3. index futures

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.

2

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

3

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... ) /

4

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 ( ... ) /

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


STOCK NUMBER 2 SPLIT 3 TO 1.

(30 + 40 + 40 + 36 + 50)/D = 40


6

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

7

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

8

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

11

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:

13

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.

14

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

15

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

16

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

18

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

20

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

21

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

22

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

23


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.

24

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

25


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:

26

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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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

3. index futures

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