STOCK INDEX FUTURES

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STOCK INDEX FUTURES A STOCK INDEX IS A SINGLE
NUMBER BASED ON INFORMATION ASSOCIATED WITH A
BASKET OF STOCK PRICES AND QUANTITIES. A STOCK
INDEX IS SOME KIND OF AN AVERAGE OF THE PRICES
AND THE QUANTITIES OF THE STOCKS THAT ARE
INCLUDED IN THE BASKET. THE MOST USED INDEXES
ARE A SIMPLE PRICE AVERAGE AND A VALUE
WEIGHTED AVERAGE.
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STOCK INDEXES THE CASH MARKET AVERAGE PRICE
INDEXES DJIA, MMI. DJIA DOW JONES INDUSTRIAL
AVERAGE. MMI MAJOR MARKET INDEX. N The
number of stocks in the index. D
Divisor. Pi i-th Stock market price. INITIALLY
D N AND THE INDEX IS SET AT A GIVEN LEVEL.
TO ASSURE INDEX CONTINUITY, THE DIVISOR IS
CHANGED OVER TIME.
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EXAMPLES STOCK SPLITS 1. 2. 1. (30 40
50 60 20) /5 40 I 40 and D 5. 2.
(30 20 50 60 20)/D 40 The index
remains 40 and the new divisor is D 4.5
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CHANGE OF STOCKS IN THE INDEX 1. 2. 1. (30
20 40 60 50)/5 40 I 40 and D
5. 2. (30 120 40 60 50)/D 40 The
index remains 40 and the new divisor is D 7.5.
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STOCK 4 DISTRIBUTED 40 STOCK DIVIDEND (30
120 40 60 50)/D 40 D 7.5. Next, (30
120 40 36 50)/D 40 The index remains 40
and the new divisor is D 6.9 STOCK NUMBER 2
SPLIT 3 TO 1. (30 40 40 36 50)/D
40 The index remains 40 and the new
divisor is D 4.9
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• ADDITIONAL STOCKS
• 1.
• 2.
• (30 50 40 60 20)/5 40
• D 5 I 40.
• 2.
• (30 50 40 60 20 35)/D 40
• D 5.875.

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VALUE WEIGHTED INDEXES S P500, NIKKEI 250,
VALUE LINE B SOME BASIS TIME
PERIOD INITIALLY, t B. THUS, THE INITIAL INDEX
VALUE IS SOME ARBITRARILY CHOSEN VALUE M. For
example, the SP500 index base period was
1941-1943 and its initial value was set at M
10. The NYSE index base period was Dec. 31, 1965
and its initial value was set at M 50. Note
that Is the value of the portfolio used in the
index.
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THE RATE OF RETURN ON A VALUE WEIGHTED INDEX
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Conclusion The return on a value weighted index
in any period t, is the weighted average of the
individual stock returns the weights are the
dollar value of the stocks as a proportion of the
total value of the portfolio used in the index.
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THE BETA OF A PORTFOLIO THEOREM A PORTFOLIOS
BETA IS THE WEIGHTED AVERAGE OF THE BETAS OF THE
STOCKS THAT COMPRISE THE PORTFOLIO. THE WEIGHTS
ARE THE DOLLAR VALUE WEIGHTS OF THE STOCKS IN THE
PORTFOLIO. In order to prove this theorem, assume
that the index is a well diversified portfolio,
I.e., it represents the market portfolio. In the
proof, P denotes the portfolio I, denotes the
index and i denotes the individual stock i 1,
2, , N.
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Proof By definition, the portfolios ß is
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STOCK INDEX FUTURES Stock index futures are
characterized by two new features 1. The value
of one contract is (FUTURES PRICE)(MULTIPLIER)
2. There is no delivery of the underlying.
Instead, all accounts are settled by cash.
