3. index futures
DESCRIPTION
TRANSCRIPT
1
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 MARKETAVERAGE 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.N. 1,..., = i ;D
P = I i
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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
(P P P D I1 2 N 1 1 ... ) /
(P P P D I1 2 N 2 1 1
2... ) /
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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.
(P P ABC) P D I1 2 N 1 1 ( ... ) /
(P P XYZ) P D I1 2 N 2 1 ( ... ) /
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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.
1.
(30 + 50 + 40 + 60 + 20)/5 = 40
D = 5 I = 40.
2.
(30 + 50 + 40 + 60 + 20 + 35)/D = 40
D = 5.875.
(P P P D I1 2 N 1 1 ... ) /
121+NN21 ID/)PP,...,P(P
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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 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.
IN P
N Ptti ti
Bi Bi
tititP, PNV
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THE RATE OF RETURN ON A VALUE WEIGHTED INDEX
: yields,by P numerator
thedividing and gMultiplyin
.PN
)P(PN
Thus, .N N but,
;PN
PNPN
VB
PNVB
PN
VB
PN
I
II R
ti
titi
ti1i+tti
ti1i+t
titi
titi1i+t1i+t
titi
titi1i+t1i+t
t
t1+tIt
9
: thatagain, Notice, .Rw R
Finally,.RV
V
or ,R]PN
PN[
:as this Rewrite.PN
RPN
,PN
PPP
PN R
titiIt
tiI
i
tititi
titi
titi
tititi
titi
ti
ti1ittiti
It
.V
V
PN
PNw
BI
ti
BiBi
titii
10
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. .Rw R titiIt
.V
V
PN
PN w
BI
ti
BiBi
titii
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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.
R
12proof. theconcludes This
.β w )VAR(R
R,COV(Rwβ
:or ,)VAR(R
)R,COV(Rwβ
: thusoperator,linear a
is covariance e that thRecall
.)VAR(R
)R,]RwCOV([ β
,Rw R;for R ngSubstituti
.)VAR(R
)R,COV(R β
iiI
IiiP
I
IiiP
I
IiiP
iiPP
I
IPP
Proof: By definition, the portfolio’s β 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:
F = 1,041.15e = 1,041.38TH
(.032-.030)40
365
THEORETICAL = 1,041.38 < 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 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
78=($250)(1,041.15)
0$20,000,00 =NF
V =1,039
1,041.1520,000,000
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THE OPTIMAL HEDGE RATIO FOR STOCK INDEX FUTURES
RECALL THAT THE MINIMUM RISK HR IS:
F)VAR(
F)S,COV( NF
17
.F
Sβ = N
:result the yields which,F
S
)VAR(r
)r,COV(r = N
:or ,))](F[VAR(r
)(F))](Sr,[COV(r = N
: toequivalent is which,)VAR(Fr
)Fr,rCOV(S = N
:F and Sfor
N in substitute Next, .r F= ΔF F
ΔF = r
Similarly, .rS = ΔS S
ΔS = r
.S
ΔS =
S
S - S = r S -S = ΔS
0SF
0
F
FSF
2F
0FSF
F
FS0F
FF00
F
S00
S
00
01S01
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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
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
20 = 366,362.50
5,400,0001.35 = NF
E$52.9/SHAR = 100,000
$510,000 - $5,800,000
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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
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%
N = 1.12,606,250
272,500 = 11F
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STOCK PORTFOLIO HEDGE
FEDERAL MOUGUL 18.875 9,000 169,875 .044 1.00MARTIN ARIETTA 73.500 8,000 88,000 .152 .80IBM 50.875 3,500 178,063 .046 .50US WEST 43.625 5,400 235,575 .061 .70BAUSCH & LOMB 54.250 10,500 569,625 .147 1.1FIRST UNION 47.750 14,400 687,600 .178 1.1WALT DISNEY 44.500 12,500 556,250 .144 1.4DELTA AIRLINES 52.875 16,600 877,725 .227 1.2
3,862,713
β(portfolio) = .044(1.00) + .152(.8) + .046(.5)
+ .061(.7) + .147(1.1) + .178(1.1)
+ .144(1.4)+ .227(1.2)
= 1.06
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
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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
16. = (1.06)263,300
3,862,713 = NF
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BENEFICIAL CORP. 40.500 11,350 459,675 .122 .95CUMMINS ENGINES 64.500 10,950 706,275 .187 1.10GILLETTE 62.000 12,400 768,800 .203 .85KMART 33.000 5,500 181,500 .048 1.15BOEING 49.000 4,600 225,400 .059 1.15W.R.GRACE 42.625 6,750 287,719 .076 1.00ELI LILLY 87.375 11,400 996,075 .263 .85PARKER PEN 20.625 7,650 157,781 .042 .75
3,783,225
MARKET TIMING HEDGE RATIO
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
β(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 portfolio’s composition.
The Minimum Variance Hedge Ratio in our case is: NF = ß[S/F].
Assume that our position is a portfolio with current market value of S and NF futures.
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F
Sβ][βN
r)E(rS
FN]r)β[E(rr
]r)[E(rβr
:Substitue ].r)[E(rβr)E(r
.r)E(r) E(r; ]r)β[E(rr)E(r
: writecan weCAPM, theFollowing
).E(rS
FN)E(r)E(r
:HENCE;S
ΔPr
DEFINE
.F
ΔF
S
FN
S
ΔS
S
ΔFN
S
ΔS
S
ΔP
ΔFNΔSΔP
FNSP
TF
FMFFMF
FMTF
FMTFP
FMFFMFS
FFSSF
SF
FF
F
F
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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 = ßT, the number of contract should be:
.F
Sβ][βN TF
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%
N = (1.25 - .95)3,783,225
269,950 = 4F
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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. portfolio’s current beta is 1.85. The SP500I MAR futures is