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.
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. P i = 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.
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
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.
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
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 S&P500 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.
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.
THE BETA OF A PORTFOLIO THEOREM: A PORTFOLIO’S 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.
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.
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 < 1,044.00 = ACTUAL THE CONTRACT IS OVERPRICED CASH AND CARRY
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 78[1,044-1,039]$250 = $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
THE OPTIMAL HEDGE RATIO FOR STOCK INDEX FUTURES RECALL THAT THE MINIMUM RISK HR IS:
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 β = 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
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 β = 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%
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
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 portfolio’s composition. The Minimum Variance Hedge Ratio in our case is: N F = ß[S/F]. Assume that our position is a portfolio with current market value of S and N F futures.
MARKET TIMING HEDGE RATIO We just proved that in order to change the position’s beta from its current value, ß, to a Target Beta = ß T , the number of contract should be: Going back to our portfolio:
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%
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. portfolio’s current beta is 1.85. The SP500I MAR futures is