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STOCK INDEX ATBITRAGE AN ARBITRAGER FACES THE
FOLLOWING MARKET DATA NOV. 4. SP500I
1,041.15 ANNUAL DIVIDEND YIELD 3 RISK-FREE
RATE 3.2 THE DECEMBER CONTRACT EXPIRES IN 40
DAYS AND STANDS AT F 1,044. The no-arbitrage
condition is
THEORETICAL 1,041.38 lt 1,044.00 ACTUAL THE
CONTRACT IS OVERPRICED CASH AND CARRY
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DATE SPOT FUTURES NOV 4 (a) BORROW 20M (c)
SHORT 78 DEC SP500I (b) BUY 20M WORTH OF
FUTURES. F 1,044 STOCKS IN THE
SAME PROPORTIONS AS IN THE SP500I DEC
18 SP500I 1.039 FUTURES EXPIRES
1,039 CASH SETTLEMENT
19,958,699.51 781,044-1,039250
97,500 REPAY THE LOAN -20,004,384.04 P/L
19,958,699.51 - 20,004,384.04 - 45,684.53
97,500.00 51,815.47 IF
TRANSACTION COST 125 BASIS POINTS/ 20M
(.00125) 25,000 NET PROFIT
26,815.47
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THE OPTIMAL HEDGE RATIO FOR STOCK INDEX
FUTURES RECALL THAT THE MINIMUM RISK HR IS
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ANTICIPATORY HEDGE OF A TAKEOVER A firm
intends to purchase 100,000 shares of XYZ ON
DEC.17. DATE SPOT FUTURES NOV.17 S
54/SHARE MAR SP500I FUTURES IS ß 1.35 AT
1,465.45 V (54)100,000 F 1,465.45(250)
5,400,000 366,362.50 LONG 20 MAR
SP500I Fs. DEC.17 S 58/SHARE SHORT 20 MAR
SP500I Fs PURCHASE 100,000 F 1, 567.45
SHARES. PROFIT COST 5,800,000 20(1,567.45
- 1,465.45)250 510,000 ACTUAL
PURCHASING PRICE
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HEDGING A ONE STOCK PORTFOLIO SPECIFIC STOCK
INFORMATION INDICATES THAT THE STOCK SHOULD
INCREASE IN VALUE BY ABOUT 9. THE MARKET IS
EXPECTED TO DECREASE BY 10, HOWEVER. THUS, WITH
BETA 1.1 THE STOCK PRICE IS EXPECTED TO REMAIN
AT ITS CURRENT VALUE. SPECULATION ON THE
UNSYSTEMATIC RISK, WE OPEN THE FOLLOWING
STRATEGY TIME SPOT FUTURES JULY 1 OWN 150,000
SHARES DEC. IF PRICE 1,090 S 17 3/8 F
1,090(250) 272,500 V 2,606,250 ß
1.1 SHORT 11 DEC. SP500I Fs SEP.30 S
17 1/8 LONG 11 DEC SP500I Fs V
2,568,750 F 1,002. PROFIT
250(11)(1,090 - 1,002) 242,000 ACTUAL V
2,810,750 INCREASE OF ABOUT 8
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STOCK PORTFOLIO HEDGE
STOCK NAME PRICE SHARES VALUE WEIGHT
BETA
ß(portfolio) .044(1.00) .152(.8)
.046(.5) .061(.7) .147(1.1) .178(1.1)
.144(1.4) .227(1.2) 1.06
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TIME CASH FUTURES MAR.31 V 3,862,713.00
SEP SP500I FUTURES F 1,052.60(250)
263,300 SHORT 16 SEP SP500I
Fs. JUL.27 V 3,751,307.00 LONG 16 SEP
SP500I Fs F 1,026.99 PROFIT
(1,052.60 - 1,026.99)(250)(16)
102,440.00 TOTAL VALUE 3,853,747.00
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MARKET TIMING HEDGE RATIO STOCK NAME PRICE
SHARES VALUE WEIGHT BETA
ß(portfolio) .122(.95) .187(1.1)
.203(.85) .048(1.15) .059(1.15)
.076(1.0) .263(.85) .042(.75)
.95
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MARKET TIMING HEDGE RATIO When we believe that
the market trend is changing, we need to change
the beta of our portfolio. We may purchase high
beta stocks and sell low beta stocks, when we
believe that the market is turning upward or
purchase low beta stocks and sell high beta
stocks, when we believe that the market is moving
down. Instead we may try to change the beta of
our position by using the INDEX FUTURES without
changing the portfolios composition. The Minimum
Variance Hedge Ratio in our case is NF
ßS/F. Assume that our position is a portfolio
with current market value of S and NF futures.
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MARKET TIMING HEDGE RATIO We just proved that in
order to change the positions beta from its
current value, ß, to a Target Beta ßT, the
number of contract should be
Going back to our portfolio
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TIME CASH FUTURES AUG.29 V 3,783,225
DEC SP500I Fs 1,079.8(250)
269,950 LONG 4 DEC SP500I Fs NOV.29
V 4,161,500 F 1,154.53 SHORT 4 DEC
SP500I Fs PROFIT (1,154.53 -
1,079.8)(250)(4) 74,730 TOTAL PORTFOLIO
VALUE 4,236,230 THE MARKET INCREASED ABOUT 7
AND THE PORTFOLIO VALUE INCREASED ABOUT 12
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MARKET TIMING HEDGE RATIO Suppose that a
portfolio manager expect the market to decline in
the next three months from November to February
next year.The current portfolio value is
75,000,000. portfolios current beta is 1.85.
The SP500I MAR futures is