost-of-Carry Model. To the extent that they do not, traders can engage in index arbitrage.
To demonstrate how index arbitrage works, we will examine a two-stock index. The Information on the index futures and the two stocks contained in the index are presented in Table 9.5.
Suppose the data from Table 9.5 holds, but the futures price is $115 which is above the fair value. The transactions for a cash-and-carry arbitrage are presented in Table 9.6.
31.
Index Arbitrage Now suppose that all the information from Table 9.5 holds, but the futures price is $105, which is below the fair value of $111.48, so a reverse cash-and-carry arbitrage is possible. Table 9.7 shows the transactions for a reverse cash-and-carry arbitrage.
When performing index arbitrage, the investor must buy or sell all of the stocks in the index.
For example, to perform index arbitrage on the S&P 500 index, one would need to purchase or sell 500 different stocks.
Because of the difficulty in doing this, the trading is frequently done by computer. This is called program trading.
The computer will download the prices of all 500 stocks, compute the fair price of the index and compare that to the price of the futures contract.
If a cash-and-carry arbitrage is suggested, the computer will initiate trades to purchase all 500 stocks. It will also sell the futures contract.
Because of the number of stocks involved, performing a successful index arbitrage involves very large sums of money and very rapid trading. As such, institutional investors (mutual funds and the like) are the ones that typically engage in index arbitrage.
33.
Predicting Dividends Payments and Investment Rates
Dividend Amount and Timing
So far we have assumed certainty with regard to dividend amount, timing and investment rates.
In the real market, dividends are predictable, but are not certain.
To the extent that they are not predicted with certainty, the cash-and-carry index arbitrage can be frustrated.
For the DJIA with 30 stocks, dividends are relatively stable. Thus prediction can be moderately accurate.
For the SEP 500 or NYSE Indexes, many smaller companies are involved and dividend prediction becomes much less certain.
Moreover, dividends are paid in seasonal patterns as shown in Figure 9.2.
Predicting the Investment Rate
Predicting the investment rate for dividends can be done with some certainty, as it is a relatively short term investment that will occur in the near future.
35.
Market Imperfections and Stock Index Futures Prices
Recall that four market imperfections could affect the pricing of futures contracts:
Direct Transaction Costs
Unequal Borrowing and Lending Rates
Margins
Restrictions on Short Selling
Market imperfections exist and can be substantial, particularly for indexes with large numbers of stocks.
The existence of market imperfections leads to no-arbitrage bounds on index arbitrage.
So the price has to get out of sync by a good bit to cover the transaction costs and other market imperfections associated with attempting the arbitrage.
Futures contracts allow speculators to make the most straightforward speculation on the direction of the market or to enter very sophisticated spread transactions to tailor the futures position to more precise opinions about the direction of stock prices.
The low transaction costs in the futures market make the speculation much easier to undertake than similar speculation in the stock market itself.
Tables 9.8 and 9.9 illustrate two cases of stock index futures speculation, a conservative inter-commodity spread and a conservative intra-commodity spread.
37.
Speculating with Stock Index Futures A trader observe that the DJIA futures is 8603.50 and the S&P 500 futures is 999. The trader expects the DJIA to go up more rapidly than the S&P 500 index due to market conditions. To bet on her intuition the trader enters into an inter-commodity spread as indicated in Table 9.8. The spread has widened as expected and thus, the trader was able to realize a $16,447.50 profit.
38.
Speculating with Stock Index Futures In the event that a trader expects more distant contracts to be more sensitive to a market move than the nearby contracts. The trader initiates a intra-commodity spread as shown in Table 9.9. In this case, the position is so conservative that there was little difference in the price changes, producing only a $112.50 profit, despite the fact that the market moved in the predicted direction.
The identity of the underlying security Delivery procedures The contract size (100 shares) Margin The trading environment The minimum price fluctuation Daily price limits The expiration cycle Trading hours Position limits
They contain provisions for adjustments to reflect certain corporate events (e.g., stock splits and special dividends).
They expire on the 3rd Friday of the delivery month.
Single stock futures are priced using the Cost-of-Carry Model.
Example
Today, Feb 20, the current price of Wal-Mart stock is $59.45/share. The JUN futures contract for Wal-Mart will expires on June 18. Wal-Mart’s quarterly dividend is expected to be 9 cents/share on April 7. The current financing cost is assumed to be 1.6% per year.
Since there is only a single dividend payment during the life of the futures contract, the cost-of-carry relationship becomes simple:
42.
Risk Management with Security Futures Contracts: Short Hedging
Hedging with stock index futures applies directly to the management of stock portfolios. This section examines short and long hedging applications for stock index futures.
Assume that a portfolio manager has a well-diversified portfolio with the following characteristics:
Portfolio Value = $40,000,000
Portfolio Beta = 1.22 (relative to the S&P 500)
S&P 500 Index = 1060.00
The portfolio manager fears that a bear market is imminent and wishes to hedge his portfolio's value against that possibility.
The manager could use the S&P 500 stock index futures contract. By selling futures, the manager should be able to offset the effect of the bear market on the portfolio by generating gains in the futures market.
43.
Risk Management with Security Futures Contracts: Short Hedging
Assuming that the S&P index futures contract stands at 1060, the advocated futures position would be given by:
where: V P = value of the portfolio V F = value of the futures contract This strategy ignores the higher volatility of the stock portfolio relative to the S&P 500 index. Table 9.10 illustrates the potential results.
44.
Risk Management with Security Futures Contracts: Short Hedging
The manager might be able to avoid this negative result by weighting the hedge ratio by the beta of the stock portfolio.
The failure to consider the difference in volatility between the stock portfolio and index futures contract leads to suboptimal hedging results.
45.
Risk Management with Security Futures Contracts: Short Hedging
Using the following equation the manager can determine the number of contracts to trade.
Where: β P = beta of the portfolio that is being hedged. Thus, The manager would sell:
46.
Risk Management with Security Futures Contracts: Long Hedging
A pension fund manager is convinced an extended bull market in Japanese equities is about to begin. The current exchange rate is $1 per ¥140. The manager anticipates funds for investing to be ¥6 billion ( $42,857,143 ≈ $43,000,000) in 3 months. The pension fund manager trades as shown in Table 9.11.
The futures profit offsets the additional cost of purchasing stocks because of an increase in prices.
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